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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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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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Beautiful problem
luutrongphuc   1
N 18 minutes ago by aidenkim119
Let triangle $ABC$ be circumscribed about circle $(I)$, and let $H$ be the orthocenter of $\triangle ABC$. The circle $(I)$ touches line $BC$ at $D$. The tangent to the circle $(BHC)$ at $H$ meets $BC$ at $S$. Let $J$ be the midpoint of $HI$, and let the line $DJ$ meet $(I)$ again at $X$. The tangent to $(I)$ parallel to $BC$ meets the line $AX$ at $T$. Prove that $ST$ is tangent to $(I)$.
1 reply
luutrongphuc
Yesterday at 5:35 AM
aidenkim119
18 minutes ago
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inequality
pennypc123456789   4
N 23 minutes ago by sqing
Let \( x, y \) be positive real numbers satisfying \( x + y = 2 \). Prove that

\[ 3(x^{\frac{2}{3}} + y^{\frac{2}{3}}) \geq 4 + 2x^{\frac{1}{3}}y^{\frac{1}{3}}. \]
4 replies
1 viewing
pennypc123456789
Mar 24, 2025
sqing
23 minutes ago
J
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Convex lattice polygon
Oksutok   2
N 36 minutes ago by Oksutok
Let $f(n)$ be the maximal number of the vertices of a convex lattice polygon with exactly $n$ lattice points in the interior. Show that:
a) $f(n) \le 2n$ for $n \ge 3$
b)$f(n)<Cn^{1/3}$ for some constant $C \in \mathbb{R}_{>0}$.
2 replies
Oksutok
Sep 29, 2024
Oksutok
36 minutes ago
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inquequality
ngocthi0101   11
N 36 minutes ago by sqing
let $a,b,c > 0$ prove that
$\frac{a}{b} + \sqrt {\frac{b}{c}} + \sqrt[3]{{\frac{c}{a}}} > \frac{5}{2}$
11 replies
ngocthi0101
Sep 26, 2014
sqing
36 minutes ago
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Uhhhhhhhhhh
sealight2107   2
N 38 minutes ago by Primeniyazidayi
Let $x,y,z$ be reals such that $0<x,y,z<\frac{1}{2}$ and $x+y+z=1$.Prove that:
$4(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}) - \frac{1}{xyz} >8$
2 replies
sealight2107
5 hours ago
Primeniyazidayi
38 minutes ago
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Problem 4
blug   3
N an hour ago by math90
Source: Polish Math Olympiad 2025 Finals P4
A positive integer $n\geq 2$ and a set $S$ consisting of $2n$ disting positive integers smaller than $n^2$ are given. Prove that there exists a positive integer $r\in \{1, 2, ..., n\}$ that can be written in the form $r=a-b$, for $a, b\in \mathbb{S}$ in at least $3$ different ways.
3 replies
blug
Yesterday at 11:59 AM
math90
an hour ago
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Inspired by pennypc123456789
sqing   1
N an hour ago by sqing
Source: Own
Let $x, y$ be real numbers such that $|x| , |y| \le 1$. Prove that
$$ 2 \le (x^2 + 1)(y^2 + 1) + 4(x - 1)(y - 1) \le 20$$Let $x, y$ be real numbers such that $|x+y| \le 1$. Prove that
$$ 2 \le (x^2 + 1)(y^2 + 1) + 4(x - 1)(y - 1) \le \frac{169}{16}$$
1 reply
sqing
an hour ago
sqing
an hour ago
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PoP+Parallel
Solilin   4
N an hour ago by aidenkim119
Source: Titu Andreescu, Lemmas in Olympiad Geometry
Let ABC be a triangle and let D, E, F be the feet of the altitudes, with D on BC, E on CA, and F on AB. Let the parallel through D to EF meet AB at X and AC at Y. Let T be the intersection of EF with BC and let M be the midpoint of side BC. Prove that the points T, M, X, Y are concyclic.
4 replies
Solilin
Today at 6:10 AM
aidenkim119
an hour ago
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square root problem that involves geometry
kjhgyuio   6
N an hour ago by Primeniyazidayi
If x is a nonnegative real number , find the minimum value of √x^2+4 + √x^2 -24x +153

6 replies
kjhgyuio
Today at 3:56 AM
Primeniyazidayi
an hour ago
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Summing the GCD of a number and the divisors of another.
EmersonSoriano   1
N an hour ago by alexheinis
Source: 2018 Peru Cono Sur TST P8
For each pair of positive integers $m$ and $n$, we define $f_m(n)$ as follows:
$$ f_m(n) = \gcd(n, d_1) + \gcd(n, d_2) + \cdots + \gcd(n, d_k), $$where $1 = d_1 < d_2 < \cdots < d_k = m$ are all the positive divisors of $m$. For example,
$f_4(6) = \gcd(6,1) + \gcd(6,2) + \gcd(6,4) = 5$.

$a)\:$ Find all positive integers $n$ such that $f_{2017}(n) = f_n(2017)$.

$b)\:$ Find all positive integers $n$ such that $f_6(n) = f_n(6)$.
1 reply
EmersonSoriano
Apr 2, 2025
alexheinis
an hour ago
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a hard geometry problen
Tuguldur   1
N 2 hours ago by whwlqkd
Let $ABCD$ be a convex quadrilateral. Suppose that the circles with diameters $AB$ and $CD$ intersect at points $X$ and $Y$. Let $P=AC\cap BD$ and $Q=AD\cap BC$. Prove that the points $P$, $Q$, $X$ and $Y$ are concyclic.
( $AB$ and $CD$ are not the diagnols)
1 reply
Tuguldur
Yesterday at 3:56 PM
whwlqkd
2 hours ago
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Problem 6
blug   1
N 3 hours ago by atdaotlohbh
Source: Polish Math Olympiad 2025 Finals P6
A strictly decreasing function $f:(0, \infty)\Rightarrow (0, \infty)$ attaining all positive values and positive numbers $a_1\ne b_1$ are given. Numbers $a_2, b_2, a_3, b_3, ...$ satisfy
$$a_{n+1}=a_n+f(b_n),\;\;\;\;\;\;\;b_{n+1}=b_n+f(a_n)$$for every $n\geq 1$. Prove that there exists a positive integer $n$ satisfying $|a_n-b_n| >2025$.
1 reply
blug
Yesterday at 12:17 PM
atdaotlohbh
3 hours ago
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Regarding Maaths olympiad prepration
omega2007   13
N 3 hours ago by omega2007
<Hey Everyone'>
I'm 10 grader student and Im starting prepration for maths olympiad..>>> From scratch (not 2+2=4 )

Do you haves compiled resources of Handouts,
PDF,
Links,
List of books topic wise
which are shared on AOPS (and from your perspective) for maths olympiad and any useful thing, which will help me in boosting Maths olympiad prepration.
13 replies
omega2007
Yesterday at 3:13 PM
omega2007
3 hours ago
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D1010 : How it is possible ?
Dattier   16
N 3 hours ago by Dattier
Source: les dattes à Dattier
Is it true that$$\forall n \in \mathbb N^*, (24^n \times B \mod A) \mod 2 = 0 $$?

A=1728400904217815186787639216753921417860004366580219212750904
024377969478249664644267971025952530803647043121025959018172048
336953969062151534282052863307398281681465366665810775710867856
720572225880311472925624694183944650261079955759251769111321319
421445397848518597584590900951222557860592579005088853698315463
815905425095325508106272375728975

B=2275643401548081847207782760491442295266487354750527085289354
965376765188468052271190172787064418854789322484305145310707614
546573398182642923893780527037224143380886260467760991228567577
953725945090125797351518670892779468968705801340068681556238850
340398780828104506916965606659768601942798676554332768254089685
307970609932846902
16 replies
Dattier
Mar 10, 2025
Dattier
3 hours ago
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integer right triangle with area twice the perimeter
parmenides51   4
N Oct 10, 2021 by pi271828
Source: 1994 Argentina IberoAmerican Training List p30 https://artofproblemsolving.com/community/c2434876_argentina_training_lists__geom
Find all right triangles with integer sides such that their area is twice their perimeter.
4 replies
parmenides51
Oct 10, 2021
pi271828
Oct 10, 2021
J
integer right triangle with area twice the perimeter
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Reveal topic tags
geometry
Integers
right triangle
area
perimeter
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Source: 1994 Argentina IberoAmerican Training List p30 https://artofproblemsolving.com/community/c2434876_argentina_training_lists__geom
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parmenides51
30629 posts
parmenides51
#1 h  Oct 10, 2021, 5:02 PM
Y by
Find all right triangles with integer sides such that their area is twice their perimeter.
This post has been edited 1 time. Last edited by parmenides51, Oct 10, 2021, 5:21 PM
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OlympusHero
17019 posts
OlympusHero
#2 h  Oct 10, 2021, 10:05 PM
Y by
Solution
This post has been edited 2 times. Last edited by OlympusHero, Oct 10, 2021, 10:56 PM
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James_HN
51 posts
James_HN
#3 h  Oct 10, 2021, 10:40 PM
Y by
Let the sides of the triangle be $a,b,\sqrt{a^2 + b^2}$. Then we have
$\frac{ab}{2} = 2(a + b + \sqrt{a^2 + b^2})$

$(\frac{ab}{4} - a - b) = \sqrt{a^2 + b^2}$

$(\frac{ab}{4} - a - b)^2 = a^2 + b^2$

$\frac{a^2b^2}{16} + a^2 + b^2 + 2ab - \frac{a^2b}{2}- \frac{ab^2}{2} = a^2 + b^2$

$\frac{a^2b^2}{16} + 2ab - \frac{a^2b}{2}- \frac{ab^2}{2} = 0$

$a^2b^2 + 32ab - 8a^2b- 8ab^2 = 0$

$ab- 8a- 8b + 64 = 32$

$(a-8)(b-8)= 32$

hence solving for a and b we get the following triples,
$$(9,40,41),(10,24,26),(12,16,20)$$
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OlympusHero
17019 posts
OlympusHero
#4 h  Oct 10, 2021, 10:46 PM
Y by
Oh my god I'm such an idiot
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pi271828
3363 posts
pi271828
#5 h  Oct 10, 2021, 10:54 PM
Y by
OlympusHero wrote:
Oh my god I'm such an idiot

smh you reversed the condition

sketch
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