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k a May Highlights and 2025 AoPS Online Class Information
jlacosta   0
May 1, 2025
May is an exciting month! National MATHCOUNTS is the second week of May in Washington D.C. and our Founder, Richard Rusczyk will be presenting a seminar, Preparing Strong Math Students for College and Careers, on May 11th.

Are you interested in working towards MATHCOUNTS and don’t know where to start? We have you covered! If you have taken Prealgebra, then you are ready for MATHCOUNTS/AMC 8 Basics. Already aiming for State or National MATHCOUNTS and harder AMC 8 problems? Then our MATHCOUNTS/AMC 8 Advanced course is for you.

Summer camps are starting next month at the Virtual Campus in math and language arts that are 2 - to 4 - weeks in duration. Spaces are still available - don’t miss your chance to have an enriching summer experience. There are middle and high school competition math camps as well as Math Beasts camps that review key topics coupled with fun explorations covering areas such as graph theory (Math Beasts Camp 6), cryptography (Math Beasts Camp 7-8), and topology (Math Beasts Camp 8-9)!

Be sure to mark your calendars for the following upcoming events:
[list][*]May 9th, 4:30pm PT/7:30pm ET, Casework 2: Overwhelming Evidence — A Text Adventure, a game where participants will work together to navigate the map, solve puzzles, and win! All are welcome.
[*]May 19th, 4:30pm PT/7:30pm ET, What's Next After Beast Academy?, designed for students finishing Beast Academy and ready for Prealgebra 1.
[*]May 20th, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 1 Math Jam, Problems 1 to 4, join the Canada/USA Mathcamp staff for this exciting Math Jam, where they discuss solutions to Problems 1 to 4 of the 2025 Mathcamp Qualifying Quiz!
[*]May 21st, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 2 Math Jam, Problems 5 and 6, Canada/USA Mathcamp staff will discuss solutions to Problems 5 and 6 of the 2025 Mathcamp Qualifying Quiz![/list]
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0 replies
jlacosta
May 1, 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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Tangencies with cyclic quadrilateral
tapir1729   21
N an hour ago by Mathandski
Source: TSTST 2024, problem 4
Let $ABCD$ be a quadrilateral inscribed in a circle with center $O$ and $E$ be the intersection of segments $AC$ and $BD$. Let $\omega_1$ be the circumcircle of $ADE$ and $\omega_2$ be the circumcircle of $BCE$. The tangent to $\omega_1$ at $A$ and the tangent to $\omega_2$ at $C$ meet at $P$. The tangent to $\omega_1$ at $D$ and the tangent to $\omega_2$ at $B$ meet at $Q$. Show that $OP=OQ$.

Merlijn Staps
21 replies
tapir1729
Jun 24, 2024
Mathandski
an hour ago
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Binary multiples of three
tapir1729   8
N an hour ago by Mathandski
Source: TSTST 2024, problem 5
For a positive integer $k$, let $s(k)$ denote the number of $1$s in the binary representation of $k$. Prove that for any positive integer $n$,
\[\sum_{i=1}^{n}(-1)^{s(3i)} > 0.\]Holden Mui
8 replies
tapir1729
Jun 24, 2024
Mathandski
an hour ago
J
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Max and min of ab+bc+ca-abc
Tiira   6
N an hour ago by MathsII-enjoy
a, b and c are three non-negative reel numbers such that a+b+c=1.
What are the extremums of
ab+bc+ca-abc
?
6 replies
Tiira
Jan 29, 2021
MathsII-enjoy
an hour ago
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Geometry
MathsII-enjoy   3
N an hour ago by MathsII-enjoy
Given triangle $ABC$ inscribed in $(O)$ with $M$ being the midpoint of $BC$. The tangents at $B, C$ of $(O)$ intersect at $D$. Let $N$ be the projection of $O$ onto $AD$. On the perpendicular bisector of $BC$, take a point $K$ that is not on $(O)$ and different from M. Circle $(KBC)$ intersects $AK$ at $F$. Lines $NF$ and $AM$ intersect at $E$. Prove that $AEF$ is an isosceles triangle.
3 replies
MathsII-enjoy
May 15, 2025
MathsII-enjoy
an hour ago
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IMO 2018 Problem 2
juckter   98
N 2 hours ago by ezpotd
Find all integers $n \geq 3$ for which there exist real numbers $a_1, a_2, \dots a_{n + 2}$ satisfying $a_{n + 1} = a_1$, $a_{n + 2} = a_2$ and
$$a_ia_{i + 1} + 1 = a_{i + 2},$$for $i = 1, 2, \dots, n$.

Proposed by Patrik Bak, Slovakia
98 replies
1 viewing
juckter
Jul 9, 2018
ezpotd
2 hours ago
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Element sum of k others
akasht   19
N 2 hours ago by ezpotd
Source: ISL 2022 A2
Let $k\ge2$ be an integer. Find the smallest integer $n \ge k+1$ with the property that there exists a set of $n$ distinct real numbers such that each of its elements can be written as a sum of $k$ other distinct elements of the set.
19 replies
akasht
Jul 9, 2023
ezpotd
2 hours ago
J
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Least swaps to get any labeling of a regular 99-gon
Photaesthesia   9
N 2 hours ago by Blast_S1
Source: 2024 China MO, Day 2, Problem 6
Let $P$ be a regular $99$-gon. Assign integers between $1$ and $99$ to the vertices of $P$ such that each integer appears exactly once. (If two assignments coincide under rotation, treat them as the same. ) An operation is a swap of the integers assigned to a pair of adjacent vertices of $P$. Find the smallest integer $n$ such that one can achieve every other assignment from a given one with no more than $n$ operations.

Proposed by Zhenhua Qu
9 replies
Photaesthesia
Nov 29, 2023
Blast_S1
2 hours ago
J
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Angles in a triangle with integer cotangents
Stear14   0
2 hours ago
In a triangle $ABC$, the point $M$ is the midpoint of $BC$ and $N$ is a point on the side $BC$ such that $BN:NC=2:1$. The cotangents of the angles $\angle BAM$, $\angle MAN$, and $\angle NAC$ are positive integers $k,m,n$.
(a) Show that the cotangent of the angle $\angle BAC$ is also an integer and equals $m-k-n$.
(b) Show that there are infinitely many possible triples $(k,m,n)$, some of which consisting of Fibonacci numbers.
0 replies
Stear14
2 hours ago
0 replies
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R+ FE f(f(xy)+y)=(x+1)f(y)
jasperE3   1
N 3 hours ago by maromex
Source: p24734470
Find all functions $f:\mathbb R^+\to\mathbb R^+$ such that for all positive real numbers $x$ and $y$:
$$f(f(xy)+y)=(x+1)f(y).$$
1 reply
jasperE3
5 hours ago
maromex
3 hours ago
J
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Polynomial Minimization
ReticulatedPython   4
N 4 hours ago by jasperE3
Consider the polynomial $$p(x)=x^{n+1}-x^{n}$$, where $x, n \in \mathbb{R+}.$

(a) Prove that the minimum value of $p(x)$ always occurs at $x=\frac{n}{n+1}.$
4 replies
ReticulatedPython
May 6, 2025
jasperE3
4 hours ago
J
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An important lemma of isogonal conjugate points
buratinogigle   6
N 4 hours ago by buratinogigle
Source: Own
Let $P$ and $Q$ be two isogonal conjugate with respect to triangle $ABC$. Let $S$ and $T$ be two points lying on the circle $(PBC)$ such that $PS$ and $PT$ are perpendicular and parallel to bisector of $\angle BAC$, respectively. Prove that $QS$ and $QT$ bisect two arcs $BC$ containing $A$ and not containing $A$, respectively, of $(ABC)$.
6 replies
buratinogigle
Mar 23, 2025
buratinogigle
4 hours ago
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Trig Identity
gauss202   2
N 4 hours ago by MathIQ.
Simplify $\dfrac{1-\cos \theta + \sin \theta}{\sqrt{1 - \cos \theta + \sin \theta - \sin \theta \cos \theta}}$
2 replies
gauss202
May 14, 2025
MathIQ.
4 hours ago
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A difficult problem [tangent circles in right triangles]
ThAzN1   48
N 4 hours ago by Autistic_Turk
Source: IMO ShortList 1998, geometry problem 8; Yugoslav TST 1999
Let $ABC$ be a triangle such that $\angle A=90^{\circ }$ and $\angle B<\angle C$. The tangent at $A$ to the circumcircle $\omega$ of triangle $ABC$ meets the line $BC$ at $D$. Let $E$ be the reflection of $A$ in the line $BC$, let $X$ be the foot of the perpendicular from $A$ to $BE$, and let $Y$ be the midpoint of the segment $AX$. Let the line $BY$ intersect the circle $\omega$ again at $Z$.

Prove that the line $BD$ is tangent to the circumcircle of triangle $ADZ$.

comment
48 replies
ThAzN1
Oct 17, 2004
Autistic_Turk
4 hours ago
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2018 Mock ARML I --7 2^n | \prod^{2048}_{k=0} C(2k , k)
parmenides51   3
N 4 hours ago by MathIQ.
Find the largest integer $n$ such that $2^n$ divides $\prod^{2048}_{k=0} {2k \choose k}$.
3 replies
parmenides51
Jan 17, 2024
MathIQ.
4 hours ago
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geometry problem
kjhgyuio   1
N Apr 10, 2025 by kjhgyuio
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1 reply
kjhgyuio
Apr 10, 2025
kjhgyuio
Apr 10, 2025
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geometry problem
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Reveal topic tags
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kjhgyuio
#1 h  Apr 10, 2025, 11:11 AM
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69 posts
kjhgyuio
#2 h  Apr 10, 2025, 11:13 AM
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