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k a April Highlights and 2025 AoPS Online Class Information
jlacosta   0
Apr 2, 2025
Spring is in full swing and summer is right around the corner, what are your plans? At AoPS Online our schedule has new classes starting now through July, so be sure to keep your skills sharp and be prepared for the Fall school year! Check out the schedule of upcoming classes below.

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[*]April 22nd (Webinar), 4:00pm PT/7:00pm ET, Competitive Programming at AoPS (USACO).[/list]
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0 replies
jlacosta
Apr 2, 2025
0 replies
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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
J
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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
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Yet another domino problem
juckter   14
N 3 minutes ago by math-olympiad-clown
Source: EGMO 2019 Problem 2
Let $n$ be a positive integer. Dominoes are placed on a $2n \times 2n$ board in such a way that every cell of the board is adjacent to exactly one cell covered by a domino. For each $n$, determine the largest number of dominoes that can be placed in this way.
(A domino is a tile of size $2 \times 1$ or $1 \times 2$. Dominoes are placed on the board in such a way that each domino covers exactly two cells of the board, and dominoes do not overlap. Two cells are said to be adjacent if they are different and share a common side.)
14 replies
juckter
Apr 9, 2019
math-olympiad-clown
3 minutes ago
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BMO 2024 SL A3
MuradSafarli   6
N 5 minutes ago by quacksaysduck

A3.
Find all triples \((a, b, c)\) of positive real numbers that satisfy the system:
\[ \begin{aligned} 11bc - 36b - 15c &= abc \\ 12ca - 10c - 28a &= abc \\ 13ab - 21a - 6b &= abc. \end{aligned} \]
6 replies
MuradSafarli
Apr 27, 2025
quacksaysduck
5 minutes ago
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BMO 2024 SL A1
MuradSafarli   8
N an hour ago by ja.
A1.

Let \( u, v, w \) be positive reals. Prove that there is a cyclic permutation \( (x, y, z) \) of \( (u, v, w) \) such that the inequality:

\[ \frac{a}{xa + yb + zc} + \frac{b}{xb + yc + za} + \frac{c}{xc + ya + zb} \geq \frac{3}{x + y + z} \]
holds for all positive real numbers \( a, b \) and \( c \).
8 replies
MuradSafarli
Apr 27, 2025
ja.
an hour ago
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Medium geometry with AH diameter circle
v_Enhance   94
N an hour ago by alexanderchew
Source: USA TSTST 2016 Problem 2, by Evan Chen
Let $ABC$ be a scalene triangle with orthocenter $H$ and circumcenter $O$. Denote by $M$, $N$ the midpoints of $\overline{AH}$, $\overline{BC}$. Suppose the circle $\gamma$ with diameter $\overline{AH}$ meets the circumcircle of $ABC$ at $G \neq A$, and meets line $AN$ at a point $Q \neq A$. The tangent to $\gamma$ at $G$ meets line $OM$ at $P$. Show that the circumcircles of $\triangle GNQ$ and $\triangle MBC$ intersect at a point $T$ on $\overline{PN}$.

Proposed by Evan Chen
94 replies
1 viewing
v_Enhance
Jun 28, 2016
alexanderchew
an hour ago
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problem interesting
Cobedangiu   7
N an hour ago by vincentwant
Let $a=3k^2+3k+1 (a,k \in N)$
$i)$ Prove that: $a^2$ is the sum of $3$ square numbers
$ii)$ Let $b \vdots a$ and $b$ is the sum of $3$ square numbers. Prove that: $b^n$ is the sum of $3$ square numbers
7 replies
1 viewing
Cobedangiu
Yesterday at 5:06 AM
vincentwant
an hour ago
J
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another problem
kjhgyuio   1
N 2 hours ago by lpieleanu
........
1 reply
kjhgyuio
2 hours ago
lpieleanu
2 hours ago
J
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2^x+3^x = yx^2
truongphatt2668   9
N 2 hours ago by Jackson0423
Prove that the following equation has infinite integer solutions:
$$2^x+3^x = yx^2$$
9 replies
truongphatt2668
Apr 22, 2025
Jackson0423
2 hours ago
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Show that XD and AM meet on Gamma
MathStudent2002   92
N 2 hours ago by Ilikeminecraft
Source: IMO Shortlist 2016, Geometry 2
Let $ABC$ be a triangle with circumcircle $\Gamma$ and incenter $I$ and let $M$ be the midpoint of $\overline{BC}$. The points $D$, $E$, $F$ are selected on sides $\overline{BC}$, $\overline{CA}$, $\overline{AB}$ such that $\overline{ID} \perp \overline{BC}$, $\overline{IE}\perp \overline{AI}$, and $\overline{IF}\perp \overline{AI}$. Suppose that the circumcircle of $\triangle AEF$ intersects $\Gamma$ at a point $X$ other than $A$. Prove that lines $XD$ and $AM$ meet on $\Gamma$.

Proposed by Evan Chen, Taiwan
92 replies
MathStudent2002
Jul 19, 2017
Ilikeminecraft
2 hours ago
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IMO 2010 Problem 5
mavropnevma   54
N 2 hours ago by shanelin-sigma
Each of the six boxes $B_1$, $B_2$, $B_3$, $B_4$, $B_5$, $B_6$ initially contains one coin. The following operations are allowed

Type 1) Choose a non-empty box $B_j$, $1\leq j \leq 5$, remove one coin from $B_j$ and add two coins to $B_{j+1}$;

Type 2) Choose a non-empty box $B_k$, $1\leq k \leq 4$, remove one coin from $B_k$ and swap the contents (maybe empty) of the boxes $B_{k+1}$ and $B_{k+2}$.

Determine if there exists a finite sequence of operations of the allowed types, such that the five boxes $B_1$, $B_2$, $B_3$, $B_4$, $B_5$ become empty, while box $B_6$ contains exactly $2010^{2010^{2010}}$ coins.

Proposed by Hans Zantema, Netherlands
54 replies
mavropnevma
Jul 8, 2010
shanelin-sigma
2 hours ago
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3 var inequality
sqing   0
2 hours ago
Source: Own
Let $ a,b,c>0 . $ Prove that
$$ \left(1 +\frac{a}{b}\right)\left(1+\frac{b}{c}\right)\left(1+\frac{c}{a}\right )\geq \frac{8}{3}\left(1+\frac{a+b}{b+c}+ \frac{b+c}{a+b}\right)$$$$ \left(1 +\frac{a^2}{b^2}\right)\left(1+\frac{b^2}{c^2}\right)\left(1+\frac{c^2}{a^2}\right )\geq \frac{8}{3}\left( 1+\frac{a^2+bc}{b^2+ca}+\frac{b^2+ca }{a^2+bc}\right)$$
0 replies
sqing
2 hours ago
0 replies
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IMO ShortList 1998, geometry problem 5
nttu   32
N 3 hours ago by lpieleanu
Source: IMO ShortList 1998, geometry problem 5
Let $ABC$ be a triangle, $H$ its orthocenter, $O$ its circumcenter, and $R$ its circumradius. Let $D$ be the reflection of the point $A$ across the line $BC$, let $E$ be the reflection of the point $B$ across the line $CA$, and let $F$ be the reflection of the point $C$ across the line $AB$. Prove that the points $D$, $E$ and $F$ are collinear if and only if $OH=2R$.
32 replies
1 viewing
nttu
Oct 14, 2004
lpieleanu
3 hours ago
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a_n < b_n for large n
tastymath75025   11
N 3 hours ago by torch
Source: 2017 ELMO Shortlist A1
Let $0<k<\frac{1}{2}$ be a real number and let $a_0, b_0$ be arbitrary real numbers in $(0,1)$. The sequences $(a_n)_{n\ge 0}$ and $(b_n)_{n\ge 0}$ are then defined recursively by

$$a_{n+1} = \dfrac{a_n+1}{2} \text{ and } b_{n+1} = b_n^k$$
for $n\ge 0$. Prove that $a_n<b_n$ for all sufficiently large $n$.

Proposed by Michael Ma
11 replies
tastymath75025
Jul 3, 2017
torch
3 hours ago
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primes,exponentials,factorials
skellyrah   4
N 4 hours ago by aaravdodhia
find all primes p,q such that $$ \frac{p^q+q^p-p-q}{p!-q!} $$is a prime number
4 replies
skellyrah
Yesterday at 6:31 PM
aaravdodhia
4 hours ago
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Special line through antipodal
Phorphyrion   9
N 4 hours ago by ihategeo_1969
Source: 2025 Israel TST Test 1 P2
Triangle $\triangle ABC$ is inscribed in circle $\Omega$. Let $I$ denote its incenter and $I_A$ its $A$-excenter. Let $N$ denote the midpoint of arc $BAC$. Line $NI_A$ meets $\Omega$ a second time at $T$. The perpendicular to $AI$ at $I$ meets sides $AC$ and $AB$ at $E$ and $F$ respectively. The circumcircle of $\triangle BFT$ meets $BI_A$ a second time at $P$, and the circumcircle of $\triangle CET$ meets $CI_A$ a second time at $Q$. Prove that $PQ$ passes through the antipodal to $A$ on $\Omega$.
9 replies
Phorphyrion
Oct 28, 2024
ihategeo_1969
4 hours ago
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HOMC(Vietnam)
buiminhquang   2
N Jan 28, 2016 by nikolapavlovic
Let ABC be an acute-angle triangle with BC > CA. Let O, H and F be the circumcenter, orthocentre and the foot of its altitude CH, respectively. Suppose that the perpendicular to OF at F meet the side CA at P. Prove FHP = BAC.
2 replies
buiminhquang
Jan 28, 2016
nikolapavlovic
Jan 28, 2016
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HOMC(Vietnam)
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Reveal topic tags
geometry
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buiminhquang
97 posts
buiminhquang
#1 h  Jan 28, 2016, 3:03 PM • 1 Y
Y by Adventure10
Let ABC be an acute-angle triangle with BC > CA. Let O, H and F be the circumcenter, orthocentre and the foot of its altitude CH, respectively. Suppose that the perpendicular to OF at F meet the side CA at P. Prove FHP = BAC.
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mjuk
196 posts
mjuk
#2 h  Jan 28, 2016, 5:31 PM • 2 Y
Y by Adventure10, Mango247
Let $CF$ intersect $\odot ABC$ at $T$, and let $PF$ intersect $BT$ at $Q$. From Butterfly theorem we get $FP=FQ$. Also $T$ is the reflection of $H$ over $AB$, so $FH=FT$.
Therefore $\bigtriangleup FHP \cong \bigtriangleup FTQ$.
$\rightarrow \angle FHP=\angle FTQ=\angle CTB=\angle BAC$
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nikolapavlovic
1246 posts
nikolapavlovic
#3 h  Jan 28, 2016, 6:45 PM • 2 Y
Y by Adventure10, Mango247
http://www.artofproblemsolving.com/community/c6h1133p3555
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