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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
AMC- IMO preparation
asyaela.   9
N 2 minutes ago by Schintalpati
I'm a ninth grader, and I recently attempted the AMC 12, getting 18 questions correct and leaving 7 empty. I started working on Olympiad math in November and currently dedicate about two hours per day to preparation. I'm feeling a bit demotivated, but if it's possible for me to reach IMO level, I'd be willing to put in more time. How realistic is it for me to get there, and how much study would it typically take?
9 replies
+3 w
asyaela.
3 hours ago
Schintalpati
2 minutes ago
Tennessee Math Tournament (TMT) Online 2025
TennesseeMathTournament   29
N 9 minutes ago by NashvilleSC
Hello everyone! We are excited to announce a new competition, the Tennessee Math Tournament, created by the Tennessee Math Coalition! Anyone can participate in the virtual competition for free.

The testing window is from March 22nd to April 5th, 2025. Virtual competitors may participate in the competition at any time during that window.

The virtual competition consists of three rounds: Individual, Bullet, and Team. The Individual Round is 60 minutes long and consists of 30 questions (AMC 10 level). The Bullet Round is 20 minutes long and consists of 80 questions (Mathcounts Chapter level). The Team Round is 30 minutes long and consists of 16 questions (AMC 12 level). Virtual competitors may compete in teams of four, or choose to not participate in the team round.

To register and see more information, click here!

If you have any questions, please email connect@tnmathcoalition.org or reply to this thread!
29 replies
+1 w
TennesseeMathTournament
Mar 9, 2025
NashvilleSC
9 minutes ago
AIME score for college apps
Happyllamaalways   75
N 19 minutes ago by hashbrown2009
What good colleges do I have a chance of getting into with an 11 on AIME? (Any chances for Princeton)

Also idk if this has weight but I had the highest AIME score in my school.
75 replies
Happyllamaalways
Mar 13, 2025
hashbrown2009
19 minutes ago
Finally hard NT on UKR MO from NT master
mshtand1   2
N an hour ago by IAmTheHazard
Source: Ukrainian Mathematical Olympiad 2025. Day 1, Problem 11.4
A pair of positive integer numbers \((a, b)\) is given. It turns out that for every positive integer number \(n\), for which the numbers \((n - a)(n + b)\) and \(n^2 - ab\) are positive, they have the same number of divisors. Is it necessarily true that \(a = b\)?

Proposed by Oleksii Masalitin
2 replies
mshtand1
Mar 13, 2025
IAmTheHazard
an hour ago
AMC 8 discussion
Jaxman8   42
N an hour ago by mpcnotnpc
Discuss the AMC 8 below!
42 replies
Jaxman8
Jan 29, 2025
mpcnotnpc
an hour ago
IMOC 2017 G5 (<A=120 => E, F, Y,Z are concyclic, incenter related)
parmenides51   4
N an hour ago by ehuseyinyigit
Source: https://artofproblemsolving.com/community/c6h1740077p11309077
We have $\vartriangle ABC$ with $I$ as its incenter. Let $D$ be the intersection of $AI$ and $BC$ and define $E, F$ in a similar way. Furthermore, let $Y = CI \cap DE, Z = BI \cap DF$. Prove that if $\angle BAC = 120^o$, then $E, F, Y,Z$ are concyclic.
IMAGE
4 replies
parmenides51
Mar 20, 2020
ehuseyinyigit
an hour ago
Bosnia and Herzegovina JBMO TST 2013 Problem 1
gobathegreat   3
N 2 hours ago by DensSv
Source: Bosnia and Herzegovina Junior Balkan Mathematical Olympiad TST 2013
It is given $n$ positive integers. Product of any one of them with sum of remaining numbers increased by $1$ is divisible with sum of all $n$ numbers. Prove that sum of squares of all $n$ numbers is divisible with sum of all $n$ numbers
3 replies
gobathegreat
Sep 16, 2018
DensSv
2 hours ago
D1015 : A strange EF for polynomials
Dattier   0
2 hours ago
Source: les dattes à Dattier
Find all $P \in \mathbb R[x,y]$ with $P \not\in \mathbb R[x] \cup \mathbb R[y]$ and $\forall g,f$ homeomorphismes of $\mathbb R$, $P(f,g)$ is an homoemorphisme too.
0 replies
1 viewing
Dattier
2 hours ago
0 replies
P, Q,R collinear and U, R, O, V concyclic wanted, cyclic ABCD, circumcenters
parmenides51   2
N 2 hours ago by DensSv
Source: 2012 Romania JBMO TST2 P4
The quadrilateral $ABCD$ is inscribed in a circle centered at $O$, and $\{P\} = AC \cap BD, \{Q\} = AB \cap CD$. Let $R$ be the second intersection point of the circumcircles of the triangles $ABP$ and $CDP$.
a) Prove that the points $P, Q$, and $R$ are collinear.
b) If $U$ and $V$ are the circumcenters of the triangles $ABP$, and $CDP$, respectively, prove that the points $U, R, O, V$ are concyclic.
2 replies
parmenides51
May 29, 2020
DensSv
2 hours ago
Unsolved Diophantine(I think)
Nuran2010   1
N 2 hours ago by Nuran2010
Find all solutions for the equation $2^n=p+3^p$ where $n$ is a positive integer and $p$ is a prime.(Don't get mad at me,I've used the search function and did not see a correct and complete solution anywhere.)
1 reply
Nuran2010
Mar 14, 2025
Nuran2010
2 hours ago
2^a + 3^b + 1 = 6^c
togrulhamidli2011   1
N 2 hours ago by CM1910
Find all positive integers (a, b, c) such that:

\[
2^a + 3^b + 1 = 6^c
\]
1 reply
togrulhamidli2011
Today at 12:34 PM
CM1910
2 hours ago
Prove that OA and RA are perpendicular
MellowMelon   90
N 3 hours ago by ehuseyinyigit
Source: USA TSTST 2011/2012 P4
Acute triangle $ABC$ is inscribed in circle $\omega$. Let $H$ and $O$ denote its orthocenter and circumcenter, respectively. Let $M$ and $N$ be the midpoints of sides $AB$ and $AC$, respectively. Rays $MH$ and $NH$ meet $\omega$ at $P$ and $Q$, respectively. Lines $MN$ and $PQ$ meet at $R$. Prove that $OA\perp RA$.
90 replies
MellowMelon
Jul 26, 2011
ehuseyinyigit
3 hours ago
find all f
mecrazywong   8
N 3 hours ago by HamstPan38825
Source: Chinese team training 2004
Find all $f:\mathbb N\rightarrow\mathbb Z$ satisfying both of the following properties:
(1)If a,b are positive integers, then $f(ab)+f(a^2+b^2)=f(a)+f(b)$.
(2)If a,b are positive integers and a|b, then $f(a)\ge f(b)$.
8 replies
mecrazywong
Feb 16, 2005
HamstPan38825
3 hours ago
Sequence 1994
Jan   5
N 3 hours ago by AshAuktober
Source: IMO Shortlist 1994, A1
Let $ a_{0} = 1994$ and $ a_{n + 1} = \frac {a_{n}^{2}}{a_{n} + 1}$ for each nonnegative integer $ n$. Prove that $ 1994 - n$ is the greatest integer less than or equal to $ a_{n}$, $ 0 \leq n \leq 998$
5 replies
Jan
Dec 26, 2006
AshAuktober
3 hours ago
Mock AMC J
Silverfalcon   39
N Jan 31, 2005 by Silverfalcon
This is a self-taking Mock AMC. I just made this so that many people in this site and I myself can have some extra mock AMC. I don't have an answer key so if you have a right answer, post here in the spoiler.

Many questions are came from old AHSMEs or even Mock AMC's. And other contests.

1. A lattice point is a point on a coordinate plane with integer values for x and y. How many lattice points lie on a circle centered at the origin with radius 25?
A) 40
B) 20
C) 10
D) 5
E) None of above

2. Find the unit digits of the sum:

$\sum^{2005}_{k=0} k^5$
A) 1
B) 3
C) 5
D) 7
E) 9

3. How many three element subsets of the set:

{88,95,99,132,166,173}

have the property that the sum of the three elements is even?
A) 6
B) 8
C) 10
D) 12
E) 24

4. If x and y are positive integesr such that $3x+4y=100$, then $x+y=$?
A) 10
B) 12
C) 14
D) 16
E) 18

5. Let a,a',b,b' be real numbers with $a,a' \neq 0$. The solution to $ax+b=0$ is less than the solution to $a'x+b' = 0$ if and only if:
A) a'b<ab'
B) ab'<a'b
C) ab<a'b'
D) b/a<b'/a'
E) b'/a'<b/a

6. In an h-meter race, Sunny is exactly d meters ahead of Windy when Sunny sportingly starts d meters behind Windy, who is at the starting line. Both runners run at the same constant speed as they did in the first race. How many meters ahead is Sunny when Sunny finished the second race?
A) d/h
B) 0
C) $d^2/h$
D) $h^2/d$
E) $d^2/(h-d)$

7. Let $f$ be a function which $f(x/3) = x^2+x+1$. Find the sum of all values of $z$ for which $f(3z) = 7$.
A) -1/3
B) -1/9
C) 0
D) 5/9
E) 5/3

8. In an arithmetic sequence of complex numbers the first term is $3+4i$ and the second term is 7. Find the sum of the first 50 terms.
A) $5050-4700i$
B) $5000-7100i$
C) $5000+7100i$
D) $5050+7100i$
E) None of these

9. Two nonadjacent vertices of a rectangle are (4,3) and (-4,-3) and the coordinates of the other two vertices are integers. The number of such rectangles is:
A) 1
B) 2
C) 3
D) 4
E) 5

10. Inscribed in a circle is a quadrilateral having sides of lengths 25,39,52,and 60 taken consecutively. The diameter of the circle has length of:
A) 62
B) 63
C) 65
D) 66
E) 69

11. A wooden cube with edge length n units (where n is an integer > 2) is painted black all over. By slice parallel to its faces, the cube is cut into $n^3$ smaller cubes each of unit dege length. If the number of smaller of cubes with just one face painted black is equal to the number of smaller cubes completely free of paint, what is n?
A) 5
B) 6
C) 7
D) 8
E) None of these

12. An urn contains marbles of four colors: red, white, blue, and green. When four marbles are drawn without replacement, the following events are equally likely:

(a) the selection of four red marbles
(b) the selection of one white and three red marbles;
(c) the selection of one white, one blue, and two red marbles; and
(d) the selection of one marble of each color

What is the smallest number of marbles satisfying the given condition?
A) 19
B) 21
C) 46
D) 69
E) More than 69

13. In their base 10 representation, the integer $a$ consists of a sequence of 1985 eights and the integer $b$ contains a sequence of 1985 fives. What is the sum of digits of the base 10 representation of the integer $9ab$?
A) 15880
B) 17856
C) 17865
D) 17874
E) 19851

14. Let $D_n$ denote the number of diagonals plus the number of sides in a convex polygon with $n$ sides. Then:

$\frac {1}{D_4} + \frac {1}{D_5} + \frac {1}{D_6}+...\frac {1}{D_2004} = \frac {m}{n}$ where m and n are relatively prime integers. Find $m-n$.
A) -334
B) -330
C) -332
D) -328
E) None of the above

15. In a triangle ABC, angle C is a right angle and CB>CA. Point D is located on segment BC so that angle CAD is twice angle DAB. If AC/AD = 2/3, then CD/BD = m/n, where m and n are relatively prime integers. Find m+n.
A) 10
B) 14
C) 18
D) 22
E) 26

16. Evaluate:

$\frac {\sin 10 + \sin 20}{\cos 10 + \cos 20}$ where numbers are given in degrees.
A) $\tan 10 + \tan 20$
B) $\tan 30$
C) $\frac {1}{2} (\tan 10 + \tan 20)$
D) $\tan 15$
E) $\frac {1}{4} \tan 60$

17. Find the units digit in the decimal expansion of:

$(15 + \sqrt {220})^{19} + (15 + \sqrt {220})^{82}$.
A) 0
B) 2
C) 5
D) 9
E) None of the above

18. If John has 12 quarters to distribute among Akiba, Bobby, Crl, David, and Emanuel, in how many ways can he distribute the money given that Akia gets at least 50 cents?

19. If $x = \frac {-1 + i \sqrt 3}{2}$ and $y = \frac {-1 - i \sqrt 3}{2}$ where $i^2 = -1$, then which of the following is not correct?

A) $x^5+y^5 = -1$
B) $x^7+y^7 = -1$
C) $x^9+y^9 = -1$
D) $x^{11} + y^{11} = -1$
E) $x^{13} + y^{13} = -1$

20. A and C lie on a circle with radius $\sqrt {50}$. The point B is inside the circle such that <ABC = 90, AB = 6, BC = 2. Find OB.

A) 5
B) :sqrt: 26
C) :sqrt: 31
D) 11/2
E) 6

21. In triangle ABC, BC = 8 and AC = 10. If D is the midpoint of BC, and E is on AC such that EC = 3. Let EB and AD intersect at F. If DE:AF = m/n, where m and n are relatively prime integers, find m+n.
A) 28
B) 57
C) 100
D) 239
E) 87

22. How many elements in the 2002nd row of Pascal's Triangle (the one that begins with 1,2001,....) leaves remainder of 1 when divided by 3?
A) 23
B) 26
C) 28
D) 29
E) 667

23. TWo of the altitudes of the scalene triangle ABC have length 4 and 12. If the length of the third altitude is also integer, what is the biggest it can be?
A) 4
B) 5
C) 6
D) 7
E) None

24. Eight congruent equilateral triangles, each of different color, are used to construct a regular octahedron. How many distinguishable ways are there to construct the octahedron? (Two colored octahedrons are distinguishable if neither can be rotated to look just like other)
A) 210
B) 560
C) 840
D) 1260
E) 1680

25. If the base 8 representation of a perfect square is ab3c, where a is not 0, then c is:

A) 0
B) 1
C) 3
D) 4
E) Not uniquely determined

Please note that this is problem thread AND solution thread. So, post your solution so people can see about it. Also, even you're not sure about answer, please post here.

Thanks and Enjoy the problems!

P.S. I have to try this too! :lol:
39 replies
Silverfalcon
Jan 23, 2005
Silverfalcon
Jan 31, 2005
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
Silverfalcon
5006 posts
#1 • 2 Y
Y by Adventure10, Mango247
This is a self-taking Mock AMC. I just made this so that many people in this site and I myself can have some extra mock AMC. I don't have an answer key so if you have a right answer, post here in the spoiler.

Many questions are came from old AHSMEs or even Mock AMC's. And other contests.

1. A lattice point is a point on a coordinate plane with integer values for x and y. How many lattice points lie on a circle centered at the origin with radius 25?
A) 40
B) 20
C) 10
D) 5
E) None of above

2. Find the unit digits of the sum:

$\sum^{2005}_{k=0} k^5$
A) 1
B) 3
C) 5
D) 7
E) 9

3. How many three element subsets of the set:

{88,95,99,132,166,173}

have the property that the sum of the three elements is even?
A) 6
B) 8
C) 10
D) 12
E) 24

4. If x and y are positive integesr such that $3x+4y=100$, then $x+y=$?
A) 10
B) 12
C) 14
D) 16
E) 18

5. Let a,a',b,b' be real numbers with $a,a' \neq 0$. The solution to $ax+b=0$ is less than the solution to $a'x+b' = 0$ if and only if:
A) a'b<ab'
B) ab'<a'b
C) ab<a'b'
D) b/a<b'/a'
E) b'/a'<b/a

6. In an h-meter race, Sunny is exactly d meters ahead of Windy when Sunny sportingly starts d meters behind Windy, who is at the starting line. Both runners run at the same constant speed as they did in the first race. How many meters ahead is Sunny when Sunny finished the second race?
A) d/h
B) 0
C) $d^2/h$
D) $h^2/d$
E) $d^2/(h-d)$

7. Let $f$ be a function which $f(x/3) = x^2+x+1$. Find the sum of all values of $z$ for which $f(3z) = 7$.
A) -1/3
B) -1/9
C) 0
D) 5/9
E) 5/3

8. In an arithmetic sequence of complex numbers the first term is $3+4i$ and the second term is 7. Find the sum of the first 50 terms.
A) $5050-4700i$
B) $5000-7100i$
C) $5000+7100i$
D) $5050+7100i$
E) None of these

9. Two nonadjacent vertices of a rectangle are (4,3) and (-4,-3) and the coordinates of the other two vertices are integers. The number of such rectangles is:
A) 1
B) 2
C) 3
D) 4
E) 5

10. Inscribed in a circle is a quadrilateral having sides of lengths 25,39,52,and 60 taken consecutively. The diameter of the circle has length of:
A) 62
B) 63
C) 65
D) 66
E) 69

11. A wooden cube with edge length n units (where n is an integer > 2) is painted black all over. By slice parallel to its faces, the cube is cut into $n^3$ smaller cubes each of unit dege length. If the number of smaller of cubes with just one face painted black is equal to the number of smaller cubes completely free of paint, what is n?
A) 5
B) 6
C) 7
D) 8
E) None of these

12. An urn contains marbles of four colors: red, white, blue, and green. When four marbles are drawn without replacement, the following events are equally likely:

(a) the selection of four red marbles
(b) the selection of one white and three red marbles;
(c) the selection of one white, one blue, and two red marbles; and
(d) the selection of one marble of each color

What is the smallest number of marbles satisfying the given condition?
A) 19
B) 21
C) 46
D) 69
E) More than 69

13. In their base 10 representation, the integer $a$ consists of a sequence of 1985 eights and the integer $b$ contains a sequence of 1985 fives. What is the sum of digits of the base 10 representation of the integer $9ab$?
A) 15880
B) 17856
C) 17865
D) 17874
E) 19851

14. Let $D_n$ denote the number of diagonals plus the number of sides in a convex polygon with $n$ sides. Then:

$\frac {1}{D_4} + \frac {1}{D_5} + \frac {1}{D_6}+...\frac {1}{D_2004} = \frac {m}{n}$ where m and n are relatively prime integers. Find $m-n$.
A) -334
B) -330
C) -332
D) -328
E) None of the above

15. In a triangle ABC, angle C is a right angle and CB>CA. Point D is located on segment BC so that angle CAD is twice angle DAB. If AC/AD = 2/3, then CD/BD = m/n, where m and n are relatively prime integers. Find m+n.
A) 10
B) 14
C) 18
D) 22
E) 26

16. Evaluate:

$\frac {\sin 10 + \sin 20}{\cos 10 + \cos 20}$ where numbers are given in degrees.
A) $\tan 10 + \tan 20$
B) $\tan 30$
C) $\frac {1}{2} (\tan 10 + \tan 20)$
D) $\tan 15$
E) $\frac {1}{4} \tan 60$

17. Find the units digit in the decimal expansion of:

$(15 + \sqrt {220})^{19} + (15 + \sqrt {220})^{82}$.
A) 0
B) 2
C) 5
D) 9
E) None of the above

18. If John has 12 quarters to distribute among Akiba, Bobby, Crl, David, and Emanuel, in how many ways can he distribute the money given that Akia gets at least 50 cents?

19. If $x = \frac {-1 + i \sqrt 3}{2}$ and $y = \frac {-1 - i \sqrt 3}{2}$ where $i^2 = -1$, then which of the following is not correct?

A) $x^5+y^5 = -1$
B) $x^7+y^7 = -1$
C) $x^9+y^9 = -1$
D) $x^{11} + y^{11} = -1$
E) $x^{13} + y^{13} = -1$

20. A and C lie on a circle with radius $\sqrt {50}$. The point B is inside the circle such that <ABC = 90, AB = 6, BC = 2. Find OB.

A) 5
B) :sqrt: 26
C) :sqrt: 31
D) 11/2
E) 6

21. In triangle ABC, BC = 8 and AC = 10. If D is the midpoint of BC, and E is on AC such that EC = 3. Let EB and AD intersect at F. If DE:AF = m/n, where m and n are relatively prime integers, find m+n.
A) 28
B) 57
C) 100
D) 239
E) 87

22. How many elements in the 2002nd row of Pascal's Triangle (the one that begins with 1,2001,....) leaves remainder of 1 when divided by 3?
A) 23
B) 26
C) 28
D) 29
E) 667

23. TWo of the altitudes of the scalene triangle ABC have length 4 and 12. If the length of the third altitude is also integer, what is the biggest it can be?
A) 4
B) 5
C) 6
D) 7
E) None

24. Eight congruent equilateral triangles, each of different color, are used to construct a regular octahedron. How many distinguishable ways are there to construct the octahedron? (Two colored octahedrons are distinguishable if neither can be rotated to look just like other)
A) 210
B) 560
C) 840
D) 1260
E) 1680

25. If the base 8 representation of a perfect square is ab3c, where a is not 0, then c is:

A) 0
B) 1
C) 3
D) 4
E) Not uniquely determined

Please note that this is problem thread AND solution thread. So, post your solution so people can see about it. Also, even you're not sure about answer, please post here.

Thanks and Enjoy the problems!

P.S. I have to try this too! :lol:
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Silverfalcon
5006 posts
#2 • 2 Y
Y by Adventure10, Mango247
Number one: :D

My_answer
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Silverfalcon
5006 posts
#3 • 2 Y
Y by Adventure10, Mango247
Number 10

I'm not so sure about this so I'll give it a shot.

Um

Is there any better way or if I'm wrong, correct way?
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Silverfalcon
5006 posts
#4 • 2 Y
Y by Adventure10, Mango247
Number 7

My_ANSWER

I thought there was different answer on Harold Reiter's Page though.. ;)
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Silverfalcon
5006 posts
#5 • 2 Y
Y by Adventure10, Mango247
Click to reveal hidden text
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white_horse_king88
1779 posts
#6 • 2 Y
Y by Adventure10, Mango247
Silverfalcon wrote:
Click to reveal hidden text

Basically, if you line it up, you have the question: How many ways can I arrange 4 breakers/dividers, and 10 objects which I am dividing. That's just C(14,10).
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Silverfalcon
5006 posts
#7 • 2 Y
Y by Adventure10, Mango247
Hmm..

This is kinda confusing.. :?
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beta
3001 posts
#8 • 2 Y
Y by Adventure10, Mango247
#15)

Click to reveal hidden text
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eryaman
1130 posts
#9 • 2 Y
Y by Adventure10, Mango247
I don't understand how there can be one single answer to number 4. Aren't there many different possibilities?
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beta
3001 posts
#10 • 2 Y
Y by Adventure10, Mango247
eryaman wrote:
I don't understand how there can be one single answer to number 4. Aren't there many different possibilities?

Not only that, in fact none of the answers can work!

4(x+y)-x=100, so 4(x+y)>100, x+y>25! Therefore none of the choices can be correct!
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TripleM
1587 posts
#11 • 2 Y
Y by Adventure10, Mango247
Silverfalcon wrote:
Number 7

My_ANSWER

I thought there was different answer on Harold Reiter's Page though.. ;)

Mistakes in your answer
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Silverfalcon
5006 posts
#12 • 2 Y
Y by Adventure10, Mango247
It should have been 13x+4y=100

I must've typed the question wrong.

The correct answer then would be D.

13(4)+4(12) = 100 = 100 :)
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Silverfalcon
5006 posts
#13 • 2 Y
Y by Adventure10, Mango247
Hehe...

I knew I was wrong because the quesiton had another way of solving it using Sum and Product Formula (only sum part this case) but for my equation, that formula didn't work.

Somehow, it worked though. :-D

Right way would be:

$81z^2+9z-6 = (9z+3)(9z-2)$ and this also gives me a right answer.

Hehe... I got the right answer by wrong method... :lol:
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Treething
2107 posts
#14 • 2 Y
Y by Adventure10, Mango247
1. A lattice point is a point on a coordinate plane with integer values for x and y. How many lattice points lie on a circle centered at the origin with radius 25?
A) 40
B) 20
C) 10
D) 5
E) None of above

answer

2. Find the unit digits of the sum:

$\sum^{2005}_{k=0} k^5$
A) 1
B) 3
C) 5
D) 7
E) 9

answer

3. How many three element subsets of the set:

{88,95,99,132,166,173}

have the property that the sum of the three elements is even?
A) 6
B) 8
C) 10
D) 12
E) 24

answer

4. If x and y are positive integesr such that $13x+4y=100$, then $x+y=$?
A) 10
B) 12
C) 14
D) 16
E) 18

answer

1 to 4
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joml88
6343 posts
#15 • 2 Y
Y by Adventure10, Mango247
Just wanted to point a non-plug and chug method for this one:

4. If $x$ and $y$ are positive integesr such that $13x+4y=100$, then $x+y=$?

Solution

If you haven't used mods very much this might not seem like a very natural approach. Plug and chug certainly works without excruciating pain in this case but when you get harder diophantine equations (it just means an equation where you are looking for integer solutions) you will almost have to use modular arithmetic.
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1234567890
1553 posts
#16 • 2 Y
Y by Adventure10, Mango247
How would you use mods?
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NightFlarer
316 posts
#17 • 2 Y
Y by Adventure10, Mango247
This probably isn't a good explanation, but here goes:

Say you have 3 + 2 = 5
We know 5 = 0 (mod 5), and since 3 + 2 = 5, we can say 3 + 2 = 0 (mod 5), which is true.
Thus, when considering a certain mod x, the left hand side and the right hand side of the original equation must be equal to the same y (mod x)

We have 13x + 4y = 100

I considered mod 13, which wasn't the best idea, but if you do that you get
13x or 0 (mod 13) + 4y (mod 13) = 100 (mod 13)

13x is always 0 mod 13, so just ignore it for now

So 4y (mod 13) = 100 or 9 (mod 13)

4y = 9 (mod 13)

I forgot my other steps so I'll just leave it at this...
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JBL
16123 posts
#18 • 2 Y
Y by Adventure10, Mango247
The smoothest thing to do at that point is to notice that 9 = -4 (mod 13), so 4y = -4 (mod 13) so y = -1 = 12 (mod 13). (Notice that we divide -- you can only ONLY ONLY ever do this when the thing you're dividing by is relatively prime to the mod. Luckily, 13 is prime.)
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tongos
53 posts
#19 • 2 Y
Y by Adventure10, Mango247
13x+4y=100

4(25)-4y

13x= 4(25-y)

25-y=k

13x=4k (13 is prime, so k=13 and x=4)

25-y=13, y=12, x=4

12+4=16
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tongos
53 posts
#20 • 2 Y
Y by Adventure10, Mango247
quick question.

On number 14, what does d(2)004 mean, the one that is the last term of the series. ?????
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Treething
2107 posts
#21 • 2 Y
Y by Adventure10, Mango247
The main thing to notice in number 4 is that 13 and 4 are relatively prime.
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Mr.Ocax
341 posts
#22 • 2 Y
Y by Adventure10, Mango247
Um.. Number 18 doesn't have any answer choices and on number 14.. are you sure the answer isn't -335?? becuase you have -334 on there, but I'm not sure if its a typo or what. This is what i did on 14:

Click to reveal hidden text

So go figure.. maybe i did something wrong. Some help on 14 please??
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Silverfalcon
5006 posts
#23 • 2 Y
Y by Adventure10, Mango247
-335 is a correct answer.

For #18, I forgot.
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Silverfalcon
5006 posts
#24 • 2 Y
Y by Adventure10, Mango247
I'd like to get the answers for these ASAP since AMC is coming and if you can answer these 25 questions VERY EASILY, you're ready for most of AMC questions (-10 or -12).

I think exception of use of trig and log, this is harder than regular AMC and mock AMC's since it involves some hardest question on mock AMC.
Quote:
6. In an h-meter race, Sunny is exactly d meters ahead of Windy when Sunny sportingly starts d meters behind Windy, who is at the starting line. Both runners run at the same constant speed as they did in the first race. How many meters ahead is Sunny when Sunny finished the second race?

Let's call r as the rate of Sunny. Then, (h+d)/r is the time for Sunny to finish race with d meters more. To find Windy's rate, we set proportion:

Windy's rate: r = (h-d): h

Then we get (h-d)r/h as Windy's rate. Since the time for d+ race is (h+d)/r, then:

$\frac {(h-d)r}{h} * \frac {(h+d)}{r}$

Which equals h - d^2/h so Windy's ahead by d^2/h.
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Silverfalcon
5006 posts
#25 • 2 Y
Y by Adventure10, Mango247
Few answers to the problems.

#13

Since it's in form of 9*a*b, we can try in simpler case:

9*8*5 = 360
9*88*55 = 43560
9*888*555 = 4435560

So, the sum of digits equals 9n then 1985 = 9*1985 = 17865

There was a constructive method that Mr. Crawford showed although I didn't quite get it.

If anyone can explain here, I'll be thankful.

#16

I'm not sure but can I do this?

$\frac {\sin 10}{\cos 10} + \frac {\sin 20}{\cos 20} = \tan 10 + \tan 20$?

Can anyone gives some helps for the rest?

Thanks!
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Agent_00pi
4 posts
#26 • 2 Y
Y by Adventure10, Mango247
In #20, is O the center of the circle?
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joml88
6343 posts
#27 • 2 Y
Y by Adventure10, Mango247
Silverfalcon....I'm disappointed. You should know that $\frac{a+b}{c+d}\neq \frac ac+\frac bd.$

Hint
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Silverfalcon
5006 posts
#28 • 2 Y
Y by Adventure10, Mango247
That's what I thought too.

Except I wasn't sure for "TRIG" part.

No wonder it came out so nice and easy..

Hmmm...
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d343seven
237 posts
#29 • 2 Y
Y by Adventure10, Mango247
Click to reveal hidden text
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sky9073
452 posts
#30 • 2 Y
Y by Adventure10, Mango247
#11. Click to reveal hidden text

#12. Click to reveal hidden text
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Silverfalcon
5006 posts
#31 • 2 Y
Y by Adventure10, Mango247
To make this mock AMC more useful, I'll now post the keys to the answers that has posted up here.

KEYS

Answers needed:

5,8,9,10,16,17,20,21,22,24,25.


#3, I get 10. Sets that are even:

{88,95,173}
{132,95,173}
{166,95,173}
{88,99,173}
{132,99,173}
{166,99,173}
{88,95,99}
{132,95,99}
{166,95,99}
{88,132,166}.

Which one's right?

And thanks again for the answers. More answers to fill in the spaces are appreciated!
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joml88
6343 posts
#32 • 2 Y
Y by Adventure10, Mango247
Okay here are the formulas I was talking about for that trig problem:

Formulas
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litoxe2718281828
276 posts
#33 • 2 Y
Y by Adventure10, Mango247
Silverfalcon wrote:
#3, I get 10. Sets that are even:

{88,95,173}
{132,95,173}
{166,95,173}
{88,99,173}
{132,99,173}
{166,99,173}
{88,95,99}
{132,95,99}
{166,95,99}
{88,132,166}.

Which one's right?

And thanks again for the answers. More answers to fill in the spaces are appreciated!

Click to reveal hidden text

#5
Click to reveal hidden text

#8

Click to reveal hidden text
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Silverfalcon
5006 posts
#34 • 2 Y
Y by Adventure10, Mango247
Thanks to joml88 for the formula. I think I can figure out this.

See if this is right.

\[\frac {\sin 10 + \sin 20}{\cos 10 + \cos 20}\\
= \frac {2 \cdot \sin \frac {(10+20)}{2} \cdot \cos \frac {(10-20)}{2}}{2 \cdot \cos \frac {(10+20)}{2} \cdot \cos \frac {(10-20)}{2}}\\
= \frac {\sin 15}{\cos 15} = \tan 15\]

\[\text {So, our final answer is D.}\]
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Silverfalcon
5006 posts
#35 • 2 Y
Y by Adventure10, Mango247
The question about the expansion:

Find the units digit of:

\[\displaystyle (15 + \sqrt {220})^{19} + (15 + \sqrt {220})^{82} \]

I noticed that this question is extremely hard since it was #30 on one of old AHSMEs on 80's but how do you do it?

Do you expand it? Use conjugates? Would this help?

\[(15+\sqrt {220})^{19} \cdot [1+(15+\sqrt {220})^{63}]\]

Applying binomial theorem is the key to this problem but in what way?

P.S. Rereading all these problems again, I noticed why there aren't many replies
> This is a set of DIFFICULT PROBLEMS!
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zabelman
1072 posts
#36 • 2 Y
Y by Adventure10, Mango247
Thanks for this great mock test! Not having the time to look at the whole thing, I worked only on the ones we needed answers for. Of course I was triple tasking when I did them, so my sols might be flawed. Let me know if I made any mistakes, or if there are problems I missed! And I think you should recheck problem 21.

9
Click to reveal hidden text
E

10
Click to reveal hidden text

16
Click to reveal hidden text

17
Click to reveal hidden text

20
Click to reveal hidden text

21
I think there's an error in this problem.

22
Click to reveal hidden text

24
Click to reveal hidden text

25
Click to reveal hidden text
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zabelman
1072 posts
#37 • 2 Y
Y by Adventure10, Mango247
Silverfalcon wrote:
This is a set of DIFFICULT PROBLEMS!

Indeed! I say this is a little harder than the hardest AMC would be. But it's still very good review! (Luckily, I've seen one or two of these problems before :D )
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litoxe2718281828
276 posts
#38 • 2 Y
Y by Adventure10, Mango247
Many thanks to Silverfalcon for this mock AMC!

Also, zabelman, your #17 solution is irresistably good. Puts me to shame. :blush:
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zabelman
1072 posts
#39 • 2 Y
Y by Adventure10, Mango247
litoxe2718281828 wrote:
Also, zabelman, your #17 solution is irresistably good.
Thank you! It just goes to show that simple conjugates can go a long way! :P

litoxe2718281828 wrote:
Puts me to shame. :blush:
Why should it do this? There's no shame in struggling with a difficult problem. That's what problem solving's all about! And indeed, that problem is difficult.
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Silverfalcon
5006 posts
#40 • 2 Y
Y by Adventure10, Mango247
Wow, new solution!

:) Thanks for the great replies!
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