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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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Similarity
AHZOLFAGHARI   17
N 3 minutes ago by ariopro1387
Source: Iran Second Round 2015 - Problem 3 Day 1
Consider a triangle $ABC$ . The points $D,E$ are on sides $AB,AC$ such that $BDEC$ is a cyclic quadrilateral. Let $P$ be the intersection of $BE$ and $CD$. $H$ is a point on $AC$ such that $\angle PHA = 90^{\circ}$. Let $M,N$ be the midpoints of $AP,BC$. Prove that: $ ACD \sim MNH $.
17 replies
1 viewing
AHZOLFAGHARI
May 7, 2015
ariopro1387
3 minutes ago
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sequence infinitely similar to central sequence
InterLoop   0
14 minutes ago
Source: EGMO 2025/2
An infinite increasing sequence $a_1 < a_2 < a_3 < \dots$ of positive integers is called central if for every positive integer $n$, the arithmetic mean of the first $a_n$ terms of the sequence is equal to $a_n$.

Show that there exists an infinite sequence $b_1$, $b_2$, $b_3$, $\dots$ of positive integers such that for every central sequence $a_1$, $a_2$, $a_3$, $\dots$, there are infinitely many positive integers $n$ with $a_n = b_n$.
0 replies
InterLoop
14 minutes ago
0 replies
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one cyclic formed by two cyclic
CrazyInMath   0
15 minutes ago
Source: EGMO 2025/3
Let $ABC$ be an acute triangle. Points $B, D, E$, and $C$ lie on a line in this order and satisfy $BD = DE = EC$. Let $M$ and $N$ be the midpoints of $AD$ and $AE$, respectively. Suppose triangle $ADE$ is acute, and let $H$ be its orthocentre. Points $P$ and $Q$ lie on lines $BM$ and $CN$, respectively, such that $D, H, M,$ and $P$ are concyclic and pairwise different, and $E, H, N,$ and $Q$ are concyclic and pairwise different. Prove that $P, Q, N,$ and $M$ are concyclic.
0 replies
+3 w
CrazyInMath
15 minutes ago
0 replies
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pairwise coprime sum gcd
InterLoop   0
18 minutes ago
Source: EGMO 2025/1
For a positive integer $N$, let $c_1 < c_2 < \dots < c_n$ be all the positive integers smaller than $N$ that are coprime to $N$. Find all $N \ge 3$ such that
$$\gcd(N, c_i + c_{i+1}) \neq 1$$for all $1 \le i \le n - 1$.
0 replies
InterLoop
18 minutes ago
0 replies
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A problem with non-negative a,b,c
KhuongTrang   3
N 23 minutes ago by KhuongTrang
Source: own
Problem. Let $a,b,c$ be non-negative real variables with $ab+bc+ca\neq 0.$ Prove that$$\color{blue}{\sqrt{\frac{8a^{2}+\left(b-c\right)^{2}}{\left(b+c\right)^{2}}}+\sqrt{\frac{8b^{2}+\left(c-a\right)^{2}}{\left(c+a\right)^{2}}}+\sqrt{\frac{8c^{2}+\left(a-b\right)^{2}}{\left(a+b\right)^{2}}}\ge \sqrt{\frac{18(a^{2}+b^{2}+c^{2})}{ab+bc+ca}}.}$$Equality holds iff $(a,b,c)\sim(t,t,t)$ or $(a,b,c)\sim(t,t,0)$ where $t>0.$
3 replies
KhuongTrang
Mar 4, 2025
KhuongTrang
23 minutes ago
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Number Theory Chain!
JetFire008   52
N 25 minutes ago by Anto0110
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
52 replies
JetFire008
Apr 7, 2025
Anto0110
25 minutes ago
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Convex quad
MithsApprentice   81
N 29 minutes ago by LeYohan
Source: USAMO 1993
Let $\, ABCD \,$ be a convex quadrilateral such that diagonals $\, AC \,$ and $\, BD \,$ intersect at right angles, and let $\, E \,$ be their intersection. Prove that the reflections of $\, E \,$ across $\, AB, \, BC, \, CD, \, DA \,$ are concyclic.
81 replies
MithsApprentice
Oct 27, 2005
LeYohan
29 minutes ago
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Inspired by A_E_R
sqing   0
34 minutes ago
Source: Own
Let $ a,b,c,d>0 $ and $ a(b^2+c^2)\geq 4bcd.$ Prove that$$ (a^2+b^2+c^2+d^2)(\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}+\frac{1}{d^2})\geq\frac{84}{4}$$Let $ a,b,c,d>0 $ and $ a(b^2+c^2)\geq 3bcd.$ Prove that$$ (a^2+b^2+c^2+d^2)(\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}+\frac{1}{d^2})\geq\frac{625}{36}$$
0 replies
sqing
34 minutes ago
0 replies
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Interesting inequality
A_E_R   1
N an hour ago by sqing
Let a,b,c,d are positive real numbers, if the following inequality holds ab^2+ac^2>=5bcd. Find the minimum value of explanation: (a^2+b^2+c^2+d^2)(1/a^2+1/b^2+1/c^2+1/d^2)
1 reply
A_E_R
3 hours ago
sqing
an hour ago
J
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Maximum area of the triangle
adityaguharoy   1
N an hour ago by Mathzeus1024
If in some triangle $\triangle ABC$ we are given :
$\sqrt{3} \cdot \sin(C)=\frac{2- \sin A}{\cos A}$ and one side length of the triangle equals $2$, then under these conditions find the maximum area of the triangle $ABC$.
1 reply
adityaguharoy
Jan 19, 2017
Mathzeus1024
an hour ago
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Concurrent lines
BR1F1SZ   4
N an hour ago by NicoN9
Source: 2025 CJMO P2
Let $ABCD$ be a trapezoid with parallel sides $AB$ and $CD$, where $BC\neq DA$. A circle passing through $C$ and $D$ intersects $AC, AD, BC, BD$ again at $W, X, Y, Z$ respectively. Prove that $WZ, XY, AB$ are concurrent.
4 replies
BR1F1SZ
Mar 7, 2025
NicoN9
an hour ago
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Inspired by lgx57
sqing   6
N an hour ago by sqing
Source: Own
Let $ a,b\geq 0 $ and $\frac{1}{a^2+b}+\frac{1}{b^2+a}=1. $ Prove that
$$a^2+b^2-a-b\leq 1$$$$a^3+b^3-a-b\leq \frac{3+\sqrt 5}{2}$$$$a^3+b^3-a^2-b^2\leq \frac{1+\sqrt 5}{2}$$
6 replies
sqing
3 hours ago
sqing
an hour ago
J
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Arithmetic progression
BR1F1SZ   2
N an hour ago by NicoN9
Source: 2025 CJMO P1
Suppose an infinite non-constant arithmetic progression of integers contains $1$ in it. Prove that there are an infinite number of perfect cubes in this progression. (A perfect cube is an integer of the form $k^3$, where $k$ is an integer. For example, $-8$, $0$ and $1$ are perfect cubes.)
2 replies
BR1F1SZ
Mar 7, 2025
NicoN9
an hour ago
J
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Injective arithmetic comparison
adityaguharoy   1
N 2 hours ago by Mathzeus1024
Source: Own .. probably own
Show or refute :
For every injective function $f: \mathbb{N} \to \mathbb{N}$ there are elements $a,b,c$ in an arithmetic progression in the order $a<b<c$ such that $f(a)<f(b)<f(c)$ .
1 reply
adityaguharoy
Jan 16, 2017
Mathzeus1024
2 hours ago
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Polynomial
EtacticToe   3
N Apr 2, 2025 by EmersonSoriano
Source: Own
Let $f(x)$ be a monic polynomial with integer coefficient. And suppose there exist 4 distinct integer $a,b,c,d$ such that $f(a)=…=f(d)=5$.

Find all $k$ such that $f(k)=8$
3 replies
EtacticToe
Dec 14, 2024
EmersonSoriano
Apr 2, 2025
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Polynomial
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Reveal topic tags
algebra
polynomial
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Source: Own
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EtacticToe
82 posts
EtacticToe
#1 h  Dec 14, 2024, 6:43 PM
Y by
Let $f(x)$ be a monic polynomial with integer coefficient. And suppose there exist 4 distinct integer $a,b,c,d$ such that $f(a)=…=f(d)=5$.

Find all $k$ such that $f(k)=8$
This post has been edited 1 time. Last edited by EtacticToe, Dec 14, 2024, 6:44 PM
Reason: The row
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RagvaloD
4900 posts
RagvaloD
#2 h  Dec 15, 2024, 2:32 AM
Y by
$f(x)=(x-a)(x-b)(x-c)(x-d)g(x)+5$
$f(k)=(k-a)(k-b)(k-c)(k-d)g(k)+5=8 \to (k-a)(k-b)(k-c)(k-d) | 3$
But as $k-a,k-b,k-c,k-d$ should be different integers, there is no such $k$
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yuribogomolov
1 post
yuribogomolov
#3 h  Apr 2, 2025, 3:00 PM
Y by
Quote:
RagvaloD wrote:
there is no such k

3 has four integer divisors (-3, -1, 1, 3).
Z K Y
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EmersonSoriano
43 posts
EmersonSoriano
#4 h  Apr 2, 2025, 8:46 PM
Y by
yuribogomolov wrote:
Quote:
RagvaloD wrote:
there is no such k

3 has four integer divisors (-3, -1, 1, 3).

But it would have to be fulfilled that the product of those four integers is a divisor of 3, which cannot happen, since the product of those four numbers would be 9.
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