https://artofproblemsolving.com/wiki/api.php?action=feedcontributions&user=Bissue&feedformat=atomAoPS Wiki - User contributions [en]2024-03-29T07:23:17ZUser contributionsMediaWiki 1.31.1https://artofproblemsolving.com/wiki/index.php?title=User:Bissue&diff=160976User:Bissue2021-08-26T17:15:53Z<p>Bissue: Replaced content with "farley orz :heartbear:"</p>
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<div>farley orz :heartbear:</div>Bissuehttps://artofproblemsolving.com/wiki/index.php?title=User:Panda_2&diff=157627User:Panda 22021-07-09T23:05:50Z<p>Bissue: </p>
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<div>mangie orz :heart_eyes_cat:<br />
<br />
mangie orz :heart_eyes_cat:</div>Bissuehttps://artofproblemsolving.com/wiki/index.php?title=2019_IMO&diff=1575892019 IMO2021-07-09T04:54:23Z<p>Bissue: </p>
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<div>==Problem 1==<br />
Let <math>\mathbb{Z}</math> be the set of integers. Determine all functions <math>f : \mathbb{Z} \to \mathbb{Z}</math> such that, for all<br />
integers <math>a</math> and <math>b</math>, <cmath>f(2a) + 2f(b) = f(f(a + b)).</cmath>''<br />
<br />
[[2019 IMO Problems/Problem 1|Solution]]<br />
<br />
==Problem 2==<br />
In triangle <math>ABC</math>, point <math>A_1</math> lies on side <math>BC</math> and point <math>B_1</math> lies on side <math>AC</math>. Let <math>P</math> and <math>Q</math> be points on segments <math>AA_1</math> and <math>BB_1</math>, respectively, such that <math>PQ</math> is parallel to <math>AB</math>. Let <math>P_1</math> be a point on line <math>PB_1</math>, such that <math>B_1</math> lies strictly between <math>P</math> and <math>P_1</math>, and <math>\angle PP_1C=\angle BAC</math>. Similarly, let <math>Q_1</math> be the point on line <math>QA_1</math>, such that <math>A_1</math> lies strictly between <math>Q</math> and <math>Q_1</math>, and <math>\angle CQ_1Q=\angle CBA</math>.<br />
<br />
Prove that points <math>P,Q,P_1</math>, and <math>Q_1</math> are concyclic.<br />
<br />
[[2019 IMO Problems/Problem 2|Solution]]<br />
<br />
==Problem 3==<br />
A social network has <math>2019</math> users, some pairs of whom are friends. Whenever user <math>A</math> is friends with user <math>B</math>, user <math>B</math> is also friends with user <math>A</math>. Events of the following kind may happen repeatedly, one at a time:<br />
Three users <math>A</math>, <math>B</math>, and <math>C</math> such that <math>A</math> is friends with both <math>B</math> and <math>C</math>, but <math>B</math> and <math>C</math> are not friends, change their friendship statuses such that <math>B</math> and <math>C</math> are now friends, but <math>A</math> is no longer friends with <math>B</math>, and no longer friends with <math>C</math>. All other friendship statuses are unchanged.<br />
Initially, <math>1010</math> users have <math>1009</math> friends each, and <math>1009</math> users have <math>1010</math> friends each. Prove that there exists a sequence of such events after which each user is friends with at most one other user.<br />
<br />
[[2019 IMO Problems/Problem 3|Solution]]<br />
<br />
==Problem 4==<br />
Find all pairs <math>(k,n)</math> of positive integers such that <br />
<br />
<cmath>k!=(2^n-1)(2^n-2)(2^n-4)\dots(2^n-2^{n-1}).</cmath><br />
<br />
[[2019 IMO Problems/Problem 4|Solution]]<br />
<br />
==Problem 5==<br />
The Bank of Bath issues coins with an <math>H</math> on one side and a <math>T</math> on the other. Harry has <math>n</math> of these coins arranged in a line from left to right. He repeatedly performs the following operation:<br />
<br />
If there are exactly <math>k > 0</math> coins showing <math>H</math>, then he turns over the <math>k^{th}</math> coin from the left; otherwise, all coins show <math>T</math> and he stops. For example, if <math>n = 3</math> the process starting with the configuration <math>THT</math> would be <math>THT \rightarrow HHT \rightarrow HTT \rightarrow TTT</math>, which stops after three operations.<br />
<br />
(a) Show that, for each initial configuration, Harry stops after a finite number of operations.<br />
<br />
(b) For each initial configuration <math>C</math>, let <math>L(C)</math> be the number of operations before Harry stops. For<br />
example, <math>L(THT) = 3</math> and <math>L(TTT) = 0</math>. Determine the average value of <math>L(C)</math> over all <math>2^n</math><br />
possible initial configurations <math>C</math>.<br />
<br />
[[2019 IMO Problems/Problem 5|Solution]]<br />
<br />
==Problem 6==<br />
Let <math>I</math> be the incenter of acute triangle <math>ABC</math> with <math>AB \neq AC</math>. The incircle <math>\omega</math> of <math>ABC</math> is tangent to sides <math>BC</math>, <math>CA</math>, and <math>AB</math> at <math>D</math>, <math>E</math>, and <math>F</math>, respectively. The line through <math>D</math> perpendicular to <math>EF</math> meets ω again at <math>R</math>. Line <math>AR</math> meets ω again at <math>P</math>. The circumcircles of triangles <math>PCE</math> and <math>PBF</math> meet again at <math>Q</math>.<br />
Prove that lines <math>DI</math> and <math>PQ</math> meet on the line through <math>A</math> perpendicular to <math>AI</math>.<br />
<br />
[[2019 IMO Problems/Problem 6|Solution]]</div>Bissuehttps://artofproblemsolving.com/wiki/index.php?title=2019_IMO&diff=1575682019 IMO2021-07-08T22:24:13Z<p>Bissue: </p>
<hr />
<div>==Problem 1==<br />
''Let <math>\mathbb{Z}</math> be the set of integers. Determine all functions <math>f : \mathbb{Z} \to \mathbb{Z}</math> such that, for all<br />
''integers <math>a</math> and <math>b</math>, <cmath>f(2a) + 2f(b) = f(f(a + b)).</cmath>''<br />
<br />
[[2019 IMO Problems/Problem 1|Solution]]<br />
<br />
==Problem 2==<br />
In triangle <math>ABC</math>, point <math>A_1</math> lies on side <math>BC</math> and point <math>B_1</math> lies on side <math>AC</math>. Let <math>P</math> and <math>Q</math> be points on segments <math>AA_1</math> and <math>BB_1</math>, respectively, such that <math>PQ</math> is parallel to <math>AB</math>. Let <math>P_1</math> be a point on line <math>PB_1</math>, such that <math>B_1</math> lies strictly between <math>P</math> and <math>P_1</math>, and <math>\angle PP_1C=\angle BAC</math>. Similarly, let <math>Q_1</math> be the point on line <math>QA_1</math>, such that <math>A_1</math> lies strictly between <math>Q</math> and <math>Q_1</math>, and <math>\angle CQ_1Q=\angle CBA</math>.<br />
<br />
Prove that points <math>P,Q,P_1</math>, and <math>Q_1</math> are concyclic.<br />
<br />
[[2019 IMO Problems/Problem 2|Solution]]<br />
<br />
==Problem 3==<br />
A social network has <math>2019</math> users, some pairs of whom are friends. Whenever user <math>A</math> is friends with user <math>B</math>, user <math>B</math> is also friends with user <math>A</math>. Events of the following kind may happen repeatedly, one at a time:<br />
Three users <math>A</math>, <math>B</math>, and <math>C</math> such that <math>A</math> is friends with both <math>B</math> and <math>C</math>, but <math>B</math> and <math>C</math> are not friends, change their friendship statuses such that <math>B</math> and <math>C</math> are now friends, but <math>A</math> is no longer friends with <math>B</math>, and no longer friends with <math>C</math>. All other friendship statuses are unchanged.<br />
Initially, <math>1010</math> users have <math>1009</math> friends each, and <math>1009</math> users have <math>1010</math> friends each. Prove that there exists a sequence of such events after which each user is friends with at most one other user.<br />
<br />
[[2019 IMO Problems/Problem 3|Solution]]<br />
<br />
==Problem 4==<br />
Find all pairs <math>(k,n)</math> of positive integers such that <br />
<br />
<cmath>k!=(2^n-1)(2^n-2)(2^n-4)\dots(2^n-2^{n-1}).</cmath><br />
<br />
[[2019 IMO Problems/Problem 4|Solution]]<br />
<br />
==Problem 5==<br />
The Bank of Bath issues coins with an <math>H</math> on one side and a <math>T</math> on the other. Harry has <math>n</math> of these coins arranged in a line from left to right. He repeatedly performs the following operation:<br />
<br />
If there are exactly <math>k > 0</math> coins showing <math>H</math>, then he turns over the <math>k^{th}</math> coin from the left; otherwise, all coins show <math>T</math> and he stops. For example, if <math>n = 3</math> the process starting with the configuration <math>THT</math> would be <math>THT \rightarrow HHT \rightarrow HTT \rightarrow TTT</math>, which stops after three operations.<br />
<br />
(a) Show that, for each initial configuration, Harry stops after a finite number of operations.<br />
<br />
(b) For each initial configuration <math>C</math>, let <math>L(C)</math> be the number of operations before Harry stops. For<br />
example, <math>L(THT) = 3</math> and <math>L(TTT) = 0</math>. Determine the average value of <math>L(C)</math> over all <math>2^n</math><br />
possible initial configurations <math>C</math>.<br />
<br />
[[2019 IMO Problems/Problem 5|Solution]]<br />
<br />
==Problem 6==<br />
Let <math>I</math> be the incenter of acute triangle <math>ABC</math> with <math>AB \neq AC</math>. The incircle <math>\omega</math> of <math>ABC</math> is tangent to sides <math>BC</math>, <math>CA</math>, and <math>AB</math> at <math>D</math>, <math>E</math>, and <math>F</math>, respectively. The line through <math>D</math> perpendicular to <math>EF</math> meets ω again at <math>R</math>. Line <math>AR</math> meets ω again at <math>P</math>. The circumcircles of triangles <math>PCE</math> and <math>PBF</math> meet again at <math>Q</math>.<br />
Prove that lines <math>DI</math> and <math>PQ</math> meet on the line through <math>A</math> perpendicular to <math>AI</math>.<br />
<br />
[[2019 IMO Problems/Problem 6|Solution]]</div>Bissuehttps://artofproblemsolving.com/wiki/index.php?title=2019_IMO&diff=1575672019 IMO2021-07-08T22:24:00Z<p>Bissue: Undo revision 133452 by Muhaboug (talk)</p>
<hr />
<div>mathlete6453 was here<br />
<br />
==Problem 1==<br />
''Let <math>\mathbb{Z}</math> be the set of integers. Determine all functions <math>f : \mathbb{Z} \to \mathbb{Z}</math> such that, for all<br />
''integers <math>a</math> and <math>b</math>, <cmath>f(2a) + 2f(b) = f(f(a + b)).</cmath>''<br />
<br />
[[2019 IMO Problems/Problem 1|Solution]]<br />
<br />
==Problem 2==<br />
In triangle <math>ABC</math>, point <math>A_1</math> lies on side <math>BC</math> and point <math>B_1</math> lies on side <math>AC</math>. Let <math>P</math> and <math>Q</math> be points on segments <math>AA_1</math> and <math>BB_1</math>, respectively, such that <math>PQ</math> is parallel to <math>AB</math>. Let <math>P_1</math> be a point on line <math>PB_1</math>, such that <math>B_1</math> lies strictly between <math>P</math> and <math>P_1</math>, and <math>\angle PP_1C=\angle BAC</math>. Similarly, let <math>Q_1</math> be the point on line <math>QA_1</math>, such that <math>A_1</math> lies strictly between <math>Q</math> and <math>Q_1</math>, and <math>\angle CQ_1Q=\angle CBA</math>.<br />
<br />
Prove that points <math>P,Q,P_1</math>, and <math>Q_1</math> are concyclic.<br />
<br />
[[2019 IMO Problems/Problem 2|Solution]]<br />
<br />
==Problem 3==<br />
A social network has <math>2019</math> users, some pairs of whom are friends. Whenever user <math>A</math> is friends with user <math>B</math>, user <math>B</math> is also friends with user <math>A</math>. Events of the following kind may happen repeatedly, one at a time:<br />
Three users <math>A</math>, <math>B</math>, and <math>C</math> such that <math>A</math> is friends with both <math>B</math> and <math>C</math>, but <math>B</math> and <math>C</math> are not friends, change their friendship statuses such that <math>B</math> and <math>C</math> are now friends, but <math>A</math> is no longer friends with <math>B</math>, and no longer friends with <math>C</math>. All other friendship statuses are unchanged.<br />
Initially, <math>1010</math> users have <math>1009</math> friends each, and <math>1009</math> users have <math>1010</math> friends each. Prove that there exists a sequence of such events after which each user is friends with at most one other user.<br />
<br />
[[2019 IMO Problems/Problem 3|Solution]]<br />
<br />
==Problem 4==<br />
Find all pairs <math>(k,n)</math> of positive integers such that <br />
<br />
<cmath>k!=(2^n-1)(2^n-2)(2^n-4)\dots(2^n-2^{n-1}).</cmath><br />
<br />
[[2019 IMO Problems/Problem 4|Solution]]<br />
<br />
==Problem 5==<br />
The Bank of Bath issues coins with an <math>H</math> on one side and a <math>T</math> on the other. Harry has <math>n</math> of these coins arranged in a line from left to right. He repeatedly performs the following operation:<br />
<br />
If there are exactly <math>k > 0</math> coins showing <math>H</math>, then he turns over the <math>k^{th}</math> coin from the left; otherwise, all coins show <math>T</math> and he stops. For example, if <math>n = 3</math> the process starting with the configuration <math>THT</math> would be <math>THT \rightarrow HHT \rightarrow HTT \rightarrow TTT</math>, which stops after three operations.<br />
<br />
(a) Show that, for each initial configuration, Harry stops after a finite number of operations.<br />
<br />
(b) For each initial configuration <math>C</math>, let <math>L(C)</math> be the number of operations before Harry stops. For<br />
example, <math>L(THT) = 3</math> and <math>L(TTT) = 0</math>. Determine the average value of <math>L(C)</math> over all <math>2^n</math><br />
possible initial configurations <math>C</math>.<br />
<br />
[[2019 IMO Problems/Problem 5|Solution]]<br />
<br />
==Problem 6==<br />
Let <math>I</math> be the incenter of acute triangle <math>ABC</math> with <math>AB \neq AC</math>. The incircle <math>\omega</math> of <math>ABC</math> is tangent to sides <math>BC</math>, <math>CA</math>, and <math>AB</math> at <math>D</math>, <math>E</math>, and <math>F</math>, respectively. The line through <math>D</math> perpendicular to <math>EF</math> meets ω again at <math>R</math>. Line <math>AR</math> meets ω again at <math>P</math>. The circumcircles of triangles <math>PCE</math> and <math>PBF</math> meet again at <math>Q</math>.<br />
Prove that lines <math>DI</math> and <math>PQ</math> meet on the line through <math>A</math> perpendicular to <math>AI</math>.<br />
<br />
[[2019 IMO Problems/Problem 6|Solution]]</div>Bissuehttps://artofproblemsolving.com/wiki/index.php?title=User:Natsuki&diff=147731User:Natsuki2021-02-22T16:30:57Z<p>Bissue: </p>
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<div>Hi!!! This is Natsuki :)<br />
<br />
<br />
<br />
<br />
hello - bussie</div>Bissuehttps://artofproblemsolving.com/wiki/index.php?title=User_talk:Piphi&diff=144244User talk:Piphi2021-02-01T00:45:17Z<p>Bissue: /* Other Users using this user page style */</p>
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<div>{{User:Piphi/Template:Header}}<br />
<br />
== Talk to me when I'm postbanned here ==<br />
hi there --[[User:Piphi|<font style="font-family: Segoe Print;font-size: 30px">πφ</font>]] 21:05, 9 January 2021 (EST)<br />
<br />
What happened? ~[https://www.youtube.com/channel/UCM5jK-6oKK2eXB2HHjZGnCw/ Kmath1234] 21:35, 10 January 2021 (EST)<br />
<br />
We got banned for the leaderboard scheme --[[User:Piphi|<font style="font-family: Segoe Print;font-size: 30px">πφ</font>]] 17:34, 11 January 2021 (EST)<br />
<br />
you can check it out here: [https://cdn.artofproblemsolving.com/attachments/4/1/a7051ede34dcda31c8318a644dc59c48253e5a.png], there's also a SS thread about it that's full of lies, [https://artofproblemsolving.com/community/c10h2412706 here]. There is no proof that we used bots, we just all had the day off and decided to post all day. We did nothing against the ToS so idk why we're even banned. Oh and btw for everyone who thinks that peace09 did this to frame me that's completely untrue. --[[User:Piphi|<font style="font-family: Segoe Print;font-size: 30px">πφ</font>]] 17:53, 11 January 2021 (EST)<br />
<br />
Oh, nice. That sucks though. ~[https://www.youtube.com/channel/UCM5jK-6oKK2eXB2HHjZGnCw/ Kmath1234] 19:59, 11 January 2021 (EST)<br />
<br />
For those of you who know ''The Hack'', go to room 2 right now. --[[User:Piphi|<font style="font-family: Segoe Print;font-size: 30px">πφ</font>]] 20:15, 11 January 2021 (EST)<br />
<br />
Nice greed control pick yesterday, you managed to pass jjaops :D [[User:Gca|Gca]] ([[User talk:Gca|talk]]) 11:11, 12 January 2021 (EST)<br />
<br />
Thanks, I'm basically at the same spot you were yesterday but now I'm past the halfway point! --[[User:Piphi|<font style="font-family: Segoe Print;font-size: 30px">πφ</font>]] 17:34, 12 January 2021 (EST)<br />
<br />
How long is your postban? [[User:Gca|Gca]] ([[User talk:Gca|talk]]) 12:14, 21 January 2021 (EST)<br />
<br />
idk, 2 weeks to a month probably. I think I might end GC 3rd, berrybear has probably found a mostly unbeatable formula and your number is probably better than mine. --[[User:Piphi|<font style="font-family: Segoe Print;font-size: 30px">πφ</font>]] 16:15, 21 January 2021 (EST)<br />
<br />
At least the postban gives you a nice break from community... You're definitely closing the gap between 2nd and 3rd. szheng0312 was trying to stop BerryBear, but I was talking to him the other day and he hasn't been keeping track of BerryBear's picks, so that probably factors into the 13 point lead he's racked up over the past 4-5 days. I should have taken a screenshot of when I was in first, but got prideful and thought I'd be able to stay there :P [[User:Gca|Gca]] ([[User talk:Gca|talk]]) 12:00, 22 January 2021 (EST)<br />
<br />
Niiice you're one point behind. I hope you win now :) [[User:Gca|Gca]] ([[User talk:Gca|talk]]) 01:23, 23 January 2021 (EST)<br />
<br />
Yes! 1st place! --[[User:Piphi|<font style="font-family: Segoe Print;font-size: 30px">πφ</font>]] 16:17, 24 January 2021 (EST)<br />
<br />
<br />
It's looking like you might manage to pull away from BerryBear assuming he doesnt get one of those insane <math>\tfrac{n}{1}</math> picks like he did after I got first... [[User:Gca|Gca]] ([[User talk:Gca|talk]]) 23:41, 25 January 2021 (EST)<br />
<br />
Wow, this is brilliant, wish I had thought of this when I was postbanned. Anyways congrats on causing some excitement around here,keep it up. - Piper<br />
<br />
Hi piphi! -Supernova283<br />
<br />
@GCA, huh, jjaops didn't pick a number yesterday. Maybe they're not using a bot but they've found a good formula?? Or maybe his bot stopped working?? idk<br />
@Supernova, hello!<br />
@piper hehe, thx<br />
--[[User:Piphi|<font style="font-family: Segoe Print;font-size: 30px">πφ</font>]] 20:43, 28 January 2021 (EST)<br />
<br />
idk if this will work but hello -Tethystide<br />
<br />
@piphi yeah it's extremely strange. He has never been picking every day which absolutely confuses me. It's almost as if he doesn't want to have too much attention on him by jumping out to a gigantic lead in the early midgame. They've found an extremely good formula and don't want to use it every day. Once you're unpostbanned (assuming it's before the end of the game) send me a PM so we can talk about it more [[User:Gca|Gca]] ([[User talk:Gca|talk]]) 22:10, 28 January 2021 (EST)<br />
<br />
== Other Users using this user page style ==<br />
# [[User:Edud_looc]]<br />
# [[User:Rusczyk]]<br />
# [[User:Aray10]]<br />
# [[User:OlympusHero]]<br />
# [[User:Nattie]]<br />
# [[User:Ssbgm9002]]<br />
# [[User:Rohanqv]]<br />
# [[User:Prabh1512]]<br />
# [[User:GoldAlicorn9]]<br />
# [[User:Niqhtwolf]]<br />
# [[User:IntelligentElephant2010]]<br />
# [[User:Objectz]]<br />
# [[User:Palaashgang]]<br />
# [[User:Cozzmo]]<br />
# [[User:Enderramsby]]<br />
# [[User:Sugar rush]]<br />
# [[User:bissue]]</div>Bissuehttps://artofproblemsolving.com/wiki/index.php?title=User:Bissue&diff=144240User:Bissue2021-02-01T00:43:34Z<p>Bissue: </p>
<hr />
<div>Here is where I'll fully "publish" mock tests I've written. So far there's only one, but there will definitely be more to come.<br />
<br />
I'll also add a user count here. It won't change whether the action occurs, but instead how I feel about the action.<br />
<br />
(am clout chaser)<br />
<br />
__NOTOC__<div style="border:2px solid black; -webkit-border-radius: 10px; background:#F0F2F3"><br />
<br />
==<font color="black" style="font-family: ITC Avant Garde Gothic Std, Verdana"><div style="margin-left:10px">User Count</div></font>==<br />
<br />
<div style="margin-left: 10px; margin-bottom:10px"><font color="black">Increase the user count below if this is your first time discovering this page.</font></div><br />
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<center><font size="100px"><br />
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0<br />
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</font></center><br />
</div><br />
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<br />
__NOTOC__<div style="border:2px solid black; -webkit-border-radius: 10px; background:#F0F2F3"><br />
<br />
==<font color="black" style="font-family: ITC Avant Garde Gothic Std, Verdana"><div style="margin-left:10px">User Count</div></font>==<br />
<br />
<div style="margin-left: 10px; margin-bottom:10px"><font color="black">Increase the user count below if this is your first time discovering this page <b>and</b> you have used these problems as practice.</font></div><br />
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<center><font size="100px"><br />
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0<br />
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---------------------<br />
<br />
==Apocalyptic AMC 8 (2020)==<br />
This contest took place between September 8th and November 6th. All problems were written by me <b>(bissue)</b> and testsolved by <b>nikenissan, bobthegod78, knightime1010, truffle, cw357,</b> and <b>ApraTrip.</b><br />
<br />
The mock was separated into two sections: The AMC 8 and The Tiebreakers.<br />
<br />
Standard AMC 8 rules were used. All participants had 40 minutes to complete as many of the 25 problems as they could. Correct answers were worth 1 point each, while incorrect or blank answers were worth 0 points each.<br />
<br />
The Tiebreakers were used to break ties between participants with the same score on the AMC 8. The rules were the same as those used for the ARML tiebreaker. For more information, see the original post in the AoPS Mock Contests Forum here:<br />
<br />
https://artofproblemsolving.com/community/c594864h2255517<br />
<br />
Full statistics and discussion threads can be found using the link above as well.<br />
<br />
==Problem 1==<br />
<br />
To walk up a single floor in her eighteen floor apartment building, Sarah needs to take nine steps up a flight of stairs. If Sarah starts on Floor <math>3</math> and walks up <math>100</math> steps, she would end up on the flight of stairs connecting which two floors?<br />
<br />
<math>\textbf{(A)} ~ \mbox{11 and 12} \qquad \textbf{(B)} ~ \mbox{12 and 13} \qquad \textbf{(C)} ~ \mbox{13 and 14} \qquad \textbf{(D)} ~ \mbox{14 and 15} \qquad \textbf{(E)} ~ \mbox{15 and 16}</math><br />
<br />
==Problem 2==<br />
<br />
Abby, Barb, and Carlos each have <math>35</math>, <math>42</math>, and <math>31</math> trading cards respectively. If they share their trading cards equally between each other, how many more trading cards would Carlos have than before?<br />
<br />
<math>\textbf{(A)} ~ 4 \qquad \textbf{(B)} ~ 5 \qquad \textbf{(C)} ~ 6 \qquad \textbf{(D)} ~ 9 \qquad \textbf{(E)} ~ 11</math><br />
<br />
==Problem 3==<br />
<br />
In triangle <math>ABC</math> the measure of angle <math>\angle A</math> is the average of the measures of angles <math>\angle B</math> and <math>\angle C</math>. What is the measure of angle <math>\angle A</math>?<br />
<br />
<math>\textbf{(A)} ~ 45^{\circ} \qquad \textbf{(B)} ~ 60^{\circ} \qquad \textbf{(C)} ~ 75^{\circ} \qquad \textbf{(D)} ~ 90^{\circ} \qquad \textbf{(E)} ~ 120^{\circ}</math><br />
<br />
==Problem 4==<br />
<br />
A spruce tree grows by <math>25</math> feet, increasing its height by <math>25 \%</math>. If the tree grows for a second time by <math>25</math> feet, by what percent would its height increase?<br />
<br />
<math>\textbf{(A)} ~ 5 \% \qquad \textbf{(B)} ~ 15 \% \qquad \textbf{(C)} ~ 20 \% \qquad \textbf{(D)} ~ 25 \% \qquad \textbf{(E)} ~ 30 \% </math><br />
<br />
==Problem 5==<br />
<br />
Find the sum of the digits of <math>\dfrac{5 \times 10^{2020}}{2}</math>.<br />
<br />
<math>\textbf{(A)} ~ 1 \qquad \textbf{(B)} ~ 2 \qquad \textbf{(C)} ~ 5 \qquad \textbf{(D)} ~ 7 \qquad \textbf{(E)} ~ 8</math><br />
<br />
==Problem 6==<br />
<br />
Square <math>B</math> with side length three is attached to a side of square <math>A</math> with side length four, as shown in the figure below. Find the area of the shaded region.<br />
<asy><br />
size(150);<br />
draw((0, 0)--(4, 0)--(4, 4)--(0, 4)--cycle);<br />
draw((4, 1)--(7, 1)--(7, 4)--(4, 4)--cycle);<br />
filldraw((0, 0)--(4, 0)--(4, 2.285714)--cycle, grey);<br />
filldraw((4, 1)--(7, 1)--(7, 4)--(4, 2.285714)--cycle, grey);<br />
label("A", (2, 2));<br />
label("B", (5.5, 2.5));<br />
</asy><br />
<math>\textbf{(A)} ~ 10 \qquad \textbf{(B)} ~ 10 \frac{1}{2} \qquad \textbf{(C)} ~ 11 \qquad \textbf{(D)} ~ 14 \qquad \textbf{(E)} ~ 14 \frac{1}{2}</math><br />
<br />
==Problem 7==<br />
<br />
When expressed as a decimal rounded to the nearest ten-thousandth, what is the value of <math>\dfrac{125+3}{125 \times 3}</math>?<br />
<br />
<math>\textbf{(A)} ~ 0.3412 \qquad \textbf{(B)} ~ 0.3413 \qquad \textbf{(C)} ~ 0.3414 \qquad \textbf{(D)} ~ 0.3415 \qquad \textbf{(E)} ~ 0.3416</math><br />
<br />
==Problem 8==<br />
<br />
What is the value of<br />
<cmath>(1+2+3)-(2+3+4)+(3+4+5)-\cdots -(98+99+100)?</cmath><br />
<math>\textbf{(A)} ~ -150 \qquad \textbf{(B)} ~ -147 \qquad \textbf{(C)} ~ -144 \qquad \textbf{(D)} ~ 147 \qquad \textbf{(E)} ~ 150</math><br />
<br />
==Problem 9==<br />
<br />
Kayla writes down the first <math>N</math> positive integers. What is the sum of all possible values of <math>N</math> if Kayla writes five multiples of <math>13</math> and six multiples of <math>12</math>?<br />
<br />
<math>\textbf{(A)} ~ 447 \qquad \textbf{(B)} ~ 453 \qquad \textbf{(C)} ~ 518 \qquad \textbf{(D)} ~ 525 \qquad \textbf{(E)} ~ 548</math><br />
<br />
==Problem 10==<br />
<br />
In Murphy's seventh grade homeroom, <math>\frac{7}{12}</math> of the students like tennis, <math>\frac{2}{3}</math> of the students like badminton, and <math>\frac{1}{12}</math> of the students like neither. What is the minimum possible number of students who like both tennis and badminton?<br />
<br />
<math>\textbf{(A)} ~ 1 \qquad \textbf{(B)} ~ 2 \qquad \textbf{(C)} ~ 3 \qquad \textbf{(D)} ~ 4 \qquad \textbf{(E)} ~ 6</math><br />
<br />
==Problem 11==<br />
<br />
For how many values of <math>N</math> does there exist a regular <math>N</math> sided polygon whose vertices all lie on the vertices of a regular <math>24</math> sided polygon?<br />
<br />
<math>\textbf{(A)} ~ 6 \qquad \textbf{(B)} ~ 7 \qquad \textbf{(C)} ~ 8 \qquad \textbf{(D)} ~ 9 \qquad \textbf{(E)} ~ 10</math><br />
<br />
==Problem 12==<br />
<br />
Quadrilateral <math>WXYZ</math> has its vertices on the sides of rectangle <math>ABCD</math> with <math>AB=7</math> and <math>BC=5</math>, as shown below. What is the area of quadrilateral <math>WXYZ</math>?<br />
<asy><br />
size(150);<br />
draw((0, 0)--(7, 0)--(7, 5)--(0, 5)--cycle);<br />
label("A", (0, 0), SW);<br />
label("B", (7, 0), SE);<br />
label("C", (7, 5), NE);<br />
label("D", (0, 5), NW);<br />
filldraw((0, 1)--(4, 0)--(7, 3)--(4, 5)--cycle, grey);<br />
label("W", (0, 1), W);<br />
label("X", (4, 0), S);<br />
label("Y", (7, 3), E);<br />
label("Z", (4, 5), N);<br />
label("4", (2, -0.5));<br />
label("3", (5.5, -0.5));<br />
label("4", (2, 5.5));<br />
label("3", (5.5, 5.5));<br />
</asy><br />
<math>\textbf{(A)} ~ 15 \dfrac{1}{2} \qquad \textbf{(B)} ~ 16 \qquad \textbf{(C)} ~ 16 \dfrac{1}{2} \qquad \textbf{(D)} ~ 17 \qquad \textbf{(E)} ~ 17 \dfrac{1}{2}</math><br />
<br />
==Problem 13==<br />
<br />
To drive to the supermarket, Mable drives for <math>m</math> miles, then drives <math>12</math> miles per hour faster for the remaining <math>\frac{4}{3}m</math> miles. The amount of time Mable spent driving at each of the two speeds was equal. What was Mable's average speed during her drive to the supermarket, in miles per hour?<br />
<br />
<math>\textbf{(A)} ~ \dfrac{81}{2} \qquad \textbf{(B)} ~ \dfrac{288}{7} \qquad \textbf{(C)} ~ 42 \qquad \textbf{(D)} ~ \dfrac{300}{7} \qquad \textbf{(E)} ~ 50</math><br />
<br />
==Problem 14==<br />
<br />
Six circles of radius one are cut out of the rectangle below. What is the area of the shaded region?<br />
<asy><br />
size(150);<br />
filldraw((0, 0)--(6, 0)--(6, 4)--(0, 4)--cycle, grey);<br />
filldraw(circle((1, 1), 1), white);<br />
filldraw(circle((3, 1), 1), white);<br />
filldraw(circle((5, 1), 1), white);<br />
filldraw(circle((1, 3), 1), white);<br />
filldraw(circle((3, 3), 1), white);<br />
filldraw(circle((5, 3), 1), white);<br />
</asy><br />
<math>\textbf{(A)} ~ 20-6\pi \qquad \textbf{(B)} ~ 24-6\pi \qquad \textbf{(C)} ~ 28-6\pi \qquad \textbf{(D)} ~ 30-6\pi \qquad \textbf{(E)} ~ 32-6\pi</math><br />
<br />
==Problem 15==<br />
<br />
One metronome beeps at a steady rate of <math>72</math> beeps per minute, while another metronome beeps at a steady rate of <math>96</math> beeps per minute. If both metronomes beep at the same time once, how long will it take, in seconds, until they first beep at the same time again?<br />
<br />
<math>\textbf{(A)} ~ 2 \dfrac{1}{2} \qquad \textbf{(B)} ~ 5 \qquad \textbf{(C)} ~ 10 \qquad \textbf{(D)} ~ 18 \qquad \textbf{(E)} ~ 24</math><br />
<br />
==Problem 16==<br />
<br />
A square with side length two is placed on a table, forming a <math>30</math> degree angle with the table's surface. How much higher is the top vertex of the square than the table?<br />
<asy><br />
size(150);<br />
draw((0, 0)--(0.882, 0.4714)--(0.4106, 1.3534)--(-0.4714, 0.882)--cycle);<br />
draw((-0.5, 0)--(1, 0), linewidth(3));<br />
draw((-0.75, 1.3534)--(-0.65, 1.3534));<br />
draw((-0.7, 1.3534)--(-0.7, 0));<br />
draw((-0.75, 0)--(-0.65, 0));<br />
</asy><br />
<math>\textbf{(A)} ~ \dfrac{5}{2} \qquad \textbf{(B)} ~ \sqrt{3}+1 \qquad \textbf{(C)} ~ \dfrac{4\sqrt{3}}{3} \qquad \textbf{(D)} ~ 3 \qquad \textbf{(E)} ~ \dfrac{3\sqrt{3}}{2}+1</math><br />
<br />
==Problem 17==<br />
<br />
Kurtis' school schedule is made up of four classes, followed by lunch, followed by three more classes. In how many ways can Kurtis arrange his schedule if two of his classes (Reading and Writing) must occur one immediately after the other?<br />
<br />
<math>\textbf{(A)} ~ 600 \qquad \textbf{(B)} ~ 840 \qquad \textbf{(C)} ~ 1200 \qquad \textbf{(D)} ~ 1440 \qquad \textbf{(E)} ~ 1680</math><br />
<br />
==Problem 18==<br />
<br />
When the number <math>25</math> is added to a list of numbers with total sum <math>S</math>, the average of all the numbers increases by one. What is the sum of the digits of the greatest possible value of <math>S</math>?<br />
<br />
<math>\textbf{(A)} ~ 6 \qquad \textbf{(B)} ~ 7 \qquad \textbf{(C)} ~ 8 \qquad \textbf{(D)} ~ 9 \qquad \textbf{(E)} ~ 12</math><br />
<br />
==Problem 19==<br />
<br />
A magician randomly picks a three digit positive integer to put into her hat and pulls out the same number with its digits in reverse order. (For example <math>496</math> would become <math>694</math> and <math>720</math> would become <math>27</math>.) What is the probability the magician pulls out a multiple of <math>22</math>?<br />
<br />
<math>\textbf{(A)} ~ \dfrac{1}{15} \qquad \textbf{(B)} ~ \dfrac{1}{18} \qquad \textbf{(C)} ~ \dfrac{1}{20} \qquad \textbf{(D)} ~ \dfrac{1}{25} \qquad \textbf{(E)} ~ \dfrac{1}{30}</math><br />
<br />
==Problem 20==<br />
<br />
Tyrone has three books to read in six days. He reads one-half of a single book every day. In how many ways can he finish all the books if he may not read the same book two days in a row?<br />
<br />
<math>\textbf{(A)} ~ 12 \qquad \textbf{(B)} ~ 18 \qquad \textbf{(C)} ~ 24 \qquad \textbf{(D)} ~ 30 \qquad \textbf{(E)} ~ 36</math><br />
<br />
==Problem 21==<br />
<br />
There exists a circle that is tangent to <math>\overline{AB}</math> and <math>\overline{BC}</math> at <math>A</math> and <math>C</math>, respectively. If <math>AB=BC=13</math> and <math>AC=10</math>, what is the radius of the circle?<br />
<asy><br />
size(150);<br />
draw((-5, 0)--(5, 0)--(0, -12)--cycle);<br />
draw(circle((0, 2.08333), 5.41666));<br />
label("A", (-5, 0), W);<br />
label("C", (5, 0), E);<br />
label("B", (0, -12), S);<br />
label("13", (-2.7, -6), W);<br />
label("13", (2.7, -6), E);<br />
label("10", (0, 0.2), N);<br />
</asy><br />
<math>\textbf{(A)} ~ \dfrac{60}{13} \qquad \textbf{(B)} ~ 5 \qquad \textbf{(C)} ~ \dfrac{26}{5} \qquad \textbf{(D)} ~ \dfrac{65}{12} \qquad \textbf{(E)} ~ \dfrac{156}{25}</math><br />
<br />
==Problem 22==<br />
<br />
For each of the distinct sets of numbers containing only positive integers between <math>1</math> and <math>9</math> inclusive, Jordan writes the sum of the numbers in that set. What is the sum of the numbers Jordan writes?<br />
<br />
<math>\textbf{(A)} ~ 11520 \qquad \textbf{(B)} ~ 11565 \qquad \textbf{(C)} ~ 11610 \qquad \textbf{(D)} ~ 11655 \qquad \textbf{(E)} ~ 11700</math><br />
<br />
==Problem 23==<br />
<br />
In rectangle <math>ABCD</math>, the perpendicular from <math>B</math> to diagonal <math>\overline{AC}</math> bisects segment <math>\overline{CD}</math>. Which of the following is closest to <math>\frac{AB}{BC}</math>?<br />
<br />
<math>\textbf{(A)} ~ \dfrac{5}{4} \qquad \textbf{(B)} ~ \dfrac{4}{3} \qquad \textbf{(C)} ~ \dfrac{7}{5} \qquad \textbf{(D)} ~ \dfrac{3}{2} \qquad \textbf{(E)} ~ \dfrac{8}{5}</math><br />
<br />
==Problem 24==<br />
<br />
How many ordered triples of positive integers <math>(a, b, c)</math> satisfy <math>\text{gcd}(a, b, c)=20</math> and <math>\text{lcm}(a, b, c)=240</math>?<br />
<br />
<math>\textbf{(A)} ~ 18 \qquad \textbf{(B)} ~ 24 \qquad \textbf{(C)} ~ 36 \qquad \textbf{(D)} ~ 54 \qquad \textbf{(E)} ~ 72 </math><br />
<br />
==Problem 25==<br />
<br />
Cheyanne rolls two standard six sided dice, then repeatedly rerolls all dice which show an odd number and stops as soon as all dice show an even number. What is the probability Cheyanne stops after exactly four rounds of rerolling?<br />
<br />
<math>\textbf{(A)} ~ \dfrac{61}{1024} \qquad \textbf{(B)} ~ \dfrac{1}{16} \qquad \textbf{(C)} ~ \dfrac{67}{1024} \qquad \textbf{(D)} ~ \dfrac{9}{128} \qquad \textbf{(E)} ~ \dfrac{29}{256}</math><br />
<br />
------------------<br />
==Tiebreaker Problem 1==<br />
<br />
A whiteboard has positive real numbers <math>1</math> and <math>m</math> written on it. Every second, if the numbers <math>x</math> and <math>y</math> are on the whiteboard, a ghost will replace those numbers with <math>|x^2-y^2|</math> and <math>2xy</math>. The ghost stops once one number on the whiteboard is <math>m</math> times the other. For how many positive real numbers <math>m</math> does the ghost stop after exactly <math>16</math> seconds?<br />
<br />
==Tiebreaker Problem 2==<br />
<br />
The perpendicular bisectors of triangle <math>ABC</math> can be described in the coordinate plane as lines <math>y=0</math>, <math>y=x</math>, and <math>y=\sqrt{3}x</math>. Given that triangle <math>ABC</math> has circumradius <math>1</math>, find its area.<br />
<br />
==Tiebreaker Problem 3==<br />
<br />
The diagram below is constructed by attaching an equilateral triangle, a square, a regular pentagon, and a regular hexagon together. Compute the measure of the obtuse angle formed by the three red vertices.<br />
<asy><br />
import graph; size(10cm); <br />
real labelscalefactor = 0.5; /* changes label-to-point distance */<br />
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ <br />
pen dotstyle = black; /* point style */ <br />
real xmin = -29.36, xmax = -9.8, ymin = 4.78, ymax = 17.66; /* image dimensions */<br />
<br />
draw((-22,12)--(-19,12)--(-20.5,14.598076211353318)--cycle, linewidth(2)); <br />
draw((-20.5,14.598076211353318)--(-19,12)--(-16.401923788646684,13.5)--(-17.90192378864668,16.098076211353316)--cycle, linewidth(2)); <br />
draw((-16.401923788646684,13.5)--(-19,12)--(-18.376264927546725,9.065557197798585)--(-15.392699241441907,8.751971807995622)--(-14.172489312214504,11.492608180923423)--cycle, linewidth(2)); <br />
draw((-18.376264927546725,9.065557197798585)--(-19,12)--(-21.85316954888546,12.927050983124847)--(-24.082604025317647,10.919659164048271)--(-23.45886895286437,7.985216361846854)--(-20.60569940397891,7.058165378722009)--cycle, linewidth(2)); <br />
Label laxis; laxis.p = fontsize(10); <br />
string blank(real x) {return "";} <br />
xaxis(xmin, xmax, Ticks(laxis, blank, Step = 1, Size = 2, NoZero),EndArrow(6), above = true); <br />
yaxis(ymin, ymax, Ticks(laxis, blank, Step = 1, Size = 2, NoZero),EndArrow(6), above = true); /* draws axes; NoZero hides '0' label */ <br />
/* draw figures */<br />
draw((-22,12)--(-19,12), linewidth(2)); <br />
draw((-19,12)--(-20.5,14.598076211353318), linewidth(2)); <br />
draw((-20.5,14.598076211353318)--(-22,12), linewidth(2)); <br />
draw((-20.5,14.598076211353318)--(-19,12), linewidth(2)); <br />
draw((-19,12)--(-16.401923788646684,13.5), linewidth(2)); <br />
draw((-16.401923788646684,13.5)--(-17.90192378864668,16.098076211353316), linewidth(2)); <br />
draw((-17.90192378864668,16.098076211353316)--(-20.5,14.598076211353318), linewidth(2)); <br />
draw((-16.401923788646684,13.5)--(-19,12), linewidth(2)); <br />
draw((-19,12)--(-18.376264927546725,9.065557197798585), linewidth(2)); <br />
draw((-18.376264927546725,9.065557197798585)--(-15.392699241441907,8.751971807995622), linewidth(2)); <br />
draw((-15.392699241441907,8.751971807995622)--(-14.172489312214504,11.492608180923423), linewidth(2)); <br />
draw((-14.172489312214504,11.492608180923423)--(-16.401923788646684,13.5), linewidth(2)); <br />
draw((-18.376264927546725,9.065557197798585)--(-19,12), linewidth(2)); <br />
draw((-19,12)--(-21.85316954888546,12.927050983124847), linewidth(2)); <br />
draw((-21.85316954888546,12.927050983124847)--(-24.082604025317647,10.919659164048271), linewidth(2)); <br />
draw((-24.082604025317647,10.919659164048271)--(-23.45886895286437,7.985216361846854), linewidth(2)); <br />
draw((-23.45886895286437,7.985216361846854)--(-20.60569940397891,7.058165378722009), linewidth(2)); <br />
draw((-20.60569940397891,7.058165378722009)--(-18.376264927546725,9.065557197798585), linewidth(2)); <br />
/* dots and labels */<br />
dot((-22,12),red); <br />
dot((-19,12),red); <br />
dot((-21.526119073220972,12.820785841919117),linewidth(4pt) + dotstyle+red); <br />
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); <br />
</asy><br />
------------------------------<br />
==Answer Key==<br />
<br />
AMC 8: DBBCD / CBBAD / AECBA / BCDDD / DACEA<br />
<br />
Tiebreakers: (<math>65280</math>, <math>\dfrac{3-\sqrt{3}}{4}</math>, <math>102</math>)</div>Bissuehttps://artofproblemsolving.com/wiki/index.php?title=Mock_AMC&diff=142987Mock AMC2021-01-22T20:50:08Z<p>Bissue: fixed a formatting issue (which has somehow been here for a while? why has nobody fixed this yet?)</p>
<hr />
<div>A '''Mock AMC''' is a contest intended to mimic an actual [[AMC]] (American Mathematics Competitions 8, 10, or 12) exam. A number of '''Mock AMC''' competitions have been hosted on the [[Art of Problem Solving]] message boards. They are generally made by one community member and then administered for any of the other community members to take. Sometimes, the administrator may ask other people to sign up to write problems for the contest.<br />
<br />
Mock AMCs are usually very popular in the months leading up to the actual [[AMC]] competition. There is no guarantee that community members will make Mock AMCs in any given year, but there probably will be one.<br />
<br />
Feel free to remove mock tests that are not high quality or to recommend others.<br />
<br />
Ongoing mock AMCs, as well as other mock contests, can be found in the [https://artofproblemsolving.com/community/c594864_aops_mock_contests AoPS Mock Contests forum].<br />
<br />
== Tips for Writing a Mock AMC ==<br />
Anyone can write a Mock AMC and administer it. If you are interested in writing one, here are some tips:<br />
<br />
* Look at past [[AMC]]/[[AHSME]] tests to get a feel for what kind of problems you should write and what difficulty level they should be.<br />
* Look at famous theorems and formulas and see if there's any way you can make a good problem out of them.<br />
* If you are running out of creative juice and decide to pull problems from contests, try using problems from obscure contests first, if possible. This way, even the more experienced test takers will hopefully find problems that they do not already know how to do.<br />
* Pair up with another user on AoPS and write it together. Two minds are much better than one. With just one person, the problems may be biased toward one subject, but with two people, the chances of this happening are smaller.<br />
<br />
== Past Mock AMCs ==<br />
<br />
Listed below are the [higher-quality] Mock AMCs that have been hosted over AoPS in the past. Feel free to add, remove, or recommend.<br />
<br />
Note that the "level" column represents the originally intended difficulty. In other words, if a person makes a mock [[AMC 12]], the level would be "12", even if the problems themselves are much easier. Recommended mock tests are starred.<br />
<br />
=== Mock AMC 12 ===<br />
<br />
{| class="wikitable" style="text-align:center;width:100%"<br />
|-<br />
|<br />
! scope="col" | '''Author'''<br />
! scope="col" | '''Year'''<br />
! scope="col" | '''Initial Discussion'''<br />
! scope="col" | '''Problems'''<br />
! scope="col" width=80 | '''Answers'''<br />
! scope="col" | '''Results/Discussion'''<br />
|-<br />
! scope="row" | '''Mock AMC #1'''<br />
| mathfanatic<br />
| 2003<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=9321 Initial Discussion]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=9353 Problems]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=9572 1-5]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?t=9573 6-10]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?t=9574 11-15]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?t=9575 16-20]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?t=9576 21-25]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=9365 Results]<br />
|-<br />
! scope="row" | '''Mock AMC #2'''<br />
| mathfanatic<br />
| 2004<br />
| [http://www.artofproblemsolving.com/community/c5h10497p67837 Initial Discussion]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=10497 Problems]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=10497 Solutions]<br />
| n/a<br />
|-<br />
! scope="row" | '''Mock AMC A'''<br />
| JSRosen3<br />
| 2004<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=14138 Initial Discussion]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=14361 Problems]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=102483#p102483 Answers] [http://www.artofproblemsolving.com/community/c5h14516 Solutions]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=14489 Results / Discussion]<br />
|-<br />
! scope="row" | '''Mock AMC B'''<br />
| beta<br />
| 2004<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=14735 Initial Discussion]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=14764 Problems]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=14894 Answers] [http://www.artofproblemsolving.com/community/c5h14884 Solutions]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=105741#p105741 Results / Discussion]<br />
|-<br />
! scope="row" | '''Mock AMC C'''<br />
| JGeneson<br />
| 2004<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=15001 Initial Discussion]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=15134 Problems]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=15251 Answers]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=15251 Results / Discussion]<br />
|-<br />
! scope="row" | '''Mock AMC D'''<br />
| joml88<br />
| 2004<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=16886 Initial Discussion]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=17888 Problems]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=17891 Answers]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=17891 Results / Discussion]<br />
|-<br />
! scope="row" | '''Mock AMC E'''<br />
| Silverfalcon<br />
| 2004<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=21997 Initial Discussion]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=22141 Problems]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=22344 Answers]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=22344 Results / Discussion]<br />
|-<br />
! scope="row" | '''Mock AMC F'''<br />
| joml88<br />
| 2005<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=22049 Initial Discussion]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=23163 Problems]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=23177 Answers]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=23177 Results / Discussion]<br />
|-<br />
! scope="row" | '''Mock AMC G'''<br />
| Lucky707<br />
| 2005<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=24355 Initial Discussion]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=24974 Problems]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=25087 Answers]<br />
| [http://www.artofproblemsolving.com/community/c5h25087p157983 Results / Discussion]<br />
|-<br />
! scope="row" | '''Mock AMC H'''<br />
| Silverfalcon<br />
| 2005<br />
| [http://www.artofproblemsolving.com/community/c5h24437p154514 Initial Discussion]<br />
| [http://www.artofproblemsolving.com/community/c5h24437p154514 Problems]<br />
| throughout thread<br />
| n/a<br />
|-<br />
! scope="row" | '''Mock AMC I'''<br />
| white_horse_king88<br />
| 2005<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=21280 Initial Discussion]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=25181 Problems]<br />
| n/a<br />
| n/a<br />
|-<br />
! scope="row" | '''Mock AMC J'''<br />
| Silverfalcon<br />
| 2005<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=47625 Initial Discussion]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=48129 Problems]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=304246#p304246 Answers]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=48132 Results / Discussion]<br />
|-<br />
! scope="row" | '''Mock AMC K'''<br />
| amirhtlusa<br />
| 2005<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=49958 Initial Discussion]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=50515 Problems]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=321102#p321102 Answers]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=321102#p321102 Results / Discussion]<br />
|-<br />
! scope="row" | '''Mock AMC L'''<br />
| amirhtlusa<br />
| 2005<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=61330 Initial Discussion]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=63041 Problems]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=63258 Answers]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=63258 Results / Discussion]<br />
|-<br />
! scope="row" | '''Mock AMC M'''<br />
| Silverfalcon<br />
| 2006<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=63542 Initial Discussion]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=78982 Problems]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=79749 Answers]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=79749 Discussion]<br />
|-<br />
! scope="row" | '''Mock AMC N'''<br />
| chess64<br />
| 2006<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=98894 Initial Discussion]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=99307 Problems]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=99344 Answers]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=99344 Results] / [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=99566 Discussion]<br />
|-<br />
! scope="row" | '''Mock AMC O'''<br />
| mustafa<br />
| 2006<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=121312 Initial Discussion]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=122126 Problems]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=126071 Answers]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=126071 Results/Discussion]<br />
|-<br />
! scope="row" | '''Mock AMC P'''<br />
| Anirudh<br />
| 2006<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=125029 Initial Discussion]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=709655#p709655 Problems]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=716240#p716240 Answers]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=716262#p716262 Results]/[http://www.artofproblemsolving.com/Forum/viewtopic.php?t=125029 Discussion]<br />
|-<br />
! scope="row" | '''Mock AMC Q'''<br />
| calc rulz<br />
| 2006<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=125194 Initial Discussion]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=125886 Problems]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=128102 Answers]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=128102 Results/Discussion]<br />
|-<br />
! scope="row" | '''Mock AMC R'''<br />
| rnwang2<br />
| 2006<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=126107 Initial Discussion]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=715597#p715597 Problems]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=127421 Answers]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=127421 Results/Discussion]<br />
|-<br />
! scope="row" | '''Mock AMC S'''<br />
| mysmartmouth<br />
| 2007<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=127221 Initial Discussion]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=128689 Problems]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=131403 Answers]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=131403 Results/Discussion]<br />
|-<br />
! scope="row" | '''Mock AMC T'''<br />
| paladin8<br />
| 2007<br />
| [http://www.artofproblemsolving.com/community/c5h127979p726003 Initial Discussion]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=127979 Problems]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=129159 Answers]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=129159 Results/Discussion]<br />
|-<br />
! scope="row" | '''Mock AMC U'''<br />
| Silverfalcon<br />
| 2008<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=184067 Initial Discussion]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=185233 Problems]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=185238 Answers]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=185236 Results]/[http://www.artofproblemsolving.com/Forum/viewtopic.php?t=185238 Discussion]<br />
|-<br />
! scope="row" | '''Mock AMC V'''<br />
| gfour84<br />
| 2009<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=298452 Initial Discussion]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=1634175#p1634175 Problems]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=1637429#p1637429 Answers]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=1637006#p1637006 Results]/[http://www.artofproblemsolving.com/Forum/viewtopic.php?t=302030 Discussion]<br />
|-<br />
! scope="row" | '''Mock AMC 12'''<br />
| MathTwo<br />
| 2010<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=319184 Initial Discussion]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=1759276#p1759276 Problems]<br />
| n/a<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=1778080#p1778080 Results]/[http://www.artofproblemsolving.com/Forum/viewtopic.php?t=328510 Discussion]<br />
|-<br />
! scope="row" | '''Mock AMC 2/11'''<br />
| Caelestor<br />
| 2011<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=390907 Initial Discussion]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=2183293#p2183293 Problems]<br />
| n/a<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=390907 Discussion]<br />
|-<br />
! scope="row" | '''Mock AMC 1/12'''<br />
| Lord.of.AMC<br />
| 2012<br />
| [http://www.artofproblemsolving.com/community/c5h456321p2563731 Initial Discussion]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=2605537#p2605537 Problems]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=2610048#p2610048 Answers]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=2610048#p2610048 Results]<br />
|-<br />
! scope="row" | '''Mock AMC 12'''<br />
| Diehard<br />
| 2012<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=459149 Initial Discussion]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=459710 Problems]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=2580327#p2580327 Answers]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=459710 Discussion]<br />
|-<br />
! scope="row" | '''Mock AMC 12 2012'''<br />
| python123<br />
| 2012<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=456256 Initial Discussion]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=2576205#p2576205 Problems]<br />
| [http://www.artofproblemsolving.com/community/c5h456256p2580610 Answers]<br />
| [http://www.artofproblemsolving.com/community/c5h456256p2580610 Results / Discussion]<br />
|-<br />
! scope="row" | '''Mock AMC A'''<br />
| Binomial-theorem<br />
| 2012<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=473867 Initial Discussion]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=2692082#p2692082 Problems]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=2717578#p2717578 Solutions]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=2707954#p2707954 Results]<br />
|-<br />
! scope = "row" | '''Almost 2016 Mock AMC 11.5'''<br />
| whatshisbucket<br />
| 2015<br />
| [http://www.artofproblemsolving.com/community/c5h1175087p5665743 Initial Discussion]<br />
| [http://www.artofproblemsolving.com/community/c5h1175087p5665743 Problems]<br />
| [http://artofproblemsolving.com/community/c209194_almost_2016_mock_amc_11.5 Solutions]<br />
| [http://www.artofproblemsolving.com/community/c5h1175087p5665743 Results / Discussion]<br />
|-<br />
! scope="row" | '''hnkevin42 Mock AMC 12'''<br />
| hnkevin42<br />
| 2016<br />
| [http://www.artofproblemsolving.com/community/c5h1187484p5778754 Initial Discussion]<br />
| [http://www.artofproblemsolving.com/community/c5h1187484p5778754 Problems]<br />
| [http://www.artofproblemsolving.com/community/c5h1187484p5913294 Answers]<br />
| [http://www.artofproblemsolving.com/community/c5h1187484p5778754 Results/Discussion]<br />
|-<br />
! scope="row" | '''Last-Minute Mock AMC 12'''<br />
| CountofMC<br />
| 2018<br />
| [https://artofproblemsolving.com/community/c594864h1582044_lastminute_mock_amc_12 Initial Discussion]<br />
| [https://artofproblemsolving.com/community/c594864h1582044_lastminute_mock_amc_12 Problems]<br />
| [https://artofproblemsolving.com/community/c594864h1582044_lastminute_mock_amc_12 Answers]<br />
| [https://artofproblemsolving.com/community/c4t334444f4_lastminute_mock_amc_12 Results/Discussion]<br />
|-<br />
!scope="row" | Christmas Math Competitions Year 2*<br />
| CMC Committee<br />
| 2018-2019<br />
| [https://artofproblemsolving.com/community/c594864h1747367_aime_ii_released_christmas_mathematics_competition_cmc_year_2 Initial Discussion]<br />
| [https://drive.google.com/file/d/1uS3YqAK10jB1RkCQCLQgeBHUACydJph-/view CMC 12A] [https://drive.google.com/file/d/1txKy-MfZCPPnUNTLCIad8dbatV-EGu71/view CMC 12B]<br />
| [https://artofproblemsolving.com/community/c798404h1760024_cmc_10a12a_amp_10b12b_year_2__problems_and_answer_keys Answers]<br />
| [https://artofproblemsolving.com/community/c594864h1747367p11379904 Results] / [https://artofproblemsolving.com/community/c798404_christmas_mathematics_competitions_year_2 Discussion]<br />
|-<br />
!scope="row" | 2019 Mock AMC 12B*<br />
| fidgetboss_4000<br />
| 2019<br />
| [https://artofproblemsolving.com/community/c594864h1897633p13160252 Initial Discussion]<br />
| [https://artofproblemsolving.com/community/c594864h1897633p13160252 Problems] (or click here: [[Mock AMC 12B Problems]]<br />
| [https://artofproblemsolving.com/community/c963448h1919463_answer_keys_d Answers]<br />
| N/A / N/A<br />
|-<br />
!scope="row" | Christmas Math Competitions Year 3*<br />
| CMC Committee<br />
| 2019-2020<br />
| [https://artofproblemsolving.com/community/c594864h1967029_cime_ii_released_christmas_math_competition_cmc_year_3 Initial Discussion]<br />
| [http://cmc.ericshen.net/CMC-2020/CMC-2020-12A.pdf CMC 12A] [http://cmc.ericshen.net/CMC-2020/CMC-2020-12B-booklet.pdf CMC 12B]<br />
| [https://artofproblemsolving.com/community/c1035147h1980361p13760203 Answers]<br />
| [https://artofproblemsolving.com/community/c594864h1967029p13623910 Results] / [https://artofproblemsolving.com/community/c1035147_christmas_mathematics_competitions_year_3 Discussion]<br />
|-<br />
!scope="row" | January 2020 Mock AMC 10/12<br />
| P_Groudon<br />
| 2020<br />
| <br />
[https://artofproblemsolving.com/community/c594864h1984167_concluded_january_2020_mock_amc_1012 Initial Discussion]<br />
| [https://artofproblemsolving.com/community/c594864h1984167_concluded_january_2020_mock_amc_1012 Problems]<br />
| [https://artofproblemsolving.com/community/c1035224h1984207_january_2020_answer_keys Answers]<br />
| [https://artofproblemsolving.com/community/c594864h1984167p13963794 Results]<br />
|-<br />
!scope="row" | OTSS 2020<br />
| kevinmathz<br />
| 2020<br />
| <br />
[https://artofproblemsolving.com/community/c594864h2066002_otss_olympiad_test_spring_series Initial Discussion]<br />
| [https://drive.google.com/file/d/1JfuEvDGw9i99XeQYACPUDzHebxyrcmtE/edit TMC 10A] [https://drive.google.com/file/d/1FkKjAgehDwpmmNOj_mjvYuCCV1r7p08p/edit TMC 12A]<br />
| [https://artofproblemsolving.com/community/c1130807h2094651_tmc_1012_a_solutions Answers]<br />
| [https://artofproblemsolving.com/community/c1130807h2081571_links_to_mocks_solutions_and_leaderboards Results]<br />
|-<br />
!scope="row" | FMC 2020<br />
| fidgetboss_4000<br />
| 2020<br />
| <br />
[https://artofproblemsolving.com/community/c594864h2069998 Initial Discussion]<br />
| [https://drive.google.com/file/d/17jscyJzVCFDlV6YL0OpPKczyk04Tltsx/view FMC 10A] [https://drive.google.com/file/d/1pMdXlGy9F5hbWSP8CGANAhAA9_ct7pdS/view FMC 12A]<br />
| [https://artofproblemsolving.com/community/c1177836_2020_fmc_public_discussion_forum_uwu Answers]<br />
| [https://artofproblemsolving.com/community/q2h2069998p15525666 Results]<br />
|-<br />
!scope="row" | 2020 Mock AMC 12<br />
| scrabbler94<br />
| 2020<br />
| <br />
[https://artofproblemsolving.com/community/c594864h2159345_contest_over_2020_mock_amc_1012_scrabbler94 Initial Discussion]<br />
| [https://artofproblemsolving.com/community/c594864h2159345_contest_over_2020_mock_amc_1012_scrabbler94 Problems]<br />
| [https://artofproblemsolving.com/community/c594864h2159345p16460394 Answers]<br />
| [https://artofproblemsolving.com/community/c594864h2159345p16460394 Results]<br />
|-<br />
|}<br />
<br />
=== Mock AMC 10 ===<br />
<br />
{| class="wikitable" style="text-align:center;width:100%"<br />
|-<br />
|<br />
! scope="col" | '''Author'''<br />
! scope="col" | '''Year'''<br />
! scope="col" | '''Initial Discussion'''<br />
! scope="col" | '''Problems'''<br />
! scope="col" | '''Answers'''<br />
! scope="col" | '''Results/Discussion'''<br />
|-<br />
! scope="row" | '''Mock AMC 10'''<br />
| #H34N1<br />
| 2008<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=212730 Initial Discussion]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=214081 Problems]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=213727 Answers]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=213732 Results/Discussion]<br />
|-<br />
! scope="row" | '''agent's Mock AMC 10'''<br />
| agentcx<br />
| 2009<br />
| [http://www.artofproblemsolving.com/community/c5h331823p1775678 Initial Discussion]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=1781208#p1781208 Problems]<br />
| n/a<br />
| [http://www.artofproblemsolving.com/community/c5h331823p1782769 Results / Discussion]<br />
|-<br />
! scope="row" | '''Mock AMC 10 Set'''<br />
| AwesomeToad<br />
| 2010<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=311120 Initial Discussion]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=1762829#p1762829 Problems]<br />
| n/a<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=1778740#p1778740 Results]<br />
|-<br />
! scope="row" | '''Mock AMC 10/12'''<br />
| djmathman<br />
| 2013<br />
| [http://www.artofproblemsolving.com/community/c5h556673s1_first_mock_amcs_of_the_20132014_season Initial Discussion]<br />
| [http://www.artofproblemsolving.com/community/c5h556673p3294854 Problems]<br />
| [http://www.artofproblemsolving.com/community/c5h556673p3324794 Answers]<br />
| [http://www.artofproblemsolving.com/community/c5h556673p3324794 Results / Discussion]<br />
|-<br />
! scope="row" | '''Mock AMC 10 2014-2015*'''<br />
| AlcumusGuy<br />
| 2014<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=618080&hilit=AlcumusGuy%27s+mock+amc+10 Initial Discussion]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=618080&hilit=AlcumusGuy%27s+mock+amc+10 Problems]<br />
| [http://www.artofproblemsolving.com/community/c5h618080p3710795 Answers/Results]<br />
| [http://artofproblemsolving.com/community/c56018 Problem Discussion]<br />
|-<br />
! scope="row" | '''Kelvin the Frog v2015'''<br />
| BOGTRO<br />
| 2015<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=623373&hilit=Kelvin+the+frog Initial Discussion]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=623373&hilit=Kelvin+the+frog Problems]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=3731642#p3731642 Answers]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=3731642#p3731642 Results]<br />
|-<br />
! scope="row" | '''Mock AMC 10/12'''<br />
| joey8189681<br />
| 2015<br />
| [http://www.artofproblemsolving.com/community/c5h624714p3741743 Initial Discussion]<br />
| [https://www.dropbox.com/s/1c8hhl6yt2awpix/Mock%20AMC%2010.pdf?dl=0 Problems]<br />
| [http://www.artofproblemsolving.com/community/c5h624714p3754593 Answers]<br />
| [http://www.artofproblemsolving.com/community/q2h626092p3756574 Results]<br />
|-<br />
! scope="row" | '''May Mock AMC 10 Contest'''<br />
| azmath333<br />
| 2015<br />
| [http://www.artofproblemsolving.com/community/c5h1082684_mock_amc_1012_contest Initial Discussion]<br />
| [http://www.artofproblemsolving.com/community/c5h1082684_mock_amc_1012_contest Problems]<br />
| [http://www.artofproblemsolving.com/community/c5h1082684p4812575 Answers] [http://www.artofproblemsolving.com/community/c5h1082684p4825662 Solutions]<br />
| [http://www.artofproblemsolving.com/community/c5h1082684p4812575 Results / Discussion] <br />
|-<br />
! scope="row" | '''2015 Mock AMC 10*'''<br />
| droid347<br />
| 2015<br />
| [http://www.artofproblemsolving.com/community/c5h1094505p4893513 Initial Discussion]<br />
| [http://www.artofproblemsolving.com/community/c5h1094505p4893513 Problems]<br />
| [http://www.artofproblemsolving.com/community/c5h1094505p5051006 Answers]<br />
| [http://www.artofproblemsolving.com/community/c5h1094505p5045295 Results / Discussion]<br />
|-<br />
! scope="row" | '''July Mock AMC 10 Contest'''<br />
| akshaygowrish<br />
| 2015<br />
| [http://artofproblemsolving.com/community/c5h1104125_july_mock_amc_10_contest Initial Discussion]<br />
| [https://docs.google.com/document/d/12OXd-mmS_T7SyMcqw1hfvykNlcj-4TR4mMXlRJ09uBg/edit?usp=sharing Problems]<br />
| [https://docs.google.com/document/d/1-Q_w5XNKcRaffvpCLSdoJ51TpRpcDAgYLmV9OlczBCs/edit?usp=sharing Answer Key]<br />
| [http://artofproblemsolving.com/community/c5h1104125p5154283 Results / Discussion]<br />
|-<br />
! scope = "row" |'''August Mock AMC 10'''<br />
| azmath333<br />
| 2015<br />
| [http://www.artofproblemsolving.com/community/c5h1115462_august_mock_amc_10 Initial Discussion]<br />
| [http://latex.artofproblemsolving.com/miscpdf/augustmockamc10.pdf Problems]<br />
| [http://www.artofproblemsolving.com/community/c5h1115462p5270430 Answers]<br />
| [http://www.artofproblemsolving.com/community/c5h1115462p5270430 Results] / [http://www.artofproblemsolving.com/community/c121880_august_mock_amc_10_discussion_forum Discussion]<br />
|-<br />
! scope = "row" | '''September Mock AMC 10'''<br />
| cpma213<br />
| 2015<br />
| [http://artofproblemsolving.com/community/c5h1137447p5320386 Initial Discussion]<br />
| [http://artofproblemsolving.com/community/c5h1137447p5320386 Problems]<br />
| n/a<br />
| [http://artofproblemsolving.com/community/c5h1137447p5479954 Results / Discussion]<br />
|-<br />
! scope = "row" | '''December 2015 Mock AMC 1^3*(3+7)'''<br />
| mathisawesome2169<br />
| 2015<br />
| [http://www.artofproblemsolving.com/community/c5h1173465p5647504 Initial Discussion]<br />
| [http://www.artofproblemsolving.com/community/c5h1173465p5647504 Problems]<br />
| n/a<br />
| [http://www.artofproblemsolving.com/community/c5h1173465p5707101 Results / Discussion]<br />
|-<br />
! scope = "row" | '''New Year's Mock AMC10*'''<br />
| checkmatetang<br />
| 2015<br />
| [http://artofproblemsolving.com/community/c5h1171366p5627439 Initial Discussion]<br />
| [http://artofproblemsolving.com/community/c5h1171366p5627439 Problems]<br />
| [http://artofproblemsolving.com/community/c5h1171366p5711384 Answers]<br />
| [http://artofproblemsolving.com/community/c5h1171366p5711384 Results] / [http://artofproblemsolving.com/community/c202656_new_years_mock_amc10_discussion Discussion]<br />
|-<br />
! scope = "row" | '''2015-2016 Mock AMC 10*'''<br />
| PersonPsychopath<br />
| 2015<br />
| [http://artofproblemsolving.com/community/c5h1168398p5595105 Initial Discussion]<br />
| [http://artofproblemsolving.com/community/c5h1168398p5595105 Problems]<br />
| [http://artofproblemsolving.com/community/c147536h1182005_mock_amc10_wrapup Answers]<br />
| [http://artofproblemsolving.com/community/c5h1168398p5595105 Results / Discussion]<br />
|-<br />
! scope = "row" | '''Mock AMC 10 2015-2016*'''<br />
| AlcumusGuy<br />
| 2015<br />
| [http://www.artofproblemsolving.com/community/c5h1177413p5687990 Initial Discussion]<br />
| [http://artofproblemsolving.com/community/c5h1177413_mock_amc_10_20152016_released Problems]<br />
| [http://artofproblemsolving.com/community/c212844_mock_amc_10_20152016 Solutions]<br />
| [http://artofproblemsolving.com/community/c5h1177413p5769839 Results / Discussion]<br />
|-<br />
! scope = "row" | '''January 2016 Mock AMC 10*'''<br />
| atmchallenge<br />
| 2015<br />
| [http://www.artofproblemsolving.com/community/c5h1178971p5700377 Initial Discussion]<br />
| [http://www.artofproblemsolving.com/community/c5h1178971p5700377 Problems]<br />
| [http://www.artofproblemsolving.com/community/c5h1178971p5838468 Answers]<br />
| [http://www.artofproblemsolving.com/community/c5h1178971p5838468 Results/Discussion]<br />
|-<br />
! scope = "row" | '''2015-2016 Mock AMC 10* #2'''<br />
| PersonPsychopath<br />
| 2015<br />
| [http://artofproblemsolving.com/community/c5h1188467 Initial Discussion]<br />
| [http://artofproblemsolving.com/community/c5h1188467 Problems]<br />
| [http://artofproblemsolving.com/community/c147536h1175930 Solutions]<br />
| [http://artofproblemsolving.com/community/c5h1188467 Results / Discussion]<br />
|-<br />
<br />
! scope="row" | '''MeepyMeepMeep Mock AMC 10'''<br />
| MeepyMeepMeep, speck<br />
| 2016<br />
| [http://artofproblemsolving.com/community/c5h1195432p5852002 Initial Discussion]<br />
| [https://www.dropbox.com/s/uw0herhuqk8zbp5/Mock%20AMC%2010.pdf?dl=0 Problems]<br />
| n/a<br />
| n/a<br />
|- <br />
<br />
!scope="row" | eisirrational's Mock AMC 10<br />
| eisirrational, illogical_21, whatshisbucket<br />
| 2017<br />
| [https://artofproblemsolving.com/community/c5h1372991_eisirrationals_mock_amc_10 Initial Discussion]<br />
| [https://artofproblemsolving.com/community/c5h1372991_eisirrationals_mock_amc_10 Problems]<br />
| [http://artofproblemsolving.com/community/c418591h1392426p7811150 Answer Key]<br />
| [https://artofproblemsolving.com/community/c418591_mock_amc_10_discussion Discussion]<br />
|-<br />
<br />
!scope="row" | Summer Mock AMC 10<br />
| Rowechen, OmicronGamma, FedeX333X, KenV, kvedula2004, ItzVineeth<br />
| 2017<br />
| [https://artofproblemsolving.com/community/c5h1471834_summer_mock_amc_series Initial Discussion]<br />
| [https://drive.google.com/file/d/0B3M-fxa6QG0_ODNpSnJZcEt3djQ/view Problems]<br />
| [http://latex.artofproblemsolving.com/miscpdf/hehdrwpi.pdf?t=1500270651030 Answers]<br />
| [https://artofproblemsolving.com/community/c481237_2017_summer_mock_amc_10_discussion_forum Statistics / Discussion]<br />
|-<br />
<br />
!scope="row" | scrabbler94's Mock AMC 10<br />
| scrabbler94<br />
| 2018<br />
| [https://artofproblemsolving.com/community/c5h1569848_mock_2018_amc_10_test_scrabbler94 Initial Discussion]<br />
| [https://artofproblemsolving.com/community/c5h1569848_mock_2018_amc_10_test_scrabbler94 Problems]<br />
| [https://artofproblemsolving.com/community/c5h1569848_mock_2018_amc_10_test_scrabbler94 Answers]<br />
| [http://artofproblemsolving.com/community/c5h1569848p9704902 Results] / [https://artofproblemsolving.com/community/c593716_scrabbler94s_mock_amc_10_discussion Discussion]<br />
|-<br />
<br />
! scope="row" | '''2018 Mock AMC 10'''<br />
| blue8931<br />
| 2018<br />
| [https://artofproblemsolving.com/community/c594864h1606066 Initial Discussion]<br />
| [https://artofproblemsolving.com/downloads/printable_post_collections/628064 Problems]<br />
| [https://artofproblemsolving.com/community/c628116h1632436_official_answer_key Answer Key]<br />
| [https://artofproblemsolving.com/community/c628116_2018_mock_amc_10_discussion_forum Results / Discussion]<br />
|-<br />
<br />
!scope="row" | 2018 Memorial Day Mock AMC 10<br />
| QIDb602<br />
| 2018<br />
| [https://artofproblemsolving.com/community/c594864h1645806_released_2018_memorial_day_mock_amc_10 Initial Discussion]<br />
| [https://drive.google.com/file/d/1paAHspy5PH1J9fMoNYTb7cIEH3Di72K5/view Problems]<br />
| [https://drive.google.com/file/d/1tipZuHE11jmmOfnSDnhkwKnvIOnbsbaL/view?usp=sharing Answers]<br />
| [https://artofproblemsolving.com/community/c594864h1645806_released_2018_memorial_day_mock_amc_10 Results] / [https://artofproblemsolving.com/community/c671484 Discussion]<br />
|-<br />
<br />
!scope="row" | Autumn Mock AMC 10<br />
| Krypton36, AlastorMoody, alphaone001, dchenmathcounts, InternetPerson10, kootrapali, mathdragon2000, MathGeek2018, Stormersyle<br />
| 2018<br />
| [http://artofproblemsolving.com/community/c594864h1710983p11032970 Initial Discussion]<br />
| [http://artofproblemsolving.com/community/c594864h1710983p11032970 Problems]<br />
| [http://artofproblemsolving.com/community/c594864h1710983p11331788 Answers]<br />
| [http://artofproblemsolving.com/community/c594864h1710983p11332566 Results] / [https://artofproblemsolving.com/community/c761744_autumn_mock_amc_10_discussion_forum_d Discussion]<br />
|-<br />
<br />
!scope="row" | 2018 December Mock AMC 10<br />
| mathchampion1, kcbhatraju, kootrapali, chocolatelover111<br />
| 2018<br />
| [https://artofproblemsolving.com/community/c594864h1743064_2018_december_mock_amc_10_released Initial Discussion]<br />
| [https://artofproblemsolving.com/community/c594864h1743064_2018_december_mock_amc_10_released Problems]<br />
| [https://artofproblemsolving.com/community/c784638_december_mock_amc_10_2018_discussion_forum Answers]<br />
| [https://artofproblemsolving.com/community/c784638_december_mock_amc_10_2018_discussion_forum Results] / [https://artofproblemsolving.com/community/c784638_december_mock_amc_10_2018_discussion_forum Discussion]<br />
|-<br />
<br />
!scope="row" | Christmas Math Competitions Year 2*<br />
| CMC Committee<br />
| 2018-2019<br />
| [https://artofproblemsolving.com/community/c594864h1747367_aime_ii_released_christmas_mathematics_competition_cmc_year_2 Initial Discussion]<br />
| [https://drive.google.com/file/d/15heOR_6rnH70joxJRfZBhQXa8Wezb2AW/view CMC 10A] [https://drive.google.com/file/d/1ZNayjDykKYj4889nXtOb7k9KBu_bqXki/view CMC 10B]<br />
| [https://artofproblemsolving.com/community/c798404h1760024_cmc_10a12a_amp_10b12b_year_2__problems_and_answer_keys Answers]<br />
| [https://artofproblemsolving.com/community/c594864h1747367p11379904 Results] / [https://artofproblemsolving.com/community/c798404_christmas_mathematics_competitions_year_2 Discussion]<br />
|-<br />
<br />
!scope="row" | 2019 Mock AMC 10C*<br />
| fidgetboss_4000<br />
| 2019<br />
| [https://artofproblemsolving.com/community/c594864h1786263_released_2019_amc_10c Initial Discussion]<br />
| [https://artofproblemsolving.com/community/c594864h1786263_released_2019_amc_10c Problems] (or click here: [[2019 AMC 10C Problems]]<br />
| [https://artofproblemsolving.com/community/c594864h1786263_released_2019_amc_10c Answers - In Thread]<br />
| N/A / N/A<br />
|-<br />
!scope="row" | 2019 Mock AMC 10B*<br />
| fidgetboss_4000<br />
| 2019<br />
| [https://artofproblemsolving.com/community/c594864h1897633p13160252 Initial Discussion]<br />
| [https://artofproblemsolving.com/community/c594864h1897633p13160252 Problems] (or click here: [[Mock AMC 10B Problems]]<br />
| [https://artofproblemsolving.com/community/c963448h1919463_answer_keys_d Answers]<br />
| N/A / N/A<br />
|-<br />
!scope="row" | Christmas Math Competitions Year 3*<br />
| CMC Committee<br />
| 2019-2020<br />
| [https://artofproblemsolving.com/community/c594864h1967029_cime_ii_released_christmas_math_competition_cmc_year_3 Initial Discussion]<br />
| [http://cmc.ericshen.net/CMC-2020/CMC-2020-10A.pdf CMC 10A] [http://cmc.ericshen.net/CMC-2020/CMC-2020-10B-booklet.pdf CMC 10B]<br />
| [https://artofproblemsolving.com/community/c1035147h1980361p13760203 Answers]<br />
| [https://artofproblemsolving.com/community/c594864h1967029p13623910 Results] / [https://artofproblemsolving.com/community/c1035147_christmas_mathematics_competitions_year_3 Discussion]<br />
|-<br />
!scope="row" | Stormersyle's Mock AMC 10*<br />
| Stormersyle<br />
| 2019-2020<br />
| [https://artofproblemsolving.com/community/c594864h1974214p13694448 Initial Discussion]<br />
| [https://artofproblemsolving.com/community/c594864h1974214p13694448 Problems]<br />
| [https://artofproblemsolving.com/community/c594864h1974214p13694448 Answers]<br />
| [https://artofproblemsolving.com/community/c594864h1974214p13694448 Results] / [https://artofproblemsolving.com/community/c594864h1974214p13694448 Discussion]<br />
|-<br />
!scope="row" | January 2020 Mock AMC 10/12<br />
| P_Groudon<br />
| 2020<br />
| <br />
[https://artofproblemsolving.com/community/c594864h1984167_concluded_january_2020_mock_amc_1012 Initial Discussion]<br />
| [https://artofproblemsolving.com/community/c594864h1984167_concluded_january_2020_mock_amc_1012 Problems]<br />
| [https://artofproblemsolving.com/community/c1035224h1984207_january_2020_answer_keys Answers]<br />
| [https://artofproblemsolving.com/community/c594864h1984167p13963794 Results]<br />
|-<br />
!scope="row" | 2020 Mock Combo AMC 10<br />
| fidgetboss_4000<br />
| 2020<br />
| <br />
[https://artofproblemsolving.com/community/c594864h2005582_amc_1012_released_mock_combo_amc_1012_ii Initial Discussion]<br />
| [https://artofproblemsolving.com/wiki/index.php?title=2020_Mock_Combo_AMC_10&action=view Problems]<br />
| [https://artofproblemsolving.com/community/c1121049h2053291_answer_key Answers]<br />
| [https://artofproblemsolving.com/community/c594864h2005582_amc_1012_released_mock_combo_amc_1012_ii Results]<br />
|-<br />
!scope="row" | OTSS 2020<br />
| kevinmathz<br />
| 2020<br />
| <br />
[https://artofproblemsolving.com/community/c594864h2066002_otss_olympiad_test_spring_series Initial Discussion]<br />
| [https://drive.google.com/file/d/1JfuEvDGw9i99XeQYACPUDzHebxyrcmtE/edit TMC 10A] [https://drive.google.com/file/d/1FkKjAgehDwpmmNOj_mjvYuCCV1r7p08p/edit TMC 12A]<br />
| [https://artofproblemsolving.com/community/c1130807h2094651_tmc_1012_a_solutions Answers]<br />
| [https://artofproblemsolving.com/community/c1130807h2081571_links_to_mocks_solutions_and_leaderboards Results]<br />
|-<br />
!scope="row" | FMC 2020<br />
| fidgetboss_4000<br />
| 2020<br />
| <br />
[https://artofproblemsolving.com/community/c594864h2069998 Initial Discussion]<br />
| [https://drive.google.com/file/d/17jscyJzVCFDlV6YL0OpPKczyk04Tltsx/view FMC 10A] [https://drive.google.com/file/d/1pMdXlGy9F5hbWSP8CGANAhAA9_ct7pdS/view FMC 12A]<br />
| [https://artofproblemsolving.com/community/c1177836_2020_fmc_public_discussion_forum_uwu Answers]<br />
| [https://artofproblemsolving.com/community/q2h2069998p15525666 Results]<br />
|-<br />
!scope="row" | 2020 Mock AMC 10*<br />
| scrabbler94<br />
| 2020<br />
| <br />
[https://artofproblemsolving.com/community/c594864h2159345_contest_over_2020_mock_amc_1012_scrabbler94 Initial Discussion]<br />
| [https://artofproblemsolving.com/community/c594864h2159345_contest_over_2020_mock_amc_1012_scrabbler94 Problems]<br />
| [https://artofproblemsolving.com/community/c594864h2159345p16460394 Answers]<br />
| [https://artofproblemsolving.com/community/c594864h2159345p16460394 Results]<br />
|-<br />
!scope="row" | JMC 2020*<br />
| skyscraper<br />
| 2020<br />
| <br />
[https://artofproblemsolving.com/community/c594864h2164953_jmc_10_concluded_july_mock_amc Initial Discussion]<br />
| [https://artofproblemsolving.com/community/c594864h2164953_jmc_10_concluded_july_mock_amc Problems]<br />
| [https://artofproblemsolving.com/community/c594864h2164953p16895447 Answers]<br />
| [https://artofproblemsolving.com/community/c594864h2164953p16895447 Results]<br />
|-<br />
!scope="row" | PMC 2020<br />
| kred9<br />
| 2020<br />
| <br />
[https://artofproblemsolving.com/community/c594864h2229291 Initial Discussion]<br />
| [https://artofproblemsolving.com/community/c594864h2229291 Problems]<br />
| [https://artofproblemsolving.com/community/c594864h2229291p17672697 Answers]<br />
| [https://artofproblemsolving.com/community/c594864h2229291p17672697 Results]<br />
|-<br />
!scope="row" | DMC 2020<br />
| DeToasty3<br />
| 2020<br />
| <br />
[https://artofproblemsolving.com/community/q1h2282481p17891592 Initial Discussion]<br />
| [http://detoasty3.gq/DMC/2020_DMC_10.pdf?i=1 Problems]<br />
| [http://detoasty3.gq/DMC/2020_DMC_10_Solutions.pdf Answers]<br />
| [https://artofproblemsolving.com/community/c594864h2282481p18910036 Results]<br />
|-<br />
!scope="row" | InternetAMC10<br />
| InternetPerson10<br />
| 2020<br />
| <br />
[https://artofproblemsolving.com/community/c594864h2358167_internetamc10_deadline_passed Initial Discussion]<br />
| [https://drive.google.com/file/d/1OkWr99DDG51OaFoqAEzeUU4IOCtNH7n4/view Problems]<br />
| [https://artofproblemsolving.com/community/c1711319h2411880_answer_key Answers]<br />
| [https://artofproblemsolving.com/community/c594864h2358167_internetamc10_deadline_passed Results] /<br />
[https://artofproblemsolving.com/community/c1711319_internetamc10_public_discussion_forum Discussion]<br />
|-<br />
|}<br />
<br />
=== Mock AMC 8 ===<br />
<br />
{| class="wikitable" style="text-align:center;width:100%"<br />
|-<br />
|<br />
! scope="col" | '''Author'''<br />
! scope="col" | '''Year'''<br />
! scope="col" | '''Initial Discussion'''<br />
! scope="col" | '''Problems'''<br />
! scope="col" width=80 | '''Answers'''<br />
! scope="col" | '''Results/Discussion'''<br />
|-<br />
! scope="row" | '''Mock AMC 8'''<br />
| mathfanatic<br />
| 2004<br />
| [http://www.artofproblemsolving.com/community/c5h15878p111575 Initial Discussion]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=15878 Problems]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=114070#p114070 Answers]<br />
| [http://www.artofproblemsolving.com/community/c5h15878p114404 Results / Discussion]<br />
|-<br />
! scope="row" | '''Mock AMC 8'''<br />
| 13375P34K43V312<br />
| 2006<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=107507 Initial Discussion]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=108455 Problems]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=630802#p630802 Answers]<br />
| [http://www.artofproblemsolving.com/community/c5h108455p623522 Results/Discussion]<br />
|-<br />
! scope="row" | '''Mock AMC 8'''<br />
| ahaanomegas<br />
| 2012<br />
| [http://www.artofproblemsolving.com/community/c5h459346p2577870 Initial Discussion]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=2598111#p2598111 Problems]<br />
| n/a<br />
| n/a<br />
|-<br />
! scope="row" | '''Mock AMC 8'''<br />
| utahjazz<br />
| 2012<br />
| [http://www.artofproblemsolving.com/community/c5h485041p2717878 Initial Discussion]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=2728186#p2728186 Problems]<br />
| n/a<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=2731391#p2731391 Results]<br />
|-<br />
! scope="row" | '''Mock AMC 8'''<br />
| iNomOnCountdown<br />
| 2014<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?f=132&t=590451&hilit=iNomonCountdown%27s+mock+amc+8 Initial Discussion]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?f=132&t=590451&hilit=iNomonCountdown%27s+mock+amc+8 Problems]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?f=132&t=590451&start=0 Answers]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?f=132&t=590451&start=0 Results / Discussion]<br />
|-<br />
! scope = "row" | '''Mock AMC 8'''<br />
| Tan<br />
| 2014<br />
| [http://www.artofproblemsolving.com/community/c5h613297p3648226 Initial Discussion]<br />
| [http://www.artofproblemsolving.com/community/c5h613297p3648227 Problems]<br />
| [https://docs.google.com/spreadsheets/d/1U1QJ9r4r2xFbRByk9gXOhEVV48ScjoLWZp4IFanbKF0/edit#gid=1475864782 Answers]<br />
| [https://docs.google.com/spreadsheets/d/1U1QJ9r4r2xFbRByk9gXOhEVV48ScjoLWZp4IFanbKF0/edit#gid=1475864782 Results / Discussion]<br />
|-<br />
! scope = "row" | '''2015 Hard Mock AMC 8*'''<br />
| Not_a_Username, 8invalid8<br />
| 2015<br />
| [http://artofproblemsolving.com/community/q1h1128391p5227860 Initial Discussion]<br />
| [http://latex.artofproblemsolving.com/miscpdf/kashimyo.pdf Problems]<br />
| [http://www.artofproblemsolving.com/community/c5h1128391p5423312 Answers]<br />
| [http://www.artofproblemsolving.com/community/c5h1128391p5423312 Results / Discussion]<br />
|-<br />
! scope = "row" | '''Mock AMC 8'''<br />
| PersonPsychopath<br />
| 2015<br />
| [http://artofproblemsolving.com/community/c3h1152332p5457736 Initial Discussion]<br />
| [http://artofproblemsolving.com/downloads/printable_post_collections/163667 Problems]<br />
| [http://artofproblemsolving.com/community/category-admin/165328 Forum]<br />
| [http://artofproblemsolving.com/community/c165328h1155882p5484131 Results / Discussion]<br />
|-<br />
! scope = "row" | '''Mock AMC 8 #2'''<br />
| PersonPsychopath<br />
| 2015<br />
| [http://artofproblemsolving.com/community/c3h1167425p5586244 Initial Discussion]<br />
| [http://artofproblemsolving.com/downloads/printable_post_collections/165838 Problems]<br />
| [http://artofproblemsolving.com/community/c147536_the_mock_amc_8_forum Forum]<br />
| [http://artofproblemsolving.com/community/c3h1167425p5586244 Results / Discussion]<br />
|-<br />
! scope = "row" | '''Mock AMC 8 2015'''<br />
| Alberty44<br />
| 2015<br />
| [http://artofproblemsolving.com/community/c194276_discusssion Initial Discussion]<br />
| [http://artofproblemsolving.com/community/c5h1168822p5674003 Problems]<br />
| n/a<br />
| [http://artofproblemsolving.com/community/c194276h1175928p5673996 Results/Discussion]<br />
|-<br />
! scope = "row" | '''Christmas AMC 8'''<br />
| Mudkipswims42<br />
| 2015<br />
| [http://artofproblemsolving.com/community/c5h1177489p5688588 Initial Discussion]<br />
| [http://artofproblemsolving.com/community/c5h1177489p5688588 Problems]<br />
| [http://www.artofproblemsolving.com/community/c229455_christmas_amc8_discussion_forum Forum]<br />
| [http://artofproblemsolving.com/community/c5h1177489p5880200 Results/Discussion]<br />
|- <br />
<br />
! scope = "row" | '''Mock AMC 8!'''<br />
| eisirrational, pretzel, AOPS12142015, Th3Numb3rThr33, e_power_pi_times_i<br />
| 2017<br />
| [https://artofproblemsolving.com/community/c5h1526187_mock_amc_8 Initial Discussion]<br />
| [http://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvYi84LzFkNzUyMWYwZTViOTQ1MzRmZmU3ODE0NmI2MzIzMGUxNjA2MTcwLnBkZg==&rn=TW9jayBBTUMgOCB2Ny5wZGY= Problems]<br />
| [http://artofproblemsolving.com/community/c3h1545004p9368333 Answer Key/Solutions]<br />
| [https://artofproblemsolving.com/community/c561318_mock_amc_8_2017_discussion Discussion]<br />
|-<br />
<br />
! scope = "row" | '''Mock Summ(er)ation AMC 8!'''<br />
| mathchamp1, kevinmathz, Gali, mathdragon2000, reddragon644, ShreyJ, Quadrastic, Mathnerd1223334444<br />
| 2018<br />
| [https://artofproblemsolving.com/community/c594864h1672811_2018_summeration_mock_amc_8 Initial Discussion]<br />
| [https://math-adventures.weebly.com/uploads/1/1/8/8/118819793/amc_8_problems.pdf Problems]<br />
| n/a<br />
| n/a<br />
|-<br />
<br />
! scope = "row" | '''popcorn1's AMC 8 2018'''<br />
| popcorn1<br />
| 2018<br />
| [https://artofproblemsolving.com/community/c594864h1715072_popcorn1s_amc_8_2018 Initial Discussion]<br />
| [http://artofproblemsolving.com/community/c594864h1715072p11074872 Problems]<br />
| Answer Key / Solutions - not released<br />
| Discussion - not released<br />
|-<br />
<br />
! scope = "row" | ''' Stormersyle's Mock AMC 8<br />
| Stormersyle <br />
| 2018<br />
| [http://artofproblemsolving.com/community/c594864h1722790p11146826 Initial Discussion]<br />
| [http://artofproblemsolving.com/community/c594864h1722790p11146826 Problems]<br />
| [https://latex.artofproblemsolving.com/miscpdf/qgdafjlp.pdf?t=1544494957792 Answer Key/Solutions]<br />
| Discussion - not released<br />
|-<br />
<br />
! scope = "row" | '''mathchampion1's Christmas Mock AMC 8'''<br />
| mathchampion1<br />
| 2018<br />
| [https://artofproblemsolving.com/community/c594864h1752494_released_mock_amc_8_christmas_edition Initial Discussion]<br />
| [https://artofproblemsolving.com/community/c594864h1752494_released_mock_amc_8_christmas_edition Problems]<br />
| Answer Key / Solutions - not released<br />
| Discussion - not released<br />
|-<br />
<br />
! scope = "row" | '''June 2019 Mock AMC 8'''<br />
| fidgetboss_4000<br />
| 2019<br />
| [https://artofproblemsolving.com/community/c594864h1853912_june_2019_mock_amc_8 Initial Discussion]<br />
| [https://artofproblemsolving.com/community/c594864h1853912_june_2019_mock_amc_8 Problems]<br />
| Answer Key / Solutions - not released<br />
| Discussion - not released<br />
|-<br />
<br />
! scope = "row" | '''popcorn1's AMC 8 2019'''<br />
| popcorn1<br />
| 2019<br />
| [https://artofproblemsolving.com/community/c594864h1896786 Initial Discussion]<br />
| [https://artofproblemsolving.com/community/c594864h1896786 Problems]<br />
| [https://artofproblemsolving.com/community/c947561h1950746_answer_keys Answer Key/Solutions]<br />
| [https://artofproblemsolving.com/community/c947561_popcorn1s_amc_8_2019_discussion_forum Discussion]<br />
|-<br />
<br />
! scope = "row" | '''lethan3's Mock AMC 8'''<br />
| lethan3<br />
| 2020<br />
| [https://artofproblemsolving.com/community/c594864h2216496_lethan3s_mock_amc_8_ended Initial Discussion]<br />
| [https://artofproblemsolving.com/community/c594864h2216496_lethan3s_mock_amc_8_ended Problems]<br />
| [https://artofproblemsolving.com/community/c594864h2216496_lethan3s_mock_amc_8_ended Answer Key/Solutions]<br />
| [https://artofproblemsolving.com/community/c1266375_lethan3s_mock_amc_8_discussion_forum Discussion]<br />
|-<br />
<br />
! scope = "row" | '''AAMC 8 Year 1'''<br />
| bissue<br />
| 2020<br />
| [https://artofproblemsolving.com/community/c594864h2255517 Initial Discussion]<br />
| [https://artofproblemsolving.com/community/q1h2255517p17556358 Problems]<br />
| [https://artofproblemsolving.com/community/q1h2255517p18736140 Answer Key]<br />
| [https://artofproblemsolving.com/community/q1h2255517p18736140 Discussion]<br />
|}<br />
<br />
== See also ==<br />
* [[American Mathematics Competitions]]<br />
* [[Math books]]<br />
* [[Mathematics competitions]]<br />
* [[Mock AIME]]<br />
* [[Mock MathCounts]]<br />
* [[Mock USAMO]]<br />
* [[Mock USAJMO]]<br />
* [[Resources for mathematics competitions]]<br />
* [[AoPS Past Contests]]</div>Bissuehttps://artofproblemsolving.com/wiki/index.php?title=Mock_AMC&diff=141437Mock AMC2021-01-03T19:46:44Z<p>Bissue: added AAMC 8 (Year 1) and fixed spacing</p>
<hr />
<div>A '''Mock AMC''' is a contest intended to mimic an actual [[AMC]] (American Mathematics Competitions 8, 10, or 12) exam. A number of '''Mock AMC''' competitions have been hosted on the [[Art of Problem Solving]] message boards. They are generally made by one community member and then administered for any of the other community members to take. Sometimes, the administrator may ask other people to sign up to write problems for the contest.<br />
<br />
Mock AMCs are usually very popular in the months leading up to the actual [[AMC]] competition. There is no guarantee that community members will make Mock AMCs in any given year, but there probably will be one.<br />
<br />
Feel free to remove mock tests that are not high quality or to recommend others.<br />
<br />
Ongoing mock AMCs, as well as other mock contests, can be found in the [https://artofproblemsolving.com/community/c594864_aops_mock_contests AoPS Mock Contests forum].<br />
<br />
== Tips for Writing a Mock AMC ==<br />
Anyone can write a Mock AMC and administer it. If you are interested in writing one, here are some tips:<br />
<br />
* Look at past [[AMC]]/[[AHSME]] tests to get a feel for what kind of problems you should write and what difficulty level they should be.<br />
* Look at famous theorems and formulas and see if there's any way you can make a good problem out of them.<br />
* If you are running out of creative juice and decide to pull problems from contests, try using problems from obscure contests first, if possible. This way, even the more experienced test takers will hopefully find problems that they do not already know how to do.<br />
* Pair up with another user on AoPS and write it together. Two minds are much better than one. With just one person, the problems may be biased toward one subject, but with two people, the chances of this happening are smaller.<br />
<br />
== Past Mock AMCs ==<br />
<br />
Listed below are the [higher-quality] Mock AMCs that have been hosted over AoPS in the past. Feel free to add, remove, or recommend.<br />
<br />
Note that the "level" column represents the originally intended difficulty. In other words, if a person makes a mock [[AMC 12]], the level would be "12", even if the problems themselves are much easier. Recommended mock tests are starred.<br />
<br />
=== Mock AMC 12 ===<br />
<br />
{| class="wikitable" style="text-align:center;width:100%"<br />
|-<br />
|<br />
! scope="col" | '''Author'''<br />
! scope="col" | '''Year'''<br />
! scope="col" | '''Initial Discussion'''<br />
! scope="col" | '''Problems'''<br />
! scope="col" width=80 | '''Answers'''<br />
! scope="col" | '''Results/Discussion'''<br />
|-<br />
! scope="row" | '''Mock AMC #1'''<br />
| mathfanatic<br />
| 2003<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=9321 Initial Discussion]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=9353 Problems]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=9572 1-5]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?t=9573 6-10]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?t=9574 11-15]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?t=9575 16-20]<br />
[http://www.artofproblemsolving.com/Forum/viewtopic.php?t=9576 21-25]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=9365 Results]<br />
|-<br />
! scope="row" | '''Mock AMC #2'''<br />
| mathfanatic<br />
| 2004<br />
| [http://www.artofproblemsolving.com/community/c5h10497p67837 Initial Discussion]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=10497 Problems]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=10497 Solutions]<br />
| n/a<br />
|-<br />
! scope="row" | '''Mock AMC A'''<br />
| JSRosen3<br />
| 2004<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=14138 Initial Discussion]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=14361 Problems]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=102483#p102483 Answers] [http://www.artofproblemsolving.com/community/c5h14516 Solutions]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=14489 Results / Discussion]<br />
|-<br />
! scope="row" | '''Mock AMC B'''<br />
| beta<br />
| 2004<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=14735 Initial Discussion]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=14764 Problems]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=14894 Answers] [http://www.artofproblemsolving.com/community/c5h14884 Solutions]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=105741#p105741 Results / Discussion]<br />
|-<br />
! scope="row" | '''Mock AMC C'''<br />
| JGeneson<br />
| 2004<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=15001 Initial Discussion]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=15134 Problems]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=15251 Answers]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=15251 Results / Discussion]<br />
|-<br />
! scope="row" | '''Mock AMC D'''<br />
| joml88<br />
| 2004<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=16886 Initial Discussion]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=17888 Problems]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=17891 Answers]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=17891 Results / Discussion]<br />
|-<br />
! scope="row" | '''Mock AMC E'''<br />
| Silverfalcon<br />
| 2004<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=21997 Initial Discussion]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=22141 Problems]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=22344 Answers]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=22344 Results / Discussion]<br />
|-<br />
! scope="row" | '''Mock AMC F'''<br />
| joml88<br />
| 2005<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=22049 Initial Discussion]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=23163 Problems]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=23177 Answers]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=23177 Results / Discussion]<br />
|-<br />
! scope="row" | '''Mock AMC G'''<br />
| Lucky707<br />
| 2005<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=24355 Initial Discussion]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=24974 Problems]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=25087 Answers]<br />
| [http://www.artofproblemsolving.com/community/c5h25087p157983 Results / Discussion]<br />
|-<br />
! scope="row" | '''Mock AMC H'''<br />
| Silverfalcon<br />
| 2005<br />
| [http://www.artofproblemsolving.com/community/c5h24437p154514 Initial Discussion]<br />
| [http://www.artofproblemsolving.com/community/c5h24437p154514 Problems]<br />
| throughout thread<br />
| n/a<br />
|-<br />
! scope="row" | '''Mock AMC I'''<br />
| white_horse_king88<br />
| 2005<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=21280 Initial Discussion]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=25181 Problems]<br />
| n/a<br />
| n/a<br />
|-<br />
! scope="row" | '''Mock AMC J'''<br />
| Silverfalcon<br />
| 2005<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=47625 Initial Discussion]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=48129 Problems]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=304246#p304246 Answers]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=48132 Results / Discussion]<br />
|-<br />
! scope="row" | '''Mock AMC K'''<br />
| amirhtlusa<br />
| 2005<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=49958 Initial Discussion]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=50515 Problems]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=321102#p321102 Answers]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=321102#p321102 Results / Discussion]<br />
|-<br />
! scope="row" | '''Mock AMC L'''<br />
| amirhtlusa<br />
| 2005<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=61330 Initial Discussion]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=63041 Problems]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=63258 Answers]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=63258 Results / Discussion]<br />
|-<br />
! scope="row" | '''Mock AMC M'''<br />
| Silverfalcon<br />
| 2006<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=63542 Initial Discussion]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=78982 Problems]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=79749 Answers]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=79749 Discussion]<br />
|-<br />
! scope="row" | '''Mock AMC N'''<br />
| chess64<br />
| 2006<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=98894 Initial Discussion]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=99307 Problems]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=99344 Answers]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=99344 Results] / [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=99566 Discussion]<br />
|-<br />
! scope="row" | '''Mock AMC O'''<br />
| mustafa<br />
| 2006<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=121312 Initial Discussion]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=122126 Problems]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=126071 Answers]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=126071 Results/Discussion]<br />
|-<br />
! scope="row" | '''Mock AMC P'''<br />
| Anirudh<br />
| 2006<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=125029 Initial Discussion]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=709655#p709655 Problems]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=716240#p716240 Answers]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=716262#p716262 Results]/[http://www.artofproblemsolving.com/Forum/viewtopic.php?t=125029 Discussion]<br />
|-<br />
! scope="row" | '''Mock AMC Q'''<br />
| calc rulz<br />
| 2006<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=125194 Initial Discussion]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=125886 Problems]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=128102 Answers]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=128102 Results/Discussion]<br />
|-<br />
! scope="row" | '''Mock AMC R'''<br />
| rnwang2<br />
| 2006<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=126107 Initial Discussion]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=715597#p715597 Problems]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=127421 Answers]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=127421 Results/Discussion]<br />
|-<br />
! scope="row" | '''Mock AMC S'''<br />
| mysmartmouth<br />
| 2007<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=127221 Initial Discussion]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=128689 Problems]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=131403 Answers]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=131403 Results/Discussion]<br />
|-<br />
! scope="row" | '''Mock AMC T'''<br />
| paladin8<br />
| 2007<br />
| [http://www.artofproblemsolving.com/community/c5h127979p726003 Initial Discussion]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=127979 Problems]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=129159 Answers]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=129159 Results/Discussion]<br />
|-<br />
! scope="row" | '''Mock AMC U'''<br />
| Silverfalcon<br />
| 2008<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=184067 Initial Discussion]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=185233 Problems]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=185238 Answers]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=185236 Results]/[http://www.artofproblemsolving.com/Forum/viewtopic.php?t=185238 Discussion]<br />
|-<br />
! scope="row" | '''Mock AMC V'''<br />
| gfour84<br />
| 2009<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=298452 Initial Discussion]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=1634175#p1634175 Problems]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=1637429#p1637429 Answers]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=1637006#p1637006 Results]/[http://www.artofproblemsolving.com/Forum/viewtopic.php?t=302030 Discussion]<br />
|-<br />
! scope="row" | '''Mock AMC 12'''<br />
| MathTwo<br />
| 2010<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=319184 Initial Discussion]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=1759276#p1759276 Problems]<br />
| n/a<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=1778080#p1778080 Results]/[http://www.artofproblemsolving.com/Forum/viewtopic.php?t=328510 Discussion]<br />
|-<br />
! scope="row" | '''Mock AMC 2/11'''<br />
| Caelestor<br />
| 2011<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=390907 Initial Discussion]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=2183293#p2183293 Problems]<br />
| n/a<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=390907 Discussion]<br />
|-<br />
! scope="row" | '''Mock AMC 1/12'''<br />
| Lord.of.AMC<br />
| 2012<br />
| [http://www.artofproblemsolving.com/community/c5h456321p2563731 Initial Discussion]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=2605537#p2605537 Problems]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=2610048#p2610048 Answers]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=2610048#p2610048 Results]<br />
|-<br />
! scope="row" | '''Mock AMC 12'''<br />
| Diehard<br />
| 2012<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=459149 Initial Discussion]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=459710 Problems]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=2580327#p2580327 Answers]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=459710 Discussion]<br />
|-<br />
! scope="row" | '''Mock AMC 12 2012'''<br />
| python123<br />
| 2012<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=456256 Initial Discussion]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=2576205#p2576205 Problems]<br />
| [http://www.artofproblemsolving.com/community/c5h456256p2580610 Answers]<br />
| [http://www.artofproblemsolving.com/community/c5h456256p2580610 Results / Discussion]<br />
|-<br />
! scope="row" | '''Mock AMC A'''<br />
| Binomial-theorem<br />
| 2012<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=473867 Initial Discussion]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=2692082#p2692082 Problems]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=2717578#p2717578 Solutions]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=2707954#p2707954 Results]<br />
|-<br />
! scope = "row" | '''Almost 2016 Mock AMC 11.5'''<br />
| whatshisbucket<br />
| 2015<br />
| [http://www.artofproblemsolving.com/community/c5h1175087p5665743 Initial Discussion]<br />
| [http://www.artofproblemsolving.com/community/c5h1175087p5665743 Problems]<br />
| [http://artofproblemsolving.com/community/c209194_almost_2016_mock_amc_11.5 Solutions]<br />
| [http://www.artofproblemsolving.com/community/c5h1175087p5665743 Results / Discussion]<br />
|-<br />
! scope="row" | '''hnkevin42 Mock AMC 12'''<br />
| hnkevin42<br />
| 2016<br />
| [http://www.artofproblemsolving.com/community/c5h1187484p5778754 Initial Discussion]<br />
| [http://www.artofproblemsolving.com/community/c5h1187484p5778754 Problems]<br />
| [http://www.artofproblemsolving.com/community/c5h1187484p5913294 Answers]<br />
| [http://www.artofproblemsolving.com/community/c5h1187484p5778754 Results/Discussion]<br />
|-<br />
! scope="row" | '''Last-Minute Mock AMC 12'''<br />
| CountofMC<br />
| 2018<br />
| [https://artofproblemsolving.com/community/c594864h1582044_lastminute_mock_amc_12 Initial Discussion]<br />
| [https://artofproblemsolving.com/community/c594864h1582044_lastminute_mock_amc_12 Problems]<br />
| [https://artofproblemsolving.com/community/c594864h1582044_lastminute_mock_amc_12 Answers]<br />
| [https://artofproblemsolving.com/community/c4t334444f4_lastminute_mock_amc_12 Results/Discussion]<br />
|-<br />
!scope="row" | Christmas Math Competitions Year 2*<br />
| CMC Committee<br />
| 2018-2019<br />
| [https://artofproblemsolving.com/community/c594864h1747367_aime_ii_released_christmas_mathematics_competition_cmc_year_2 Initial Discussion]<br />
| [https://drive.google.com/file/d/1uS3YqAK10jB1RkCQCLQgeBHUACydJph-/view CMC 12A] [https://drive.google.com/file/d/1txKy-MfZCPPnUNTLCIad8dbatV-EGu71/view CMC 12B]<br />
| [https://artofproblemsolving.com/community/c798404h1760024_cmc_10a12a_amp_10b12b_year_2__problems_and_answer_keys Answers]<br />
| [https://artofproblemsolving.com/community/c594864h1747367p11379904 Results] / [https://artofproblemsolving.com/community/c798404_christmas_mathematics_competitions_year_2 Discussion]<br />
|-<br />
!scope="row" | 2019 Mock AMC 12B*<br />
| fidgetboss_4000<br />
| 2019<br />
| [https://artofproblemsolving.com/community/c594864h1897633p13160252 Initial Discussion]<br />
| [https://artofproblemsolving.com/community/c594864h1897633p13160252 Problems] (or click here: [[Mock AMC 12B Problems]]<br />
| [https://artofproblemsolving.com/community/c963448h1919463_answer_keys_d Answers]<br />
| N/A / N/A<br />
|-<br />
!scope="row" | Christmas Math Competitions Year 3*<br />
| CMC Committee<br />
| 2019-2020<br />
| [https://artofproblemsolving.com/community/c594864h1967029_cime_ii_released_christmas_math_competition_cmc_year_3 Initial Discussion]<br />
| [http://cmc.ericshen.net/CMC-2020/CMC-2020-12A.pdf CMC 12A] [http://cmc.ericshen.net/CMC-2020/CMC-2020-12B-booklet.pdf CMC 12B]<br />
| [https://artofproblemsolving.com/community/c1035147h1980361p13760203 Answers]<br />
| [https://artofproblemsolving.com/community/c594864h1967029p13623910 Results] / [https://artofproblemsolving.com/community/c1035147_christmas_mathematics_competitions_year_3 Discussion]<br />
|-<br />
!scope="row" | January 2020 Mock AMC 10/12<br />
| P_Groudon<br />
| 2020<br />
| <br />
[https://artofproblemsolving.com/community/c594864h1984167_concluded_january_2020_mock_amc_1012 Initial Discussion]<br />
| [https://artofproblemsolving.com/community/c594864h1984167_concluded_january_2020_mock_amc_1012 Problems]<br />
| [https://artofproblemsolving.com/community/c1035224h1984207_january_2020_answer_keys Answers]<br />
| [https://artofproblemsolving.com/community/c594864h1984167p13963794 Results]<br />
|-<br />
!scope="row" | OTSS 2020<br />
| kevinmathz<br />
| 2020<br />
| <br />
[https://artofproblemsolving.com/community/c594864h2066002_otss_olympiad_test_spring_series Initial Discussion]<br />
| [https://drive.google.com/file/d/1JfuEvDGw9i99XeQYACPUDzHebxyrcmtE/edit TMC 10A] [https://drive.google.com/file/d/1FkKjAgehDwpmmNOj_mjvYuCCV1r7p08p/edit TMC 12A]<br />
| [https://artofproblemsolving.com/community/c1130807h2094651_tmc_1012_a_solutions Answers]<br />
| [https://artofproblemsolving.com/community/c1130807h2081571_links_to_mocks_solutions_and_leaderboards Results]<br />
|-<br />
!scope="row" | FMC 2020<br />
| fidgetboss_4000<br />
| 2020<br />
| <br />
[https://artofproblemsolving.com/community/c594864h2069998 Initial Discussion]<br />
| [https://drive.google.com/file/d/17jscyJzVCFDlV6YL0OpPKczyk04Tltsx/view FMC 10A] [https://drive.google.com/file/d/1pMdXlGy9F5hbWSP8CGANAhAA9_ct7pdS/view FMC 12A]<br />
| [https://artofproblemsolving.com/community/c1177836_2020_fmc_public_discussion_forum_uwu Answers]<br />
| [https://artofproblemsolving.com/community/q2h2069998p15525666 Results]<br />
|-<br />
!scope="row" | 2020 Mock AMC 12<br />
| scrabbler94<br />
| 2020<br />
| <br />
[https://artofproblemsolving.com/community/c594864h2159345_contest_over_2020_mock_amc_1012_scrabbler94 Initial Discussion]<br />
| [https://artofproblemsolving.com/community/c594864h2159345_contest_over_2020_mock_amc_1012_scrabbler94 Problems]<br />
| [https://artofproblemsolving.com/community/c594864h2159345p16460394 Answers]<br />
| [https://artofproblemsolving.com/community/c594864h2159345p16460394 Results]<br />
|-<br />
|}<br />
<br />
=== Mock AMC 10 ===<br />
<br />
{| class="wikitable" style="text-align:center;width:100%"<br />
|-<br />
|<br />
! scope="col" | '''Author'''<br />
! scope="col" | '''Year'''<br />
! scope="col" | '''Initial Discussion'''<br />
! scope="col" | '''Problems'''<br />
! scope="col" | '''Answers'''<br />
! scope="col" | '''Results/Discussion'''<br />
|-<br />
! scope="row" | '''Mock AMC 10'''<br />
| #H34N1<br />
| 2008<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=212730 Initial Discussion]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=214081 Problems]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=213727 Answers]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=213732 Results/Discussion]<br />
|-<br />
! scope="row" | '''agent's Mock AMC 10'''<br />
| agentcx<br />
| 2009<br />
| [http://www.artofproblemsolving.com/community/c5h331823p1775678 Initial Discussion]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=1781208#p1781208 Problems]<br />
| n/a<br />
| [http://www.artofproblemsolving.com/community/c5h331823p1782769 Results / Discussion]<br />
|-<br />
! scope="row" | '''Mock AMC 10 Set'''<br />
| AwesomeToad<br />
| 2010<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=311120 Initial Discussion]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=1762829#p1762829 Problems]<br />
| n/a<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=1778740#p1778740 Results]<br />
|-<br />
! scope="row" | '''Mock AMC 10/12'''<br />
| djmathman<br />
| 2013<br />
| [http://www.artofproblemsolving.com/community/c5h556673s1_first_mock_amcs_of_the_20132014_season Initial Discussion]<br />
| [http://www.artofproblemsolving.com/community/c5h556673p3294854 Problems]<br />
| [http://www.artofproblemsolving.com/community/c5h556673p3324794 Answers]<br />
| [http://www.artofproblemsolving.com/community/c5h556673p3324794 Results / Discussion]<br />
|-<br />
! scope="row" | '''Mock AMC 10 2014-2015*'''<br />
| AlcumusGuy<br />
| 2014<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=618080&hilit=AlcumusGuy%27s+mock+amc+10 Initial Discussion]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=618080&hilit=AlcumusGuy%27s+mock+amc+10 Problems]<br />
| [http://www.artofproblemsolving.com/community/c5h618080p3710795 Answers/Results]<br />
| [http://artofproblemsolving.com/community/c56018 Problem Discussion]<br />
|-<br />
! scope="row" | '''Kelvin the Frog v2015'''<br />
| BOGTRO<br />
| 2015<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=623373&hilit=Kelvin+the+frog Initial Discussion]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?f=133&t=623373&hilit=Kelvin+the+frog Problems]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=3731642#p3731642 Answers]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=3731642#p3731642 Results]<br />
|-<br />
! scope="row" | '''Mock AMC 10/12'''<br />
| joey8189681<br />
| 2015<br />
| [http://www.artofproblemsolving.com/community/c5h624714p3741743 Initial Discussion]<br />
| [https://www.dropbox.com/s/1c8hhl6yt2awpix/Mock%20AMC%2010.pdf?dl=0 Problems]<br />
| [http://www.artofproblemsolving.com/community/c5h624714p3754593 Answers]<br />
| [http://www.artofproblemsolving.com/community/q2h626092p3756574 Results]<br />
|-<br />
! scope="row" | '''May Mock AMC 10 Contest'''<br />
| azmath333<br />
| 2015<br />
| [http://www.artofproblemsolving.com/community/c5h1082684_mock_amc_1012_contest Initial Discussion]<br />
| [http://www.artofproblemsolving.com/community/c5h1082684_mock_amc_1012_contest Problems]<br />
| [http://www.artofproblemsolving.com/community/c5h1082684p4812575 Answers] [http://www.artofproblemsolving.com/community/c5h1082684p4825662 Solutions]<br />
| [http://www.artofproblemsolving.com/community/c5h1082684p4812575 Results / Discussion] <br />
|-<br />
! scope="row" | '''2015 Mock AMC 10*'''<br />
| droid347<br />
| 2015<br />
| [http://www.artofproblemsolving.com/community/c5h1094505p4893513 Initial Discussion]<br />
| [http://www.artofproblemsolving.com/community/c5h1094505p4893513 Problems]<br />
| [http://www.artofproblemsolving.com/community/c5h1094505p5051006 Answers]<br />
| [http://www.artofproblemsolving.com/community/c5h1094505p5045295 Results / Discussion]<br />
|-<br />
! scope="row" | '''July Mock AMC 10 Contest'''<br />
| akshaygowrish<br />
| 2015<br />
| [http://artofproblemsolving.com/community/c5h1104125_july_mock_amc_10_contest Initial Discussion]<br />
| [https://docs.google.com/document/d/12OXd-mmS_T7SyMcqw1hfvykNlcj-4TR4mMXlRJ09uBg/edit?usp=sharing Problems]<br />
| [https://docs.google.com/document/d/1-Q_w5XNKcRaffvpCLSdoJ51TpRpcDAgYLmV9OlczBCs/edit?usp=sharing Answer Key]<br />
| [http://artofproblemsolving.com/community/c5h1104125p5154283 Results / Discussion]<br />
|-<br />
! scope = "row" |'''August Mock AMC 10'''<br />
| azmath333<br />
| 2015<br />
| [http://www.artofproblemsolving.com/community/c5h1115462_august_mock_amc_10 Initial Discussion]<br />
| [http://latex.artofproblemsolving.com/miscpdf/augustmockamc10.pdf Problems]<br />
| [http://www.artofproblemsolving.com/community/c5h1115462p5270430 Answers]<br />
| [http://www.artofproblemsolving.com/community/c5h1115462p5270430 Results] / [http://www.artofproblemsolving.com/community/c121880_august_mock_amc_10_discussion_forum Discussion]<br />
|-<br />
! scope = "row" | '''September Mock AMC 10'''<br />
| cpma213<br />
| 2015<br />
| [http://artofproblemsolving.com/community/c5h1137447p5320386 Initial Discussion]<br />
| [http://artofproblemsolving.com/community/c5h1137447p5320386 Problems]<br />
| n/a<br />
| [http://artofproblemsolving.com/community/c5h1137447p5479954 Results / Discussion]<br />
|-<br />
! scope = "row" | '''December 2015 Mock AMC 1^3*(3+7)'''<br />
| mathisawesome2169<br />
| 2015<br />
| [http://www.artofproblemsolving.com/community/c5h1173465p5647504 Initial Discussion]<br />
| [http://www.artofproblemsolving.com/community/c5h1173465p5647504 Problems]<br />
| n/a<br />
| [http://www.artofproblemsolving.com/community/c5h1173465p5707101 Results / Discussion]<br />
|-<br />
! scope = "row" | '''New Year's Mock AMC10*'''<br />
| checkmatetang<br />
| 2015<br />
| [http://artofproblemsolving.com/community/c5h1171366p5627439 Initial Discussion]<br />
| [http://artofproblemsolving.com/community/c5h1171366p5627439 Problems]<br />
| [http://artofproblemsolving.com/community/c5h1171366p5711384 Answers]<br />
| [http://artofproblemsolving.com/community/c5h1171366p5711384 Results] / [http://artofproblemsolving.com/community/c202656_new_years_mock_amc10_discussion Discussion]<br />
|-<br />
! scope = "row" | '''2015-2016 Mock AMC 10*'''<br />
| PersonPsychopath<br />
| 2015<br />
| [http://artofproblemsolving.com/community/c5h1168398p5595105 Initial Discussion]<br />
| [http://artofproblemsolving.com/community/c5h1168398p5595105 Problems]<br />
| [http://artofproblemsolving.com/community/c147536h1182005_mock_amc10_wrapup Answers]<br />
| [http://artofproblemsolving.com/community/c5h1168398p5595105 Results / Discussion]<br />
|-<br />
! scope = "row" | '''Mock AMC 10 2015-2016*'''<br />
| AlcumusGuy<br />
| 2015<br />
| [http://www.artofproblemsolving.com/community/c5h1177413p5687990 Initial Discussion]<br />
| [http://artofproblemsolving.com/community/c5h1177413_mock_amc_10_20152016_released Problems]<br />
| [http://artofproblemsolving.com/community/c212844_mock_amc_10_20152016 Solutions]<br />
| [http://artofproblemsolving.com/community/c5h1177413p5769839 Results / Discussion]<br />
|-<br />
! scope = "row" | '''January 2016 Mock AMC 10*'''<br />
| atmchallenge<br />
| 2015<br />
| [http://www.artofproblemsolving.com/community/c5h1178971p5700377 Initial Discussion]<br />
| [http://www.artofproblemsolving.com/community/c5h1178971p5700377 Problems]<br />
| [http://www.artofproblemsolving.com/community/c5h1178971p5838468 Answers]<br />
| [http://www.artofproblemsolving.com/community/c5h1178971p5838468 Results/Discussion]<br />
|-<br />
! scope = "row" | '''2015-2016 Mock AMC 10* #2'''<br />
| PersonPsychopath<br />
| 2015<br />
| [http://artofproblemsolving.com/community/c5h1188467 Initial Discussion]<br />
| [http://artofproblemsolving.com/community/c5h1188467 Problems]<br />
| [http://artofproblemsolving.com/community/c147536h1175930 Solutions]<br />
| [http://artofproblemsolving.com/community/c5h1188467 Results / Discussion]<br />
|-<br />
<br />
! scope="row" | '''MeepyMeepMeep Mock AMC 10'''<br />
| MeepyMeepMeep, speck<br />
| 2016<br />
| [http://artofproblemsolving.com/community/c5h1195432p5852002 Initial Discussion]<br />
| [https://www.dropbox.com/s/uw0herhuqk8zbp5/Mock%20AMC%2010.pdf?dl=0 Problems]<br />
| n/a<br />
| n/a<br />
|- <br />
<br />
!scope="row" | eisirrational's Mock AMC 10<br />
| eisirrational, illogical_21, whatshisbucket<br />
| 2017<br />
| [https://artofproblemsolving.com/community/c5h1372991_eisirrationals_mock_amc_10 Initial Discussion]<br />
| [https://artofproblemsolving.com/community/c5h1372991_eisirrationals_mock_amc_10 Problems]<br />
| [http://artofproblemsolving.com/community/c418591h1392426p7811150 Answer Key]<br />
| [https://artofproblemsolving.com/community/c418591_mock_amc_10_discussion Discussion]<br />
|-<br />
<br />
!scope="row" | Summer Mock AMC 10<br />
| Rowechen, OmicronGamma, FedeX333X, KenV, kvedula2004, ItzVineeth<br />
| 2017<br />
| [https://artofproblemsolving.com/community/c5h1471834_summer_mock_amc_series Initial Discussion]<br />
| [https://drive.google.com/file/d/0B3M-fxa6QG0_ODNpSnJZcEt3djQ/view Problems]<br />
| [http://latex.artofproblemsolving.com/miscpdf/hehdrwpi.pdf?t=1500270651030 Answers]<br />
| [https://artofproblemsolving.com/community/c481237_2017_summer_mock_amc_10_discussion_forum Statistics / Discussion]<br />
|-<br />
<br />
!scope="row" | scrabbler94's Mock AMC 10<br />
| scrabbler94<br />
| 2018<br />
| [https://artofproblemsolving.com/community/c5h1569848_mock_2018_amc_10_test_scrabbler94 Initial Discussion]<br />
| [https://artofproblemsolving.com/community/c5h1569848_mock_2018_amc_10_test_scrabbler94 Problems]<br />
| [https://artofproblemsolving.com/community/c5h1569848_mock_2018_amc_10_test_scrabbler94 Answers]<br />
| [http://artofproblemsolving.com/community/c5h1569848p9704902 Results] / [https://artofproblemsolving.com/community/c593716_scrabbler94s_mock_amc_10_discussion Discussion]<br />
|-<br />
<br />
! scope="row" | '''2018 Mock AMC 10'''<br />
| blue8931<br />
| 2018<br />
| [https://artofproblemsolving.com/community/c594864h1606066 Initial Discussion]<br />
| [https://artofproblemsolving.com/downloads/printable_post_collections/628064 Problems]<br />
| [https://artofproblemsolving.com/community/c628116h1632436_official_answer_key Answer Key]<br />
| [https://artofproblemsolving.com/community/c628116_2018_mock_amc_10_discussion_forum Results / Discussion]<br />
|-<br />
<br />
!scope="row" | 2018 Memorial Day Mock AMC 10<br />
| QIDb602<br />
| 2018<br />
| [https://artofproblemsolving.com/community/c594864h1645806_released_2018_memorial_day_mock_amc_10 Initial Discussion]<br />
| [https://drive.google.com/file/d/1paAHspy5PH1J9fMoNYTb7cIEH3Di72K5/view Problems]<br />
| [https://drive.google.com/file/d/1tipZuHE11jmmOfnSDnhkwKnvIOnbsbaL/view?usp=sharing Answers]<br />
| [https://artofproblemsolving.com/community/c594864h1645806_released_2018_memorial_day_mock_amc_10 Results] / [https://artofproblemsolving.com/community/c671484 Discussion]<br />
|-<br />
<br />
!scope="row" | Autumn Mock AMC 10<br />
| Krypton36, AlastorMoody, alphaone001, dchenmathcounts, InternetPerson10, kootrapali, mathdragon2000, MathGeek2018, Stormersyle<br />
| 2018<br />
| [http://artofproblemsolving.com/community/c594864h1710983p11032970 Initial Discussion]<br />
| [http://artofproblemsolving.com/community/c594864h1710983p11032970 Problems]<br />
| [http://artofproblemsolving.com/community/c594864h1710983p11331788 Answers]<br />
| [http://artofproblemsolving.com/community/c594864h1710983p11332566 Results] / [https://artofproblemsolving.com/community/c761744_autumn_mock_amc_10_discussion_forum_d Discussion]<br />
|-<br />
<br />
!scope="row" | 2018 December Mock AMC 10<br />
| mathchampion1, kcbhatraju, kootrapali, chocolatelover111<br />
| 2018<br />
| [https://artofproblemsolving.com/community/c594864h1743064_2018_december_mock_amc_10_released Initial Discussion]<br />
| [https://artofproblemsolving.com/community/c594864h1743064_2018_december_mock_amc_10_released Problems]<br />
| [https://artofproblemsolving.com/community/c784638_december_mock_amc_10_2018_discussion_forum Answers]<br />
| [https://artofproblemsolving.com/community/c784638_december_mock_amc_10_2018_discussion_forum Results] / [https://artofproblemsolving.com/community/c784638_december_mock_amc_10_2018_discussion_forum Discussion]<br />
|-<br />
<br />
!scope="row" | Christmas Math Competitions Year 2*<br />
| CMC Committee<br />
| 2018-2019<br />
| [https://artofproblemsolving.com/community/c594864h1747367_aime_ii_released_christmas_mathematics_competition_cmc_year_2 Initial Discussion]<br />
| [https://drive.google.com/file/d/15heOR_6rnH70joxJRfZBhQXa8Wezb2AW/view CMC 10A] [https://drive.google.com/file/d/1ZNayjDykKYj4889nXtOb7k9KBu_bqXki/view CMC 10B]<br />
| [https://artofproblemsolving.com/community/c798404h1760024_cmc_10a12a_amp_10b12b_year_2__problems_and_answer_keys Answers]<br />
| [https://artofproblemsolving.com/community/c594864h1747367p11379904 Results] / [https://artofproblemsolving.com/community/c798404_christmas_mathematics_competitions_year_2 Discussion]<br />
|-<br />
<br />
!scope="row" | 2019 Mock AMC 10C*<br />
| fidgetboss_4000<br />
| 2019<br />
| [https://artofproblemsolving.com/community/c594864h1786263_released_2019_amc_10c Initial Discussion]<br />
| [https://artofproblemsolving.com/community/c594864h1786263_released_2019_amc_10c Problems] (or click here: [[2019 AMC 10C Problems]]<br />
| [https://artofproblemsolving.com/community/c594864h1786263_released_2019_amc_10c Answers - In Thread]<br />
| N/A / N/A<br />
|-<br />
!scope="row" | 2019 Mock AMC 10B*<br />
| fidgetboss_4000<br />
| 2019<br />
| [https://artofproblemsolving.com/community/c594864h1897633p13160252 Initial Discussion]<br />
| [https://artofproblemsolving.com/community/c594864h1897633p13160252 Problems] (or click here: [[Mock AMC 10B Problems]]<br />
| [https://artofproblemsolving.com/community/c963448h1919463_answer_keys_d Answers]<br />
| N/A / N/A<br />
|-<br />
!scope="row" | Christmas Math Competitions Year 3*<br />
| CMC Committee<br />
| 2019-2020<br />
| [https://artofproblemsolving.com/community/c594864h1967029_cime_ii_released_christmas_math_competition_cmc_year_3 Initial Discussion]<br />
| [http://cmc.ericshen.net/CMC-2020/CMC-2020-10A.pdf CMC 10A] [http://cmc.ericshen.net/CMC-2020/CMC-2020-10B-booklet.pdf CMC 10B]<br />
| [https://artofproblemsolving.com/community/c1035147h1980361p13760203 Answers]<br />
| [https://artofproblemsolving.com/community/c594864h1967029p13623910 Results] / [https://artofproblemsolving.com/community/c1035147_christmas_mathematics_competitions_year_3 Discussion]<br />
|-<br />
!scope="row" | Stormersyle's Mock AMC 10*<br />
| Stormersyle<br />
| 2019-2020<br />
| [https://artofproblemsolving.com/community/c594864h1974214p13694448 Initial Discussion]<br />
| [https://artofproblemsolving.com/community/c594864h1974214p13694448 Problems]<br />
| [https://artofproblemsolving.com/community/c594864h1974214p13694448 Answers]<br />
| [https://artofproblemsolving.com/community/c594864h1974214p13694448 Results] / [https://artofproblemsolving.com/community/c594864h1974214p13694448 Discussion]<br />
|-<br />
!scope="row" | January 2020 Mock AMC 10/12<br />
| P_Groudon<br />
| 2020<br />
| <br />
[https://artofproblemsolving.com/community/c594864h1984167_concluded_january_2020_mock_amc_1012 Initial Discussion]<br />
| [https://artofproblemsolving.com/community/c594864h1984167_concluded_january_2020_mock_amc_1012 Problems]<br />
| [https://artofproblemsolving.com/community/c1035224h1984207_january_2020_answer_keys Answers]<br />
| [https://artofproblemsolving.com/community/c594864h1984167p13963794 Results]<br />
|-<br />
!scope="row" | 2020 Mock Combo AMC 10<br />
| fidgetboss_4000<br />
| 2020<br />
| <br />
[https://artofproblemsolving.com/community/c594864h2005582_amc_1012_released_mock_combo_amc_1012_ii Initial Discussion]<br />
| [https://artofproblemsolving.com/wiki/index.php?title=2020_Mock_Combo_AMC_10&action=view Problems]<br />
| [https://artofproblemsolving.com/community/c1121049h2053291_answer_key Answers]<br />
| [https://artofproblemsolving.com/community/c594864h2005582_amc_1012_released_mock_combo_amc_1012_ii Results]<br />
|-<br />
!scope="row" | OTSS 2020<br />
| kevinmathz<br />
| 2020<br />
| <br />
[https://artofproblemsolving.com/community/c594864h2066002_otss_olympiad_test_spring_series Initial Discussion]<br />
| [https://drive.google.com/file/d/1JfuEvDGw9i99XeQYACPUDzHebxyrcmtE/edit TMC 10A] [https://drive.google.com/file/d/1FkKjAgehDwpmmNOj_mjvYuCCV1r7p08p/edit TMC 12A]<br />
| [https://artofproblemsolving.com/community/c1130807h2094651_tmc_1012_a_solutions Answers]<br />
| [https://artofproblemsolving.com/community/c1130807h2081571_links_to_mocks_solutions_and_leaderboards Results]<br />
|-<br />
!scope="row" | FMC 2020<br />
| fidgetboss_4000<br />
| 2020<br />
| <br />
[https://artofproblemsolving.com/community/c594864h2069998 Initial Discussion]<br />
| [https://drive.google.com/file/d/17jscyJzVCFDlV6YL0OpPKczyk04Tltsx/view FMC 10A] [https://drive.google.com/file/d/1pMdXlGy9F5hbWSP8CGANAhAA9_ct7pdS/view FMC 12A]<br />
| [https://artofproblemsolving.com/community/c1177836_2020_fmc_public_discussion_forum_uwu Answers]<br />
| [https://artofproblemsolving.com/community/q2h2069998p15525666 Results]<br />
|-<br />
!scope="row" | 2020 Mock AMC 10*<br />
| scrabbler94<br />
| 2020<br />
| <br />
[https://artofproblemsolving.com/community/c594864h2159345_contest_over_2020_mock_amc_1012_scrabbler94 Initial Discussion]<br />
| [https://artofproblemsolving.com/community/c594864h2159345_contest_over_2020_mock_amc_1012_scrabbler94 Problems]<br />
| [https://artofproblemsolving.com/community/c594864h2159345p16460394 Answers]<br />
| [https://artofproblemsolving.com/community/c594864h2159345p16460394 Results]<br />
|-<br />
!scope="row" | JMC 2020*<br />
| skyscraper<br />
| 2020<br />
| <br />
[https://artofproblemsolving.com/community/c594864h2164953_jmc_10_concluded_july_mock_amc Initial Discussion]<br />
| [https://artofproblemsolving.com/community/c594864h2164953_jmc_10_concluded_july_mock_amc Problems]<br />
| [https://artofproblemsolving.com/community/c594864h2164953p16895447 Answers]<br />
| [https://artofproblemsolving.com/community/c594864h2164953p16895447 Results]<br />
|-<br />
!scope="row" | PMC 2020<br />
| kred9<br />
| 2020<br />
| <br />
[https://artofproblemsolving.com/community/c594864h2229291 Initial Discussion]<br />
| [https://artofproblemsolving.com/community/c594864h2229291 Problems]<br />
| [https://artofproblemsolving.com/community/c594864h2229291p17672697 Answers]<br />
| [https://artofproblemsolving.com/community/c594864h2229291p17672697 Results]<br />
|-<br />
!scope="row" | DMC 2020<br />
| DeToasty3<br />
| 2020<br />
| <br />
[https://artofproblemsolving.com/community/q1h2282481p17891592 Initial Discussion]<br />
| [http://detoasty3.gq/DMC/2020_DMC_10.pdf?i=1 Problems]<br />
| [http://detoasty3.gq/DMC/2020_DMC_10_Solutions.pdf Answers]<br />
| [https://artofproblemsolving.com/community/c594864h2282481p18910036 Results]<br />
|-<br />
|}<br />
<br />
=== Mock AMC 8 ===<br />
<br />
{| class="wikitable" style="text-align:center;width:100%"<br />
|-<br />
|<br />
! scope="col" | '''Author'''<br />
! scope="col" | '''Year'''<br />
! scope="col" | '''Initial Discussion'''<br />
! scope="col" | '''Problems'''<br />
! scope="col" width=80 | '''Answers'''<br />
! scope="col" | '''Results/Discussion'''<br />
|-<br />
! scope="row" | '''Mock AMC 8'''<br />
| mathfanatic<br />
| 2004<br />
| [http://www.artofproblemsolving.com/community/c5h15878p111575 Initial Discussion]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=15878 Problems]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=114070#p114070 Answers]<br />
| [http://www.artofproblemsolving.com/community/c5h15878p114404 Results / Discussion]<br />
|-<br />
! scope="row" | '''Mock AMC 8'''<br />
| 13375P34K43V312<br />
| 2006<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=107507 Initial Discussion]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=108455 Problems]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=630802#p630802 Answers]<br />
| [http://www.artofproblemsolving.com/community/c5h108455p623522 Results/Discussion]<br />
|-<br />
! scope="row" | '''Mock AMC 8'''<br />
| ahaanomegas<br />
| 2012<br />
| [http://www.artofproblemsolving.com/community/c5h459346p2577870 Initial Discussion]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=2598111#p2598111 Problems]<br />
| n/a<br />
| n/a<br />
|-<br />
! scope="row" | '''Mock AMC 8'''<br />
| utahjazz<br />
| 2012<br />
| [http://www.artofproblemsolving.com/community/c5h485041p2717878 Initial Discussion]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=2728186#p2728186 Problems]<br />
| n/a<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=2731391#p2731391 Results]<br />
|-<br />
! scope="row" | '''Mock AMC 8'''<br />
| iNomOnCountdown<br />
| 2014<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?f=132&t=590451&hilit=iNomonCountdown%27s+mock+amc+8 Initial Discussion]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?f=132&t=590451&hilit=iNomonCountdown%27s+mock+amc+8 Problems]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?f=132&t=590451&start=0 Answers]<br />
| [http://www.artofproblemsolving.com/Forum/viewtopic.php?f=132&t=590451&start=0 Results / Discussion]<br />
|-<br />
! scope = "row" | '''Mock AMC 8'''<br />
| Tan<br />
| 2014<br />
| [http://www.artofproblemsolving.com/community/c5h613297p3648226 Initial Discussion]<br />
| [http://www.artofproblemsolving.com/community/c5h613297p3648227 Problems]<br />
| [https://docs.google.com/spreadsheets/d/1U1QJ9r4r2xFbRByk9gXOhEVV48ScjoLWZp4IFanbKF0/edit#gid=1475864782 Answers]<br />
| [https://docs.google.com/spreadsheets/d/1U1QJ9r4r2xFbRByk9gXOhEVV48ScjoLWZp4IFanbKF0/edit#gid=1475864782 Results / Discussion]<br />
|-<br />
! scope = "row" | '''2015 Hard Mock AMC 8*'''<br />
| Not_a_Username, 8invalid8<br />
| 2015<br />
| [http://artofproblemsolving.com/community/q1h1128391p5227860 Initial Discussion]<br />
| [http://latex.artofproblemsolving.com/miscpdf/kashimyo.pdf Problems]<br />
| [http://www.artofproblemsolving.com/community/c5h1128391p5423312 Answers]<br />
| [http://www.artofproblemsolving.com/community/c5h1128391p5423312 Results / Discussion]<br />
|-<br />
! scope = "row" | '''Mock AMC 8'''<br />
| PersonPsychopath<br />
| 2015<br />
| [http://artofproblemsolving.com/community/c3h1152332p5457736 Initial Discussion]<br />
| [http://artofproblemsolving.com/downloads/printable_post_collections/163667 Problems]<br />
| [http://artofproblemsolving.com/community/category-admin/165328 Forum]<br />
| [http://artofproblemsolving.com/community/c165328h1155882p5484131 Results / Discussion]<br />
|-<br />
! scope = "row" | '''Mock AMC 8 #2'''<br />
| PersonPsychopath<br />
| 2015<br />
| [http://artofproblemsolving.com/community/c3h1167425p5586244 Initial Discussion]<br />
| [http://artofproblemsolving.com/downloads/printable_post_collections/165838 Problems]<br />
| [http://artofproblemsolving.com/community/c147536_the_mock_amc_8_forum Forum]<br />
| [http://artofproblemsolving.com/community/c3h1167425p5586244 Results / Discussion]<br />
|-<br />
! scope = "row" | '''Mock AMC 8 2015'''<br />
| Alberty44<br />
| 2015<br />
| [http://artofproblemsolving.com/community/c194276_discusssion Initial Discussion]<br />
| [http://artofproblemsolving.com/community/c5h1168822p5674003 Problems]<br />
| n/a<br />
| [http://artofproblemsolving.com/community/c194276h1175928p5673996 Results/Discussion]<br />
|-<br />
! scope = "row" | '''Christmas AMC 8'''<br />
| Mudkipswims42<br />
| 2015<br />
| [http://artofproblemsolving.com/community/c5h1177489p5688588 Initial Discussion]<br />
| [http://artofproblemsolving.com/community/c5h1177489p5688588 Problems]<br />
| [http://www.artofproblemsolving.com/community/c229455_christmas_amc8_discussion_forum Forum]<br />
| [http://artofproblemsolving.com/community/c5h1177489p5880200 Results/Discussion]<br />
|- <br />
<br />
! scope = "row" | '''Mock AMC 8!'''<br />
| eisirrational, pretzel, AOPS12142015, Th3Numb3rThr33, e_power_pi_times_i<br />
| 2017<br />
| [https://artofproblemsolving.com/community/c5h1526187_mock_amc_8 Initial Discussion]<br />
| [http://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvYi84LzFkNzUyMWYwZTViOTQ1MzRmZmU3ODE0NmI2MzIzMGUxNjA2MTcwLnBkZg==&rn=TW9jayBBTUMgOCB2Ny5wZGY= Problems]<br />
| [http://artofproblemsolving.com/community/c3h1545004p9368333 Answer Key/Solutions]<br />
| [https://artofproblemsolving.com/community/c561318_mock_amc_8_2017_discussion Discussion]<br />
|-<br />
<br />
! scope = "row" | '''Mock Summ(er)ation AMC 8!'''<br />
| mathchamp1, kevinmathz, Gali, mathdragon2000, reddragon644, ShreyJ, Quadrastic, Mathnerd1223334444<br />
| 2018<br />
| [https://artofproblemsolving.com/community/c594864h1672811_2018_summeration_mock_amc_8 Initial Discussion]<br />
| [https://math-adventures.weebly.com/uploads/1/1/8/8/118819793/amc_8_problems.pdf Problems]<br />
| n/a<br />
| n/a<br />
|-<br />
<br />
! scope = "row" | '''popcorn1's AMC 8 2018'''<br />
| popcorn1<br />
| 2018<br />
| [https://artofproblemsolving.com/community/c594864h1715072_popcorn1s_amc_8_2018 Initial Discussion]<br />
| [http://artofproblemsolving.com/community/c594864h1715072p11074872 Problems]<br />
| Answer Key / Solutions - not released<br />
| Discussion - not released<br />
|-<br />
<br />
! scope = "row" | ''' Stormersyle's Mock AMC 8<br />
| Stormersyle <br />
| 2018<br />
| [http://artofproblemsolving.com/community/c594864h1722790p11146826 Initial Discussion]<br />
| [http://artofproblemsolving.com/community/c594864h1722790p11146826 Problems]<br />
| [https://latex.artofproblemsolving.com/miscpdf/qgdafjlp.pdf?t=1544494957792 Answer Key/Solutions]<br />
| Discussion - not released<br />
|-<br />
<br />
! scope = "row" | '''mathchampion1's Christmas Mock AMC 8'''<br />
| mathchampion1<br />
| 2018<br />
| [https://artofproblemsolving.com/community/c594864h1752494_released_mock_amc_8_christmas_edition Initial Discussion]<br />
| [https://artofproblemsolving.com/community/c594864h1752494_released_mock_amc_8_christmas_edition Problems]<br />
| Answer Key / Solutions - not released<br />
| Discussion - not released<br />
|-<br />
<br />
! scope = "row" | '''June 2019 Mock AMC 8'''<br />
| fidgetboss_4000<br />
| 2019<br />
| [https://artofproblemsolving.com/community/c594864h1853912_june_2019_mock_amc_8 Initial Discussion]<br />
| [https://artofproblemsolving.com/community/c594864h1853912_june_2019_mock_amc_8 Problems]<br />
| Answer Key / Solutions - not released<br />
| Discussion - not released<br />
|-<br />
<br />
! scope = "row" | '''popcorn1's AMC 8 2019'''<br />
| popcorn1<br />
| 2019<br />
| [https://artofproblemsolving.com/community/c594864h1896786 Initial Discussion]<br />
| [https://artofproblemsolving.com/community/c594864h1896786 Problems]<br />
| [https://artofproblemsolving.com/community/c947561h1950746_answer_keys Answer Key/Solutions]<br />
| [https://artofproblemsolving.com/community/c947561_popcorn1s_amc_8_2019_discussion_forum Discussion]<br />
|-<br />
<br />
! scope = "row" | '''lethan3's Mock AMC 8'''<br />
| lethan3<br />
| 2020<br />
| [https://artofproblemsolving.com/community/c594864h2216496_lethan3s_mock_amc_8_ended Initial Discussion]<br />
| [https://artofproblemsolving.com/community/c594864h2216496_lethan3s_mock_amc_8_ended Problems]<br />
| [https://artofproblemsolving.com/community/c594864h2216496_lethan3s_mock_amc_8_ended Answer Key/Solutions]<br />
| [https://artofproblemsolving.com/community/c1266375_lethan3s_mock_amc_8_discussion_forum Discussion]<br />
|-<br />
<br />
! scope = "row" | '''AAMC 8 Year 1'''<br />
| bissue<br />
| 2020<br />
| [https://artofproblemsolving.com/community/c594864h2255517 Initial Discussion]<br />
| [https://artofproblemsolving.com/community/q1h2255517p17556358 Problems]<br />
| [https://artofproblemsolving.com/community/q1h2255517p18736140 Answer Key]<br />
| [https://artofproblemsolving.com/community/q1h2255517p18736140 Discussion]<br />
|-<br />
<br />
! scope = "row" | '''OMC 8 2021'''<br />
| ObjectZ<br />
| 2021<br />
| [https://artofproblemsolving.com/community/c594864h2303233_omc_8_mock_amc_8 Initial Discussion]<br />
| Problems - not released<br />
| Answer Key / Solutions - not released<br />
| Discussion - not released<br />
|-<br />
<br />
}<br />
<br />
== See also ==<br />
* [[American Mathematics Competitions]]<br />
* [[Math books]]<br />
* [[Mathematics competitions]]<br />
* [[Mock AIME]]<br />
* [[Mock MathCounts]]<br />
* [[Mock USAMO]]<br />
* [[Mock USAJMO]]<br />
* [[Resources for mathematics competitions]]<br />
* [[AoPS Past Contests]]</div>Bissuehttps://artofproblemsolving.com/wiki/index.php?title=User:Bissue&diff=141018User:Bissue2020-12-30T06:15:18Z<p>Bissue: o another "visit count" user page o</p>
<hr />
<div>Here is where I'll fully "publish" mock tests I've written. So far there's only one, but there may be more to come.<br />
<br />
also maybe I'll make one of those "visit counters" so I know whether people actually care<br />
<br />
honestly the amount of "visits" I get won't change anything about the action<br />
<br />
just about how I feel as a result of the action.<br />
<br />
<b>A total of <u>0</u> other people have found this user page.</b><br />
<br />
<b>A total of <u>0</u> other people have used these problems as practice.</b><br />
---------------------<br />
<br />
==Apocalyptic AMC 8 (2020)==<br />
This contest took place between September 8th and November 6th. All problems were written by me <b>(bissue)</b> and testsolved by <b>nikenissan, bobthegod78, knightime1010, truffle, cw357,</b> and <b>ApraTrip.</b><br />
<br />
The mock was separated into two sections: The AMC 8 and The Tiebreakers.<br />
<br />
Standard AMC 8 rules were used. All participants had 40 minutes to complete as many of the 25 problems as they could. Correct answers were worth 1 point each, while incorrect or blank answers were worth 0 points each.<br />
<br />
The Tiebreakers were used to break ties between participants with the same score on the AMC 8. The rules were the same as those used for the ARML tiebreaker. For more information, see the original post in the AoPS Mock Contests Forum here:<br />
<br />
https://artofproblemsolving.com/community/c594864h2255517<br />
<br />
Full statistics and discussion threads can be found using the link above as well.<br />
<br />
==Problem 1==<br />
<br />
To walk up a single floor in her eighteen floor apartment building, Sarah needs to take nine steps up a flight of stairs. If Sarah starts on Floor <math>3</math> and walks up <math>100</math> steps, she would end up on the flight of stairs connecting which two floors?<br />
<br />
<math>\textbf{(A)} ~ \mbox{11 and 12} \qquad \textbf{(B)} ~ \mbox{12 and 13} \qquad \textbf{(C)} ~ \mbox{13 and 14} \qquad \textbf{(D)} ~ \mbox{14 and 15} \qquad \textbf{(E)} ~ \mbox{15 and 16}</math><br />
<br />
==Problem 2==<br />
<br />
Abby, Barb, and Carlos each have <math>35</math>, <math>42</math>, and <math>31</math> trading cards respectively. If they share their trading cards equally between each other, how many more trading cards would Carlos have than before?<br />
<br />
<math>\textbf{(A)} ~ 4 \qquad \textbf{(B)} ~ 5 \qquad \textbf{(C)} ~ 6 \qquad \textbf{(D)} ~ 9 \qquad \textbf{(E)} ~ 11</math><br />
<br />
==Problem 3==<br />
<br />
In triangle <math>ABC</math> the measure of angle <math>\angle A</math> is the average of the measures of angles <math>\angle B</math> and <math>\angle C</math>. What is the measure of angle <math>\angle A</math>?<br />
<br />
<math>\textbf{(A)} ~ 45^{\circ} \qquad \textbf{(B)} ~ 60^{\circ} \qquad \textbf{(C)} ~ 75^{\circ} \qquad \textbf{(D)} ~ 90^{\circ} \qquad \textbf{(E)} ~ 120^{\circ}</math><br />
<br />
==Problem 4==<br />
<br />
A spruce tree grows by <math>25</math> feet, increasing its height by <math>25 \%</math>. If the tree grows for a second time by <math>25</math> feet, by what percent would its height increase?<br />
<br />
<math>\textbf{(A)} ~ 5 \% \qquad \textbf{(B)} ~ 15 \% \qquad \textbf{(C)} ~ 20 \% \qquad \textbf{(D)} ~ 25 \% \qquad \textbf{(E)} ~ 30 \% </math><br />
<br />
==Problem 5==<br />
<br />
Find the sum of the digits of <math>\dfrac{5 \times 10^{2020}}{2}</math>.<br />
<br />
<math>\textbf{(A)} ~ 1 \qquad \textbf{(B)} ~ 2 \qquad \textbf{(C)} ~ 5 \qquad \textbf{(D)} ~ 7 \qquad \textbf{(E)} ~ 8</math><br />
<br />
==Problem 6==<br />
<br />
Square <math>B</math> with side length three is attached to a side of square <math>A</math> with side length four, as shown in the figure below. Find the area of the shaded region.<br />
<asy><br />
size(150);<br />
draw((0, 0)--(4, 0)--(4, 4)--(0, 4)--cycle);<br />
draw((4, 1)--(7, 1)--(7, 4)--(4, 4)--cycle);<br />
filldraw((0, 0)--(4, 0)--(4, 2.285714)--cycle, grey);<br />
filldraw((4, 1)--(7, 1)--(7, 4)--(4, 2.285714)--cycle, grey);<br />
label("A", (2, 2));<br />
label("B", (5.5, 2.5));<br />
</asy><br />
<math>\textbf{(A)} ~ 10 \qquad \textbf{(B)} ~ 10 \frac{1}{2} \qquad \textbf{(C)} ~ 11 \qquad \textbf{(D)} ~ 14 \qquad \textbf{(E)} ~ 14 \frac{1}{2}</math><br />
<br />
==Problem 7==<br />
<br />
When expressed as a decimal rounded to the nearest ten-thousandth, what is the value of <math>\dfrac{125+3}{125 \times 3}</math>?<br />
<br />
<math>\textbf{(A)} ~ 0.3412 \qquad \textbf{(B)} ~ 0.3413 \qquad \textbf{(C)} ~ 0.3414 \qquad \textbf{(D)} ~ 0.3415 \qquad \textbf{(E)} ~ 0.3416</math><br />
<br />
==Problem 8==<br />
<br />
What is the value of<br />
<cmath>(1+2+3)-(2+3+4)+(3+4+5)-\cdots -(98+99+100)?</cmath><br />
<math>\textbf{(A)} ~ -150 \qquad \textbf{(B)} ~ -147 \qquad \textbf{(C)} ~ -144 \qquad \textbf{(D)} ~ 147 \qquad \textbf{(E)} ~ 150</math><br />
<br />
==Problem 9==<br />
<br />
Kayla writes down the first <math>N</math> positive integers. What is the sum of all possible values of <math>N</math> if Kayla writes five multiples of <math>13</math> and six multiples of <math>12</math>?<br />
<br />
<math>\textbf{(A)} ~ 447 \qquad \textbf{(B)} ~ 453 \qquad \textbf{(C)} ~ 518 \qquad \textbf{(D)} ~ 525 \qquad \textbf{(E)} ~ 548</math><br />
<br />
==Problem 10==<br />
<br />
In Murphy's seventh grade homeroom, <math>\frac{7}{12}</math> of the students like tennis, <math>\frac{2}{3}</math> of the students like badminton, and <math>\frac{1}{12}</math> of the students like neither. What is the minimum possible number of students who like both tennis and badminton?<br />
<br />
<math>\textbf{(A)} ~ 1 \qquad \textbf{(B)} ~ 2 \qquad \textbf{(C)} ~ 3 \qquad \textbf{(D)} ~ 4 \qquad \textbf{(E)} ~ 6</math><br />
<br />
==Problem 11==<br />
<br />
For how many values of <math>N</math> does there exist a regular <math>N</math> sided polygon whose vertices all lie on the vertices of a regular <math>24</math> sided polygon?<br />
<br />
<math>\textbf{(A)} ~ 6 \qquad \textbf{(B)} ~ 7 \qquad \textbf{(C)} ~ 8 \qquad \textbf{(D)} ~ 9 \qquad \textbf{(E)} ~ 10</math><br />
<br />
==Problem 12==<br />
<br />
Quadrilateral <math>WXYZ</math> has its vertices on the sides of rectangle <math>ABCD</math> with <math>AB=7</math> and <math>BC=5</math>, as shown below. What is the area of quadrilateral <math>WXYZ</math>?<br />
<asy><br />
size(150);<br />
draw((0, 0)--(7, 0)--(7, 5)--(0, 5)--cycle);<br />
label("A", (0, 0), SW);<br />
label("B", (7, 0), SE);<br />
label("C", (7, 5), NE);<br />
label("D", (0, 5), NW);<br />
filldraw((0, 1)--(4, 0)--(7, 3)--(4, 5)--cycle, grey);<br />
label("W", (0, 1), W);<br />
label("X", (4, 0), S);<br />
label("Y", (7, 3), E);<br />
label("Z", (4, 5), N);<br />
label("4", (2, -0.5));<br />
label("3", (5.5, -0.5));<br />
label("4", (2, 5.5));<br />
label("3", (5.5, 5.5));<br />
</asy><br />
<math>\textbf{(A)} ~ 15 \dfrac{1}{2} \qquad \textbf{(B)} ~ 16 \qquad \textbf{(C)} ~ 16 \dfrac{1}{2} \qquad \textbf{(D)} ~ 17 \qquad \textbf{(E)} ~ 17 \dfrac{1}{2}</math><br />
<br />
==Problem 13==<br />
<br />
To drive to the supermarket, Mable drives for <math>m</math> miles, then drives <math>12</math> miles per hour faster for the remaining <math>\frac{4}{3}m</math> miles. The amount of time Mable spent driving at each of the two speeds was equal. What was Mable's average speed during her drive to the supermarket, in miles per hour?<br />
<br />
<math>\textbf{(A)} ~ \dfrac{81}{2} \qquad \textbf{(B)} ~ \dfrac{288}{7} \qquad \textbf{(C)} ~ 42 \qquad \textbf{(D)} ~ \dfrac{300}{7} \qquad \textbf{(E)} ~ 50</math><br />
<br />
==Problem 14==<br />
<br />
Six circles of radius one are cut out of the rectangle below. What is the area of the shaded region?<br />
<asy><br />
size(150);<br />
filldraw((0, 0)--(6, 0)--(6, 4)--(0, 4)--cycle, grey);<br />
filldraw(circle((1, 1), 1), white);<br />
filldraw(circle((3, 1), 1), white);<br />
filldraw(circle((5, 1), 1), white);<br />
filldraw(circle((1, 3), 1), white);<br />
filldraw(circle((3, 3), 1), white);<br />
filldraw(circle((5, 3), 1), white);<br />
</asy><br />
<math>\textbf{(A)} ~ 20-6\pi \qquad \textbf{(B)} ~ 24-6\pi \qquad \textbf{(C)} ~ 28-6\pi \qquad \textbf{(D)} ~ 30-6\pi \qquad \textbf{(E)} ~ 32-6\pi</math><br />
<br />
==Problem 15==<br />
<br />
One metronome beeps at a steady rate of <math>72</math> beeps per minute, while another metronome beeps at a steady rate of <math>96</math> beeps per minute. If both metronomes beep at the same time once, how long will it take, in seconds, until they first beep at the same time again?<br />
<br />
<math>\textbf{(A)} ~ 2 \dfrac{1}{2} \qquad \textbf{(B)} ~ 5 \qquad \textbf{(C)} ~ 10 \qquad \textbf{(D)} ~ 18 \qquad \textbf{(E)} ~ 24</math><br />
<br />
==Problem 16==<br />
<br />
A square with side length two is placed on a table, forming a <math>30</math> degree angle with the table's surface. How much higher is the top vertex of the square than the table?<br />
<asy><br />
size(150);<br />
draw((0, 0)--(0.882, 0.4714)--(0.4106, 1.3534)--(-0.4714, 0.882)--cycle);<br />
draw((-0.5, 0)--(1, 0), linewidth(3));<br />
draw((-0.75, 1.3534)--(-0.65, 1.3534));<br />
draw((-0.7, 1.3534)--(-0.7, 0));<br />
draw((-0.75, 0)--(-0.65, 0));<br />
</asy><br />
<math>\textbf{(A)} ~ \dfrac{5}{2} \qquad \textbf{(B)} ~ \sqrt{3}+1 \qquad \textbf{(C)} ~ \dfrac{4\sqrt{3}}{3} \qquad \textbf{(D)} ~ 3 \qquad \textbf{(E)} ~ \dfrac{3\sqrt{3}}{2}+1</math><br />
<br />
==Problem 17==<br />
<br />
Kurtis' school schedule is made up of four classes, followed by lunch, followed by three more classes. In how many ways can Kurtis arrange his schedule if two of his classes (Reading and Writing) must occur one immediately after the other?<br />
<br />
<math>\textbf{(A)} ~ 600 \qquad \textbf{(B)} ~ 840 \qquad \textbf{(C)} ~ 1200 \qquad \textbf{(D)} ~ 1440 \qquad \textbf{(E)} ~ 1680</math><br />
<br />
==Problem 18==<br />
<br />
When the number <math>25</math> is added to a list of numbers with total sum <math>S</math>, the average of all the numbers increases by one. What is the sum of the digits of the greatest possible value of <math>S</math>?<br />
<br />
<math>\textbf{(A)} ~ 6 \qquad \textbf{(B)} ~ 7 \qquad \textbf{(C)} ~ 8 \qquad \textbf{(D)} ~ 9 \qquad \textbf{(E)} ~ 12</math><br />
<br />
==Problem 19==<br />
<br />
A magician randomly picks a three digit positive integer to put into her hat and pulls out the same number with its digits in reverse order. (For example <math>496</math> would become <math>694</math> and <math>720</math> would become <math>27</math>.) What is the probability the magician pulls out a multiple of <math>22</math>?<br />
<br />
<math>\textbf{(A)} ~ \dfrac{1}{15} \qquad \textbf{(B)} ~ \dfrac{1}{18} \qquad \textbf{(C)} ~ \dfrac{1}{20} \qquad \textbf{(D)} ~ \dfrac{1}{25} \qquad \textbf{(E)} ~ \dfrac{1}{30}</math><br />
<br />
==Problem 20==<br />
<br />
Tyrone has three books to read in six days. He reads one-half of a single book every day. In how many ways can he finish all the books if he may not read the same book two days in a row?<br />
<br />
<math>\textbf{(A)} ~ 12 \qquad \textbf{(B)} ~ 18 \qquad \textbf{(C)} ~ 24 \qquad \textbf{(D)} ~ 30 \qquad \textbf{(E)} ~ 36</math><br />
<br />
==Problem 21==<br />
<br />
There exists a circle that is tangent to <math>\overline{AB}</math> and <math>\overline{BC}</math> at <math>A</math> and <math>C</math>, respectively. If <math>AB=BC=13</math> and <math>AC=10</math>, what is the radius of the circle?<br />
<asy><br />
size(150);<br />
draw((-5, 0)--(5, 0)--(0, -12)--cycle);<br />
draw(circle((0, 2.08333), 5.41666));<br />
label("A", (-5, 0), W);<br />
label("C", (5, 0), E);<br />
label("B", (0, -12), S);<br />
label("13", (-2.7, -6), W);<br />
label("13", (2.7, -6), E);<br />
label("10", (0, 0.2), N);<br />
</asy><br />
<math>\textbf{(A)} ~ \dfrac{60}{13} \qquad \textbf{(B)} ~ 5 \qquad \textbf{(C)} ~ \dfrac{26}{5} \qquad \textbf{(D)} ~ \dfrac{65}{12} \qquad \textbf{(E)} ~ \dfrac{156}{25}</math><br />
<br />
==Problem 22==<br />
<br />
For each of the distinct sets of numbers containing only positive integers between <math>1</math> and <math>9</math> inclusive, Jordan writes the sum of the numbers in that set. What is the sum of the numbers Jordan writes?<br />
<br />
<math>\textbf{(A)} ~ 11520 \qquad \textbf{(B)} ~ 11565 \qquad \textbf{(C)} ~ 11610 \qquad \textbf{(D)} ~ 11655 \qquad \textbf{(E)} ~ 11700</math><br />
<br />
==Problem 23==<br />
<br />
In rectangle <math>ABCD</math>, the perpendicular from <math>B</math> to diagonal <math>\overline{AC}</math> bisects segment <math>\overline{CD}</math>. Which of the following is closest to <math>\frac{AB}{BC}</math>?<br />
<br />
<math>\textbf{(A)} ~ \dfrac{5}{4} \qquad \textbf{(B)} ~ \dfrac{4}{3} \qquad \textbf{(C)} ~ \dfrac{7}{5} \qquad \textbf{(D)} ~ \dfrac{3}{2} \qquad \textbf{(E)} ~ \dfrac{8}{5}</math><br />
<br />
==Problem 24==<br />
<br />
How many ordered triples of positive integers <math>(a, b, c)</math> satisfy <math>\text{gcd}(a, b, c)=20</math> and <math>\text{lcm}(a, b, c)=240</math>?<br />
<br />
<math>\textbf{(A)} ~ 18 \qquad \textbf{(B)} ~ 24 \qquad \textbf{(C)} ~ 36 \qquad \textbf{(D)} ~ 54 \qquad \textbf{(E)} ~ 72 </math><br />
<br />
==Problem 25==<br />
<br />
Cheyanne rolls two standard six sided dice, then repeatedly rerolls all dice which show an odd number and stops as soon as all dice show an even number. What is the probability Cheyanne stops after exactly four rounds of rerolling?<br />
<br />
<math>\textbf{(A)} ~ \dfrac{61}{1024} \qquad \textbf{(B)} ~ \dfrac{1}{16} \qquad \textbf{(C)} ~ \dfrac{67}{1024} \qquad \textbf{(D)} ~ \dfrac{9}{128} \qquad \textbf{(E)} ~ \dfrac{29}{256}</math><br />
<br />
------------------<br />
==Tiebreaker Problem 1==<br />
<br />
A whiteboard has positive real numbers <math>1</math> and <math>m</math> written on it. Every second, if the numbers <math>x</math> and <math>y</math> are on the whiteboard, a ghost will replace those numbers with <math>|x^2-y^2|</math> and <math>2xy</math>. The ghost stops once one number on the whiteboard is <math>m</math> times the other. For how many positive real numbers <math>m</math> does the ghost stop after exactly <math>16</math> seconds?<br />
<br />
==Tiebreaker Problem 2==<br />
<br />
The perpendicular bisectors of triangle <math>ABC</math> can be described in the coordinate plane as lines <math>y=0</math>, <math>y=x</math>, and <math>y=\sqrt{3}x</math>. Given that triangle <math>ABC</math> has circumradius <math>1</math>, find its area.<br />
<br />
==Tiebreaker Problem 3==<br />
<br />
The diagram below is constructed by attaching an equilateral triangle, a square, a regular pentagon, and a regular hexagon together. Compute the measure of the obtuse angle formed by the three red vertices.<br />
<asy><br />
import graph; size(10cm); <br />
real labelscalefactor = 0.5; /* changes label-to-point distance */<br />
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ <br />
pen dotstyle = black; /* point style */ <br />
real xmin = -29.36, xmax = -9.8, ymin = 4.78, ymax = 17.66; /* image dimensions */<br />
<br />
draw((-22,12)--(-19,12)--(-20.5,14.598076211353318)--cycle, linewidth(2)); <br />
draw((-20.5,14.598076211353318)--(-19,12)--(-16.401923788646684,13.5)--(-17.90192378864668,16.098076211353316)--cycle, linewidth(2)); <br />
draw((-16.401923788646684,13.5)--(-19,12)--(-18.376264927546725,9.065557197798585)--(-15.392699241441907,8.751971807995622)--(-14.172489312214504,11.492608180923423)--cycle, linewidth(2)); <br />
draw((-18.376264927546725,9.065557197798585)--(-19,12)--(-21.85316954888546,12.927050983124847)--(-24.082604025317647,10.919659164048271)--(-23.45886895286437,7.985216361846854)--(-20.60569940397891,7.058165378722009)--cycle, linewidth(2)); <br />
Label laxis; laxis.p = fontsize(10); <br />
string blank(real x) {return "";} <br />
xaxis(xmin, xmax, Ticks(laxis, blank, Step = 1, Size = 2, NoZero),EndArrow(6), above = true); <br />
yaxis(ymin, ymax, Ticks(laxis, blank, Step = 1, Size = 2, NoZero),EndArrow(6), above = true); /* draws axes; NoZero hides '0' label */ <br />
/* draw figures */<br />
draw((-22,12)--(-19,12), linewidth(2)); <br />
draw((-19,12)--(-20.5,14.598076211353318), linewidth(2)); <br />
draw((-20.5,14.598076211353318)--(-22,12), linewidth(2)); <br />
draw((-20.5,14.598076211353318)--(-19,12), linewidth(2)); <br />
draw((-19,12)--(-16.401923788646684,13.5), linewidth(2)); <br />
draw((-16.401923788646684,13.5)--(-17.90192378864668,16.098076211353316), linewidth(2)); <br />
draw((-17.90192378864668,16.098076211353316)--(-20.5,14.598076211353318), linewidth(2)); <br />
draw((-16.401923788646684,13.5)--(-19,12), linewidth(2)); <br />
draw((-19,12)--(-18.376264927546725,9.065557197798585), linewidth(2)); <br />
draw((-18.376264927546725,9.065557197798585)--(-15.392699241441907,8.751971807995622), linewidth(2)); <br />
draw((-15.392699241441907,8.751971807995622)--(-14.172489312214504,11.492608180923423), linewidth(2)); <br />
draw((-14.172489312214504,11.492608180923423)--(-16.401923788646684,13.5), linewidth(2)); <br />
draw((-18.376264927546725,9.065557197798585)--(-19,12), linewidth(2)); <br />
draw((-19,12)--(-21.85316954888546,12.927050983124847), linewidth(2)); <br />
draw((-21.85316954888546,12.927050983124847)--(-24.082604025317647,10.919659164048271), linewidth(2)); <br />
draw((-24.082604025317647,10.919659164048271)--(-23.45886895286437,7.985216361846854), linewidth(2)); <br />
draw((-23.45886895286437,7.985216361846854)--(-20.60569940397891,7.058165378722009), linewidth(2)); <br />
draw((-20.60569940397891,7.058165378722009)--(-18.376264927546725,9.065557197798585), linewidth(2)); <br />
/* dots and labels */<br />
dot((-22,12),red); <br />
dot((-19,12),red); <br />
dot((-21.526119073220972,12.820785841919117),linewidth(4pt) + dotstyle+red); <br />
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); <br />
</asy><br />
------------------------------<br />
==Answer Key==<br />
<br />
AMC 8: DBBCD / CBBAD / AECBA / BCDDD / DACEA<br />
<br />
Tiebreakers: (<math>65280</math>, <math>\dfrac{3-\sqrt{3}}{4}</math>, <math>102</math>)</div>Bissuehttps://artofproblemsolving.com/wiki/index.php?title=The_Devil%27s_Triangle&diff=140954The Devil's Triangle2020-12-29T23:22:35Z<p>Bissue: </p>
<hr />
<div>=Definition=<br />
==Generalized Wooga Looga Theorem (The Devil's Triangle)==<br />
For any triangle <math>\triangle ABC</math>, let <math>D, E</math> and <math>F</math> be points on <math>BC, AC</math> and <math>AB</math> respectively. The Generalizwed Wooga Looga Theorem or the Devil's Triangle Theorem states that if <math>\frac{BD}{CD}=r, \frac{CE}{AE}=s</math> and <math>\frac{AF}{BF}=t</math>, then <math>\frac{[DEF]}{[ABC]}=1-\frac{r(s+1)+s(t+1)+t(r+1)}{(r+1)(s+1)(t+1)}=\frac{rst+1}{(r+1)(s+1)(t+1)}</math>.<br />
<br />
(*Simplification found by @Gogobao)<br />
<br />
=Proofs=<br />
==Proof 1==<br />
Proof by CoolJupiter:<br />
<br />
We have the following ratios:<br />
<math>\frac{BD}{BC}=\frac{r}{r+1}, \frac{CD}{BC}=\frac{1}{r+1},\frac{CE}{AC}=\frac{s}{s+1}, \frac{AE}{AC}=\frac{1}{s+1},\frac{AF}{AB}=\frac{t}{t+1}, \frac{BF}{AB}=\frac{1}{t+1}</math>.<br />
<br />
Now notice that <math>[DEF]=[ABC]-([BDF]+[CDE]+[AEF])</math>.<br />
<br />
We attempt to find the area of each of the smaller triangles. <br />
<br />
<br />
Notice that <math>\frac{[BDF]}{[ABC]}=\frac{BF}{AB}\times \frac{BD}{BC}=\frac{r}{(r+1)(t+1)}</math> using the ratios derived earlier.<br />
<br />
<br />
Similarly, <math>\frac{[CDE]}{[ABC]}=\frac{s}{(r+1)(s+1)}</math> and <math>\frac{[AEF]}{[ABC]}=\frac{t}{(s+1)(t+1)}</math>.<br />
<br />
<br />
Thus, <math>\frac{[BDF]+[CDE]+[AEF]}{[ABC]}=\frac{r}{(r+1)(t+1)}+\frac{s}{(r+1)(s+1)}+\frac{t}{(s+1)(t+1)}=\frac{r(s+1)+s(t+1)+t(r+1)}{(r+1)(s+1)(t+1)}</math>.<br />
<br />
Finally, we have <math>\frac{[DEF]}{[ABC]}=1-\frac{r(s+1)+s(t+1)+t(r+1)}{(r+1)(s+1)(t+1)}=\boxed{\frac{rst+1}{(r+1)(s+1)(t+1)}}</math>.<br />
<br />
~@CoolJupiter<br />
==Proof 2==<br />
Proof by math_comb01<br />
Apply Barycentrics <math>\triangle ABC</math>. Then <math>A=(1,0,0),B=(0,1,0),C=(0,0,1)</math>. also <math>D=\left(0,\tfrac {1}{r+1},\tfrac {r}{r+1}\right),E=\left(\tfrac {s}{s+1},0,\tfrac {1}{s+1}\right),F=\left(\tfrac {1}{t+1},\tfrac {t}{t+1},0\right)</math><br />
In the barycentrics, the area formula is <math>[XYZ]=\begin{vmatrix} x_{1} &y_{1} &z_{1} \\ x_{2} &y_{2} &z_{2} \\ x_{3}& y_{3} & z_{3} \end{vmatrix}\cdot [ABC]</math> where <math>\triangle XYZ</math> is a random triangle and <math>\triangle ABC</math> is the reference triangle. Using this, we <cmath>\frac{[DEF]}{[ABC]}</cmath>=<math> \begin{vmatrix} 0&\tfrac {1}{r+1}&\tfrac {r}{r+1} \\ \tfrac {s}{s+1}&0&\tfrac {1}{s+1}\\ \tfrac {1}{t+1}&\tfrac {t}{t+1}&0 \end{vmatrix}</math>=<math>\frac{1}{[s+1][r+1][t+1]}</math><math>+\frac{rst}{([s+1][r+1][t+1]}</math>=<math>\frac{rst+1}{([s+1][r+1][t+1]}</math><br />
~@Math_comb01<br />
<br />
=Other Remarks=<br />
This theorem is a generalization of the Wooga Looga Theorem, which @RedFireTruck claims to have "rediscovered". The link to the theorem can be found here:<br />
https://artofproblemsolving.com/wiki/index.php/Wooga_Looga_Theorem<br />
<br />
Essentially, Wooga Looga is a special case of this, specifically when <math>r=s=t</math>.<br />
<br />
=Testimonials=<br />
<br />
This is Routh's theorem isn't it~ Ilovepizza2020<br />
<br />
Wow this generalization of my theorem is amazing. good job. - Foogle and Hoogle, Members of the Ooga Booga Tribe of The Caveman Society<br />
<br />
trivial by <math>\frac{1}{2}ab\sin(C)</math> but ok ~ bissue</div>Bissuehttps://artofproblemsolving.com/wiki/index.php?title=User:Bissue&diff=140608User:Bissue2020-12-25T23:19:09Z<p>Bissue: </p>
<hr />
<div>Here is where I'll fully "publish" mock tests I've written. So far there's only one, but there may be more to come.<br />
---------------------<br />
<br />
==Apocalyptic AMC 8 (2020)==<br />
This contest took place between September 8th and November 6th. All problems were written by me <b>(bissue)</b> and testsolved by <b>nikenissan, bobthegod78, knightime1010, truffle, cw357,</b> and <b>ApraTrip.</b><br />
<br />
The mock was separated into two sections: The AMC 8 and The Tiebreakers.<br />
<br />
Standard AMC 8 rules were used. All participants had 40 minutes to complete as many of the 25 problems as they could. Correct answers were worth 1 point each, while incorrect or blank answers were worth 0 points each.<br />
<br />
The Tiebreakers were used to break ties between participants with the same score on the AMC 8. The rules were the same as those used for the ARML tiebreaker. For more information, see the original post in the AoPS Mock Contests Forum here:<br />
<br />
https://artofproblemsolving.com/community/c594864h2255517<br />
<br />
Full statistics and discussion threads can be found using the link above as well.<br />
<br />
==Problem 1==<br />
<br />
To walk up a single floor in her eighteen floor apartment building, Sarah needs to take nine steps up a flight of stairs. If Sarah starts on Floor <math>3</math> and walks up <math>100</math> steps, she would end up on the flight of stairs connecting which two floors?<br />
<br />
<math>\textbf{(A)} ~ \mbox{11 and 12} \qquad \textbf{(B)} ~ \mbox{12 and 13} \qquad \textbf{(C)} ~ \mbox{13 and 14} \qquad \textbf{(D)} ~ \mbox{14 and 15} \qquad \textbf{(E)} ~ \mbox{15 and 16}</math><br />
<br />
==Problem 2==<br />
<br />
Abby, Barb, and Carlos each have <math>35</math>, <math>42</math>, and <math>31</math> trading cards respectively. If they share their trading cards equally between each other, how many more trading cards would Carlos have than before?<br />
<br />
<math>\textbf{(A)} ~ 4 \qquad \textbf{(B)} ~ 5 \qquad \textbf{(C)} ~ 6 \qquad \textbf{(D)} ~ 9 \qquad \textbf{(E)} ~ 11</math><br />
<br />
==Problem 3==<br />
<br />
In triangle <math>ABC</math> the measure of angle <math>\angle A</math> is the average of the measures of angles <math>\angle B</math> and <math>\angle C</math>. What is the measure of angle <math>\angle A</math>?<br />
<br />
<math>\textbf{(A)} ~ 45^{\circ} \qquad \textbf{(B)} ~ 60^{\circ} \qquad \textbf{(C)} ~ 75^{\circ} \qquad \textbf{(D)} ~ 90^{\circ} \qquad \textbf{(E)} ~ 120^{\circ}</math><br />
<br />
==Problem 4==<br />
<br />
A spruce tree grows by <math>25</math> feet, increasing its height by <math>25 \%</math>. If the tree grows for a second time by <math>25</math> feet, by what percent would its height increase?<br />
<br />
<math>\textbf{(A)} ~ 5 \% \qquad \textbf{(B)} ~ 15 \% \qquad \textbf{(C)} ~ 20 \% \qquad \textbf{(D)} ~ 25 \% \qquad \textbf{(E)} ~ 30 \% </math><br />
<br />
==Problem 5==<br />
<br />
Find the sum of the digits of <math>\dfrac{5 \times 10^{2020}}{2}</math>.<br />
<br />
<math>\textbf{(A)} ~ 1 \qquad \textbf{(B)} ~ 2 \qquad \textbf{(C)} ~ 5 \qquad \textbf{(D)} ~ 7 \qquad \textbf{(E)} ~ 8</math><br />
<br />
==Problem 6==<br />
<br />
Square <math>B</math> with side length three is attached to a side of square <math>A</math> with side length four, as shown in the figure below. Find the area of the shaded region.<br />
<asy><br />
size(150);<br />
draw((0, 0)--(4, 0)--(4, 4)--(0, 4)--cycle);<br />
draw((4, 1)--(7, 1)--(7, 4)--(4, 4)--cycle);<br />
filldraw((0, 0)--(4, 0)--(4, 2.285714)--cycle, grey);<br />
filldraw((4, 1)--(7, 1)--(7, 4)--(4, 2.285714)--cycle, grey);<br />
label("A", (2, 2));<br />
label("B", (5.5, 2.5));<br />
</asy><br />
<math>\textbf{(A)} ~ 10 \qquad \textbf{(B)} ~ 10 \frac{1}{2} \qquad \textbf{(C)} ~ 11 \qquad \textbf{(D)} ~ 14 \qquad \textbf{(E)} ~ 14 \frac{1}{2}</math><br />
<br />
==Problem 7==<br />
<br />
When expressed as a decimal rounded to the nearest ten-thousandth, what is the value of <math>\dfrac{125+3}{125 \times 3}</math>?<br />
<br />
<math>\textbf{(A)} ~ 0.3412 \qquad \textbf{(B)} ~ 0.3413 \qquad \textbf{(C)} ~ 0.3414 \qquad \textbf{(D)} ~ 0.3415 \qquad \textbf{(E)} ~ 0.3416</math><br />
<br />
==Problem 8==<br />
<br />
What is the value of<br />
<cmath>(1+2+3)-(2+3+4)+(3+4+5)-\cdots -(98+99+100)?</cmath><br />
<math>\textbf{(A)} ~ -150 \qquad \textbf{(B)} ~ -147 \qquad \textbf{(C)} ~ -144 \qquad \textbf{(D)} ~ 147 \qquad \textbf{(E)} ~ 150</math><br />
<br />
==Problem 9==<br />
<br />
Kayla writes down the first <math>N</math> positive integers. What is the sum of all possible values of <math>N</math> if Kayla writes five multiples of <math>13</math> and six multiples of <math>12</math>?<br />
<br />
<math>\textbf{(A)} ~ 447 \qquad \textbf{(B)} ~ 453 \qquad \textbf{(C)} ~ 518 \qquad \textbf{(D)} ~ 525 \qquad \textbf{(E)} ~ 548</math><br />
<br />
==Problem 10==<br />
<br />
In Murphy's seventh grade homeroom, <math>\frac{7}{12}</math> of the students like tennis, <math>\frac{2}{3}</math> of the students like badminton, and <math>\frac{1}{12}</math> of the students like neither. What is the minimum possible number of students who like both tennis and badminton?<br />
<br />
<math>\textbf{(A)} ~ 1 \qquad \textbf{(B)} ~ 2 \qquad \textbf{(C)} ~ 3 \qquad \textbf{(D)} ~ 4 \qquad \textbf{(E)} ~ 6</math><br />
<br />
==Problem 11==<br />
<br />
For how many values of <math>N</math> does there exist a regular <math>N</math> sided polygon whose vertices all lie on the vertices of a regular <math>24</math> sided polygon?<br />
<br />
<math>\textbf{(A)} ~ 6 \qquad \textbf{(B)} ~ 7 \qquad \textbf{(C)} ~ 8 \qquad \textbf{(D)} ~ 9 \qquad \textbf{(E)} ~ 10</math><br />
<br />
==Problem 12==<br />
<br />
Quadrilateral <math>WXYZ</math> has its vertices on the sides of rectangle <math>ABCD</math> with <math>AB=7</math> and <math>BC=5</math>, as shown below. What is the area of quadrilateral <math>WXYZ</math>?<br />
<asy><br />
size(150);<br />
draw((0, 0)--(7, 0)--(7, 5)--(0, 5)--cycle);<br />
label("A", (0, 0), SW);<br />
label("B", (7, 0), SE);<br />
label("C", (7, 5), NE);<br />
label("D", (0, 5), NW);<br />
filldraw((0, 1)--(4, 0)--(7, 3)--(4, 5)--cycle, grey);<br />
label("W", (0, 1), W);<br />
label("X", (4, 0), S);<br />
label("Y", (7, 3), E);<br />
label("Z", (4, 5), N);<br />
label("4", (2, -0.5));<br />
label("3", (5.5, -0.5));<br />
label("4", (2, 5.5));<br />
label("3", (5.5, 5.5));<br />
</asy><br />
<math>\textbf{(A)} ~ 15 \dfrac{1}{2} \qquad \textbf{(B)} ~ 16 \qquad \textbf{(C)} ~ 16 \dfrac{1}{2} \qquad \textbf{(D)} ~ 17 \qquad \textbf{(E)} ~ 17 \dfrac{1}{2}</math><br />
<br />
==Problem 13==<br />
<br />
To drive to the supermarket, Mable drives for <math>m</math> miles, then drives <math>12</math> miles per hour faster for the remaining <math>\frac{4}{3}m</math> miles. The amount of time Mable spent driving at each of the two speeds was equal. What was Mable's average speed during her drive to the supermarket, in miles per hour?<br />
<br />
<math>\textbf{(A)} ~ \dfrac{81}{2} \qquad \textbf{(B)} ~ \dfrac{288}{7} \qquad \textbf{(C)} ~ 42 \qquad \textbf{(D)} ~ \dfrac{300}{7} \qquad \textbf{(E)} ~ 50</math><br />
<br />
==Problem 14==<br />
<br />
Six circles of radius one are cut out of the rectangle below. What is the area of the shaded region?<br />
<asy><br />
size(150);<br />
filldraw((0, 0)--(6, 0)--(6, 4)--(0, 4)--cycle, grey);<br />
filldraw(circle((1, 1), 1), white);<br />
filldraw(circle((3, 1), 1), white);<br />
filldraw(circle((5, 1), 1), white);<br />
filldraw(circle((1, 3), 1), white);<br />
filldraw(circle((3, 3), 1), white);<br />
filldraw(circle((5, 3), 1), white);<br />
</asy><br />
<math>\textbf{(A)} ~ 20-6\pi \qquad \textbf{(B)} ~ 24-6\pi \qquad \textbf{(C)} ~ 28-6\pi \qquad \textbf{(D)} ~ 30-6\pi \qquad \textbf{(E)} ~ 32-6\pi</math><br />
<br />
==Problem 15==<br />
<br />
One metronome beeps at a steady rate of <math>72</math> beeps per minute, while another metronome beeps at a steady rate of <math>96</math> beeps per minute. If both metronomes beep at the same time once, how long will it take, in seconds, until they first beep at the same time again?<br />
<br />
<math>\textbf{(A)} ~ 2 \dfrac{1}{2} \qquad \textbf{(B)} ~ 5 \qquad \textbf{(C)} ~ 10 \qquad \textbf{(D)} ~ 18 \qquad \textbf{(E)} ~ 24</math><br />
<br />
==Problem 16==<br />
<br />
A square with side length two is placed on a table, forming a <math>30</math> degree angle with the table's surface. How much higher is the top vertex of the square than the table?<br />
<asy><br />
size(150);<br />
draw((0, 0)--(0.882, 0.4714)--(0.4106, 1.3534)--(-0.4714, 0.882)--cycle);<br />
draw((-0.5, 0)--(1, 0), linewidth(3));<br />
draw((-0.75, 1.3534)--(-0.65, 1.3534));<br />
draw((-0.7, 1.3534)--(-0.7, 0));<br />
draw((-0.75, 0)--(-0.65, 0));<br />
</asy><br />
<math>\textbf{(A)} ~ \dfrac{5}{2} \qquad \textbf{(B)} ~ \sqrt{3}+1 \qquad \textbf{(C)} ~ \dfrac{4\sqrt{3}}{3} \qquad \textbf{(D)} ~ 3 \qquad \textbf{(E)} ~ \dfrac{3\sqrt{3}}{2}+1</math><br />
<br />
==Problem 17==<br />
<br />
Kurtis' school schedule is made up of four classes, followed by lunch, followed by three more classes. In how many ways can Kurtis arrange his schedule if two of his classes (Reading and Writing) must occur one immediately after the other?<br />
<br />
<math>\textbf{(A)} ~ 600 \qquad \textbf{(B)} ~ 840 \qquad \textbf{(C)} ~ 1200 \qquad \textbf{(D)} ~ 1440 \qquad \textbf{(E)} ~ 1680</math><br />
<br />
==Problem 18==<br />
<br />
When the number <math>25</math> is added to a list of numbers with total sum <math>S</math>, the average of all the numbers increases by one. What is the sum of the digits of the greatest possible value of <math>S</math>?<br />
<br />
<math>\textbf{(A)} ~ 6 \qquad \textbf{(B)} ~ 7 \qquad \textbf{(C)} ~ 8 \qquad \textbf{(D)} ~ 9 \qquad \textbf{(E)} ~ 12</math><br />
<br />
==Problem 19==<br />
<br />
A magician randomly picks a three digit positive integer to put into her hat and pulls out the same number with its digits in reverse order. (For example <math>496</math> would become <math>694</math> and <math>720</math> would become <math>27</math>.) What is the probability the magician pulls out a multiple of <math>22</math>?<br />
<br />
<math>\textbf{(A)} ~ \dfrac{1}{15} \qquad \textbf{(B)} ~ \dfrac{1}{18} \qquad \textbf{(C)} ~ \dfrac{1}{20} \qquad \textbf{(D)} ~ \dfrac{1}{25} \qquad \textbf{(E)} ~ \dfrac{1}{30}</math><br />
<br />
==Problem 20==<br />
<br />
Tyrone has three books to read in six days. He reads one-half of a single book every day. In how many ways can he finish all the books if he may not read the same book two days in a row?<br />
<br />
<math>\textbf{(A)} ~ 12 \qquad \textbf{(B)} ~ 18 \qquad \textbf{(C)} ~ 24 \qquad \textbf{(D)} ~ 30 \qquad \textbf{(E)} ~ 36</math><br />
<br />
==Problem 21==<br />
<br />
There exists a circle that is tangent to <math>\overline{AB}</math> and <math>\overline{BC}</math> at <math>A</math> and <math>C</math>, respectively. If <math>AB=BC=13</math> and <math>AC=10</math>, what is the radius of the circle?<br />
<asy><br />
size(150);<br />
draw((-5, 0)--(5, 0)--(0, -12)--cycle);<br />
draw(circle((0, 2.08333), 5.41666));<br />
label("A", (-5, 0), W);<br />
label("C", (5, 0), E);<br />
label("B", (0, -12), S);<br />
label("13", (-2.7, -6), W);<br />
label("13", (2.7, -6), E);<br />
label("10", (0, 0.2), N);<br />
</asy><br />
<math>\textbf{(A)} ~ \dfrac{60}{13} \qquad \textbf{(B)} ~ 5 \qquad \textbf{(C)} ~ \dfrac{26}{5} \qquad \textbf{(D)} ~ \dfrac{65}{12} \qquad \textbf{(E)} ~ \dfrac{156}{25}</math><br />
<br />
==Problem 22==<br />
<br />
For each of the distinct sets of numbers containing only positive integers between <math>1</math> and <math>9</math> inclusive, Jordan writes the sum of the numbers in that set. What is the sum of the numbers Jordan writes?<br />
<br />
<math>\textbf{(A)} ~ 11520 \qquad \textbf{(B)} ~ 11565 \qquad \textbf{(C)} ~ 11610 \qquad \textbf{(D)} ~ 11655 \qquad \textbf{(E)} ~ 11700</math><br />
<br />
==Problem 23==<br />
<br />
In rectangle <math>ABCD</math>, the perpendicular from <math>B</math> to diagonal <math>\overline{AC}</math> bisects segment <math>\overline{CD}</math>. Which of the following is closest to <math>\frac{AB}{BC}</math>?<br />
<br />
<math>\textbf{(A)} ~ \dfrac{5}{4} \qquad \textbf{(B)} ~ \dfrac{4}{3} \qquad \textbf{(C)} ~ \dfrac{7}{5} \qquad \textbf{(D)} ~ \dfrac{3}{2} \qquad \textbf{(E)} ~ \dfrac{8}{5}</math><br />
<br />
==Problem 24==<br />
<br />
How many ordered triples of positive integers <math>(a, b, c)</math> satisfy <math>\text{gcd}(a, b, c)=20</math> and <math>\text{lcm}(a, b, c)=240</math>?<br />
<br />
<math>\textbf{(A)} ~ 18 \qquad \textbf{(B)} ~ 24 \qquad \textbf{(C)} ~ 36 \qquad \textbf{(D)} ~ 54 \qquad \textbf{(E)} ~ 72 </math><br />
<br />
==Problem 25==<br />
<br />
Cheyanne rolls two standard six sided dice, then repeatedly rerolls all dice which show an odd number and stops as soon as all dice show an even number. What is the probability Cheyanne stops after exactly four rounds of rerolling?<br />
<br />
<math>\textbf{(A)} ~ \dfrac{61}{1024} \qquad \textbf{(B)} ~ \dfrac{1}{16} \qquad \textbf{(C)} ~ \dfrac{67}{1024} \qquad \textbf{(D)} ~ \dfrac{9}{128} \qquad \textbf{(E)} ~ \dfrac{29}{256}</math><br />
<br />
------------------<br />
==Tiebreaker Problem 1==<br />
<br />
A whiteboard has positive real numbers <math>1</math> and <math>m</math> written on it. Every second, if the numbers <math>x</math> and <math>y</math> are on the whiteboard, a ghost will replace those numbers with <math>|x^2-y^2|</math> and <math>2xy</math>. The ghost stops once one number on the whiteboard is <math>m</math> times the other. For how many positive real numbers <math>m</math> does the ghost stop after exactly <math>16</math> seconds?<br />
<br />
==Tiebreaker Problem 2==<br />
<br />
The perpendicular bisectors of triangle <math>ABC</math> can be described in the coordinate plane as lines <math>y=0</math>, <math>y=x</math>, and <math>y=\sqrt{3}x</math>. Given that triangle <math>ABC</math> has circumradius <math>1</math>, find its area.<br />
<br />
==Tiebreaker Problem 3==<br />
<br />
The diagram below is constructed by attaching an equilateral triangle, a square, a regular pentagon, and a regular hexagon together. Compute the measure of the obtuse angle formed by the three red vertices.<br />
<asy><br />
import graph; size(10cm); <br />
real labelscalefactor = 0.5; /* changes label-to-point distance */<br />
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ <br />
pen dotstyle = black; /* point style */ <br />
real xmin = -29.36, xmax = -9.8, ymin = 4.78, ymax = 17.66; /* image dimensions */<br />
<br />
draw((-22,12)--(-19,12)--(-20.5,14.598076211353318)--cycle, linewidth(2)); <br />
draw((-20.5,14.598076211353318)--(-19,12)--(-16.401923788646684,13.5)--(-17.90192378864668,16.098076211353316)--cycle, linewidth(2)); <br />
draw((-16.401923788646684,13.5)--(-19,12)--(-18.376264927546725,9.065557197798585)--(-15.392699241441907,8.751971807995622)--(-14.172489312214504,11.492608180923423)--cycle, linewidth(2)); <br />
draw((-18.376264927546725,9.065557197798585)--(-19,12)--(-21.85316954888546,12.927050983124847)--(-24.082604025317647,10.919659164048271)--(-23.45886895286437,7.985216361846854)--(-20.60569940397891,7.058165378722009)--cycle, linewidth(2)); <br />
Label laxis; laxis.p = fontsize(10); <br />
string blank(real x) {return "";} <br />
xaxis(xmin, xmax, Ticks(laxis, blank, Step = 1, Size = 2, NoZero),EndArrow(6), above = true); <br />
yaxis(ymin, ymax, Ticks(laxis, blank, Step = 1, Size = 2, NoZero),EndArrow(6), above = true); /* draws axes; NoZero hides '0' label */ <br />
/* draw figures */<br />
draw((-22,12)--(-19,12), linewidth(2)); <br />
draw((-19,12)--(-20.5,14.598076211353318), linewidth(2)); <br />
draw((-20.5,14.598076211353318)--(-22,12), linewidth(2)); <br />
draw((-20.5,14.598076211353318)--(-19,12), linewidth(2)); <br />
draw((-19,12)--(-16.401923788646684,13.5), linewidth(2)); <br />
draw((-16.401923788646684,13.5)--(-17.90192378864668,16.098076211353316), linewidth(2)); <br />
draw((-17.90192378864668,16.098076211353316)--(-20.5,14.598076211353318), linewidth(2)); <br />
draw((-16.401923788646684,13.5)--(-19,12), linewidth(2)); <br />
draw((-19,12)--(-18.376264927546725,9.065557197798585), linewidth(2)); <br />
draw((-18.376264927546725,9.065557197798585)--(-15.392699241441907,8.751971807995622), linewidth(2)); <br />
draw((-15.392699241441907,8.751971807995622)--(-14.172489312214504,11.492608180923423), linewidth(2)); <br />
draw((-14.172489312214504,11.492608180923423)--(-16.401923788646684,13.5), linewidth(2)); <br />
draw((-18.376264927546725,9.065557197798585)--(-19,12), linewidth(2)); <br />
draw((-19,12)--(-21.85316954888546,12.927050983124847), linewidth(2)); <br />
draw((-21.85316954888546,12.927050983124847)--(-24.082604025317647,10.919659164048271), linewidth(2)); <br />
draw((-24.082604025317647,10.919659164048271)--(-23.45886895286437,7.985216361846854), linewidth(2)); <br />
draw((-23.45886895286437,7.985216361846854)--(-20.60569940397891,7.058165378722009), linewidth(2)); <br />
draw((-20.60569940397891,7.058165378722009)--(-18.376264927546725,9.065557197798585), linewidth(2)); <br />
/* dots and labels */<br />
dot((-22,12),red); <br />
dot((-19,12),red); <br />
dot((-21.526119073220972,12.820785841919117),linewidth(4pt) + dotstyle+red); <br />
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); <br />
</asy><br />
------------------------------<br />
==Answer Key==<br />
<br />
AMC 8: DBBCD / CBBAD / AECBA / BCDDD / DACEA<br />
<br />
Tiebreakers: (<math>65280</math>, <math>\dfrac{3-\sqrt{3}}{4}</math>, <math>102</math>)</div>Bissuehttps://artofproblemsolving.com/wiki/index.php?title=User:Bissue&diff=140588User:Bissue2020-12-25T21:03:34Z<p>Bissue: </p>
<hr />
<div>Here is where I'll fully publish mock tests I've written. So far there's only one, but there may be more to come.<br />
<br />
==Apocalyptic AMC 8 (2020)==<br />
<br />
==Problem 1==<br />
<br />
To walk up a single floor in her eighteen floor apartment building, Sarah needs to take nine steps up a flight of stairs. If Sarah starts on Floor <math>3</math> and walks up <math>100</math> steps, she would end up on the flight of stairs connecting which two floors?<br />
<br />
<math>\textbf{(A)} ~ \mbox{11 and 12} \qquad \textbf{(B)} ~ \mbox{12 and 13} \qquad \textbf{(C)} ~ \mbox{13 and 14} \qquad \textbf{(D)} ~ \mbox{14 and 15} \qquad \textbf{(E)} ~ \mbox{15 and 16}</math><br />
<br />
==Problem 2==<br />
<br />
Abby, Barb, and Carlos each have <math>35</math>, <math>42</math>, and <math>31</math> trading cards respectively. If they share their trading cards equally between each other, how many more trading cards would Carlos have than before?<br />
<br />
<math>\textbf{(A)} ~ 4 \qquad \textbf{(B)} ~ 5 \qquad \textbf{(C)} ~ 6 \qquad \textbf{(D)} ~ 9 \qquad \textbf{(E)} ~ 11</math><br />
<br />
==Problem 3==<br />
<br />
In triangle <math>ABC</math> the measure of angle <math>\angle A</math> is the average of the measures of angles <math>\angle B</math> and <math>\angle C</math>. What is the measure of angle <math>\angle A</math>?<br />
<br />
<math>\textbf{(A)} ~ 45^{\circ} \qquad \textbf{(B)} ~ 60^{\circ} \qquad \textbf{(C)} ~ 75^{\circ} \qquad \textbf{(D)} ~ 90^{\circ} \qquad \textbf{(E)} ~ 120^{\circ}</math><br />
<br />
==Problem 4==<br />
<br />
A spruce tree grows by <math>25</math> feet, increasing its height by <math>25 \%</math>. If the tree grows for a second time by <math>25</math> feet, by what percent would its height increase?<br />
<br />
<math>\textbf{(A)} ~ 5 \% \qquad \textbf{(B)} ~ 15 \% \qquad \textbf{(C)} ~ 20 \% \qquad \textbf{(D)} ~ 25 \% \qquad \textbf{(E)} ~ 30 \% </math><br />
<br />
==Problem 5==<br />
<br />
Find the sum of the digits of <math>\dfrac{5 \times 10^{2020}}{2}</math>.<br />
<br />
<math>\textbf{(A)} ~ 1 \qquad \textbf{(B)} ~ 2 \qquad \textbf{(C)} ~ 5 \qquad \textbf{(D)} ~ 7 \qquad \textbf{(E)} ~ 8</math><br />
<br />
==Problem 6==<br />
<br />
Square <math>B</math> with side length three is attached to a side of square <math>A</math> with side length four, as shown in the figure below. Find the area of the shaded region.<br />
<asy><br />
size(150);<br />
draw((0, 0)--(4, 0)--(4, 4)--(0, 4)--cycle);<br />
draw((4, 1)--(7, 1)--(7, 4)--(4, 4)--cycle);<br />
filldraw((0, 0)--(4, 0)--(4, 2.285714)--cycle, grey);<br />
filldraw((4, 1)--(7, 1)--(7, 4)--(4, 2.285714)--cycle, grey);<br />
label("A", (2, 2));<br />
label("B", (5.5, 2.5));<br />
</asy><br />
<math>\textbf{(A)} ~ 10 \qquad \textbf{(B)} ~ 10 \frac{1}{2} \qquad \textbf{(C)} ~ 11 \qquad \textbf{(D)} ~ 14 \qquad \textbf{(E)} ~ 14 \frac{1}{2}</math><br />
<br />
==Problem 7==<br />
<br />
When expressed as a decimal rounded to the nearest ten-thousandth, what is the value of <math>\dfrac{125+3}{125 \times 3}</math>?<br />
<br />
<math>\textbf{(A)} ~ 0.3412 \qquad \textbf{(B)} ~ 0.3413 \qquad \textbf{(C)} ~ 0.3414 \qquad \textbf{(D)} ~ 0.3415 \qquad \textbf{(E)} ~ 0.3416</math><br />
<br />
==Problem 8==<br />
<br />
What is the value of<br />
<cmath>(1+2+3)-(2+3+4)+(3+4+5)-\cdots -(98+99+100)?</cmath><br />
<math>\textbf{(A)} ~ -150 \qquad \textbf{(B)} ~ -147 \qquad \textbf{(C)} ~ -144 \qquad \textbf{(D)} ~ 147 \qquad \textbf{(E)} ~ 150</math><br />
<br />
==Problem 9==<br />
<br />
Kayla writes down the first <math>N</math> positive integers. What is the sum of all possible values of <math>N</math> if Kayla writes five multiples of <math>13</math> and six multiples of <math>12</math>?<br />
<br />
<math>\textbf{(A)} ~ 447 \qquad \textbf{(B)} ~ 453 \qquad \textbf{(C)} ~ 518 \qquad \textbf{(D)} ~ 525 \qquad \textbf{(E)} ~ 548</math><br />
<br />
==Problem 10==<br />
<br />
In Murphy's seventh grade homeroom, <math>\frac{7}{12}</math> of the students like tennis, <math>\frac{2}{3}</math> of the students like badminton, and <math>\frac{1}{12}</math> of the students like neither. What is the minimum possible number of students who like both tennis and badminton?<br />
<br />
<math>\textbf{(A)} ~ 1 \qquad \textbf{(B)} ~ 2 \qquad \textbf{(C)} ~ 3 \qquad \textbf{(D)} ~ 4 \qquad \textbf{(E)} ~ 6</math><br />
<br />
==Problem 11==<br />
<br />
For how many values of <math>N</math> does there exist a regular <math>N</math> sided polygon whose vertices all lie on the vertices of a regular <math>24</math> sided polygon?<br />
<br />
<math>\textbf{(A)} ~ 6 \qquad \textbf{(B)} ~ 7 \qquad \textbf{(C)} ~ 8 \qquad \textbf{(D)} ~ 9 \qquad \textbf{(E)} ~ 10</math><br />
<br />
==Problem 12==<br />
<br />
Quadrilateral <math>WXYZ</math> has its vertices on the sides of rectangle <math>ABCD</math> with <math>AB=7</math> and <math>BC=5</math>, as shown below. What is the area of quadrilateral <math>WXYZ</math>?<br />
<asy><br />
size(150);<br />
draw((0, 0)--(7, 0)--(7, 5)--(0, 5)--cycle);<br />
label("A", (0, 0), SW);<br />
label("B", (7, 0), SE);<br />
label("C", (7, 5), NE);<br />
label("D", (0, 5), NW);<br />
filldraw((0, 1)--(4, 0)--(7, 3)--(4, 5)--cycle, grey);<br />
label("W", (0, 1), W);<br />
label("X", (4, 0), S);<br />
label("Y", (7, 3), E);<br />
label("Z", (4, 5), N);<br />
label("4", (2, -0.5));<br />
label("3", (5.5, -0.5));<br />
label("4", (2, 5.5));<br />
label("3", (5.5, 5.5));<br />
</asy><br />
<math>\textbf{(A)} ~ 15 \dfrac{1}{2} \qquad \textbf{(B)} ~ 16 \qquad \textbf{(C)} ~ 16 \dfrac{1}{2} \qquad \textbf{(D)} ~ 17 \qquad \textbf{(E)} ~ 17 \dfrac{1}{2}</math><br />
<br />
==Problem 13==<br />
<br />
To drive to the supermarket, Mable drives for <math>m</math> miles, then drives <math>12</math> miles per hour faster for the remaining <math>\frac{4}{3}m</math> miles. The amount of time Mable spent driving at each of the two speeds was equal. What was Mable's average speed during her drive to the supermarket, in miles per hour?<br />
<br />
<math>\textbf{(A)} ~ \dfrac{81}{2} \qquad \textbf{(B)} ~ \dfrac{288}{7} \qquad \textbf{(C)} ~ 42 \qquad \textbf{(D)} ~ \dfrac{300}{7} \qquad \textbf{(E)} ~ 50</math><br />
<br />
==Problem 14==<br />
<br />
Six circles of radius one are cut out of the rectangle below. What is the area of the shaded region?<br />
<asy><br />
size(150);<br />
filldraw((0, 0)--(6, 0)--(6, 4)--(0, 4)--cycle, grey);<br />
filldraw(circle((1, 1), 1), white);<br />
filldraw(circle((3, 1), 1), white);<br />
filldraw(circle((5, 1), 1), white);<br />
filldraw(circle((1, 3), 1), white);<br />
filldraw(circle((3, 3), 1), white);<br />
filldraw(circle((5, 3), 1), white);<br />
</asy><br />
<math>\textbf{(A)} ~ 20-6\pi \qquad \textbf{(B)} ~ 24-6\pi \qquad \textbf{(C)} ~ 28-6\pi \qquad \textbf{(D)} ~ 30-6\pi \qquad \textbf{(E)} ~ 32-6\pi</math><br />
<br />
==Problem 15==<br />
<br />
One metronome beeps at a steady rate of <math>72</math> beeps per minute, while another metronome beeps at a steady rate of <math>96</math> beeps per minute. If both metronomes beep at the same time once, how long will it take, in seconds, until they first beep at the same time again?<br />
<br />
<math>\textbf{(A)} ~ 2 \dfrac{1}{2} \qquad \textbf{(B)} ~ 5 \qquad \textbf{(C)} ~ 10 \qquad \textbf{(D)} ~ 18 \qquad \textbf{(E)} ~ 24</math><br />
<br />
==Problem 16==<br />
<br />
A square with side length two is placed on a table, forming a <math>30</math> degree angle with the table's surface. How much higher is the top vertex of the square than the table?<br />
<asy><br />
size(150);<br />
draw((0, 0)--(0.882, 0.4714)--(0.4106, 1.3534)--(-0.4714, 0.882)--cycle);<br />
draw((-0.5, 0)--(1, 0), linewidth(3));<br />
draw((-0.75, 1.3534)--(-0.65, 1.3534));<br />
draw((-0.7, 1.3534)--(-0.7, 0));<br />
draw((-0.75, 0)--(-0.65, 0));<br />
</asy><br />
<math>\textbf{(A)} ~ \dfrac{5}{2} \qquad \textbf{(B)} ~ \sqrt{3}+1 \qquad \textbf{(C)} ~ \dfrac{4\sqrt{3}}{3} \qquad \textbf{(D)} ~ 3 \qquad \textbf{(E)} ~ \dfrac{3\sqrt{3}}{2}+1</math><br />
<br />
==Problem 17==<br />
<br />
Kurtis' school schedule is made up of four classes, followed by lunch, followed by three more classes. In how many ways can Kurtis arrange his schedule if two of his classes (Reading and Writing) must occur one immediately after the other?<br />
<br />
<math>\textbf{(A)} ~ 600 \qquad \textbf{(B)} ~ 840 \qquad \textbf{(C)} ~ 1200 \qquad \textbf{(D)} ~ 1440 \qquad \textbf{(E)} ~ 1680</math><br />
<br />
==Problem 18==<br />
<br />
When the number <math>25</math> is added to a list of numbers with total sum <math>S</math>, the average of all the numbers increases by one. What is the sum of the digits of the greatest possible value of <math>S</math>?<br />
<br />
<math>\textbf{(A)} ~ 6 \qquad \textbf{(B)} ~ 7 \qquad \textbf{(C)} ~ 8 \qquad \textbf{(D)} ~ 9 \qquad \textbf{(E)} ~ 12</math><br />
<br />
==Problem 19==<br />
<br />
A magician randomly picks a three digit positive integer to put into her hat and pulls out the same number with its digits in reverse order. (For example <math>496</math> would become <math>694</math> and <math>720</math> would become <math>27</math>.) What is the probability the magician pulls out a multiple of <math>22</math>?<br />
<br />
<math>\textbf{(A)} ~ \dfrac{1}{15} \qquad \textbf{(B)} ~ \dfrac{1}{18} \qquad \textbf{(C)} ~ \dfrac{1}{20} \qquad \textbf{(D)} ~ \dfrac{1}{25} \qquad \textbf{(E)} ~ \dfrac{1}{30}</math><br />
<br />
==Problem 20==<br />
<br />
Tyrone has three books to read in six days. He reads one-half of a single book every day. In how many ways can he finish all the books if he may not read the same book two days in a row?<br />
<br />
<math>\textbf{(A)} ~ 12 \qquad \textbf{(B)} ~ 18 \qquad \textbf{(C)} ~ 24 \qquad \textbf{(D)} ~ 30 \qquad \textbf{(E)} ~ 36</math><br />
<br />
==Problem 21==<br />
<br />
There exists a circle that is tangent to <math>\overline{AB}</math> and <math>\overline{BC}</math> at <math>A</math> and <math>C</math>, respectively. If <math>AB=BC=13</math> and <math>AC=10</math>, what is the radius of the circle?<br />
<asy><br />
size(150);<br />
draw((-5, 0)--(5, 0)--(0, -12)--cycle);<br />
draw(circle((0, 2.08333), 5.41666));<br />
label("A", (-5, 0), W);<br />
label("C", (5, 0), E);<br />
label("B", (0, -12), S);<br />
label("13", (-2.7, -6), W);<br />
label("13", (2.7, -6), E);<br />
label("10", (0, 0.2), N);<br />
</asy><br />
<math>\textbf{(A)} ~ \dfrac{60}{13} \qquad \textbf{(B)} ~ 5 \qquad \textbf{(C)} ~ \dfrac{26}{5} \qquad \textbf{(D)} ~ \dfrac{65}{12} \qquad \textbf{(E)} ~ \dfrac{156}{25}</math><br />
<br />
==Problem 22==<br />
<br />
For each of the distinct sets of numbers containing only positive integers between <math>1</math> and <math>9</math> inclusive, Jordan writes the sum of the numbers in that set. What is the sum of the numbers Jordan writes?<br />
<br />
<math>\textbf{(A)} ~ 11520 \qquad \textbf{(B)} ~ 11565 \qquad \textbf{(C)} ~ 11610 \qquad \textbf{(D)} ~ 11655 \qquad \textbf{(E)} ~ 11700</math><br />
<br />
==Problem 23==<br />
<br />
In rectangle <math>ABCD</math>, the perpendicular from <math>B</math> to diagonal <math>\overline{AC}</math> bisects segment <math>\overline{CD}</math>. Which of the following is closest to <math>\frac{AB}{BC}</math>?<br />
<br />
<math>\textbf{(A)} ~ \dfrac{5}{4} \qquad \textbf{(B)} ~ \dfrac{4}{3} \qquad \textbf{(C)} ~ \dfrac{7}{5} \qquad \textbf{(D)} ~ \dfrac{3}{2} \qquad \textbf{(E)} ~ \dfrac{8}{5}</math><br />
<br />
==Problem 24==<br />
<br />
How many ordered triples of positive integers <math>(a, b, c)</math> satisfy <math>\text{gcd}(a, b, c)=20</math> and <math>\text{lcm}(a, b, c)=240</math>?<br />
<br />
<math>\textbf{(A)} ~ 18 \qquad \textbf{(B)} ~ 24 \qquad \textbf{(C)} ~ 36 \qquad \textbf{(D)} ~ 54 \qquad \textbf{(E)} ~ 72 </math><br />
<br />
==Problem 25==<br />
<br />
Cheyanne rolls two standard six sided dice, then repeatedly rerolls all dice which show an odd number and stops as soon as all dice show an even number. What is the probability Cheyanne stops after exactly four rounds of rerolling?<br />
<br />
<math>\textbf{(A)} ~ \dfrac{61}{1024} \qquad \textbf{(B)} ~ \dfrac{1}{16} \qquad \textbf{(C)} ~ \dfrac{67}{1024} \qquad \textbf{(D)} ~ \dfrac{9}{128} \qquad \textbf{(E)} ~ \dfrac{29}{256}</math><br />
<br />
==Tiebreaker Problem 1==<br />
<br />
A whiteboard has positive real numbers <math>1</math> and <math>m</math> written on it. Every second, if the numbers <math>x</math> and <math>y</math> are on the whiteboard, a ghost will replace those numbers with <math>|x^2-y^2|</math> and <math>2xy</math>. The ghost stops once one number on the whiteboard is <math>m</math> times the other. For how many positive real numbers <math>m</math> does the ghost stop after exactly <math>16</math> seconds?<br />
<br />
==Tiebreaker Problem 2==<br />
<br />
The perpendicular bisectors of triangle <math>ABC</math> can be described in the coordinate plane as lines <math>y=0</math>, <math>y=x</math>, and <math>y=\sqrt{3}x</math>. Given that triangle <math>ABC</math> has circumradius <math>1</math>, find its area.<br />
<br />
==Tiebreaker Problem 3==<br />
<br />
The diagram below is constructed by attaching an equilateral triangle, a square, a regular pentagon, and a regular hexagon together. Compute the measure of the obtuse angle formed by the three red vertices.<br />
<asy><br />
import graph; size(10cm); <br />
real labelscalefactor = 0.5; /* changes label-to-point distance */<br />
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ <br />
pen dotstyle = black; /* point style */ <br />
real xmin = -29.36, xmax = -9.8, ymin = 4.78, ymax = 17.66; /* image dimensions */<br />
<br />
draw((-22,12)--(-19,12)--(-20.5,14.598076211353318)--cycle, linewidth(2)); <br />
draw((-20.5,14.598076211353318)--(-19,12)--(-16.401923788646684,13.5)--(-17.90192378864668,16.098076211353316)--cycle, linewidth(2)); <br />
draw((-16.401923788646684,13.5)--(-19,12)--(-18.376264927546725,9.065557197798585)--(-15.392699241441907,8.751971807995622)--(-14.172489312214504,11.492608180923423)--cycle, linewidth(2)); <br />
draw((-18.376264927546725,9.065557197798585)--(-19,12)--(-21.85316954888546,12.927050983124847)--(-24.082604025317647,10.919659164048271)--(-23.45886895286437,7.985216361846854)--(-20.60569940397891,7.058165378722009)--cycle, linewidth(2)); <br />
Label laxis; laxis.p = fontsize(10); <br />
string blank(real x) {return "";} <br />
xaxis(xmin, xmax, Ticks(laxis, blank, Step = 1, Size = 2, NoZero),EndArrow(6), above = true); <br />
yaxis(ymin, ymax, Ticks(laxis, blank, Step = 1, Size = 2, NoZero),EndArrow(6), above = true); /* draws axes; NoZero hides '0' label */ <br />
/* draw figures */<br />
draw((-22,12)--(-19,12), linewidth(2)); <br />
draw((-19,12)--(-20.5,14.598076211353318), linewidth(2)); <br />
draw((-20.5,14.598076211353318)--(-22,12), linewidth(2)); <br />
draw((-20.5,14.598076211353318)--(-19,12), linewidth(2)); <br />
draw((-19,12)--(-16.401923788646684,13.5), linewidth(2)); <br />
draw((-16.401923788646684,13.5)--(-17.90192378864668,16.098076211353316), linewidth(2)); <br />
draw((-17.90192378864668,16.098076211353316)--(-20.5,14.598076211353318), linewidth(2)); <br />
draw((-16.401923788646684,13.5)--(-19,12), linewidth(2)); <br />
draw((-19,12)--(-18.376264927546725,9.065557197798585), linewidth(2)); <br />
draw((-18.376264927546725,9.065557197798585)--(-15.392699241441907,8.751971807995622), linewidth(2)); <br />
draw((-15.392699241441907,8.751971807995622)--(-14.172489312214504,11.492608180923423), linewidth(2)); <br />
draw((-14.172489312214504,11.492608180923423)--(-16.401923788646684,13.5), linewidth(2)); <br />
draw((-18.376264927546725,9.065557197798585)--(-19,12), linewidth(2)); <br />
draw((-19,12)--(-21.85316954888546,12.927050983124847), linewidth(2)); <br />
draw((-21.85316954888546,12.927050983124847)--(-24.082604025317647,10.919659164048271), linewidth(2)); <br />
draw((-24.082604025317647,10.919659164048271)--(-23.45886895286437,7.985216361846854), linewidth(2)); <br />
draw((-23.45886895286437,7.985216361846854)--(-20.60569940397891,7.058165378722009), linewidth(2)); <br />
draw((-20.60569940397891,7.058165378722009)--(-18.376264927546725,9.065557197798585), linewidth(2)); <br />
/* dots and labels */<br />
dot((-22,12),red); <br />
dot((-19,12),red); <br />
dot((-21.526119073220972,12.820785841919117),linewidth(4pt) + dotstyle+red); <br />
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); <br />
</asy><br />
<br />
==Answer Key==<br />
<br />
AMC 8: DBBCD / CBBAD / AECBA / BCDDD / DACEA<br />
<br />
Tiebreakers: (<math>65280</math>, <math>\dfrac{3-\sqrt{3}}{4}</math>, <math>102</math>)</div>Bissuehttps://artofproblemsolving.com/wiki/index.php?title=User:Bissue&diff=140585User:Bissue2020-12-25T20:53:21Z<p>Bissue: </p>
<hr />
<div>i suppose ill use this page to store all the mock contests i write<br />
<br />
so far there's only one<br />
<br />
maybe more will come<br />
<br />
==Apocalyptic AMC 8 (2020)==<br />
<br />
==Problem 1==<br />
<br />
To walk up a single floor in her eighteen floor apartment building, Sarah needs to take nine steps up a flight of stairs. If Sarah starts on Floor <math>3</math> and walks up <math>100</math> steps, she would end up on the flight of stairs connecting which two floors?<br />
<br />
<math>\textbf{(A)} ~ \mbox{11 and 12} \qquad \textbf{(B)} ~ \mbox{12 and 13} \qquad \textbf{(C)} ~ \mbox{13 and 14} \qquad \textbf{(D)} ~ \mbox{14 and 15} \qquad \textbf{(E)} ~ \mbox{15 and 16}</math><br />
<br />
==Problem 2==<br />
<br />
Abby, Barb, and Carlos each have <math>35</math>, <math>42</math>, and <math>31</math> trading cards respectively. If they share their trading cards equally between each other, how many more trading cards would Carlos have than before?<br />
<br />
<math>\textbf{(A)} ~ 4 \qquad \textbf{(B)} ~ 5 \qquad \textbf{(C)} ~ 6 \qquad \textbf{(D)} ~ 9 \qquad \textbf{(E)} ~ 11</math><br />
<br />
==Problem 3==<br />
<br />
In triangle <math>ABC</math> the measure of angle <math>A</math> is the average of the measures of angles <math>B</math> and <math>C</math>. What is the measure of angle <math>A</math>?<br />
<br />
<math>\textbf{(A)} ~ 45^{\circ} \qquad \textbf{(B)} ~ 60^{\circ} \qquad \textbf{(C)} ~ 75^{\circ} \qquad \textbf{(D)} ~ 90^{\circ} \qquad \textbf{(E)} ~ 120^{\circ}</math><br />
<br />
==Problem 4==<br />
<br />
A spruce tree grows by <math>25</math> feet, increasing its height by <math>25 \%</math>. If the tree grows for a second time by <math>25</math> feet, by what percent would its height increase?<br />
<br />
<math>\textbf{(A)} ~ 5 \% \qquad \textbf{(B)} ~ 15 \% \qquad \textbf{(C)} ~ 20 \% \qquad \textbf{(D)} ~ 25 \% \qquad \textbf{(E)} ~ 30 \% </math><br />
<br />
==Problem 5==<br />
<br />
Find the sum of the digits of <math>\dfrac{5 \times 10^{2020}}{2}</math>.<br />
<br />
<math>\textbf{(A)} ~ 1 \qquad \textbf{(B)} ~ 2 \qquad \textbf{(C)} ~ 5 \qquad \textbf{(D)} ~ 7 \qquad \textbf{(E)} ~ 8</math><br />
<br />
==Problem 6==<br />
<br />
Square <math>B</math> with side length three is attached to a side of square <math>A</math> with side length four, as shown in the figure below. Find the area of the shaded region.<br />
<asy><br />
size(150);<br />
draw((0, 0)--(4, 0)--(4, 4)--(0, 4)--cycle);<br />
draw((4, 1)--(7, 1)--(7, 4)--(4, 4)--cycle);<br />
filldraw((0, 0)--(4, 0)--(4, 2.285714)--cycle, grey);<br />
filldraw((4, 1)--(7, 1)--(7, 4)--(4, 2.285714)--cycle, grey);<br />
label("A", (2, 2));<br />
label("B", (5.5, 2.5));<br />
</asy><br />
<math>\textbf{(A)} ~ 10 \qquad \textbf{(B)} ~ 10 \frac{1}{2} \qquad \textbf{(C)} ~ 11 \qquad \textbf{(D)} ~ 14 \qquad \textbf{(E)} ~ 14 \frac{1}{2}</math><br />
<br />
==Problem 7==<br />
<br />
When expressed as a decimal rounded to the nearest ten-thousandth, what is the value of <math>\dfrac{125+3}{125 \times 3}</math>?<br />
<br />
<math>\textbf{(A)} ~ 0.3412 \qquad \textbf{(B)} ~ 0.3413 \qquad \textbf{(C)} ~ 0.3414 \qquad \textbf{(D)} ~ 0.3415 \qquad \textbf{(E)} ~ 0.3416</math><br />
<br />
==Problem 8==<br />
<br />
What is the value of<br />
<cmath>(1+2+3)-(2+3+4)+(3+4+5)-\cdots -(98+99+100)?</cmath><br />
<math>\textbf{(A)} ~ -150 \qquad \textbf{(B)} ~ -147 \qquad \textbf{(C)} ~ -144 \qquad \textbf{(D)} ~ 147 \qquad \textbf{(E)} ~ 150</math><br />
<br />
==Problem 9==<br />
<br />
Kayla writes down the first <math>N</math> positive integers. What is the sum of all possible values of <math>N</math> if Kayla writes five multiples of <math>13</math> and six multiples of <math>12</math>?<br />
<br />
<math>\textbf{(A)} ~ 447 \qquad \textbf{(B)} ~ 453 \qquad \textbf{(C)} ~ 518 \qquad \textbf{(D)} ~ 525 \qquad \textbf{(E)} ~ 548</math><br />
<br />
==Problem 10==<br />
<br />
In Murphy's seventh grade homeroom, <math>\frac{7}{12}</math> of the students like tennis, <math>\frac{2}{3}</math> of the students like badminton, and <math>\frac{1}{12}</math> of the students like neither. What is the minimum possible number of students who like both tennis and badminton?<br />
<br />
<math>\textbf{(A)} ~ 1 \qquad \textbf{(B)} ~ 2 \qquad \textbf{(C)} ~ 3 \qquad \textbf{(D)} ~ 4 \qquad \textbf{(E)} ~ 6</math><br />
<br />
==Problem 11==<br />
<br />
For how many values of <math>N</math> does there exist a regular <math>N</math> sided polygon whose vertices all lie on the vertices of a regular <math>24</math> sided polygon?<br />
<br />
<math>\textbf{(A)} ~ 6 \qquad \textbf{(B)} ~ 7 \qquad \textbf{(C)} ~ 8 \qquad \textbf{(D)} ~ 9 \qquad \textbf{(E)} ~ 10</math><br />
<br />
==Problem 12==<br />
<br />
Quadrilateral <math>WXYZ</math> has its vertices on the sides of rectangle <math>ABCD</math> with <math>AB=7</math> and <math>BC=5</math>, as shown below. What is the area of quadrilateral <math>WXYZ</math>?<br />
<asy><br />
size(150);<br />
draw((0, 0)--(7, 0)--(7, 5)--(0, 5)--cycle);<br />
label("A", (0, 0), SW);<br />
label("B", (7, 0), SE);<br />
label("C", (7, 5), NE);<br />
label("D", (0, 5), NW);<br />
filldraw((0, 1)--(4, 0)--(7, 3)--(4, 5)--cycle, grey);<br />
label("W", (0, 1), W);<br />
label("X", (4, 0), S);<br />
label("Y", (7, 3), E);<br />
label("Z", (4, 5), N);<br />
label("4", (2, -0.5));<br />
label("3", (5.5, -0.5));<br />
label("4", (2, 5.5));<br />
label("3", (5.5, 5.5));<br />
</asy><br />
<math>\textbf{(A)} ~ 15 \dfrac{1}{2} \qquad \textbf{(B)} ~ 16 \qquad \textbf{(C)} ~ 16 \dfrac{1}{2} \qquad \textbf{(D)} ~ 17 \qquad \textbf{(E)} ~ 17 \dfrac{1}{2}</math><br />
<br />
==Problem 13==<br />
<br />
To drive to the supermarket, Mable drives for <math>m</math> miles, then drives <math>12</math> miles per hour faster for the remaining <math>\frac{4}{3}m</math> miles. The amount of time Mable spent driving at each of the two speeds was equal. What was Mable's average speed during her drive to the supermarket, in miles per hour?<br />
<br />
<math>\textbf{(A)} ~ \dfrac{81}{2} \qquad \textbf{(B)} ~ \dfrac{288}{7} \qquad \textbf{(C)} ~ 42 \qquad \textbf{(D)} ~ \dfrac{300}{7} \qquad \textbf{(E)} ~ 50</math><br />
<br />
==Problem 14==<br />
<br />
Six circles of radius one are cut out of the rectangle below. What is the area of the shaded region?<br />
<asy><br />
size(150);<br />
filldraw((0, 0)--(6, 0)--(6, 4)--(0, 4)--cycle, grey);<br />
filldraw(circle((1, 1), 1), white);<br />
filldraw(circle((3, 1), 1), white);<br />
filldraw(circle((5, 1), 1), white);<br />
filldraw(circle((1, 3), 1), white);<br />
filldraw(circle((3, 3), 1), white);<br />
filldraw(circle((5, 3), 1), white);<br />
</asy><br />
<math>\textbf{(A)} ~ 20-6\pi \qquad \textbf{(B)} ~ 24-6\pi \qquad \textbf{(C)} ~ 28-6\pi \qquad \textbf{(D)} ~ 30-6\pi \qquad \textbf{(E)} ~ 32-6\pi</math><br />
<br />
==Problem 15==<br />
<br />
One metronome beeps at a steady rate of <math>72</math> beeps per minute, while another metronome beeps at a steady rate of <math>96</math> beeps per minute. If both metronomes beep at the same time once, how long will it take, in seconds, until they first beep at the same time again?<br />
<br />
<math>\textbf{(A)} ~ 2 \dfrac{1}{2} \qquad \textbf{(B)} ~ 5 \qquad \textbf{(C)} ~ 10 \qquad \textbf{(D)} ~ 18 \qquad \textbf{(E)} ~ 24</math><br />
<br />
==Problem 16==<br />
<br />
A square with side length two is placed on a table, forming a <math>30</math> degree angle with the table's surface. How much higher is the top vertex of the square than the table?<br />
<asy><br />
size(150);<br />
draw((0, 0)--(0.882, 0.4714)--(0.4106, 1.3534)--(-0.4714, 0.882)--cycle);<br />
draw((-0.5, 0)--(1, 0), linewidth(3));<br />
draw((-0.75, 1.3534)--(-0.65, 1.3534));<br />
draw((-0.7, 1.3534)--(-0.7, 0));<br />
draw((-0.75, 0)--(-0.65, 0));<br />
</asy><br />
<math>\textbf{(A)} ~ \dfrac{5}{2} \qquad \textbf{(B)} ~ \sqrt{3}+1 \qquad \textbf{(C)} ~ \dfrac{4\sqrt{3}}{3} \qquad \textbf{(D)} ~ 3 \qquad \textbf{(E)} ~ \dfrac{3\sqrt{3}}{2}+1</math><br />
<br />
==Problem 17==<br />
<br />
Kurtis' school schedule is made up of four classes, followed by lunch, followed by three more classes. In how many ways can Kurtis arrange his schedule if two of his classes (Reading and Writing) must occur one immediately after the other?<br />
<br />
<math>\textbf{(A)} ~ 600 \qquad \textbf{(B)} ~ 840 \qquad \textbf{(C)} ~ 1200 \qquad \textbf{(D)} ~ 1440 \qquad \textbf{(E)} ~ 1680</math><br />
<br />
==Problem 18==<br />
<br />
When the number <math>25</math> is added to a list of numbers with total sum <math>S</math>, the average of all the numbers increases by one. What is the sum of the digits of the greatest possible value of <math>S</math>?<br />
<br />
<math>\textbf{(A)} ~ 6 \qquad \textbf{(B)} ~ 7 \qquad \textbf{(C)} ~ 8 \qquad \textbf{(D)} ~ 9 \qquad \textbf{(E)} ~ 12</math><br />
<br />
==Problem 19==<br />
<br />
A magician randomly picks a three digit positive integer to put into her hat and pulls out the same number with its digits in reverse order. (For example <math>496</math> would become <math>694</math> and <math>720</math> would become <math>27</math>.) What is the probability the magician pulls out a multiple of <math>22</math>?<br />
<br />
<math>\textbf{(A)} ~ \dfrac{1}{15} \qquad \textbf{(B)} ~ \dfrac{1}{18} \qquad \textbf{(C)} ~ \dfrac{1}{20} \qquad \textbf{(D)} ~ \dfrac{1}{25} \qquad \textbf{(E)} ~ \dfrac{1}{30}</math><br />
<br />
==Problem 20==<br />
<br />
Tyrone has three books to read in six days. He reads one-half of a single book every day. In how many ways can he finish all the books if he may not read the same book two days in a row?<br />
<br />
<math>\textbf{(A)} ~ 12 \qquad \textbf{(B)} ~ 18 \qquad \textbf{(C)} ~ 24 \qquad \textbf{(D)} ~ 30 \qquad \textbf{(E)} ~ 36</math><br />
<br />
==Problem 21==<br />
<br />
There exists a circle that is tangent to <math>AB</math> and <math>BC</math> at <math>A</math> and <math>C</math>, respectively. If <math>AB=BC=13</math> and <math>AC=10</math>, what is the radius of the circle?<br />
<asy><br />
size(150);<br />
draw((-5, 0)--(5, 0)--(0, -12)--cycle);<br />
draw(circle((0, 2.08333), 5.41666));<br />
label("A", (-5, 0), W);<br />
label("C", (5, 0), E);<br />
label("B", (0, -12), S);<br />
label("13", (-2.7, -6), W);<br />
label("13", (2.7, -6), E);<br />
label("10", (0, 0.2), N);<br />
</asy><br />
<math>\textbf{(A)} ~ \dfrac{60}{13} \qquad \textbf{(B)} ~ 5 \qquad \textbf{(C)} ~ \dfrac{26}{5} \qquad \textbf{(D)} ~ \dfrac{65}{12} \qquad \textbf{(E)} ~ \dfrac{156}{25}</math><br />
<br />
==Problem 22==<br />
<br />
For each of the distinct sets of numbers containing only positive integers between <math>1</math> and <math>9</math> inclusive, Jordan writes the sum of the numbers in that set. What is the sum of the numbers Jordan writes?<br />
<br />
<math>\textbf{(A)} ~ 11520 \qquad \textbf{(B)} ~ 11565 \qquad \textbf{(C)} ~ 11610 \qquad \textbf{(D)} ~ 11655 \qquad \textbf{(E)} ~ 11700</math><br />
<br />
==Problem 23==<br />
<br />
In rectangle <math>ABCD</math>, the perpendicular from <math>B</math> to diagonal <math>AC</math> bisects segment <math>CD</math>. Which of the following is closest to <math>\frac{AB}{BC}</math>?<br />
<br />
<math>\textbf{(A)} ~ \dfrac{5}{4} \qquad \textbf{(B)} ~ \dfrac{4}{3} \qquad \textbf{(C)} ~ \dfrac{7}{5} \qquad \textbf{(D)} ~ \dfrac{3}{2} \qquad \textbf{(E)} ~ \dfrac{8}{5}</math><br />
<br />
==Problem 24==<br />
<br />
How many ordered triples of positive integers <math>(a, b, c)</math> satisfy <math>\text{gcd}(a, b, c)=20</math> and <math>\text{lcm}(a, b, c)=240</math>?<br />
<br />
<math>\textbf{(A)} ~ 18 \qquad \textbf{(B)} ~ 24 \qquad \textbf{(C)} ~ 36 \qquad \textbf{(D)} ~ 54 \qquad \textbf{(E)} ~ 72 </math><br />
<br />
==Problem 25==<br />
<br />
Cheyanne rolls two standard six sided dice, then repeatedly rerolls all dice which show an odd number and stops as soon as all dice show an even number. What is the probability Cheyanne stops after exactly four rounds of rerolling?<br />
<br />
<math>\textbf{(A)} ~ \dfrac{61}{1024} \qquad \textbf{(B)} ~ \dfrac{1}{16} \qquad \textbf{(C)} ~ \dfrac{67}{1024} \qquad \textbf{(D)} ~ \dfrac{9}{128} \qquad \textbf{(E)} ~ \dfrac{29}{256}</math></div>Bissuehttps://artofproblemsolving.com/wiki/index.php?title=User:Bissue&diff=140583User:Bissue2020-12-25T20:47:03Z<p>Bissue: </p>
<hr />
<div>i suppose i have a user page now<br />
<br />
2020 Apocalyptic AMC 8:<br />
<br />
==Problem 1==<br />
<br />
To walk up a single floor in her eighteen floor apartment building, Sarah needs to take nine steps up a flight of stairs. If Sarah starts on Floor <math>3</math> and walks up <math>100</math> steps, she would end up on the flight of stairs connecting which two floors?<br />
<br />
<math>\textbf{(A)} ~ \mbox{11 and 12} \qquad \textbf{(B)} ~ \mbox{12 and 13} \qquad \textbf{(C)} ~ \mbox{13 and 14} \qquad \textbf{(D)} ~ \mbox{14 and 15} \qquad \textbf{(E)} ~ \mbox{15 and 16}</math><br />
<br />
==Problem 2==<br />
<br />
Abby, Barb, and Carlos each have <math>35</math>, <math>42</math>, and <math>31</math> trading cards respectively. If they share their trading cards equally between each other, how many more trading cards would Carlos have than before?<br />
<br />
<math>\textbf{(A)} ~ 4 \qquad \textbf{(B)} ~ 5 \qquad \textbf{(C)} ~ 6 \qquad \textbf{(D)} ~ 9 \qquad \textbf{(E)} ~ 11</math><br />
<br />
==Problem 3==<br />
<br />
In triangle <math>ABC</math> the measure of angle <math>A</math> is the average of the measures of angles <math>B</math> and <math>C</math>. What is the measure of angle <math>A</math>?<br />
<br />
<math>\textbf{(A)} ~ 45^{\circ} \qquad \textbf{(B)} ~ 60^{\circ} \qquad \textbf{(C)} ~ 75^{\circ} \qquad \textbf{(D)} ~ 90^{\circ} \qquad \textbf{(E)} ~ 120^{\circ}</math><br />
<br />
==Problem 4==<br />
<br />
A spruce tree grows by <math>25</math> feet, increasing its height by <math>25 \%</math>. If the tree grows for a second time by <math>25</math> feet, by what percent would its height increase?<br />
<br />
<math>\textbf{(A)} ~ 5 \% \qquad \textbf{(B)} ~ 15 \% \qquad \textbf{(C)} ~ 20 \% \qquad \textbf{(D)} ~ 25 \% \qquad \textbf{(E)} ~ 30 \% </math><br />
<br />
==Problem 5==<br />
<br />
Find the sum of the digits of <math>\dfrac{5 \times 10^{2020}}{2}</math>.<br />
<br />
<math>\textbf{(A)} ~ 1 \qquad \textbf{(B)} ~ 2 \qquad \textbf{(C)} ~ 5 \qquad \textbf{(D)} ~ 7 \qquad \textbf{(E)} ~ 8</math><br />
<br />
==Problem 6==<br />
<br />
Square <math>B</math> with side length three is attached to a side of square <math>A</math> with side length four, as shown in the figure below. Find the area of the shaded region.<br />
<asy><br />
size(150);<br />
draw((0, 0)--(4, 0)--(4, 4)--(0, 4)--cycle);<br />
draw((4, 1)--(7, 1)--(7, 4)--(4, 4)--cycle);<br />
filldraw((0, 0)--(4, 0)--(4, 2.285714)--cycle, grey);<br />
filldraw((4, 1)--(7, 1)--(7, 4)--(4, 2.285714)--cycle, grey);<br />
label("A", (2, 2));<br />
label("B", (5.5, 2.5));<br />
</asy><br />
<math>\textbf{(A)} ~ 10 \qquad \textbf{(B)} ~ 10 \frac{1}{2} \qquad \textbf{(C)} ~ 11 \qquad \textbf{(D)} ~ 14 \qquad \textbf{(E)} ~ 14 \frac{1}{2}</math><br />
<br />
==Problem 7==<br />
<br />
When expressed as a decimal rounded to the nearest ten-thousandth, what is the value of <math>\dfrac{125+3}{125 \times 3}</math>?<br />
<br />
<math>\textbf{(A)} ~ 0.3412 \qquad \textbf{(B)} ~ 0.3413 \qquad \textbf{(C)} ~ 0.3414 \qquad \textbf{(D)} ~ 0.3415 \qquad \textbf{(E)} ~ 0.3416</math><br />
<br />
==Problem 8==<br />
<br />
What is the value of<br />
<cmath>(1+2+3)-(2+3+4)+(3+4+5)-\cdots -(98+99+100)?</cmath><br />
<math>\textbf{(A)} ~ -150 \qquad \textbf{(B)} ~ -147 \qquad \textbf{(C)} ~ -144 \qquad \textbf{(D)} ~ 147 \qquad \textbf{(E)} ~ 150</math><br />
<br />
==Problem 9==<br />
<br />
Kayla writes down the first <math>N</math> positive integers. What is the sum of all possible values of <math>N</math> if Kayla writes five multiples of <math>13</math> and six multiples of <math>12</math>?<br />
<br />
<math>\textbf{(A)} ~ 447 \qquad \textbf{(B)} ~ 453 \qquad \textbf{(C)} ~ 518 \qquad \textbf{(D)} ~ 525 \qquad \textbf{(E)} ~ 548</math><br />
<br />
==Problem 10==<br />
<br />
In Murphy's seventh grade homeroom, <math>\frac{7}{12}</math> of the students like tennis, <math>\frac{2}{3}</math> of the students like badminton, and <math>\frac{1}{12}</math> of the students like neither. What is the minimum possible number of students who like both tennis and badminton?<br />
<br />
<math>\textbf{(A)} ~ 1 \qquad \textbf{(B)} ~ 2 \qquad \textbf{(C)} ~ 3 \qquad \textbf{(D)} ~ 4 \qquad \textbf{(E)} ~ 6</math><br />
<br />
==Problem 11==<br />
<br />
For how many values of <math>N</math> does there exist a regular <math>N</math> sided polygon whose vertices all lie on the vertices of a regular <math>24</math> sided polygon?<br />
<br />
<math>\textbf{(A)} ~ 6 \qquad \textbf{(B)} ~ 7 \qquad \textbf{(C)} ~ 8 \qquad \textbf{(D)} ~ 9 \qquad \textbf{(E)} ~ 10</math><br />
<br />
==Problem 12==<br />
<br />
Quadrilateral <math>WXYZ</math> has its vertices on the sides of rectangle <math>ABCD</math> with <math>AB=7</math> and <math>BC=5</math>, as shown below. What is the area of quadrilateral <math>WXYZ</math>?<br />
<asy><br />
size(150);<br />
draw((0, 0)--(7, 0)--(7, 5)--(0, 5)--cycle);<br />
label("A", (0, 0), SW);<br />
label("B", (7, 0), SE);<br />
label("C", (7, 5), NE);<br />
label("D", (0, 5), NW);<br />
filldraw((0, 1)--(4, 0)--(7, 3)--(4, 5)--cycle, grey);<br />
label("W", (0, 1), W);<br />
label("X", (4, 0), S);<br />
label("Y", (7, 3), E);<br />
label("Z", (4, 5), N);<br />
label("4", (2, -0.5));<br />
label("3", (5.5, -0.5));<br />
label("4", (2, 5.5));<br />
label("3", (5.5, 5.5));<br />
</asy><br />
<math>\textbf{(A)} ~ 15 \dfrac{1}{2} \qquad \textbf{(B)} ~ 16 \qquad \textbf{(C)} ~ 16 \dfrac{1}{2} \qquad \textbf{(D)} ~ 17 \qquad \textbf{(E)} ~ 17 \dfrac{1}{2}</math><br />
<br />
==Problem 13==<br />
<br />
To drive to the supermarket, Mable drives for <math>m</math> miles, then drives <math>12</math> miles per hour faster for the remaining <math>\frac{4}{3}m</math> miles. The amount of time Mable spent driving at each of the two speeds was equal. What was Mable's average speed during her drive to the supermarket, in miles per hour?<br />
<br />
<math>\textbf{(A)} ~ \dfrac{81}{2} \qquad \textbf{(B)} ~ \dfrac{288}{7} \qquad \textbf{(C)} ~ 42 \qquad \textbf{(D)} ~ \dfrac{300}{7} \qquad \textbf{(E)} ~ 50</math><br />
<br />
==Problem 14==<br />
<br />
Six circles of radius one are cut out of the rectangle below. What is the area of the shaded region?<br />
<asy><br />
size(150);<br />
filldraw((0, 0)--(6, 0)--(6, 4)--(0, 4)--cycle, grey);<br />
filldraw(circle((1, 1), 1), white);<br />
filldraw(circle((3, 1), 1), white);<br />
filldraw(circle((5, 1), 1), white);<br />
filldraw(circle((1, 3), 1), white);<br />
filldraw(circle((3, 3), 1), white);<br />
filldraw(circle((5, 3), 1), white);<br />
</asy><br />
<math>\textbf{(A)} ~ 20-6\pi \qquad \textbf{(B)} ~ 24-6\pi \qquad \textbf{(C)} ~ 28-6\pi \qquad \textbf{(D)} ~ 30-6\pi \qquad \textbf{(E)} ~ 32-6\pi</math><br />
<br />
==Problem 15==<br />
<br />
One metronome beeps at a steady rate of <math>72</math> beeps per minute, while another metronome beeps at a steady rate of <math>96</math> beeps per minute. If both metronomes beep at the same time once, how long will it take, in seconds, until they first beep at the same time again?<br />
<br />
<math>\textbf{(A)} ~ 2 \dfrac{1}{2} \qquad \textbf{(B)} ~ 5 \qquad \textbf{(C)} ~ 10 \qquad \textbf{(D)} ~ 18 \qquad \textbf{(E)} ~ 24</math><br />
<br />
==Problem 16==<br />
<br />
A square with side length two is placed on a table, forming a <math>30</math> degree angle with the table's surface. How much higher is the top vertex of the square than the table?<br />
<asy><br />
size(150);<br />
draw((0, 0)--(0.882, 0.4714)--(0.4106, 1.3534)--(-0.4714, 0.882)--cycle);<br />
draw((-0.5, 0)--(1, 0), linewidth(3));<br />
draw((-0.75, 1.3534)--(-0.65, 1.3534));<br />
draw((-0.7, 1.3534)--(-0.7, 0));<br />
draw((-0.75, 0)--(-0.65, 0));<br />
</asy><br />
<math>\textbf{(A)} ~ \dfrac{5}{2} \qquad \textbf{(B)} ~ \sqrt{3}+1 \qquad \textbf{(C)} ~ \dfrac{4\sqrt{3}}{3} \qquad \textbf{(D)} ~ 3 \qquad \textbf{(E)} ~ \dfrac{3\sqrt{3}}{2}+1</math><br />
<br />
==Problem 17==<br />
<br />
Kurtis' school schedule is made up of four classes, followed by lunch, followed by three more classes. In how many ways can Kurtis arrange his schedule if two of his classes (Reading and Writing) must occur one immediately after the other?<br />
<br />
<math>\textbf{(A)} ~ 600 \qquad \textbf{(B)} ~ 840 \qquad \textbf{(C)} ~ 1200 \qquad \textbf{(D)} ~ 1440 \qquad \textbf{(E)} ~ 1680</math><br />
<br />
==Problem 18==<br />
<br />
When the number <math>25</math> is added to a list of numbers with total sum <math>S</math>, the average of all the numbers increases by one. What is the sum of the digits of the greatest possible value of <math>S</math>?<br />
<br />
<math>\textbf{(A)} ~ 6 \qquad \textbf{(B)} ~ 7 \qquad \textbf{(C)} ~ 8 \qquad \textbf{(D)} ~ 9 \qquad \textbf{(E)} ~ 12</math><br />
<br />
==Problem 19==<br />
<br />
A magician randomly picks a three digit positive integer to put into her hat and pulls out the same number with its digits in reverse order. (For example <math>496</math> would become <math>694</math> and <math>720</math> would become <math>27</math>.) What is the probability the magician pulls out a multiple of <math>22</math>?<br />
<br />
<math>\textbf{(A)} ~ \dfrac{1}{15} \qquad \textbf{(B)} ~ \dfrac{1}{18} \qquad \textbf{(C)} ~ \dfrac{1}{20} \qquad \textbf{(D)} ~ \dfrac{1}{25} \qquad \textbf{(E)} ~ \dfrac{1}{30}</math><br />
<br />
==Problem 20==<br />
<br />
Tyrone has three books to read in six days. He reads one-half of a single book every day. In how many ways can he finish all the books if he may not read the same book two days in a row?<br />
<br />
<math>\textbf{(A)} ~ 12 \qquad \textbf{(B)} ~ 18 \qquad \textbf{(C)} ~ 24 \qquad \textbf{(D)} ~ 30 \qquad \textbf{(E)} ~ 36</math><br />
<br />
==Problem 21==<br />
<br />
There exists a circle that is tangent to <math>AB</math> and <math>BC</math> at <math>A</math> and <math>C</math>, respectively. If <math>AB=BC=13</math> and <math>AC=10</math>, what is the radius of the circle?<br />
<asy><br />
size(150);<br />
draw((-5, 0)--(5, 0)--(0, -12)--cycle);<br />
draw(circle((0, 2.08333), 5.41666));<br />
label("A", (-5, 0), W);<br />
label("C", (5, 0), E);<br />
label("B", (0, -12), S);<br />
label("13", (-2.7, -6), W);<br />
label("13", (2.7, -6), E);<br />
label("10", (0, 0.2), N);<br />
</asy><br />
<math>\textbf{(A)} ~ \dfrac{60}{13} \qquad \textbf{(B)} ~ 5 \qquad \textbf{(C)} ~ \dfrac{26}{5} \qquad \textbf{(D)} ~ \dfrac{65}{12} \qquad \textbf{(E)} ~ \dfrac{156}{25}</math><br />
<br />
==Problem 22==<br />
<br />
For each of the distinct sets of numbers containing only positive integers between <math>1</math> and <math>9</math> inclusive, Jordan writes the sum of the numbers in that set. What is the sum of the numbers Jordan writes?<br />
<br />
<math>\textbf{(A)} ~ 11520 \qquad \textbf{(B)} ~ 11565 \qquad \textbf{(C)} ~ 11610 \qquad \textbf{(D)} ~ 11655 \qquad \textbf{(E)} ~ 11700</math><br />
<br />
==Problem 23==<br />
<br />
In rectangle <math>ABCD</math>, the perpendicular from <math>B</math> to diagonal <math>AC</math> bisects segment <math>CD</math>. Which of the following is closest to <math>\frac{AB}{BC}</math>?<br />
<br />
<math>\textbf{(A)} ~ \dfrac{5}{4} \qquad \textbf{(B)} ~ \dfrac{4}{3} \qquad \textbf{(C)} ~ \dfrac{7}{5} \qquad \textbf{(D)} ~ \dfrac{3}{2} \qquad \textbf{(E)} ~ \dfrac{8}{5}</math><br />
<br />
==Problem 24==<br />
<br />
How many ordered triples of positive integers <math>(a, b, c)</math> satisfy <math>\text{gcd}(a, b, c)=20</math> and <math>\text{lcm}(a, b, c)=240</math>?<br />
<br />
<math>\textbf{(A)} ~ 18 \qquad \textbf{(B)} ~ 24 \qquad \textbf{(C)} ~ 36 \qquad \textbf{(D)} ~ 54 \qquad \textbf{(E)} ~ 72 </math><br />
<br />
==Problem 25==<br />
<br />
Cheyanne rolls two standard six sided dice, then repeatedly rerolls all dice which show an odd number and stops as soon as all dice show an even number. What is the probability Cheyanne stops after exactly four rounds of rerolling?<br />
<br />
<math>\textbf{(A)} ~ \dfrac{61}{1024} \qquad \textbf{(B)} ~ \dfrac{1}{16} \qquad \textbf{(C)} ~ \dfrac{67}{1024} \qquad \textbf{(D)} ~ \dfrac{9}{128} \qquad \textbf{(E)} ~ \dfrac{29}{256}</math></div>Bissuehttps://artofproblemsolving.com/wiki/index.php?title=User:Bissue&diff=140581User:Bissue2020-12-25T20:41:35Z<p>Bissue: </p>
<hr />
<div>i suppose i have a user page now<br />
<br />
2020 Apocalyptic AMC 8:<br />
<br />
1. To walk up a single floor in her eighteen floor apartment building, Sarah needs to take nine steps up a flight of stairs. If Sarah starts on Floor <math>3</math> and walks up <math>100</math> steps, she would end up on the flight of stairs connecting which two floors?<br />
<br />
<math>\textbf{(A)} ~ \mbox{11 and 12} \qquad \textbf{(B)} ~ \mbox{12 and 13} \qquad \textbf{(C)} ~ \mbox{13 and 14} \qquad \textbf{(D)} ~ \mbox{14 and 15} \qquad \textbf{(E)} ~ \mbox{15 and 16}</math><br />
<br />
2. Abby, Barb, and Carlos each have <math>35</math>, <math>42</math>, and <math>31</math> trading cards respectively. If they share their trading cards equally between each other, how many more trading cards would Carlos have than before?<br />
<br />
<math>\textbf{(A)} ~ 4 \qquad \textbf{(B)} ~ 5 \qquad \textbf{(C)} ~ 6 \qquad \textbf{(D)} ~ 9 \qquad \textbf{(E)} ~ 11</math><br />
<br />
3. In triangle <math>ABC</math> the measure of angle <math>A</math> is the average of the measures of angles <math>B</math> and <math>C</math>. What is the measure of angle <math>A</math>?<br />
<br />
<math>\textbf{(A)} ~ 45^{\circ} \qquad \textbf{(B)} ~ 60^{\circ} \qquad \textbf{(C)} ~ 75^{\circ} \qquad \textbf{(D)} ~ 90^{\circ} \qquad \textbf{(E)} ~ 120^{\circ}</math><br />
<br />
4. A spruce tree grows by <math>25</math> feet, increasing its height by <math>25 \%</math>. If the tree grows for a second time by <math>25</math> feet, by what percent would its height increase?<br />
<br />
<math>\textbf{(A)} ~ 5 \% \qquad \textbf{(B)} ~ 15 \% \qquad \textbf{(C)} ~ 20 \% \qquad \textbf{(D)} ~ 25 \% \qquad \textbf{(E)} ~ 30 \% </math><br />
<br />
5. Find the sum of the digits of <math>\dfrac{5 \times 10^{2020}}{2}</math>.<br />
<br />
<math>\textbf{(A)} ~ 1 \qquad \textbf{(B)} ~ 2 \qquad \textbf{(C)} ~ 5 \qquad \textbf{(D)} ~ 7 \qquad \textbf{(E)} ~ 8</math><br />
<br />
6. Square <math>B</math> with side length three is attached to a side of square <math>A</math> with side length four, as shown in the figure below. Find the area of the shaded region.<br />
[center]<br />
[asy]<br />
size(150);<br />
draw((0, 0)--(4, 0)--(4, 4)--(0, 4)--cycle);<br />
draw((4, 1)--(7, 1)--(7, 4)--(4, 4)--cycle);<br />
filldraw((0, 0)--(4, 0)--(4, 2.285714)--cycle, grey);<br />
filldraw((4, 1)--(7, 1)--(7, 4)--(4, 2.285714)--cycle, grey);<br />
label("A", (2, 2));<br />
label("B", (5.5, 2.5));<br />
[/asy]<br />
[/center]<br />
<math>\textbf{(A)} ~ 10 \qquad \textbf{(B)} ~ 10 \frac{1}{2} \qquad \textbf{(C)} ~ 11 \qquad \textbf{(D)} ~ 14 \qquad \textbf{(E)} ~ 14 \frac{1}{2}</math><br />
<br />
7. When expressed as a decimal rounded to the nearest ten-thousandth, what is the value of <math>\dfrac{125+3}{125 \times 3}</math>?<br />
<br />
<math>\textbf{(A)} ~ 0.3412 \qquad \textbf{(B)} ~ 0.3413 \qquad \textbf{(C)} ~ 0.3414 \qquad \textbf{(D)} ~ 0.3415 \qquad \textbf{(E)} ~ 0.3416</math><br />
<br />
8. What is the value of<br />
<cmath>(1+2+3)-(2+3+4)+(3+4+5)-\cdots -(98+99+100)?</cmath><br />
<math>\textbf{(A)} ~ -150 \qquad \textbf{(B)} ~ -147 \qquad \textbf{(C)} ~ -144 \qquad \textbf{(D)} ~ 147 \qquad \textbf{(E)} ~ 150</math><br />
<br />
9. Kayla writes down the first <math>N</math> positive integers. What is the sum of all possible values of <math>N</math> if Kayla writes five multiples of <math>13</math> and six multiples of <math>12</math>?<br />
<br />
<math>\textbf{(A)} ~ 447 \qquad \textbf{(B)} ~ 453 \qquad \textbf{(C)} ~ 518 \qquad \textbf{(D)} ~ 525 \qquad \textbf{(E)} ~ 548</math><br />
<br />
10. In Murphy's seventh grade homeroom, <math>\frac{7}{12}</math> of the students like tennis, <math>\frac{2}{3}</math> of the students like badminton, and <math>\frac{1}{12}</math> of the students like neither. What is the minimum possible number of students who like both tennis and badminton?<br />
<br />
<math>\textbf{(A)} ~ 1 \qquad \textbf{(B)} ~ 2 \qquad \textbf{(C)} ~ 3 \qquad \textbf{(D)} ~ 4 \qquad \textbf{(E)} ~ 6</math><br />
<br />
11. For how many values of <math>N</math> does there exist a regular <math>N</math> sided polygon whose vertices all lie on the vertices of a regular <math>24</math> sided polygon?<br />
<br />
<math>\textbf{(A)} ~ 6 \qquad \textbf{(B)} ~ 7 \qquad \textbf{(C)} ~ 8 \qquad \textbf{(D)} ~ 9 \qquad \textbf{(E)} ~ 10</math><br />
<br />
12. Quadrilateral <math>WXYZ</math> has its vertices on the sides of rectangle <math>ABCD</math> with <math>AB=7</math> and <math>BC=5</math>, as shown below. What is the area of quadrilateral <math>WXYZ</math>?<br />
[center]<br />
[asy]<br />
size(150);<br />
draw((0, 0)--(7, 0)--(7, 5)--(0, 5)--cycle);<br />
label("A", (0, 0), SW);<br />
label("B", (7, 0), SE);<br />
label("C", (7, 5), NE);<br />
label("D", (0, 5), NW);<br />
filldraw((0, 1)--(4, 0)--(7, 3)--(4, 5)--cycle, grey);<br />
label("W", (0, 1), W);<br />
label("X", (4, 0), S);<br />
label("Y", (7, 3), E);<br />
label("Z", (4, 5), N);<br />
label("4", (2, -0.5));<br />
label("3", (5.5, -0.5));<br />
label("4", (2, 5.5));<br />
label("3", (5.5, 5.5));<br />
[/asy]<br />
[/center]<br />
<math>\textbf{(A)} ~ 15 \dfrac{1}{2} \qquad \textbf{(B)} ~ 16 \qquad \textbf{(C)} ~ 16 \dfrac{1}{2} \qquad \textbf{(D)} ~ 17 \qquad \textbf{(E)} ~ 17 \dfrac{1}{2}</math><br />
<br />
13. To drive to the supermarket, Mable drives for <math>m</math> miles, then drives <math>12</math> miles per hour faster for the remaining <math>\frac{4}{3}m</math> miles. The amount of time Mable spent driving at each of the two speeds was equal. What was Mable's average speed during her drive to the supermarket, in miles per hour?<br />
<br />
<math>\textbf{(A)} ~ \dfrac{81}{2} \qquad \textbf{(B)} ~ \dfrac{288}{7} \qquad \textbf{(C)} ~ 42 \qquad \textbf{(D)} ~ \dfrac{300}{7} \qquad \textbf{(E)} ~ 50</math><br />
<br />
14. Six circles of radius one are cut out of the rectangle below. What is the area of the shaded region?<br />
[center]<br />
[asy]<br />
size(150);<br />
filldraw((0, 0)--(6, 0)--(6, 4)--(0, 4)--cycle, grey);<br />
filldraw(circle((1, 1), 1), white);<br />
filldraw(circle((3, 1), 1), white);<br />
filldraw(circle((5, 1), 1), white);<br />
filldraw(circle((1, 3), 1), white);<br />
filldraw(circle((3, 3), 1), white);<br />
filldraw(circle((5, 3), 1), white);<br />
[/asy]<br />
[/center]<br />
<math>\textbf{(A)} ~ 20-6\pi \qquad \textbf{(B)} ~ 24-6\pi \qquad \textbf{(C)} ~ 28-6\pi \qquad \textbf{(D)} ~ 30-6\pi \qquad \textbf{(E)} ~ 32-6\pi</math><br />
<br />
15. One metronome beeps at a steady rate of <math>72</math> beeps per minute, while another metronome beeps at a steady rate of <math>96</math> beeps per minute. If both metronomes beep at the same time once, how long will it take, in seconds, until they first beep at the same time again?<br />
<br />
<math>\textbf{(A)} ~ 2 \dfrac{1}{2} \qquad \textbf{(B)} ~ 5 \qquad \textbf{(C)} ~ 10 \qquad \textbf{(D)} ~ 18 \qquad \textbf{(E)} ~ 24</math><br />
<br />
16. A square with side length two is placed on a table, forming a <math>30</math> degree angle with the table's surface. How much higher is the top vertex of the square than the table?<br />
[center][asy]<br />
size(150);<br />
draw((0, 0)--(0.882, 0.4714)--(0.4106, 1.3534)--(-0.4714, 0.882)--cycle);<br />
draw((-0.5, 0)--(1, 0), linewidth(3));<br />
draw((-0.75, 1.3534)--(-0.65, 1.3534));<br />
draw((-0.7, 1.3534)--(-0.7, 0));<br />
draw((-0.75, 0)--(-0.65, 0));<br />
[/asy][/center]<br />
<math>\textbf{(A)} ~ \dfrac{5}{2} \qquad \textbf{(B)} ~ \sqrt{3}+1 \qquad \textbf{(C)} ~ \dfrac{4\sqrt{3}}{3} \qquad \textbf{(D)} ~ 3 \qquad \textbf{(E)} ~ \dfrac{3\sqrt{3}}{2}+1</math><br />
<br />
17. Kurtis' school schedule is made up of four classes, followed by lunch, followed by three more classes. In how many ways can Kurtis arrange his schedule if two of his classes (Reading and Writing) must occur one immediately after the other?<br />
<br />
<math>\textbf{(A)} ~ 600 \qquad \textbf{(B)} ~ 840 \qquad \textbf{(C)} ~ 1200 \qquad \textbf{(D)} ~ 1440 \qquad \textbf{(E)} ~ 1680</math><br />
<br />
18. When the number <math>25</math> is added to a list of numbers with total sum <math>S</math>, the average of all the numbers increases by one. What is the sum of the digits of the greatest possible value of <math>S</math>?<br />
<br />
<math>\textbf{(A)} ~ 6 \qquad \textbf{(B)} ~ 7 \qquad \textbf{(C)} ~ 8 \qquad \textbf{(D)} ~ 9 \qquad \textbf{(E)} ~ 12</math><br />
<br />
19. A magician randomly picks a three digit positive integer to put into her hat and pulls out the same number with its digits in reverse order. (For example <math>496</math> would become <math>694</math> and <math>720</math> would become <math>27</math>.) What is the probability the magician pulls out a multiple of <math>22</math>?<br />
<br />
<math>\textbf{(A)} ~ \dfrac{1}{15} \qquad \textbf{(B)} ~ \dfrac{1}{18} \qquad \textbf{(C)} ~ \dfrac{1}{20} \qquad \textbf{(D)} ~ \dfrac{1}{25} \qquad \textbf{(E)} ~ \dfrac{1}{30}</math><br />
<br />
20. Tyrone has three books to read in six days. He reads one-half of a single book every day. In how many ways can he finish all the books if he may not read the same book two days in a row?<br />
<br />
<math>\textbf{(A)} ~ 12 \qquad \textbf{(B)} ~ 18 \qquad \textbf{(C)} ~ 24 \qquad \textbf{(D)} ~ 30 \qquad \textbf{(E)} ~ 36</math><br />
<br />
21. There exists a circle that is tangent to <math>AB</math> and <math>BC</math> at <math>A</math> and <math>C</math>, respectively. If <math>AB=BC=13</math> and <math>AC=10</math>, what is the radius of the circle?<br />
[center][asy]<br />
size(150);<br />
draw((-5, 0)--(5, 0)--(0, -12)--cycle);<br />
draw(circle((0, 2.08333), 5.41666));<br />
label("A", (-5, 0), W);<br />
label("C", (5, 0), E);<br />
label("B", (0, -12), S);<br />
label("13", (-2.7, -6), W);<br />
label("13", (2.7, -6), E);<br />
label("10", (0, 0.2), N);<br />
[/asy][/center]<br />
<math>\textbf{(A)} ~ \dfrac{60}{13} \qquad \textbf{(B)} ~ 5 \qquad \textbf{(C)} ~ \dfrac{26}{5} \qquad \textbf{(D)} ~ \dfrac{65}{12} \qquad \textbf{(E)} ~ \dfrac{156}{25}</math><br />
<br />
22. For each of the distinct sets of numbers containing only positive integers between <math>1</math> and <math>9</math> inclusive, Jordan writes the sum of the numbers in that set. What is the sum of the numbers Jordan writes?<br />
<br />
<math>\textbf{(A)} ~ 11520 \qquad \textbf{(B)} ~ 11565 \qquad \textbf{(C)} ~ 11610 \qquad \textbf{(D)} ~ 11655 \qquad \textbf{(E)} ~ 11700</math><br />
<br />
23. In rectangle <math>ABCD</math>, the perpendicular from <math>B</math> to diagonal <math>AC</math> bisects segment <math>CD</math>. Which of the following is closest to <math>\frac{AB}{BC}</math>?<br />
<br />
<math>\textbf{(A)} ~ \dfrac{5}{4} \qquad \textbf{(B)} ~ \dfrac{4}{3} \qquad \textbf{(C)} ~ \dfrac{7}{5} \qquad \textbf{(D)} ~ \dfrac{3}{2} \qquad \textbf{(E)} ~ \dfrac{8}{5}</math><br />
<br />
24. How many ordered triples of positive integers <math>(a, b, c)</math> satisfy <math>\text{gcd}(a, b, c)=20</math> and <math>\text{lcm}(a, b, c)=240</math>?<br />
<br />
<math>\textbf{(A)} ~ 18 \qquad \textbf{(B)} ~ 24 \qquad \textbf{(C)} ~ 36 \qquad \textbf{(D)} ~ 54 \qquad \textbf{(E)} ~ 72 </math><br />
<br />
25. Cheyanne rolls two standard six sided dice, then repeatedly rerolls all dice which show an odd number and stops as soon as all dice show an even number. What is the probability Cheyanne stops after exactly four rounds of rerolling?<br />
<br />
<math>\textbf{(A)} ~ \dfrac{61}{1024} \qquad \textbf{(B)} ~ \dfrac{1}{16} \qquad \textbf{(C)} ~ \dfrac{67}{1024} \qquad \textbf{(D)} ~ \dfrac{9}{128} \qquad \textbf{(E)} ~ \dfrac{29}{256}</math></div>Bissuehttps://artofproblemsolving.com/wiki/index.php?title=User:Bissue&diff=140580User:Bissue2020-12-25T20:34:53Z<p>Bissue: </p>
<hr />
<div>i suppose i have a user page now<br />
<br />
P7: When expressed as a decimal rounded to the nearest ten-thousandth, what is the value of <math>\dfrac{125+3}{125 \times 3}</math>?<br />
<br />
<math>\textbf{(A)} ~ 0.3412 \qquad \textbf{(B)} ~ 0.3413 \qquad \textbf{(C)} ~ 0.3414 \qquad \textbf{(D)} ~ 0.3415 \qquad \textbf{(E)} ~ 0.3416</math><br />
<br />
P9: Kayla writes down the first <math>N</math> positive integers. What is the sum of all possible values of <math>N</math> if Kayla writes five multiples of <math>13</math> and six multiples of <math>12</math>?<br />
<br />
<math>\textbf{(A)} ~ 447 \qquad \textbf{(B)} ~ 453 \qquad \textbf{(C)} ~ 518 \qquad \textbf{(D)} ~ 525 \qquad \textbf{(E)} ~ 548</math><br />
<br />
P17: Kurtis' school schedule is made up of four classes, followed by lunch, followed by three more classes. In how many ways can Kurtis arrange his schedule if two of his classes (Reading and Writing) must occur one immediately after the other?<br />
<br />
<math>\textbf{(A)} ~ 600 \qquad \textbf{(B)} ~ 840 \qquad \textbf{(C)} ~ 1200 \qquad \textbf{(D)} ~ 1440 \qquad \textbf{(E)} ~ 1680</math><br />
<br />
P18: When the number <math>25</math> is added to a list of numbers with total sum <math>S</math>, the average of all the numbers increases by one. What is the sum of the digits of the greatest possible value of <math>S</math>?<br />
<br />
<math>\textbf{(A)} ~ 6 \qquad \textbf{(B)} ~ 7 \qquad \textbf{(C)} ~ 8 \qquad \textbf{(D)} ~ 9 \qquad \textbf{(E)} ~ 12</math><br />
<br />
P19: A magician randomly picks a three digit positive integer to put into her hat and pulls out the same number with its digits in reverse order. (For example <math>496</math> would become <math>694</math> and <math>720</math> would become <math>27</math>.) What is the probability the magician pulls out a multiple of <math>22</math>?<br />
<br />
<math>\textbf{(A)} ~ \dfrac{1}{15} \qquad \textbf{(B)} ~ \dfrac{1}{18} \qquad \textbf{(C)} ~ \dfrac{1}{20} \qquad \textbf{(D)} ~ \dfrac{1}{25} \qquad \textbf{(E)} ~ \dfrac{1}{30}</math><br />
<br />
P23: In rectangle <math>ABCD</math>, the perpendicular from <math>B</math> to diagonal <math>AC</math> bisects segment <math>CD</math>. Which of the following is closest to <math>\frac{AB}{BC}</math>?<br />
<br />
<math>\textbf{(A)} ~ \dfrac{5}{4} \qquad \textbf{(B)} ~ \dfrac{4}{3} \qquad \textbf{(C)} ~ \dfrac{7}{5} \qquad \textbf{(D)} ~ \dfrac{3}{2} \qquad \textbf{(E)} ~ \dfrac{8}{5}</math><br />
<br />
P24: How many ordered triples of positive integers <math>(a, b, c)</math> satisfy <math>\text{gcd}(a, b, c)=20</math> and <math>\text{lcm}(a, b, c)=240</math>?<br />
<br />
<math>\textbf{(A)} ~ 18 \qquad \textbf{(B)} ~ 24 \qquad \textbf{(C)} ~ 36 \qquad \textbf{(D)} ~ 54 \qquad \textbf{(E)} ~ 72 </math><br />
<br />
P25: Cheyanne rolls two standard six sided dice, then repeatedly rerolls all dice which show an odd number and stops as soon as all dice show an even number. What is the probability Cheyanne stops after exactly four rounds of rerolling?<br />
<br />
<math>\textbf{(A)} ~ \dfrac{61}{1024} \qquad \textbf{(B)} ~ \dfrac{1}{16} \qquad \textbf{(C)} ~ \dfrac{67}{1024} \qquad \textbf{(D)} ~ \dfrac{9}{128} \qquad \textbf{(E)} ~ \dfrac{29}{256}</math></div>Bissuehttps://artofproblemsolving.com/wiki/index.php?title=User:Bissue&diff=140578User:Bissue2020-12-25T20:26:44Z<p>Bissue: Created page with "i suppose i have a user page now"</p>
<hr />
<div>i suppose i have a user page now</div>Bissuehttps://artofproblemsolving.com/wiki/index.php?title=User:Piphi&diff=122355User:Piphi2020-05-13T15:22:54Z<p>Bissue: 69 + 1 = 70</p>
<hr />
<div><center>[[File:Piphi-Avatar.png]]</center><br />
<br />
<div style="border:2px solid black; background:#eeeeee;"><br />
::::<font style="font-family: Verdana, sans-serif">[[User:Piphi|Userpage]] | [[User talk:Piphi|Talk]] | [[Special:Contributions/Piphi|Contributions]]</font><br />
</div><br />
<div style="border:2px solid black; background:#dddddd; align:center"><br />
==<font color="black" style="font-family: MV Boli, Verdana">User Count</font>==<br />
<font color="black">If this is your first time visting this page, edit it by incrementing the user count below by one.<br />
<br />
<center><font size="100px">69 + 1</font></center><br />
</font> <br />
</div><br />
<div style="border:2px solid black; background:#cccccc; align:center"><br />
<br />
==<font color="black" style="font-family: MV Boli, Verdana">About Me</font>==<br />
<font color="black">PM me if you want to find out about some cool things you can do with the AoPS wiki.<br />
<br />
My main project on the AoPS wiki is [[AoPS_Administrators#Current_Admins | a list of all the AoPS admins]], everyone is welcome to add more admins to the list by clicking [https://artofproblemsolving.com/wiki/index.php?title=AoPS_Administrators&action=edit&section=1 here]. I also added most of the info in the [[Reaper Archives]].</font> <br />
</div><br />
<div style="border:2px solid black; background:#bbbbbb; align:center"><br />
<br />
==<font color="black" style="font-family: MV Boli, Verdana">Asymptote</font>==<br />
<br />
Here is a list of the different drawings I have made using Asymptote.<br />
<br />
* [[User:Piphi/Firefox | Firefox Logo]] (Started April 25th, 2020, Finished April 28th, 2020)<br />
* [[User:Piphi/Eclipse | Eclipse Logo]] (Started April 29th, 2020, Finished April 29th, 2020)<br />
* [[User:Piphi/Screencast | Screencast Logo]] (Started April 29th, 2020, Finished April 29th, 2020)<br />
* [[User:Piphi/Whatsapp | Whatsapp Logo]] (Started May 2nd, 2020, Finished May 2nd, 2020)<br />
* [[User:Piphi/Office 365 | Office 365 Logo]] (Started May 4th, 2020, Finished May 4th, 2020)<br />
* [[User:Piphi/LG | LG Logo]] (Started May 5th, 2020, Finished May 6th, 2020)<br />
<br />
Work In Progress<br />
<br />
* Wolfram Logo (Not Started Yet)</div>Bissue