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k a May Highlights and 2025 AoPS Online Class Information
jlacosta   0
May 1, 2025
May is an exciting month! National MATHCOUNTS is the second week of May in Washington D.C. and our Founder, Richard Rusczyk will be presenting a seminar, Preparing Strong Math Students for College and Careers, on May 11th.

Are you interested in working towards MATHCOUNTS and don’t know where to start? We have you covered! If you have taken Prealgebra, then you are ready for MATHCOUNTS/AMC 8 Basics. Already aiming for State or National MATHCOUNTS and harder AMC 8 problems? Then our MATHCOUNTS/AMC 8 Advanced course is for you.

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[*]May 21st, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 2 Math Jam, Problems 5 and 6, Canada/USA Mathcamp staff will discuss solutions to Problems 5 and 6 of the 2025 Mathcamp Qualifying Quiz![/list]
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0 replies
jlacosta
May 1, 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
∑(a-b)(a-c)/(2a^2 + (b+c)^2) >= 0
Zhero   24
N a few seconds ago by RevolveWithMe101
Source: ELMO Shortlist 2010, A2
Let $a,b,c$ be positive reals. Prove that
\[ \frac{(a-b)(a-c)}{2a^2 + (b+c)^2} + \frac{(b-c)(b-a)}{2b^2 + (c+a)^2} + \frac{(c-a)(c-b)}{2c^2 + (a+b)^2} \geq 0. \]

Calvin Deng.
24 replies
Zhero
Jul 5, 2012
RevolveWithMe101
a few seconds ago
i am not abel to prove or disprove
frost23   8
N 2 minutes ago by frost23
Source: made on my own
Let $a_1a_1a_2a_2.............a_na_n$ be a perfect square then, is it true that it must be of the form
$10^{2(n-2)}\cdot7744$
8 replies
frost23
2 hours ago
frost23
2 minutes ago
points on sides of a triangle, intersections, extensions, ratio of areas wanted
parmenides51   1
N 11 minutes ago by FrancoGiosefAG
Source: Mexican Mathematical Olympiad 1997 OMM P5
Let $P,Q,R$ be points on the sides $BC,CA,AB$ respectively of a triangle $ABC$. Suppose that $BQ$ and $CR$ meet at $A', AP$ and $CR$ meet at $B'$, and $AP$ and $BQ$ meet at $C'$, such that $AB' = B'C', BC' =C'A'$, and $CA'= A'B'$. Compute the ratio of the area of $\triangle PQR$ to the area of $\triangle ABC$.
1 reply
parmenides51
Jul 28, 2018
FrancoGiosefAG
11 minutes ago
starting with intersecting circles, line passes through midpoint wanted
parmenides51   2
N 30 minutes ago by EmersonSoriano
Source: Peru Ibero TST 2014
Circles $C_1$ and $C_2$ intersect at different points $A$ and $B$. The straight lines tangents to $C_1$ that pass through $A$ and $B$ intersect at $T$. Let $M$ be a point on $C_1$ that is out of $C_2$. The $MT$ line intersects $C_1$ at $C$ again, the $MA$ line intersects again to $C_2$ in $K$ and the line $AC$ intersects again to the circumference $C_2$ in $L$. Prove that the $MC$ line passes through the midpoint of the $KL$ segment.
2 replies
parmenides51
Jul 23, 2019
EmersonSoriano
30 minutes ago
An inequality
Rushil   14
N 30 minutes ago by frost23
Source: Indian RMO 1994 Problem 8
If $a,b,c$ are positive real numbers such that $a+b+c = 1$, prove that \[ (1+a)(1+b)(1+c) \geq 8 (1-a)(1-b)(1-c) . \]
14 replies
Rushil
Oct 25, 2005
frost23
30 minutes ago
3 var inequality
SunnyEvan   6
N 30 minutes ago by JARP091
Let $ a,b,c \in R $ ,such that $ a^2+b^2+c^2=4(ab+bc+ca)$Prove that :$$ \frac{7-2\sqrt{14}}{48} \leq \frac{a^3b+b^3c+c^3a}{(a^2+b^2+c^2)^2} \leq \frac{7+2\sqrt{14}}{48} $$
6 replies
SunnyEvan
May 17, 2025
JARP091
30 minutes ago
collinearity as a result of perpendicularity and equality
parmenides51   2
N an hour ago by FrancoGiosefAG
Source: Mexican Mathematical Olympiad 1996 OMM P6
In a triangle $ABC$ with $AB < BC < AC$, points $A' ,B' ,C'$ are such that $AA' \perp BC$ and $AA' = BC, BB' \perp  CA$ and $BB'=CA$, and $CC' \perp AB$ and $CC'= AB$, as shown on the picture. Suppose that $\angle AC'B$ is a right angle. Prove that the points $A',B' ,C' $ are collinear.
2 replies
parmenides51
Jul 28, 2018
FrancoGiosefAG
an hour ago
3 var inequality
JARP091   6
N an hour ago by JARP091
Source: Own
Let \( x, y, z \in \mathbb{R}^+ \). Prove that
\[
\sum_{\text{cyc}} \frac{x^3}{y^2 + z^2} \geq \frac{x + y + z}{2}
\]without using the Rearrangement Inequality or Chebyshev's Inequality.
6 replies
JARP091
Today at 8:54 AM
JARP091
an hour ago
Helplooo
Bet667   1
N an hour ago by Lil_flip38
Let $ABC$ be an acute angled triangle.And altitudes $AD$ and $BE$ intersects at point $H$.Let $F$ be a point on ray $AD$ such that $DH=DF$.Circumcircle of $AEF$ intersects line $BC$ at $K$ and $L$ so prove that $BK=BL$
1 reply
Bet667
2 hours ago
Lil_flip38
an hour ago
Cyclic sum of 1/(a+1/b+1)
v_Enhance   22
N an hour ago by Rayvhs
Source: ELMO Shortlist 2013: Problem A2, by David Stoner
Prove that for all positive reals $a,b,c$,
\[\frac{1}{a+\frac{1}{b}+1}+\frac{1}{b+\frac{1}{c}+1}+\frac{1}{c+\frac{1}{a}+1}\ge \frac{3}{\sqrt[3]{abc}+\frac{1}{\sqrt[3]{abc}}+1}. \]Proposed by David Stoner
22 replies
v_Enhance
Jul 23, 2013
Rayvhs
an hour ago
xf(x + xy) = xf(x) + f(x^2)f(y)
orl   14
N an hour ago by jasperE3
Source: MEMO 2008, Team, Problem 5
Determine all functions $ f: \mathbb{R} \mapsto \mathbb{R}$ such that
\[ x f(x + xy) = x f(x) + f \left( x^2 \right) f(y) \quad  \forall  x,y \in \mathbb{R}.\]
14 replies
orl
Sep 10, 2008
jasperE3
an hour ago
Beautiful Number Theory
tastymath75025   34
N 2 hours ago by Adywastaken
Source: 2022 ISL N8
Prove that $5^n-3^n$ is not divisible by $2^n+65$ for any positive integer $n$.
34 replies
tastymath75025
Jul 9, 2023
Adywastaken
2 hours ago
Hard Functional Equation in the Complex Numbers
yaybanana   1
N 2 hours ago by jasperE3
Source: Own
Find all functions $f:\mathbb {C}\rightarrow \mathbb {C}$, s.t :

$f(xf(y)) + f(x^2+y) = f(x+y)x + f(f(y))$

for all $x,y \in \mathbb{C}$
1 reply
yaybanana
Apr 9, 2025
jasperE3
2 hours ago
Find all numbers
Rushil   11
N 2 hours ago by frost23
Source: Indian RMO 1994 Problem 3
Find all 6-digit numbers $a_1a_2a_3a_4a_5a_6$ formed by using the digits $1,2,3,4,5,6$ once each such that the number $a_1a_2a_2\ldots a_k$ is divisible by $k$ for $1 \leq k \leq 6$.
11 replies
Rushil
Oct 25, 2005
frost23
2 hours ago
Tiling rectangle with smaller rectangles.
MarkBcc168   61
N Apr 23, 2025 by YaoAOPS
Source: IMO Shortlist 2017 C1
A rectangle $\mathcal{R}$ with odd integer side lengths is divided into small rectangles with integer side lengths. Prove that there is at least one among the small rectangles whose distances from the four sides of $\mathcal{R}$ are either all odd or all even.

Proposed by Jeck Lim, Singapore
61 replies
MarkBcc168
Jul 10, 2018
YaoAOPS
Apr 23, 2025
Tiling rectangle with smaller rectangles.
G H J
G H BBookmark kLocked kLocked NReply
Source: IMO Shortlist 2017 C1
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MarkBcc168
1595 posts
#1 • 13 Y
Y by Amir Hossein, Davi-8191, rustam, Kayak, Tenee, Purple_Planet, centslordm, jhu08, megarnie, Adventure10, Mango247, lian_the_noob12, Sedro
A rectangle $\mathcal{R}$ with odd integer side lengths is divided into small rectangles with integer side lengths. Prove that there is at least one among the small rectangles whose distances from the four sides of $\mathcal{R}$ are either all odd or all even.

Proposed by Jeck Lim, Singapore
This post has been edited 2 times. Last edited by MarkBcc168, Jul 15, 2018, 12:57 PM
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MarkBcc168
1595 posts
#2 • 31 Y
Y by Amir Hossein, Wizard_32, ring_r, hocashi, vsathiam, vivoloh, jj_ca888, Pluto1708, Tenee, rashah76, Purple_Planet, Lord_Eldo_Santos, like123, Wizard0001, centslordm, mijail, jhu08, megarnie, Muaaz.SY, PRMOisTheHardestExam, hakN, sabkx, Mathlover_1, two_steps, Adventure10, Mango247, ihatemath123, Newmaths, kiyoras_2001, Sedro, Funcshun840
Solution
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RMO-prep
82 posts
#3 • 5 Y
Y by centslordm, jhu08, sabkx, Adventure10, Mango247
It was also India TST
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juckter
323 posts
#4 • 4 Y
Y by centslordm, jhu08, Adventure10, Mango247
Let $\mathcal{R}$'s dimensions be $m$ and $n$, with $m$ and $n$ odd. Label the square $(x,y)$ with $A$ if $x \equiv y \equiv 1 \bmod{2}$, $B$ if $x \equiv y \equiv 0 \bmod{2}$, and $C$ if $x \not \equiv y \bmod{2}$. Then a rectangle satisfies the problem's condition iff all of its corners are labeled with $A$ or all of them are labeled with $B$. Notice that because $m$ and $n$ are odd, there exists one less square labeled $C$ than the total number of squares labeled $A$ of $B$, and thus there must exist at least one small rectangle in which more squares are labeled $A$ or $B$. But this is easily seen to imply that all four corners of the rectangle are labeled $A$ or $B$.
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orthocentre
72 posts
#6 • 4 Y
Y by centslordm, jhu08, Adventure10, Mango247
Edit: made a really dumb mistake.
This post has been edited 1 time. Last edited by orthocentre, Apr 7, 2019, 9:49 PM
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joyce_tan
94 posts
#7 • 4 Y
Y by centslordm, jhu08, sabkx, Adventure10
jdevine wrote:
There's a much easier way to do this than with colouring arguments.

Click to reveal hidden text
There's a reason why everyone else is using coloring arguments - your solution proves a weaker statement than the problem. You've proven that a rectangle with both odd dimensions exists, and an odd dimension ensures that the opposite distances have equal parity, but you haven't shown that the parity of the distances in the two different axes are equal, i.e. you could have a rectangle that is an odd distance from the top and the bottom, but an even distance from the left and the right.
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mofumofu
179 posts
#8 • 15 Y
Y by Ankoganit, 62861, DVDthe1st, Kayak, oneplusone, rkm0959, AnArtist, AlastorMoody, skt, rashah76, centslordm, jhu08, megarnie, sabkx, Adventure10
This problem was proposed by oneplusone. :)
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test20
988 posts
#9 • 4 Y
Y by joyce_tan, centslordm, jhu08, Adventure10
jdevine wrote:
There's a much easier way to do this than with colouring arguments.

Consider the length of $\mathcal{R}$. Since it's odd, we let it be $2n+1$. Consider a smaller rectangle $\mathcal{S}$. Let the distance from the left side of $\mathcal{S}$ to the left side of $\mathcal{R}$ be $a$, and let the distance from the right side of $\mathcal{S}$ to the right side of $\mathcal{R}$ be $b$. Iff $a$ and $b$ are the same parity, then $a+b$ is even and so $2n+1-a-b$ is odd. Repeat in the other direction (up and down instead of left and right).

This means that we want to prove that there exists a rectangle whose sides are of odd length. Suppose, for sake of contradiction, that there isn't one. Then all the small rectangles have even area, but the large rectangle has an odd area. Contradiction.

I'm not sure why they put that on the IMO shortlist... it would have been much more suitable in something like this and would have been a really good problem to put in, probably near the end of the paper.

Naah, that doesn't quite work.

You have only established the existence of a rectangle $S$ whose sides are of odd length.
But $S$ might have odd distances to the northern and southern side of $\cal R$,
and even distances to the eastern and western side of $\cal R$.
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anantmudgal09
1980 posts
#10 • 4 Y
Y by Saikat002, centslordm, jhu08, Adventure10
By induction on chessboard polygons, it is easy to show that each rectangle in such a tiling can be indexed with vertices as lattice points.

Now for each rectangle of area $A$ apply the following operation. For $A \equiv 0 \pmod{2}$ tile it with dominoes and for $A \equiv 1\pmod{2}$ mark the leftmost corner and tile the remaining rectangle with dominoes. Observe that each tiling $\mathcal{T}$ maps to a tiling $\mathcal{T}’$ consisting only of dominoes and boxes such that $\mathcal{T}$ has a desired rectangle iff $\mathcal{T}’$ does.

Apply chessboard colouring with upper left corner coloured red. For $\mathcal{T}’$ notice that any box placed at red square is desired. If not then all such red squares are covered in dominoes. Since no domino contains two monochrome squares and we have more red squares, we get the desired contradiction.
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orthocentre
72 posts
#11 • 4 Y
Y by centslordm, jhu08, Adventure10, Mango247
test20 wrote:
jdevine wrote:
There's a much easier way to do this than with colouring arguments.

Consider the length of $\mathcal{R}$. Since it's odd, we let it be $2n+1$. Consider a smaller rectangle $\mathcal{S}$. Let the distance from the left side of $\mathcal{S}$ to the left side of $\mathcal{R}$ be $a$, and let the distance from the right side of $\mathcal{S}$ to the right side of $\mathcal{R}$ be $b$. Iff $a$ and $b$ are the same parity, then $a+b$ is even and so $2n+1-a-b$ is odd. Repeat in the other direction (up and down instead of left and right).

This means that we want to prove that there exists a rectangle whose sides are of odd length. Suppose, for sake of contradiction, that there isn't one. Then all the small rectangles have even area, but the large rectangle has an odd area. Contradiction.

I'm not sure why they put that on the IMO shortlist... it would have been much more suitable in something like this and would have been a really good problem to put in, probably near the end of the paper.

Naah, that doesn't quite work.

You have only established the existence of a rectangle $S$ whose sides are of odd length.
But $S$ might have odd distances to the northern and southern side of $\cal R$,
and even distances to the eastern and western side of $\cal R$.

Oh... I thought it was way too easy for IMO shortlist... :D
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Math.Is.Beautiful
850 posts
#12 • 4 Y
Y by centslordm, jhu08, Adventure10, Mango247
MarkBcc168 wrote:
A rectangle $\mathcal{R}$ with odd integer side lengths is divided into small rectangles with integer side lengths. Prove that there is at least one among the small rectangles whose distances from the four sides of $\mathcal{R}$ are either all odd or all even.

Don't know, but I feel like I have seen this problem long ago.


EDIT: No, it wasn't this one, it was the following theorem:
Theorem: If a rectangle can tiled with rectangles, each of which have at least one side of integeral length, then the rectangle which was tiled also has at least one side of integeral length.
This post has been edited 1 time. Last edited by Math.Is.Beautiful, Jul 15, 2018, 8:26 AM
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oneplusone
1459 posts
#14 • 27 Y
Y by 62861, CZRorz, DVDthe1st, Kayak, mofumofu, anantmudgal09, Wizard_32, Salman.elourdi, ring_r, AnArtist, Idea_lover, djmathman, rashah76, OlympusHero, Eliot, centslordm, jhu08, PRMOisTheHardestExam, sabkx, ihatemath123, Adventure10, Mango247, RTB, EpicBird08, Sedro, OronSH, Rijul saini
I discovered this problem when I was trying to solve the problem "there exists only finitely many ways to divide a rectangle into smaller rectangles of equal area". A subproblem is that there is no smooth deformation of the smaller rectangles that preserves all their area (turns out there is a completely different elegant proof for this). A slightly stronger statement is that if we perturb each edge slightly up/down or left/right, there is a rectangle that strictly contains its original form, or is strictly contained in its original form. This is equivalent to this C1. I was hoping to find some kind of "Brouwer fixed-pt" solution, but couldn't find one.

My original solution is to colour the rectangle black/white in a checkerboard manner, with the 4 corners black, then it suffices to find a smaller rectangle with 4 black corners. We count the number of black and white corners among all the rectangles. For each corner vertex that is not the corner of $\mathcal{R}$, there are an equal number of black and white rectangle corners at that vertex. So there are more black corners than white, hence there is a smaller rectangle with at least 3 black corners. But a rectangle with 3 black corners necessarily has the 4th corner black.
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yayups
1614 posts
#16 • 4 Y
Y by centslordm, jhu08, Adventure10, Mango247
ugh this took way too long, but I suck at combo anyways....

Call $(i,j)$ white if $2\mid i-j$, and call it black otherwise. We wish to show the existence of a rectangle with all white corners. However, a rectangle has all white corners if and only if it has more white squares than black squares. But since $\mathcal{R}$ has more whites than blacks, by PHP we are done.

Remark: In my "defense", for most of the time I was trying to show that the number of all white cornered squares was odd (this is the usual strategy), but experimentation very quickly reveals this to be FALSE. Therefore, everything I was trying beforehand was bound to fail. But I was too lazy to actually experiment till the very end.
Lesson: experiment

One more remark:
oneplusone wrote:
My original solution is to colour the rectangle black/white in a checkerboard manner, with the 4 corners black, then it suffices to find a smaller rectangle with 4 black corners. We count the number of black and white corners among all the rectangles. For each corner vertex that is not the corner of $\mathcal{R}$, there are an equal number of black and white rectangle corners at that vertex. So there are more black corners than white, hence there is a smaller rectangle with at least 3 black corners. But a rectangle with 3 black corners necessarily has the 4th corner black.
I think this solution needs slight rewording since $1\times x$ squares have at most 2 corners.
This post has been edited 2 times. Last edited by yayups, Aug 21, 2018, 7:44 AM
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DynamoBlaze
170 posts
#17 • 3 Y
Y by centslordm, jhu08, Adventure10
I need some clarification. Are the rectangles in a grid? I mean, does the large rectangle have rows and columns which intersect to make the smaller rectangles, or are the small rectangles independent? In other words, if there is a rectangle $PQRS$, will $QR$ be the side of one other small rectangle as well?
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Durjoy1729
221 posts
#18 • 8 Y
Y by Digonta1729, potentialenergy, DonaldJ.Trump, ike.chen, centslordm, jhu08, lian_the_noob12, Adventure10
Official solution
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