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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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Find all pairs (x; y)
sqing   5
N 3 minutes ago by EVKV
Source: 7th European Mathematical Cup , Junior Category, Q2
Find all pairs $ (x; y) $ of positive integers such that
$$xy | x^2 + 2y -1.$$
5 replies
sqing
Dec 25, 2018
EVKV
3 minutes ago
J
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problem....
Cobedangiu   1
N 7 minutes ago by giangtruong13
$a,b,c>0$ and $a+b+c=1$
Prove that: $Q=ab+bc+ca-2abc\le\dfrac{7}{27}$
1 reply
Cobedangiu
2 hours ago
giangtruong13
7 minutes ago
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Why is the old one deleted?
EeEeRUT   10
N 18 minutes ago by kamatadu
Source: EGMO 2025 P1
For a positive integer $N$, let $c_1 < c_2 < \cdots < c_m$ be all positive integers smaller than $N$ that are coprime to $N$. Find all $N \geqslant 3$ such that $$\gcd( N, c_i + c_{i+1}) \neq 1$$for all $1 \leqslant i \leqslant m-1$

Here $\gcd(a, b)$ is the largest positive integer that divides both $a$ and $b$. Integers $a$ and $b$ are coprime if $\gcd(a, b) = 1$.

Proposed by Paulius Aleknavičius, Lithuania
10 replies
EeEeRUT
Apr 16, 2025
kamatadu
18 minutes ago
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NMTC JUNIOR 2023 FINALS P1
Jishnu4414l   3
N 27 minutes ago by Pseudo_Matter
Source: NMTC 2023 FINALS P1 (Junior)
Find integers $m,n$ such that the sum of their cubes is equal to the square of their sum.
3 replies
Jishnu4414l
Nov 5, 2023
Pseudo_Matter
27 minutes ago
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Geometry Problem
Itoz   1
N 34 minutes ago by aidenkim119
Source: Own
Given $\triangle ABC$. Let the perpendicular line from $A$ to $BC$ meets $BC,\odot(ABC)$ at points $S,K$, respectively, and the foot from $B$ to $AC$ is $L$. $\odot (AKL)$ intersects line $AB$ at $T(\neq A)$, $\odot(AST)$ intersects line $AC$ at $M(\neq A)$, and lines $TM,CK$ intersect at $N$.

Prove that $\odot(CNM)$ is tangent to $\odot (BST)$.
1 reply
Itoz
2 hours ago
aidenkim119
34 minutes ago
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Circles tangent to AD and AB intersect on AC
gghx   2
N 44 minutes ago by zhanbolatzh
Source: SMO senior 2024 Q1
In an acute triangle $ABC$, $AC>AB$, $D$ is the point on $BC$ such that $AD=AB$. Let $\omega_1$ be the circle through $C$ tangent to $AD$ at $D$, and $\omega_2$ the circle through $C$ tangent to $AB$ at $B$. Let $F(\ne C)$ be the second intersection of $\omega_1$ and $\omega_2$. Prove that $F$ lies on $AC$.
2 replies
gghx
Aug 3, 2024
zhanbolatzh
44 minutes ago
J
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a combinatorial geometry problem
xyz123456   4
N an hour ago by xyz123456
$A_i\left(x_i{,}y_i\right){,}0\le x_i{,}y_i\le 1{,}1\le i\le 6.prove\ that:\exists \ 1\le i<j\le 6\ {,}\left|A_iA_j\right|\le \frac{\sqrt{13}}{6}$
4 replies
xyz123456
Mar 3, 2025
xyz123456
an hour ago
J
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a combinatoric problem
xyz123456   0
an hour ago
Source: Noga Alon
$A_n=\left[n\right]^3{,}f\left(n\right)=\min \left|S\right|{,}S\ satisfy\ S\subseteq \ A_nand\ \forall \ x\in \ A_n{,}\exists \ y{,}z\in \ S\ {,}x{,}y{,}zcollinear$ $prove\ that:\exists \ c_1{,}c_2>0{,}c_1n^{\frac{6}{5}}<f\left(n\right)<c_2n^{\frac{6}{5}}\ln n$
0 replies
xyz123456
an hour ago
0 replies
J
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Turbo's en route to visit each cell of the board
Lukaluce   18
N an hour ago by kamatadu
Source: EGMO 2025 P5
Let $n > 1$ be an integer. In a configuration of an $n \times n$ board, each of the $n^2$ cells contains an arrow, either pointing up, down, left, or right. Given a starting configuration, Turbo the snail starts in one of the cells of the board and travels from cell to cell. In each move, Turbo moves one square unit in the direction indicated by the arrow in her cell (possibly leaving the board). After each move, the arrows in all of the cells rotate $90^{\circ}$ counterclockwise. We call a cell good if, starting from that cell, Turbo visits each cell of the board exactly once, without leaving the board, and returns to her initial cell at the end. Determine, in terms of $n$, the maximum number of good cells over all possible starting configurations.

Proposed by Melek Güngör, Turkey
18 replies
Lukaluce
Apr 14, 2025
kamatadu
an hour ago
J
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Shortlist 2017/G4
fastlikearabbit   120
N an hour ago by Frd_19_Hsnzde
Source: Shortlist 2017, Romanian TST 2018
In triangle $ABC$, let $\omega$ be the excircle opposite to $A$. Let $D, E$ and $F$ be the points where $\omega$ is tangent to $BC, CA$, and $AB$, respectively. The circle $AEF$ intersects line $BC$ at $P$ and $Q$. Let $M$ be the midpoint of $AD$. Prove that the circle $MPQ$ is tangent to $\omega$.
120 replies
fastlikearabbit
Jul 10, 2018
Frd_19_Hsnzde
an hour ago
J
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Inequality with a,b,c,d
GeoMorocco   3
N an hour ago by ZeroHero
Source: Moroccan Training 2025
Let $ a,b,c,d$ positive real numbers such that $ a+b+c+d=3+\frac{1}{abcd}$ . Prove that :
$$ a^2+b^2+c^2+d^2+5abcd \geq 9 $$
3 replies
GeoMorocco
Apr 9, 2025
ZeroHero
an hour ago
J
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Weird function?
ItzsleepyXD   1
N an hour ago by ItzsleepyXD
Source: Own
Find all functions \( f: \mathbb{R} \rightarrow \mathbb{R} \) such that for all \( x, y \in \mathbb{R} \),
\[ f(x + f(2y)) + f(x^2 - y) = f(f(x)) f(x + 1) + 2y - f(y). \]
1 reply
ItzsleepyXD
Apr 11, 2025
ItzsleepyXD
an hour ago
J
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Quadric function
soryn   2
N 2 hours ago by soryn
If f(x)=ax^2+bx+c, a,b,c integers, |a|>=3, and M îs the set of integers x for which f(x) is a prime number and f has exactly one integer solution,prove that M has at most three elements.
2 replies
soryn
Today at 2:47 AM
soryn
2 hours ago
J
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one cyclic formed by two cyclic
CrazyInMath   34
N 2 hours ago by Assassino9931
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.
34 replies
1 viewing
CrazyInMath
Apr 13, 2025
Assassino9931
2 hours ago
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Four spheres pass through a single point [Germany BW 2000]
Amir Hossein   2
N Feb 3, 2011 by m.candales
For each vertex of a given tetrahedron, a sphere passing through that vertex and the midpoints of the edges outgoing from this vertex is constructed. Prove that these four spheres pass through a single point.
2 replies
Amir Hossein
Feb 3, 2011
m.candales
Feb 3, 2011
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Four spheres pass through a single point [Germany BW 2000]
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Reveal topic tags
geometry
3D geometry
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tetrahedron
geometry unsolved
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Amir Hossein
5452 posts
Amir Hossein
#1 h  Feb 3, 2011, 8:35 AM • 2 Y
Y by Adventure10, Mango247
For each vertex of a given tetrahedron, a sphere passing through that vertex and the midpoints of the edges outgoing from this vertex is constructed. Prove that these four spheres pass through a single point.
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nsato
15653 posts
nsato
#2 h  Feb 3, 2011, 3:45 PM • 3 Y
Y by Adventure10, Mango247, Amir Hossein
This is a specific case of Roberts' Theorem. (There is nothing special about the midpoints.)
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m.candales
186 posts
m.candales
#3 h  Feb 3, 2011, 4:01 PM • 3 Y
Y by Adventure10, Mango247, Amir Hossein
Let $ABCD$ be the tetrahedron. Let $S_a, S_b, S_c, S_d$ be the spheres through $A, B, C, D$ respectively. Let $S$ be the circumscribed sphere of $ABCD$
and let $O$ be its center. Let $B_1, C_1, D_1$ be the midpoints of $AB, AC, AD$.
We have that $B_1, C_1, D_1$ are the images of $B, C, D$ under a contraction of radius $\frac 12$ and origin at $A$. Then $S_a$ is the image of $S$ under
that contraction.
Let $P$ be the point where $AO$ intersects $S$ by second time. Let $P'$ be the image of $P$ under the above contraction. We have that $AP' = \frac 12 AP$,
then $P' = O$, and then $O\in S_a$.
Similarly $O\in S_b, S_c, S_d$.
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