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Contests & Programs AMC and other contests, summer programs, etc.
AMC and other contests, summer programs, etc.
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Contests & Programs AMC and other contests, summer programs, etc.
AMC and other contests, summer programs, etc.
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k a March Highlights and 2025 AoPS Online Class Information
jlacosta   0
Mar 2, 2025
March is the month for State MATHCOUNTS competitions! Kudos to everyone who participated in their local chapter competitions and best of luck to all going to State! Join us on March 11th for a Math Jam devoted to our favorite Chapter competition problems! Are you interested in training for MATHCOUNTS? Be sure to check out our AMC 8/MATHCOUNTS Basics and Advanced courses.

Are you ready to level up with Olympiad training? Registration is open with early bird pricing available for our WOOT programs: MathWOOT (Levels 1 and 2), CodeWOOT, PhysicsWOOT, and ChemWOOT. What is WOOT? WOOT stands for Worldwide Online Olympiad Training and is a 7-month high school math Olympiad preparation and testing program that brings together many of the best students from around the world to learn Olympiad problem solving skills. Classes begin in September!

Do you have plans this summer? There are so many options to fit your schedule and goals whether attending a summer camp or taking online classes, it can be a great break from the routine of the school year. Check out our summer courses at AoPS Online, or if you want a math or language arts class that doesn’t have homework, but is an enriching summer experience, our AoPS Virtual Campus summer camps may be just the ticket! We are expanding our locations for our AoPS Academies across the country with 15 locations so far and new campuses opening in Saratoga CA, Johns Creek GA, and the Upper West Side NY. Check out this page for summer camp information.

Be sure to mark your calendars for the following events:
[list][*]March 5th (Wednesday), 4:30pm PT/7:30pm ET, HCSSiM Math Jam 2025. Amber Verser, Assistant Director of the Hampshire College Summer Studies in Mathematics, will host an information session about HCSSiM, a summer program for high school students.
[*]March 6th (Thursday), 4:00pm PT/7:00pm ET, Free Webinar on Math Competitions from elementary through high school. Join us for an enlightening session that demystifies the world of math competitions and helps you make informed decisions about your contest journey.
[*]March 11th (Tuesday), 4:30pm PT/7:30pm ET, 2025 MATHCOUNTS Chapter Discussion MATH JAM. AoPS instructors will discuss some of their favorite problems from the MATHCOUNTS Chapter Competition. All are welcome!
[*]March 13th (Thursday), 4:00pm PT/7:00pm ET, Free Webinar about Summer Camps at the Virtual Campus. Transform your summer into an unforgettable learning adventure! From elementary through high school, we offer dynamic summer camps featuring topics in mathematics, language arts, and competition preparation - all designed to fit your schedule and ignite your passion for learning.[/list]
Our full course list for upcoming classes is below:
All classes run 7:30pm-8:45pm ET/4:30pm - 5:45pm PT unless otherwise noted.

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0 replies
jlacosta
Mar 2, 2025
0 replies
k i Adding contests to the Contest Collections
dcouchman   1
N Apr 5, 2023 by v_Enhance
Want to help AoPS remain a valuable Olympiad resource? Help us add contests to AoPS's Contest Collections.

Find instructions and a list of contests to add here: https://artofproblemsolving.com/community/c40244h1064480_contests_to_add
1 reply
dcouchman
Sep 9, 2019
v_Enhance
Apr 5, 2023
k i Zero tolerance
ZetaX   49
N May 4, 2019 by NoDealsHere
Source: Use your common sense! (enough is enough)
Some users don't want to learn, some other simply ignore advises.
But please follow the following guideline:


To make it short: ALWAYS USE YOUR COMMON SENSE IF POSTING!
If you don't have common sense, don't post.


More specifically:

For new threads:


a) Good, meaningful title:
The title has to say what the problem is about in best way possible.
If that title occured already, it's definitely bad. And contest names aren't good either.
That's in fact a requirement for being able to search old problems.

Examples:
Bad titles:
- "Hard"/"Medium"/"Easy" (if you find it so cool how hard/easy it is, tell it in the post and use a title that tells us the problem)
- "Number Theory" (hey guy, guess why this forum's named that way¿ and is it the only such problem on earth¿)
- "Fibonacci" (there are millions of Fibonacci problems out there, all posted and named the same...)
- "Chinese TST 2003" (does this say anything about the problem¿)
Good titles:
- "On divisors of a³+2b³+4c³-6abc"
- "Number of solutions to x²+y²=6z²"
- "Fibonacci numbers are never squares"


b) Use search function:
Before posting a "new" problem spend at least two, better five, minutes to look if this problem was posted before. If it was, don't repost it. If you have anything important to say on topic, post it in one of the older threads.
If the thread is locked cause of this, use search function.

Update (by Amir Hossein). The best way to search for two keywords in AoPS is to input
[code]+"first keyword" +"second keyword"[/code]
so that any post containing both strings "first word" and "second form".


c) Good problem statement:
Some recent really bad post was:
[quote]$lim_{n\to 1}^{+\infty}\frac{1}{n}-lnn$[/quote]
It contains no question and no answer.
If you do this, too, you are on the best way to get your thread deleted. Write everything clearly, define where your variables come from (and define the "natural" numbers if used). Additionally read your post at least twice before submitting. After you sent it, read it again and use the Edit-Button if necessary to correct errors.


For answers to already existing threads:


d) Of any interest and with content:
Don't post things that are more trivial than completely obvious. For example, if the question is to solve $x^{3}+y^{3}=z^{3}$, do not answer with "$x=y=z=0$ is a solution" only. Either you post any kind of proof or at least something unexpected (like "$x=1337, y=481, z=42$ is the smallest solution). Someone that does not see that $x=y=z=0$ is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
Similar, posting "I have solved this problem" but not posting anything else is not welcome; it even looks that you just want to show off what a genius you are.

e) Well written and checked answers:
Like c) for new threads, check your solutions at least twice for mistakes. And after sending, read it again and use the Edit-Button if necessary to correct errors.



To repeat it: ALWAYS USE YOUR COMMON SENSE IF POSTING!


Everything definitely out of range of common sense will be locked or deleted (exept for new users having less than about 42 posts, they are newbies and need/get some time to learn).

The above rules will be applied from next monday (5. march of 2007).
Feel free to discuss on this here.
49 replies
ZetaX
Feb 27, 2007
NoDealsHere
May 4, 2019
Lower bound for integer relatively prime to n
62861   23
N 2 hours ago by YaoAOPS
Source: USA Winter Team Selection Test #1 for IMO 2018, Problem 1
Let $n \ge 2$ be a positive integer, and let $\sigma(n)$ denote the sum of the positive divisors of $n$. Prove that the $n^{\text{th}}$ smallest positive integer relatively prime to $n$ is at least $\sigma(n)$, and determine for which $n$ equality holds.

Proposed by Ashwin Sah
23 replies
62861
Dec 11, 2017
YaoAOPS
2 hours ago
Number of quadratic residues mod $p$ up to $x$
Gauler   0
2 hours ago
Call an integer $n$ to be $y$-smooth if all of its prime factors are $\le y$. Let $\Psi(x,y)$ denote the number of $y$-smooth integers upto $x$. Let $y = y(p)$ be the least quadratic non-residue mod $p$. Show that there are at least $\Psi(x,y)$ quadratic residues mod $p$ up to $x$.
0 replies
Gauler
2 hours ago
0 replies
Property of Arithmetic function
Gauler   0
2 hours ago
If $f$ is a non-negative arithmetic function and
$$F(\sigma)=\sum_{n=1}^\infty \frac{f(n)}{n^\sigma}$$is convergent for some $0<\sigma<1$, then prove that
$$\sum_{n\le x} f(n) + x\sum_{n>x} \frac{f(n)}{n} \le x^\sigma F(\sigma).$$
0 replies
Gauler
2 hours ago
0 replies
Graph Theory in China TST
steven_zhang123   0
2 hours ago
Source: China TST Quiz 4 P3
For a positive integer \( n \geq 6 \), find the smallest integer \( S(n) \) such that any graph with \( n \) vertices and at least \( S(n) \) edges must contain at least two disjoint cycles (cycles with no common vertices).
0 replies
steven_zhang123
2 hours ago
0 replies
Directed Paths and Cycles
steven_zhang123   0
2 hours ago
Source: China TST Quiz 4 P2.2
Given distinct positive integers \( g \) and \( h \), let all integer points on the number line \( OX \) be vertices. Define a directed graph \( G \) as follows: for any integer point \( x \), \( x \rightarrow x + g \), \( x \rightarrow x - h \). For integers \( k, l (k < l) \), let \( G[k, l] \) denote the subgraph of \( G \) with vertices limited to the interval \([k, l]\). Find the largest positive integer \( \alpha \) such that for any integer \( r \), the subgraph \( G[r, r + \alpha - 1] \) of \( G \) is acyclic. Clarify the structure of subgraphs \( G[r, r + \alpha - 1] \) and \( G[r, r + \alpha] \) (i.e., how many connected components and what each component is like).
0 replies
steven_zhang123
2 hours ago
0 replies
Counting Circles and Maximizing Edges
steven_zhang123   0
2 hours ago
Source: China TST 2001 Quiz 4 P2.1
Let the vertex set \( V \) of a graph be partitioned into \( h \) parts \( (V = V_1 \cup V_2 \cup \cdots \cup V_h) \), with \(|V_1| = n_1, |V_2| = n_2, \ldots, |V_h| = n_h \). If there is an edge between any two vertices only when they belong to different parts, the graph is called a complete \( h \)-partite graph, denoted as \( k(n_1, n_2, \ldots, n_h) \). Let \( n \) and \( r \) be positive integers, \( n \geq 6 \), \( r \leq \frac{2}{3}n \). Consider the complete \( r + 1 \)-partite graph \( k\left(\underbrace{1, 1, \ldots, 1}_{r}, n - r\right) \).

Answer the following questions:
1. Find the maximum number of disjoint circles (i.e., circles with no common vertices) in this complete \( r + 1 \)-partite graph.
2. Given \( n \), for all \( r \leq \frac{2}{3}n \), find the maximum number of edges in a complete \( r + 1 \)-partite graph \( k(1, 1, \ldots, 1, n - r) \) where no more than one circle is disjoint.
0 replies
steven_zhang123
2 hours ago
0 replies
circumcenter of BJK lies on line AC, median, right angle, circumcircle related
parmenides51   22
N 3 hours ago by Ilikeminecraft
Source: 2019 RMM Shortlist G1
Let $BM$ be a median in an acute-angled triangle $ABC$. A point $K$ is chosen on the line through $C$ tangent to the circumcircle of $\vartriangle BMC$ so that $\angle KBC = 90^\circ$. The segments $AK$ and $BM$ meet at $J$. Prove that the circumcenter of $\triangle BJK$ lies on the line $AC$.

Aleksandr Kuznetsov, Russia
22 replies
parmenides51
Jun 18, 2020
Ilikeminecraft
3 hours ago
Squares on harmonic quadrilateral
Tkn   0
3 hours ago
Let $ABCD$ be a cyclic quadrilateral for which $AB\cdot CD=AD\cdot BC$. Construct two squares $BCEF$ and $DCGH$ externally on the sides $\overline{BC}$ and $\overline{DC}$ respectively.
Suppose that $\overleftrightarrow{BD}$ meets $\overleftrightarrow{AC}$ at $X$, $\overleftrightarrow{BE}$ meets $\overleftrightarrow{DG}$ at $Z$ and $O$ denotes circumcenter of $ABCD$. Prove that $(ZEG)$ and $(ZBD)$ meets again on $\overleftrightarrow{OX}$.
0 replies
Tkn
3 hours ago
0 replies
Hard hyperbolic inverse
RenheMiResembleRice   2
N 3 hours ago by LawofCosine
Let f on the real numbers be defined by
$f\left(x\right)=x^{3}+\sinh x+1$
Explain why f has a differentiable inverse g.
2 replies
RenheMiResembleRice
5 hours ago
LawofCosine
3 hours ago
Easy geometry
rcorreaa   19
N 3 hours ago by ihategeo_1969
Source: 2022 Brazilian National Mathematical Olympiad - Problem 2
Let $ABC$ be an acute triangle, with $AB<AC$. Let $K$ be the midpoint of the arch $BC$ that does not contain $A$ and let $P$ be the midpoint of $BC$. Let $I_B,I_C$ be the $B$-excenter and $C$-excenter of $ABC$, respectively. Let $Q$ be the reflection of $K$ with respect to $A$. Prove that the points $P,Q,I_B,I_C$ are concyclic.
19 replies
rcorreaa
Nov 22, 2022
ihategeo_1969
3 hours ago
2025 ROSS Program
scls140511   6
N 4 hours ago by akliu
Since the application has ended, are we now free to discuss the problems and stats? How do you think this year's problems are?
6 replies
scls140511
5 hours ago
akliu
4 hours ago
Stanford Math Tournament (SMT) Online 2025
stanford-math-tournament   6
N 4 hours ago by Vkmsd
[center]Register for Stanford Math Tournament (SMT) Online 2025[/center]


[center] :surf: Stanford Math Tournament (SMT) Online is happening on April 13, 2025! :surf:[/center]

[center]IMAGE[/center]

Register and learn more here:
https://www.stanfordmathtournament.com/competitions/smt-2025-online

When? The contest will take place April 13, 2025. The pre-contest puzzle hunt will take place on April 12, 2025 (optional, but highly encouraged!).

What? The competition features a Power, Team, Guts, General, and Subject (choose two of Algebra, Calculus, Discrete, Geometry) rounds.

Who? You!!!!! Students in high school or below, from anywhere in the world. Register in a team of 6-8 or as an individual.

Where? Online - compete from anywhere!

Check out our Instagram: https://www.instagram.com/stanfordmathtournament/

Register and learn more here:
https://www.stanfordmathtournament.com/competitions/smt-2025-online


[center]IMAGE[/center]


[center] :surf: :surf: :surf: :surf: :surf: [/center]
6 replies
stanford-math-tournament
Mar 9, 2025
Vkmsd
4 hours ago
nice geometry
zhoujef000   26
N 4 hours ago by smbellanki
Source: 2025 AIME I #14
Let $ABCDE$ be a convex pentagon with $AB=14,$ $BC=7,$ $CD=24,$ $DE=13,$ $EA=26,$ and $\angle B=\angle E=60^{\circ}.$ For each point $X$ in the plane, define $f(X)=AX+BX+CX+DX+EX.$ The least possible value of $f(X)$ can be expressed as $m+n\sqrt{p},$ where $m$ and $n$ are positive integers and $p$ is not divisible by the square of any prime. Find $m+n+p.$
26 replies
zhoujef000
Feb 7, 2025
smbellanki
4 hours ago
Convolution of order f(n)
trumpeter   71
N 5 hours ago by chenghaohu
Source: 2019 USAMO Problem 1
Let $\mathbb{N}$ be the set of positive integers. A function $f:\mathbb{N}\to\mathbb{N}$ satisfies the equation \[\underbrace{f(f(\ldots f}_{f(n)\text{ times}}(n)\ldots))=\frac{n^2}{f(f(n))}\]for all positive integers $n$. Given this information, determine all possible values of $f(1000)$.

Proposed by Evan Chen
71 replies
trumpeter
Apr 17, 2019
chenghaohu
5 hours ago
Mock AMC 12 2012
python123   27
N Feb 2, 2015 by DivideBy0
Hi all!

Contest season is coming up! To help with the preparation, I'm planning to host a mock AMC 12 soon. I would like to make it an online contest, so that you guys can submit answers and we can post scores, and so on. I'm putting the tentative dates as the weekend of Jan 21-22, during which sending in answers is allowed. Of course, problems will be available for practice after that as well.

Sounds good? Keep practicing, and stay posted! :)

UPDATE: Problems have been posted; please see below. You have until 11:59PM Pacific Time, Sunday 22nd, to PM me the answers.

As usual, 6 points for a correct answer, 1.5 points for not answering, and 0 points for a wrong answer.
27 replies
python123
Jan 4, 2012
DivideBy0
Feb 2, 2015
Mock AMC 12 2012
G H J
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python123
27 posts
#1 • 9 Y
Y by aeryde.xin, rdj5933mile5, dft, Amir Hossein, Adventure10, and 4 other users
Hi all!

Contest season is coming up! To help with the preparation, I'm planning to host a mock AMC 12 soon. I would like to make it an online contest, so that you guys can submit answers and we can post scores, and so on. I'm putting the tentative dates as the weekend of Jan 21-22, during which sending in answers is allowed. Of course, problems will be available for practice after that as well.

Sounds good? Keep practicing, and stay posted! :)

UPDATE: Problems have been posted; please see below. You have until 11:59PM Pacific Time, Sunday 22nd, to PM me the answers.

As usual, 6 points for a correct answer, 1.5 points for not answering, and 0 points for a wrong answer.
This post has been edited 4 times. Last edited by python123, Jan 19, 2012, 1:45 AM
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aeryde.xin
66 posts
#2 • 2 Y
Y by Adventure10, Mango247
Will the difficulty level be similar to past AMC 12s? There are some mock contests that are noticeably harder than the real contests.
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python123
27 posts
#3 • 2 Y
Y by aeryde.xin, Adventure10
aeryde.xin wrote:
Will the difficulty level be similar to past AMC 12s? There are some mock contests that are noticeably harder than the real contests.

Yes, it should be similar. The test will be posted this weekend! :lol:
Z K Y
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python123
27 posts
#4 • 14 Y
Y by Mrdavid445, mcrasher, rdj5933mile5, donutsupernova, aeryde.xin, VIPMaster, dft, Amir Hossein, osmosis92, Ahskerp95, jayden94941, Adventure10, Mango247, and 1 other user
See the problems here:

Click to reveal hidden text
This post has been edited 2 times. Last edited by python123, Jan 21, 2012, 6:27 PM
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Cortana
404 posts
#5 • 1 Y
Y by Adventure10
Do we need to pm our solutions too or just the letter answer?
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python123
27 posts
#6 • 2 Y
Y by Adventure10, Mango247
Cortana wrote:
Do we need to pm our solutions too or just the letter answer?

Letter answers are good enough :)
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python123
27 posts
#7 • 2 Y
Y by Adventure10, Mango247
About 1.5 more days to submit answers :)
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python123
27 posts
#8 • 2 Y
Y by Adventure10, Mango247
Contest is over. You may discuss the problems.

Answers:
DCAED EBECC DBACA DDAEE ACBEB

One common mistake worth pointing out is on #13. The problem states that $x$ and $y$ are distinct, and that makes the correct answer A rather than D.

The top 6 scorers are:
exmath89 138
yankeefan6795 130.5
benjamin7xx 130.5
donutsupernova 127.5
VIPMaster 121.5
harbinger_of_doom 117.5

Thank you all, and good luck on the real tests!
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benjamin7xx
294 posts
#9 • 2 Y
Y by Adventure10, Mango247
Would anyone happen to have solutions for the last 3 problems?
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Diehard
1374 posts
#10 • 2 Y
Y by Adventure10, Mango247
You can use the mean-value theorem for $25$.
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exmath89
2572 posts
#11 • 3 Y
Y by Dwu123, Adventure10, Mango247
@Diehard, could you explain this "mean-value theorem"?

#23 Solution

#24 Solution
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Diehard
1374 posts
#12 • 2 Y
Y by osmosis92, Adventure10
My mistake, this isn't really the mean-value theorem; it just looks like it. Since $2^{x}>x$ everywhere, all we need to do is translate $y=x$ until it's tangent to the graph of $y=2^{x}$. So essentially, the derivative at some point $(x,2^{x})$ must equal the slope of the line $y=x$, namely, $1$. Now it's easy to find $x$.
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donutsupernova
423 posts
#13 • 2 Y
Y by Adventure10, Mango247
Wow how do you come up with that solution during the test?
my solution to 23
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Diehard
1374 posts
#14 • 2 Y
Y by Adventure10, Mango247
I actually did come up with that during the test (unofficial). :wink: However, I thought this test was too easy (22 looks like a challenge problem from a school textbook) and contained too many problems from previous years.
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Cortana
404 posts
#15 • 2 Y
Y by Adventure10, Mango247
Can someone post solutions to 16 and 22? Thanks
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donutsupernova
423 posts
#16 • 2 Y
Y by Adventure10, Mango247
22
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osmosis92
1139 posts
#17 • 1 Y
Y by Adventure10
it may be easier to find the altitudes to the diagonal.
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exmath89
2572 posts
#18 • 1 Y
Y by Adventure10
Diehard wrote:
My mistake, this isn't really the mean-value theorem; it just looks like it. Since $2^{x}>x$ everywhere, all we need to do is translate $y=x$ until it's tangent to the graph of $y=2^{x}$. So essentially, the derivative at some point $(x,2^{x})$ must equal the slope of the line $y=x$, namely, $1$. Now it's easy to find $x$.

Could someone post a solution to #25 that does not require calculus?

Thanks.
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donutsupernova
423 posts
#19 • 1 Y
Y by Adventure10
Does anyone have a solution for 15, 18, 19
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harbinger_of_doom
60 posts
#20 • 2 Y
Y by Adventure10, Mango247
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Didnt get 18 or 19... :(
22, I used law of cosines and bcsina/2. Think it's nicer than herons formula.

On 10, I actually used Vieta's instead of just pluging and chugging (didn't realize the roots were 2 and 5 :wallbash_red: )

I think some of the last 5 could have been swapped with some of the problems before it
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quantumbyte
547 posts
#21 • 2 Y
Y by Adventure10 and 1 other user
I am not sure how you would do #25 without a calculator and knowledge of calculus.
Solution
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donutsupernova
423 posts
#22 • 1 Y
Y by Adventure10
I still don't get it. So mod 3 it becomes $1,0,0,0....$ which does what?
Some idea for #25
This post has been edited 1 time. Last edited by donutsupernova, Jan 24, 2012, 5:41 AM
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exmath89
2572 posts
#23 • 1 Y
Y by Adventure10
Solution for #19

And for Problem #15, note that after $1!+2!+3!$, the rest end in a $3$ mod $10$. $3$ can't be the units digit of a perfect square, so we only have $1$ and $9$.
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harbinger_of_doom
60 posts
#24 • 2 Y
Y by Adventure10, Mango247
Oh what, I must have messed up. I got that the majority of them become 2 mod 3 which is impossible. My bad! :( (talk about getting the right answer on accident)
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pr0likethis
755 posts
#25 • 2 Y
Y by Adventure10, Mango247
jsut did this on my own and got a 124.5, with 13, 17 wrong and omitting 18, 23, 24 (did the calc solution to 25)
Very much so not as well as i hope to get in a couple of weeks :(
@exmath those two solutions are awesome...i even had those ideas! i just didnt go through with them.
can someone post solutions to 13, 17, 18? 13 and 17 i'm pretty srue i just made silly mistakes, but i'm unsure of where.
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quantumbyte
547 posts
#26 • 1 Y
Y by Adventure10
@prolikethis: How did you do #25 without a calculator(unless you magically know the log of the ln of 2).
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carmelninja
47 posts
#27 • 2 Y
Y by Adventure10, Mango247
I have an issue with the answer for #17

Here is my work:
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Is there anything wrong with my work? Or should the answer be (A)?
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DivideBy0
84 posts
#28 • 1 Y
Y by Adventure10
carmelninja wrote:
Is there anything wrong with my work? Or should the answer be (A)?
pr0likethis wrote:
17 i'm pretty srue i just made silly mistakes, but i'm unsure of where.

AHSME 1996 #14, essentially this exact problem, has answer 400.
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