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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
AIME score for college apps
Happyllamaalways   76
N 7 minutes ago by PatTheKing806
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.
76 replies
+1 w
Happyllamaalways
Mar 13, 2025
PatTheKing806
7 minutes ago
AMC- IMO preparation
asyaela.   14
N 22 minutes ago by NoSignOfTheta
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?
14 replies
+1 w
asyaela.
Yesterday at 7:14 PM
NoSignOfTheta
22 minutes ago
[Registration Open] Mustang Math Tournament 2025
MustangMathTournament   26
N 37 minutes ago by Rice_Farmer
Mustang Math is excited to announce that registration for our annual tournament, MMT 2025, is open! This year, we are bringing our tournament to 9 in-person locations, as well as online!

Locations include: Colorado, Norcal, Socal, Georgia, Illinois, Massachusetts, New Jersey, Nevada, Washington, and online. For registration and more information, check out https://mustangmath.com/competitions/mmt-2025.

MMT 2025 is a math tournament run by a group of 150+ mathematically experienced high school and college students who are dedicated to providing a high-quality and enjoyable contest for middle school students. Our tournament centers around teamwork and collaboration, incentivizing students to work with their teams not only to navigate the challenging and interesting problems of the tournament but also to develop strategies to master the unique rounds. This includes a logic puzzle round, a strategy-filled hexes round, a race-like gallop round, and our trademark ‘Mystery Mare’ round!

Awards:
[list]
[*] Medals for the top teams
[*] Shirts, pins, stickers and certificates for all participants
[*] Additional awards provided by our wonderful sponsors!
[/list]

We are also holding a free MMT prep seminar from 3/15-3/16 to help students prepare for the upcoming tournament. Join the Google Classroom! https://classroom.google.com/c/NzQ5NDUyNDY2NjM1?cjc=7sogth4
26 replies
MustangMathTournament
Mar 8, 2025
Rice_Farmer
37 minutes ago
2^a + 3^b + 1 = 6^c
togrulhamidli2011   2
N 41 minutes ago by RedFireTruck
Find all positive integers (a, b, c) such that:

\[
2^a + 3^b + 1 = 6^c
\]
2 replies
togrulhamidli2011
Yesterday at 12:34 PM
RedFireTruck
41 minutes ago
Bosnia and Herzegovina JBMO TST 2013 Problem 1
gobathegreat   4
N an hour ago by RedFireTruck
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
4 replies
gobathegreat
Sep 16, 2018
RedFireTruck
an hour ago
AMC 8 discussion
Jaxman8   45
N an hour ago by Jello0211
Discuss the AMC 8 below!
45 replies
Jaxman8
Jan 29, 2025
Jello0211
an hour ago
chat gpt
fuv870   8
N an hour ago by jkim0656
The chat gpt alreadly knows how to solve the problem of IMO USAMO and AMC?
8 replies
fuv870
Yesterday at 9:51 PM
jkim0656
an hour ago
IMO 2012 P5
mathmdmb   122
N an hour ago by KevinYang2.71
Source: IMO 2012 P5
Let $ABC$ be a triangle with $\angle BCA=90^{\circ}$, and let $D$ be the foot of the altitude from $C$. Let $X$ be a point in the interior of the segment $CD$. Let $K$ be the point on the segment $AX$ such that $BK=BC$. Similarly, let $L$ be the point on the segment $BX$ such that $AL=AC$. Let $M$ be the point of intersection of $AL$ and $BK$.

Show that $MK=ML$.

Proposed by Josef Tkadlec, Czech Republic
122 replies
mathmdmb
Jul 11, 2012
KevinYang2.71
an hour ago
Variable point on the median
MarkBcc168   47
N an hour ago by HamstPan38825
Source: APMO 2019 P3
Let $ABC$ be a scalene triangle with circumcircle $\Gamma$. Let $M$ be the midpoint of $BC$. A variable point $P$ is selected in the line segment $AM$. The circumcircles of triangles $BPM$ and $CPM$ intersect $\Gamma$ again at points $D$ and $E$, respectively. The lines $DP$ and $EP$ intersect (a second time) the circumcircles to triangles $CPM$ and $BPM$ at $X$ and $Y$, respectively. Prove that as $P$ varies, the circumcircle of $\triangle AXY$ passes through a fixed point $T$ distinct from $A$.
47 replies
MarkBcc168
Jun 11, 2019
HamstPan38825
an hour ago
Can this sequence be bounded?
darij grinberg   66
N an hour ago by shendrew7
Source: German pre-TST 2005, problem 4, ISL 2004, algebra problem 2
Let $a_0$, $a_1$, $a_2$, ... be an infinite sequence of real numbers satisfying the equation $a_n=\left|a_{n+1}-a_{n+2}\right|$ for all $n\geq 0$, where $a_0$ and $a_1$ are two different positive reals.

Can this sequence $a_0$, $a_1$, $a_2$, ... be bounded?

Proposed by Mihai Bălună, Romania
66 replies
darij grinberg
Jan 19, 2005
shendrew7
an hour ago
Sequences and limit
lehungvietbao   14
N 2 hours ago by eg4334
Source: Vietnam Mathematical OLympiad 2014
Let $({{x}_{n}}),({{y}_{n}})$ be two positive sequences defined by ${{x}_{1}}=1,{{y}_{1}}=\sqrt{3}$ and
\[ \begin{cases}  {{x}_{n+1}}{{y}_{n+1}}-{{x}_{n}}=0 \\   x_{n+1}^{2}+{{y}_{n}}=2 \end{cases} \] for all $n=1,2,3,\ldots$.
Prove that they are converges and find their limits.
14 replies
lehungvietbao
Jan 3, 2014
eg4334
2 hours ago
IMO 2014 Problem 1
Amir Hossein   131
N 2 hours ago by eg4334
Let $a_0 < a_1 < a_2 < \dots$ be an infinite sequence of positive integers. Prove that there exists a unique integer $n\geq 1$ such that
\[a_n < \frac{a_0+a_1+a_2+\cdots+a_n}{n} \leq a_{n+1}.\]
Proposed by Gerhard Wöginger, Austria.
131 replies
Amir Hossein
Jul 8, 2014
eg4334
2 hours ago
One secuence satisfying condition
hatchguy   8
N 2 hours ago by jaescl
Prove that there exists only one infinite secuence of positive integers $a_1,a_2,...$ with $a_1=1$, $a_2>1$ and $a_{n+1}^3 + 1 = a_na_{n+2}$ for all positive integers $n$.
8 replies
1 viewing
hatchguy
Sep 4, 2011
jaescl
2 hours ago
Unsolved Diophantine(I think)
Nuran2010   2
N 3 hours ago by ohiorizzler1434
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.)
2 replies
Nuran2010
Mar 14, 2025
ohiorizzler1434
3 hours ago
k I was not happy while doing this problem
Awesomeness_in_a_bun   7
N Mar 20, 2024 by asdf334
Source: 2024 USAJMO problem 2
Let $m$ and $n$ be positive integers. Let $S$ be the set of integer points $(x, y)$ with $1 \leq x \leq 2m$ and $1 \leq y \leq 2n$. A configuration of $mn$ rectangles is called $happy$ if each point in $S$ is a vertex of exactly one rectangle, and all rectangles have sides parallel to the coordinate axes. Prove that the number of happy configurations is odd.
7 replies
Awesomeness_in_a_bun
Mar 20, 2024
asdf334
Mar 20, 2024
I was not happy while doing this problem
G H J
G H BBookmark kLocked kLocked NReply
Source: 2024 USAJMO problem 2
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Awesomeness_in_a_bun
472 posts
#1
Y by
Let $m$ and $n$ be positive integers. Let $S$ be the set of integer points $(x, y)$ with $1 \leq x \leq 2m$ and $1 \leq y \leq 2n$. A configuration of $mn$ rectangles is called $happy$ if each point in $S$ is a vertex of exactly one rectangle, and all rectangles have sides parallel to the coordinate axes. Prove that the number of happy configurations is odd.
Z Y
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pi_is_3.14
1437 posts
#2
Y by
The $2\times2$ case is trivial. Now, we induct proving that $f(2m + 2, 2n)$ is odd if $f(2m, 2n)$ is odd.

There are 2 cases we consider for the rectangles containing the points in the leftmost column.

Case 1: All points in the rectangles formed with points in the leftmost column lie in the center column ($m + 2$th column)


[asy]
unitsize(0.5 cm);

dot((0,0), purple);
dot((1,0), black);
dot((2,0), black);
dot((3,0), purple);
dot((4,0), black);
dot((5,0), black);

dot((0,1), red);
dot((1,1), black);
dot((2,1), black);
dot((3,1), red);
dot((4,1), black);
dot((5,1), black);

dot((0,2), orange);
dot((1,2), black);
dot((2,2), black);
dot((3,2), orange);
dot((4,2), black);
dot((5,2), black);

dot((0,3), red);
dot((1,3), black);
dot((2,3), black);
dot((3,3), red);
dot((4,3), black);
dot((5,3), black);

dot((0,4), orange);
dot((1,4), black);
dot((2,4), black);
dot((3,4), orange);
dot((4,4), black);
dot((5,4), black);

dot((0,5), purple);
dot((1,5), black);
dot((2,5), black);
dot((3,5), purple);
dot((4,5), black);
dot((5,5),black);
[/asy]


There are $$\frac{\binom{2n}{2} \cdot \binom{2n - 2}{2} \dots \binom{2}{2}}{n!}$$ways to pair up the 2n points. From here, there are $f(2m,2n)$ to partition the remaining points into rectangles. Both of these quanitities are odd so total ways in this case are odd.


Case 2: All other points do not lie in the center column

[asy]
unitsize(0.5 cm);

dot((0,0), purple);
dot((1,0), black);
dot((2,0), purple);
dot((3,0), black);
dot((4,0), black);
dot((5,0), black);

dot((0,1), red);
dot((1,1), black);
dot((2,1), black);
dot((3,1), black);
dot((4,1), red);
dot((5,1), black);

dot((0,2), orange);
dot((1,2), black);
dot((2,2), black);
dot((3,2), orange);
dot((4,2), black);
dot((5,2), black);

dot((0,3), red);
dot((1,3), black);
dot((2,3), black);
dot((3,3), black);
dot((4,3), red);
dot((5,3), black);

dot((0,4), orange);
dot((1,4), black);
dot((2,4), black);
dot((3,4), orange);
dot((4,4), black);
dot((5,4), black);

dot((0,5), purple);
dot((1,5), black);
dot((2,5), purple);
dot((3,5), black);
dot((4,5), black);
dot((5,5), black);
[/asy]


We reflect over the points over the $m + 2$th column.


[asy]
unitsize(0.5 cm);

dot((0,0), purple);
dot((1,0), black);
dot((2,0), black);
dot((3,0), black);
dot((4,0), purple);
dot((5,0), black);

dot((0,1), red);
dot((1,1), black);
dot((2,1), red);
dot((3,1), black);
dot((4,1), black);
dot((5,1), black);

dot((0,2), orange);
dot((1,2), black);
dot((2,2), black);
dot((3,2), orange);
dot((4,2), black);
dot((5,2), black);

dot((0,3), red);
dot((1,3), black);
dot((2,3), red);
dot((3,3), black);
dot((4,3), black);
dot((5,3), black);

dot((0,4), orange);
dot((1,4), black);
dot((2,4), black);
dot((3,4), orange);
dot((4,4), black);
dot((5,4), black);

dot((0,5), purple);
dot((1,5), black);
dot((2,5), black);
dot((3,5), black);
dot((4,5), purple);
dot((5,5), black);
[/asy]

Note that the number of ways to partition the remaining points into rectangles before and after the reflection is the same. This implies the number of ways in this case is even.

In total, there are an odd number of ways.
This post has been edited 1 time. Last edited by pi_is_3.14, Mar 20, 2024, 6:18 AM
Z Y
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shendrew7
788 posts
#3
Y by
So not true
This post has been edited 1 time. Last edited by shendrew7, Mar 20, 2024, 4:12 AM
Z Y
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awesomeguy856
7255 posts
#4
Y by
swap bottom two rows
then even # of cases are removed if transformed into distinct config

otherwise, the config has bottom two rows disjoint from rest
so trivial by induction
This post has been edited 1 time. Last edited by awesomeguy856, Mar 20, 2024, 4:07 AM
Z Y
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awesomeguy856
7255 posts
#5
Y by
shendrew7 wrote:
Perform a reflection about the center of the grid (that is, the point $(m+.5, n+.5)$). Only one configuration is mapped to itself - all rectangles have their center at the center of the grid, and each point in the first quadrant corresponds to 3 unique points in the other 3 quadrants.

All other configurations can then be paired with its unique image, making the total odd. $\blacksquare$

i don't think this is true?
consider all rectangles being as small as possible
then it is reflective over the center
Z Y
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KevinYang2.71
393 posts
#6 • 2 Y
Y by megarnie, deduck
this problem was already posted here
Z Y
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A_MatheMagician
2251 posts
#7 • 1 Y
Y by GrantStar
so i did the swap solution
the are an odd number configurations that swapping two rows do not change
and even number that do

using induction odd*odd=odd so it is always odd
Z Y
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asdf334
7578 posts
#8
Y by
courtesy of circleinvert

Swap each group of adjacent rows and columns, yielding $2^{m+n}$ not-necessarily-distinct configurations. The only class of configurations with odd size is the one with $mn$ small squares. $\blacksquare$
This post has been edited 1 time. Last edited by asdf334, Mar 20, 2024, 3:01 PM
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