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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
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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Why is the old one deleted?
EeEeRUT   6
N 23 minutes ago by Safal
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 $\gcd(a, b) = 1$.
6 replies
EeEeRUT
Yesterday at 1:33 AM
Safal
23 minutes ago
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Equation of exponents
shobber   4
N 25 minutes ago by zhoujef000
Source: Canada 1998
Find all real numbers $x$ such that: \[ x = \sqrt{ x - \frac{1}{x} } + \sqrt{ 1 - \frac{1}{x} } \]
4 replies
shobber
Mar 4, 2006
zhoujef000
25 minutes ago
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this hAOpefully shoudn't BE weird
popop614   47
N 26 minutes ago by Tsikaloudakis
Source: 2023 IMO Shortlist G1
Let $ABCDE$ be a convex pentagon such that $\angle ABC = \angle AED = 90^\circ$. Suppose that the midpoint of $CD$ is the circumcenter of triangle $ABE$. Let $O$ be the circumcenter of triangle $ACD$.

Prove that line $AO$ passes through the midpoint of segment $BE$.
47 replies
popop614
Jul 17, 2024
Tsikaloudakis
26 minutes ago
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Symmetric FE
Phorphyrion   9
N 27 minutes ago by MuradSafarli
Source: 2023 Israel TST Test 7 P1
Find all functions $f:\mathbb{R}\to \mathbb{R}$ such that for all $x, y\in \mathbb{R}$ the following holds:
\[f(x)+f(y)=f(xy)+f(f(x)+f(y))\]
9 replies
Phorphyrion
May 9, 2023
MuradSafarli
27 minutes ago
J
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Diagonals BD,CE concurrent with diameter AO in cyclic ABCDE
WakeUp   11
N 31 minutes ago by zhoujef000
Source: Romanian TST 2002
Let $ABCDE$ be a cyclic pentagon inscribed in a circle of centre $O$ which has angles $\angle B=120^{\circ},\angle C=120^{\circ},$ $\angle D=130^{\circ},\angle E=100^{\circ}$. Show that the diagonals $BD$ and $CE$ meet at a point belonging to the diameter $AO$.

Dinu Șerbănescu
11 replies
WakeUp
Feb 5, 2011
zhoujef000
31 minutes ago
J
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Easy Geometry Problem in Taiwan TST
chengbilly   5
N an hour ago by Tamam
Source: 2025 Taiwan TST Round 1 Independent Study 2-G
Suppose $I$ and $I_A$ are the incenter and the $A$-excenter of triangle $ABC$, respectively.
Let $M$ be the midpoint of arc $BAC$ on the circumcircle, and $D$ be the foot of the
perpendicular from $I_A$ to $BC$. The line $MI$ intersects the circumcircle again at $T$ . For
any point $X$ on the circumcircle of triangle $ABC$, let $XT$ intersect $BC$ at $Y$ . Prove
that $A, D, X, Y$ are concyclic.
5 replies
chengbilly
Mar 6, 2025
Tamam
an hour ago
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one cyclic formed by two cyclic
CrazyInMath   32
N an hour ago by kamatadu
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.
32 replies
CrazyInMath
Apr 13, 2025
kamatadu
an hour ago
J
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powers sums and triangular numbers
gaussious   3
N an hour ago by RagvaloD
prove 1^k+2^k+3^k + \cdots + n^k \text{is divisible by } \frac{n(n+1)}{2} \text{when} k \text{is odd}
3 replies
gaussious
5 hours ago
RagvaloD
an hour ago
J
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Beautiful geometry
m4thbl3nd3r   0
an hour ago
Let $\omega$ be the circumcircle of triangle $ABC$, $M$ is the midpoint of $BC$ and $E$ be the second intersection of $AM$ and $\omega$. Tangent line of $\omega$ at $E$ intersects $BC$ at $P$, let $PKL$ be a transversal of $\omega$ and $X,Y$ be intersections of $AK,AL$ with $BC$. Let $PF$ be a tangent line of $\omega$. Prove that $LYFP$ is cyclic
0 replies
m4thbl3nd3r
an hour ago
0 replies
J
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prove |a-b| is a square, given a-b=a^2c-b^2d
Alpha314159   4
N 2 hours ago by Leman_Nabiyeva
Source: Macau Inter High School Competition
Let $a, b$ be integers such that there are consecutive integers $c,d$ satisfy $$a-b=a^2 c-b^2 d$$.
Prove : $|a-b|$ is a perfect square.
4 replies
Alpha314159
Mar 7, 2020
Leman_Nabiyeva
2 hours ago
J
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Inspired by pennypc123456789
sqing   0
2 hours ago
Source: Own
Let $ a,b,c\geq 0 $and $ab+bc+ca\neq 0.$ Prove that
$$ \frac{9-8\sqrt 2+5\sqrt 5}{6}\leq\frac{a + b}{a + 2b + c} + \dfrac{b + c}{b + 2c + a}+\dfrac{c + a}{c + 2a + b}\leq \frac{9+8\sqrt 2-5\sqrt 5}{6}$$
0 replies
sqing
2 hours ago
0 replies
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The Bank of Bath
TelMarin   100
N 2 hours ago by Ihatecombin
Source: IMO 2019, problem 5
The Bank of Bath issues coins with an $H$ on one side and a $T$ on the other. Harry has $n$ of these coins arranged in a line from left to right. He repeatedly performs the following operation: if there are exactly $k>0$ coins showing $H$, then he turns over the $k$th coin from the left; otherwise, all coins show $T$ and he stops. For example, if $n=3$ the process starting with the configuration $THT$ would be $THT \to HHT \to HTT \to TTT$, which stops after three operations.

(a) Show that, for each initial configuration, Harry stops after a finite number of operations.

(b) For each initial configuration $C$, let $L(C)$ be the number of operations before Harry stops. For example, $L(THT) = 3$ and $L(TTT) = 0$. Determine the average value of $L(C)$ over all $2^n$ possible initial configurations $C$.

Proposed by David Altizio, USA
100 replies
TelMarin
Jul 17, 2019
Ihatecombin
2 hours ago
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Permutations of Integers from 1 to n
Twoisntawholenumber   74
N 2 hours ago by Maximilian113
Source: 2020 ISL C1
Let $n$ be a positive integer. Find the number of permutations $a_1$, $a_2$, $\dots a_n$ of the
sequence $1$, $2$, $\dots$ , $n$ satisfying
$$a_1 \le 2a_2\le 3a_3 \le \dots \le na_n$$.

Proposed by United Kingdom
74 replies
Twoisntawholenumber
Jul 20, 2021
Maximilian113
2 hours ago
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11^n+2^n+6=m^3
jungle_wang   5
N 2 hours ago by Rayanelba
Source: 2024IMOC
Find all positive integers $(m,n)$ such that
$$11^n+2^n+6=m^3$$
5 replies
jungle_wang
Aug 1, 2024
Rayanelba
2 hours ago
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J
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Geometry
VicKmath7   3
N Mar 28, 2020 by sunken rock
Source: 12th Russian MO "Leonhard Euler", final round
Given is a triangle ABC with 2<B=180°+<A. Point K is in the interior of the triangle, so that <ACK=2<BCK and L is on AB, so that BK=KL. Prove that CK+AL=AC
3 replies
VicKmath7
Mar 25, 2020
sunken rock
Mar 28, 2020
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Geometry
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Reveal topic tags
geometry
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G H BBookmark kLocked kLocked NReply
Source: 12th Russian MO "Leonhard Euler", final round
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VicKmath7
1388 posts
VicKmath7
#1 h  Mar 25, 2020, 7:44 PM
Y by
Given is a triangle ABC with 2<B=180°+<A. Point K is in the interior of the triangle, so that <ACK=2<BCK and L is on AB, so that BK=KL. Prove that CK+AL=AC
Z K Y
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parmenides51
30629 posts
parmenides51
#2 h  Mar 25, 2020, 8:34 PM
Y by
please lock this contest, the contest happens these days, it hasn't ended yet
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parmenides51
30629 posts
parmenides51
#3 h  Mar 28, 2020, 3:34 PM
Y by
contest has ended
Z K Y
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sunken rock
4383 posts
sunken rock
#4 h  Mar 28, 2020, 8:45 PM
Y by
My solution is quite different, although fully synthetic as well.
Pls, see it attached.

Best regards,
sunken rock
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