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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
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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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Concurrency with 10 lines
oVlad   1
N 7 minutes ago by kokcio
Source: Romania EGMO TST 2017 Day 1 P1
Consider five points on a circle. For every three of them, we draw the perpendicular from the centroid of the triangle they determine to the line through the remaining two points. Prove that the ten lines thus formed are concurrent.
1 reply
oVlad
4 hours ago
kokcio
7 minutes ago
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Advanced topics in Inequalities
va2010   21
N 10 minutes ago by Novmath
So a while ago, I compiled some tricks on inequalities. You are welcome to post solutions below!
21 replies
va2010
Mar 7, 2015
Novmath
10 minutes ago
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GCD of a sequence
oVlad   1
N 17 minutes ago by kokcio
Source: Romania EGMO TST 2017 Day 1 P2
Determine all pairs $(a,b)$ of positive integers with the following property: all of the terms of the sequence $(a^n+b^n+1)_{n\geqslant 1}$ have a greatest common divisor $d>1.$
1 reply
oVlad
4 hours ago
kokcio
17 minutes ago
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Test from Côte d'Ivoire Diophantine equation
MENELAUSS   4
N 22 minutes ago by Pal702004
determine all triplets $(x;y;z)$ of natural numbers such that
$$y \quad \text{is prime }$$
$$y \quad \text{and} \quad 3 \quad \text{does not divide} \quad z$$
$$x^3-y^3=z^2$$
4 replies
MENELAUSS
Apr 19, 2025
Pal702004
22 minutes ago
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Concurrence, Isogonality
Wictro   40
N 28 minutes ago by CatinoBarbaraCombinatoric
Source: BMO 2019, Problem 3
Let $ABC$ be an acute scalene triangle. Let $X$ and $Y$ be two distinct interior points of the segment $BC$ such that $\angle{CAX} = \angle{YAB}$. Suppose that:
$1)$ $K$ and $S$ are the feet of the perpendiculars from from $B$ to the lines $AX$ and $AY$ respectively.
$2)$ $T$ and $L$ are the feet of the perpendiculars from $C$ to the lines $AX$ and $AY$ respectively.
Prove that $KL$ and $ST$ intersect on the line $BC$.
40 replies
Wictro
May 2, 2019
CatinoBarbaraCombinatoric
28 minutes ago
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Tango course
oVlad   1
N an hour ago by kokcio
Source: Romania EGMO TST 2019 Day 1 P4
Six boys and six girls are participating at a tango course. They meet every evening for three weeks (a total of 21 times). Each evening, at least one boy-girl pair is selected to dance in front of the others. At the end of the three weeks, every boy-girl pair has been selected at least once. Prove that there exists a person who has been selected on at least 5 distinct evenings.

Note: a person can be selected twice on the same evening.
1 reply
1 viewing
oVlad
3 hours ago
kokcio
an hour ago
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Easy Number Theory
math_comb01   36
N an hour ago by anudeep
Source: INMO 2024/3
Let $p$ be an odd prime and $a,b,c$ be integers so that the integers $$a^{2023}+b^{2023},\quad b^{2024}+c^{2024},\quad a^{2025}+c^{2025}$$are divisible by $p$.
Prove that $p$ divides each of $a,b,c$.
$\quad$
Proposed by Navilarekallu Tejaswi
36 replies
math_comb01
Jan 21, 2024
anudeep
an hour ago
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A cyclic inequality
KhuongTrang   0
an hour ago
Source: own-CRUX
IMAGE
https://cms.math.ca/.../uploads/2025/04/Wholeissue_51_4.pdf
0 replies
KhuongTrang
an hour ago
0 replies
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Prove excircle is tangent to circumcircle
sarjinius   7
N an hour ago by markam
Source: Philippine Mathematical Olympiad 2025 P4
Let $ABC$ be a triangle with incenter $I$, and let $D$ be a point on side $BC$. Points $X$ and $Y$ are chosen on lines $BI$ and $CI$ respectively such that $DXIY$ is a parallelogram. Points $E$ and $F$ are chosen on side $BC$ such that $AX$ and $AY$ are the angle bisectors of angles $\angle BAE$ and $\angle CAF$ respectively. Let $\omega$ be the circle tangent to segment $EF$, the extension of $AE$ past $E$, and the extension of $AF$ past $F$. Prove that $\omega$ is tangent to the circumcircle of triangle $ABC$.
7 replies
sarjinius
Mar 9, 2025
markam
an hour ago
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Distinct Integers with Divisibility Condition
tastymath75025   15
N an hour ago by cursed_tangent1434
Source: 2017 ELMO Shortlist N3
For each integer $C>1$ decide whether there exist pairwise distinct positive integers $a_1,a_2,a_3,...$ such that for every $k\ge 1$, $a_{k+1}^k$ divides $C^ka_1a_2...a_k$.

Proposed by Daniel Liu
15 replies
tastymath75025
Jul 3, 2017
cursed_tangent1434
an hour ago
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hard problem
Cobedangiu   4
N an hour ago by Cobedangiu
Let $a,b,c>0$ and $a+b+c=3$. Prove that:
$\dfrac{4}{a+b}+\dfrac{4}{b+c}+\dfrac{4}{c+a} \le \dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}+3$
4 replies
1 viewing
Cobedangiu
3 hours ago
Cobedangiu
an hour ago
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An easy FE
oVlad   1
N an hour ago by pco
Source: Romania EGMO TST 2017 Day 1 P3
Determine all functions $f:\mathbb R\to\mathbb R$ such that \[f(xy-1)+f(x)f(y)=2xy-1,\]for any real numbers $x{}$ and $y{}.$
1 reply
oVlad
4 hours ago
pco
an hour ago
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Fractions and reciprocals
adihaya   34
N an hour ago by de-Kirschbaum
Source: 2013 BAMO-8 #4
For a positive integer $n>2$, consider the $n-1$ fractions $$\dfrac21, \dfrac32, \cdots, \dfrac{n}{n-1}$$The product of these fractions equals $n$, but if you reciprocate (i.e. turn upside down) some of the fractions, the product will change. Can you make the product equal 1? Find all values of $n$ for which this is possible and prove that you have found them all.
34 replies
adihaya
Feb 27, 2016
de-Kirschbaum
an hour ago
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GCD Functional Equation
pinetree1   60
N an hour ago by cursed_tangent1434
Source: USA TSTST 2019 Problem 7
Let $f: \mathbb Z\to \{1, 2, \dots, 10^{100}\}$ be a function satisfying
$$\gcd(f(x), f(y)) = \gcd(f(x), x-y)$$for all integers $x$ and $y$. Show that there exist positive integers $m$ and $n$ such that $f(x) = \gcd(m+x, n)$ for all integers $x$.

Ankan Bhattacharya
60 replies
pinetree1
Jun 25, 2019
cursed_tangent1434
an hour ago
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Angle Chasing
zuss77   3
N Jun 30, 2018 by zuss77
Source: Ukraine 2017 8gr p4
See attachment:
3 replies
zuss77
Jun 29, 2018
zuss77
Jun 30, 2018
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Angle Chasing
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Reveal topic tags
geometry
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Source: Ukraine 2017 8gr p4
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zuss77
520 posts
zuss77
#1 h  Jun 29, 2018, 12:07 PM • 2 Y
Y by Adventure10, Mango247
See attachment:
Attachments:
This post has been edited 1 time. Last edited by zuss77, Jun 29, 2018, 12:29 PM
Reason: corrected
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zuss77
520 posts
zuss77
#2 h  Jun 30, 2018, 5:39 AM • 2 Y
Y by Adventure10, Mango247
Not easy. I'd say it's too hard for the eighth grade.
Here's the sketch: (in attachment)
Attachments:
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skrublord420
1277 posts
skrublord420
#3 h  Jun 30, 2018, 5:48 AM • 2 Y
Y by Adventure10, Mango247
What are we supposed to find, what are the given angles and sides, and what points are the circles passing through?
Z K Y
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zuss77
520 posts
zuss77
#4 h  Jun 30, 2018, 6:36 AM • 1 Y
Y by Adventure10
We need to prove, not find.
I lazy for Latex and thought picture will be enough. OK, here's the problem:

$\triangle ABC$. $\angle C = 90^\circ$
Inside $\triangle ABC$ there is point $K$, such that $\angle AKC = 90^\circ$ and $\angle CKB = 2 \angle CAB$.
On segment $KB$ there is point $T$, such that $\angle KTC = \angle CAK$.
$P = AK \cap BC$.
Prove that $\angle TPA = \angle ABC$.
This post has been edited 1 time. Last edited by zuss77, Jun 30, 2018, 6:43 AM
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