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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
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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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IMO Problem 4
iandrei   105
N 4 minutes ago by cj13609517288
Source: IMO ShortList 2003, geometry problem 1
Let $ABCD$ be a cyclic quadrilateral. Let $P$, $Q$, $R$ be the feet of the perpendiculars from $D$ to the lines $BC$, $CA$, $AB$, respectively. Show that $PQ=QR$ if and only if the bisectors of $\angle ABC$ and $\angle ADC$ are concurrent with $AC$.
105 replies
iandrei
Jul 14, 2003
cj13609517288
4 minutes ago
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Inspired by old results
sqing   2
N 4 minutes ago by sqing
Source: Own
Let $ a,b $ be reals such that $ a^2+b^2=2$ . Prove that
$$ (a+b)\left( \frac{1}{a+1} + \frac{1}{b+1}\right)\leq 2 $$$$ (a+b)\left( \frac{a}{b+1} + \frac{b}{a+1}\right)\leq 2 $$
2 replies
1 viewing
sqing
29 minutes ago
sqing
4 minutes ago
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NEPAL TST 2025 DAY 2
Tony_stark0094   5
N 17 minutes ago by iStud
Consider an acute triangle $\Delta ABC$. Let $D$ and $E$ be the feet of the altitudes from $A$ to $BC$ and from $B$ to $AC$ respectively.

Define $D_1$ and $D_2$ as the reflections of $D$ across lines $AB$ and $AC$, respectively. Let $\Gamma$ be the circumcircle of $\Delta AD_1D_2$. Denote by $P$ the second intersection of line $D_1B$ with $\Gamma$, and by $Q$ the intersection of ray $EB$ with $\Gamma$.

If $O$ is the circumcenter of $\Delta ABC$, prove that $O$, $D$, and $Q$ are collinear if and only if quadrilateral $BCQP$ can be inscribed within a circle.

$\textbf{Proposed by Kritesh Dhakal, Nepal.}$
5 replies
Tony_stark0094
Yesterday at 8:40 AM
iStud
17 minutes ago
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prove that any quadrilateral satisfying this inequality is a trapezoid
mqoi_KOLA   2
N 40 minutes ago by mqoi_KOLA
Prove that any quadrilateral satisfying this inequality is a Trapezoid/trapzium $$ |r - p| < q + s < r + p $$where $p,r$ are lengths of parallel sides and $q,s$ are other two sides.
2 replies
1 viewing
mqoi_KOLA
Yesterday at 3:48 AM
mqoi_KOLA
40 minutes ago
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Inspired by KHOMNYO2
sqing   2
N an hour ago by sqing
Source: Own
Let $ a,b>0 $ and $ a^2+b^2=\frac{5}{2}. $ Prove that $$ 2a + 2b + \frac{1}{a} + \frac{1}{b} +\frac{ab}{\sqrt 2}\geq 5\sqrt 2$$$$ a + b +\frac{2}{a} + \frac{2}{b} + ab\geq \frac{5}{4} + \frac{13}{\sqrt 5} $$$$ a + b +\frac{2}{a} + \frac{2}{b} + \frac{ab}{\sqrt 2}\geq \frac{5}{4\sqrt 2} + \frac{13}{\sqrt 5} $$
2 replies
sqing
Mar 28, 2025
sqing
an hour ago
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USAMO 2003 Problem 4
MithsApprentice   71
N an hour ago by LeYohan
Let $ABC$ be a triangle. A circle passing through $A$ and $B$ intersects segments $AC$ and $BC$ at $D$ and $E$, respectively. Lines $AB$ and $DE$ intersect at $F$, while lines $BD$ and $CF$ intersect at $M$. Prove that $MF = MC$ if and only if $MB\cdot MD = MC^2$.
71 replies
MithsApprentice
Sep 27, 2005
LeYohan
an hour ago
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Inspired by giangtruong13
sqing   6
N an hour ago by sqing
Source: Own
Let $ a,b,c,d\geq 0 ,a-b+d=21 $ and $ a+3b+4c=101 $. Prove that
$$ 61\leq a+b+2c+d\leq \frac{265}{3}$$$$- \frac{2121}{2}\leq ab+bc-2cd+da\leq \frac{14045}{12}$$$$\frac{519506-7471\sqrt{7471}}{27}\leq ab+bc-2cd+3da\leq 33620$$
6 replies
sqing
Apr 11, 2025
sqing
an hour ago
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Inspired by Ruji2018252
sqing   4
N an hour ago by sqing
Source: Own
Let $ a,b,c $ be reals such that $ a^2+b^2+c^2-2a-4b-4c=7. $ Prove that
$$ -4\leq 2a+b+2c\leq 20$$$$5-4\sqrt 3\leq a+b+c\leq 5+4\sqrt 3$$$$ 11-4\sqrt {14}\leq a+2b+3c\leq 11+4\sqrt {14}$$
4 replies
sqing
Apr 10, 2025
sqing
an hour ago
J
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Inspired by Abelkonkurransen 2025
sqing   0
an hour ago
Source: Own
Let $ a,b,c $ be real numbers such that $ \frac{a}{bc}+\frac{4b}{ca}+\frac{c}{ab}=24. $ Prove that
$$\frac{1}{a}+\frac{1}{2b}+\frac{1}{ c}\geq -6$$$$\frac{1}{a}+\frac{1}{2b}+\frac{1}{3c}\geq 1-\sqrt{73} $$$$\frac{1}{a}+\frac{1}{4b}+\frac{1}{ c}\geq \frac{3}{2}(1-\sqrt{33} )$$
0 replies
sqing
an hour ago
0 replies
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Problem 16
Nguyenhuyen AG   37
N an hour ago by flower417477
Let $a,b,c >0 $ such that $abc\ge1 $. Prove that
$\frac{1}{a^4+b^3+c^2}+\frac{1}{b^4+c^3+a^2}+\frac{1}{c^4+a^3+b^2}\le 1$
37 replies
1 viewing
Nguyenhuyen AG
May 3, 2010
flower417477
an hour ago
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Sets With a Given Property
oVlad   3
N 2 hours ago by flower417477
Source: Romania TST 2025 Day 1 P4
Determine the sets $S{}$ of positive integers satisfying the following two conditions:
[list=a]
[*]For any positive integers $a, b, c{}$, if $ab + bc + ca{}$ is in $S$, then so are $a + b + c{}$ and $abc$; and
[*]The set $S{}$ contains an integer $N \geqslant 160$ such that $N-2$ is not divisible by $4$.
[/list]
Bogdan Blaga, United Kingdom
3 replies
oVlad
Apr 9, 2025
flower417477
2 hours ago
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Number Theory Chain!
JetFire008   39
N 2 hours ago by whwlqkd
I will post a question and someone has to answer it. Then they have to post a question and someone else will answer it and so on. We can only post questions related to Number Theory and each problem should be more difficult than the previous. Let's start!

Question 1
39 replies
JetFire008
Apr 7, 2025
whwlqkd
2 hours ago
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A cyclic problem
KhuongTrang   2
N 2 hours ago by KhuongTrang
Source: own
Problem. Given non-negative real numbers $a,b,c: ab+bc+ca>0$ then$$\frac{1}{a+kb}+\frac{1}{b+kc}+\frac{1}{c+ka}\le f(k)\cdot\frac{a+b+c}{ab+bc+ca}$$where $$f(k)=\frac{(k^2-k+1)\left(2k^2+\sqrt{k^2-k+1}+2\sqrt{k^4-k^3+k^2}\right)}{\left(k^2+\sqrt{k^4-k^3+k^2}\right)\left(k^2-k+1+\sqrt{k^4-k^3+k^2}\right)}.$$Also, $k\ge k_{0}\approx 1.874799...$ and $k_{0}$ is largest real root of the equation$$k^8 - 3 k^7 + 10 k^6 - 25 k^5 + 30 k^4 - 25 k^3 + 10 k^2 - 3 k + 1=0.$$k=2
2 replies
KhuongTrang
Sep 5, 2024
KhuongTrang
2 hours ago
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2025 Caucasus MO Seniors P2
BR1F1SZ   2
N 3 hours ago by MathLuis
Source: Caucasus MO
Let $ABC$ be a triangle, and let $B_1$ and $B_2$ be points on segment $AC$ symmetric with respect to the midpoint of $AC$. Let $\gamma_A$ denote the circle passing through $B_1$ and tangent to line $AB$ at $A$. Similarly, let $\gamma_C$ denote the circle passing through $B_1$ and tangent to line $BC$ at $C$. Let the circles $\gamma_A$ and $\gamma_C$ intersect again at point $B'$ ($B' \neq B_1$). Prove that $\angle ABB' = \angle CBB_2$.
2 replies
BR1F1SZ
Mar 26, 2025
MathLuis
3 hours ago
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triangle with prime sidelengths and integer area doesn't exist
parmenides51   3
N Oct 11, 2021 by Al3jandro0000
Source: 1995 Argentina IberoAmerican Training List p29 https://artofproblemsolving.com/community/c2434876_argentina_training_lists__geom
Prove that a triangle with prime sidelengths and integer area doesn't exist.
3 replies
parmenides51
Oct 11, 2021
Al3jandro0000
Oct 11, 2021
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triangle with prime sidelengths and integer area doesn't exist
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Reveal topic tags
geometry
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Source: 1995 Argentina IberoAmerican Training List p29 https://artofproblemsolving.com/community/c2434876_argentina_training_lists__geom
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parmenides51
30630 posts
parmenides51
#1 h  Oct 11, 2021, 7:50 PM
Y by
Prove that a triangle with prime sidelengths and integer area doesn't exist.
This post has been edited 1 time. Last edited by parmenides51, Oct 11, 2021, 7:51 PM
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Arrowandbow426
448 posts
Arrowandbow426
#2 h  Oct 11, 2021, 8:05 PM
Y by
The only way it could have an integer side length is if it has a multiple of 2. Thus 2 is one of the side lengths. The difference between two primes is always greater than 2 (Other than 2,3). (If the difference between to primes was less than two, that would mean that one would be odd and the other even.) But that would ) Since the greatest side length in a triangle can't be less than the sum of the others, 2+x < y. Since The difference between x and y, which are prime, is greater than two. 2+x<(x+k), where k, the difference, is greater than or equal to 2. Simplifying the equation, we get 2<k. This statement is false and so a triangle with prime sidelengths and integer area is impossible.

EDIT: I just realized that 2*2/2 = 2
So somehow this post is wrong.
This post has been edited 3 times. Last edited by Arrowandbow426, Oct 11, 2021, 8:10 PM
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OlympusHero
17020 posts
OlympusHero
#3 h  Oct 11, 2021, 8:31 PM • 1 Y
Y by Aopamy
Solution

bad writeup lol
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Al3jandro0000
804 posts
Al3jandro0000
#4 h  Oct 11, 2021, 10:47 PM
Y by
@above. So you have the triangle has sides $(2,p,p)$ for $p$ prime. Thus taking height $h$ wrt side $2$ we have $h\in\mathbb Z$ and $$h^2+1=p^2$$which is impossible.
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