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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.

WOOT early bird pricing is in effect, don’t miss out! If you took MathWOOT Level 2 last year, no worries, it is all new problems this year! Our Worldwide Online Olympiad Training program is for high school level competitors. AoPS designed these courses to help our top students get the deep focus they need to succeed in their specific competition goals. Check out the details at this link for all our WOOT programs in math, computer science, chemistry, and physics.

Looking for summer camps in math and language arts? Be sure to check out the video-based summer camps offered at the Virtual Campus that are 2- to 4-weeks in duration. 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)!

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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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Counting Diagonals
Riemann123   3
N 12 minutes ago by Rohit-2006
Source: UMD I #12
A square has $2$ diagonals. A regular pentagon has $5$ diagonals. $n$ is the smallest positive integer such that a regular $n$-gon has greater than or equal to $2024$ diagonals. What is the sum of the digits of $n$?
\[\mathrm a. ~10\qquad \mathrm b. ~11 \qquad \mathrm c. ~12 \qquad\mathrm d. ~13\qquad\mathrm e. ~14\]
3 replies
Riemann123
Nov 6, 2024
Rohit-2006
12 minutes ago
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Modern Olympiad Number Theory
Wizard_32   141
N 23 minutes ago by idk12345678
Hey everyone!

I have written a book on Olympiad Number Theory! This book is intended for anyone preparing for their national and international mathematics Olympiads. It covers all the fundamentals, starting from scratch, and moves to almost all the advanced topics you would find in Modern Olympiad problems.

I have taken a very conceptual approach to the subject, and gone into depth of each topic I covered. You can take this book to be a print of how my brain perceives Number Theory :P Apart from the theory, I have filled it with handpicked solved examples, some of which are classics, while some are Olympiad problems. The solutions I have discussed are with motivation and would help you develop intuition. The problem section in each chapter starts off easy, and ends at some very challenging albeit beautiful problems. One of my favourite features of this book is the Hints and Solutions system (just like in EGMO) to the problems. Some chapters also have a "special section", which is about an interesting topic related to the chapter, which might even be useful in problem solving. You can read more in the Preface of the book.

This book is my way of giving back to the Olympiad community for the world of knowledge, friends and experience it has given me. I hope you enjoy it :)

(Published on Academia.edu) Click here to read it. Current Version: V2.

EDIT: Since many people were finding it hard to download the PDF via Academia, here's the drive link. This is Version 2 of the book with (some) typos fixed.
https://drive.google.com/file/d/1BcJTLjQaelZ4w_70oHKyImC2I8zLfyrt/view?usp=sharing

EDIT 2: I am making a list of typos and corrections. Feel free to PM me any typos if you find. I will keep updating the list here: https://drive.google.com/file/d/1xBKFZ7JV4PkYLHCFjZKdXVia0zsd2I_3/view?usp=sharing

EDIT 3: There's a discussion forum for this on AoPS now (wohoo!). Here's the link:
https://artofproblemsolving.com/community/c1610044_modern_olympiad_number_theory

EDIT 4: Here's the old list of typos from the first version: https://drive.google.com/file/d/1C4Z_-utSaJ_Ag1BU9iuzcN9GGN6AHOIv/view?usp=sharing
141 replies
1 viewing
Wizard_32
Nov 16, 2020
idk12345678
23 minutes ago
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Factor of P(x)
Brut3Forc3   16
N 26 minutes ago by Rohit-2006
Source: 1976 USAMO Problem 5
If $ P(x),Q(x),R(x)$, and $ S(x)$ are all polynomials such that \[ P(x^5)+xQ(x^5)+x^2R(x^5)=(x^4+x^3+x^2+x+1)S(x),\] prove that $ x-1$ is a factor of $ P(x)$.
16 replies
Brut3Forc3
Apr 4, 2010
Rohit-2006
26 minutes ago
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m> p if the diophantine p^2 + m^2 = n^2 holds for m,n pos. integers
parmenides51   9
N an hour ago by LeYohan
Source: 2021 Austrian Mathematical Olympiad Junior Regional Competition , Problem 4
Let $p$ be a prime number and let $m$ and $n$ be positive integers with $p^2 + m^2 = n^2$.
Prove that $m> p$.

(Karl Czakler)
9 replies
parmenides51
Sep 12, 2021
LeYohan
an hour ago
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true or false statement
pennypc123456789   4
N an hour ago by GeoMorocco
if $a,b,c$ are positive real numbers, $k \ge 3$ then
$$ \frac{a + b}{a + kb + c} + \dfrac{b + c}{b + kc + a}+\dfrac{c + a}{c + ka + b} \geq \dfrac{6}{k+2}$$
4 replies
pennypc123456789
5 hours ago
GeoMorocco
an hour ago
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Spot the symmetry
FAA2533   4
N 2 hours ago by Blackbeam999
Source: BdMO 2024 Higher Secondary National P5
Let $I$ be the incenter of $\triangle ABC$ and $P$ be a point such that $PI$ is perpendicular to $BC$ and $PA$ is parallel to $BC$. Let the line parallel to $BC$, which is tangent to the incircle of $\triangle ABC$, intersect $AB$ and $AC$ at points $Q$ and $R$ respectively. Prove that $\angle BPQ = \angle CPR$.
4 replies
FAA2533
Mar 18, 2024
Blackbeam999
2 hours ago
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Question 2
Valentin Vornicu   88
N 2 hours ago by Nari_Tom
Consider five points $ A$, $ B$, $ C$, $ D$ and $ E$ such that $ ABCD$ is a parallelogram and $ BCED$ is a cyclic quadrilateral. Let $ \ell$ be a line passing through $ A$. Suppose that $ \ell$ intersects the interior of the segment $ DC$ at $ F$ and intersects line $ BC$ at $ G$. Suppose also that $ EF = EG = EC$. Prove that $ \ell$ is the bisector of angle $ DAB$.

Author: Charles Leytem, Luxembourg
88 replies
Valentin Vornicu
Jul 25, 2007
Nari_Tom
2 hours ago
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Grid of Quadratics: Root-Enabled Arrangement
mojyla222   1
N 2 hours ago by YaoAOPS
Source: IDMC 2025 P6
Is it possible to place all monic quadratic polynomials with natural coefficients in the cells of an infinite grid on the plane, such that each polynomial is placed in exactly one cell, and for every finite rectangular subgrid with an area greater than one, the sum of the polynomials within that rectangle has a real root?


Proposed by Mojtaba Zare
1 reply
mojyla222
Today at 5:09 AM
YaoAOPS
2 hours ago
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Incenter and concurrency
jenishmalla   7
N 2 hours ago by brute12
Source: 2025 Nepal ptst p3 of 4
Let the incircle of $\triangle ABC$ touch sides $BC$, $CA$, and $AB$ at points $D$, $E$, and $F$, respectively. Let $D'$ be the diametrically opposite point of $D$ with respect to the incircle. Let lines $AD'$ and $AD$ intersect the incircle again at $X$ and $Y$, respectively. Prove that the lines $DX$, $D'Y$, and $EF$ are concurrent, i.e., the lines intersect at the same point.

(Kritesh Dhakal, Nepal)
7 replies
jenishmalla
Mar 15, 2025
brute12
2 hours ago
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Dear Sqing: So Many Inequalities...
hashtagmath   32
N 2 hours ago by aiops
I have noticed thousands upon thousands of inequalities that you have posted to HSO and was wondering where you get the inspiration, imagination, and even the validation that such inequalities are true? Also, what do you find particularly appealing and important about specifically inequalities rather than other branches of mathematics? Thank you :)
32 replies
hashtagmath
Oct 30, 2024
aiops
2 hours ago
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Squares on height in right triangle
Miquel-point   1
N 2 hours ago by LiamChen
Source: Romanian NMO 2025 7.4
Consider the right-angled triangle $ABC$ with $\angle A$ right and $AD\perp BC$, $D\in BC$. On the ray $[AD$ we take two points $E$ and $H$ so that $AE=AC$ and $AH=AB$. Consider the squares $AEFG$ and $AHJI$ containing inside $C$ and $B$, respectively. If $K=EG\cap AC$ and $L=IH\cap AB$, $N=IL\cap GK$ and $M=IB\cap GC$, prove that $LK\parallel BC$ and that $A$, $N$ and $M$ are collinear.
1 reply
Miquel-point
Yesterday at 8:20 PM
LiamChen
2 hours ago
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H is incenter of DEF
Melid   1
N 3 hours ago by Melid
Source: own?
In acute scalene triangle ABC, let H be its orthocenter and O be its circumcenter. Circumcircles of triangle AHO, BHO, CHO intersect with circumcircle of triangle ABC at D, E, F, respectively. Prove that incenter of triangle DEF is H.
1 reply
Melid
3 hours ago
Melid
3 hours ago
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Inequality results about some function
CatalinBordea   2
N 3 hours ago by Rohit-2006
Source: Romania National Olympiad 2016, grade x, p.2
Let be a function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ satisfying the conditions:
$$ \left\{\begin{matrix} f(x+y) &\le & f(x)+f(y) \\ f(tx+(1-t)y) &\le & t(f(x)) +(1-t)f(y) \end{matrix}\right. , $$for all real numbers $ x,y,t $ with $ t\in [0,1] . $

Prove that:
a) $ f(b)+f(c)\le f(a)+f(d) , $ for any real numbers $ a,b,c,d $ such that $ a\le b\le c\le d $ and $ d-c=b-a. $
b) for any natural number $ n\ge 3 $ and any $ n $ real numbers $ x_1,x_2,\ldots ,x_n, $ the following inequality holds.
$$ f\left( \sum_{1\le i\le n} x_i \right) +(n-2)\sum_{1\le i\le n} f\left( x_i \right)\ge \sum_{1\le i<j\le n} f\left( x_i+x_j \right) $$
2 replies
CatalinBordea
Aug 25, 2019
Rohit-2006
3 hours ago
J
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confusing inequality
giangtruong13   4
N 3 hours ago by giangtruong13
Let $a,b,c>0$ such that: $a^2b^2+ c^2b^2+ a^2c^2=3(abc)^2$. Prove that: $$\sum \frac{b+c}{a} \geq 2\sqrt{3(ab+bc+ca)}$$
4 replies
giangtruong13
Apr 18, 2025
giangtruong13
3 hours ago
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J
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Square and equilateral triangle
m4thbl3nd3r   2
N Apr 4, 2025 by m4thbl3nd3r
Let $ABCD$ be a square and a point $X$ lies on the interior of $ABCD$ such that triangle $BDX$ is equilateral. Evaluate $\angle AXD$
2 replies
m4thbl3nd3r
Apr 4, 2025
m4thbl3nd3r
Apr 4, 2025
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Square and equilateral triangle
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Reveal topic tags
geometry
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m4thbl3nd3r
281 posts
m4thbl3nd3r
#1 h  Apr 4, 2025, 5:06 AM • 1 Y
Y by fAaAtDoOoG
Let $ABCD$ be a square and a point $X$ lies on the interior of $ABCD$ such that triangle $BDX$ is equilateral. Evaluate $\angle AXD$
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mpcnotnpc
51 posts
mpcnotnpc
#2 h  Apr 4, 2025, 5:10 AM • 1 Y
Y by fAaAtDoOoG
what? is this oly?
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m4thbl3nd3r
281 posts
m4thbl3nd3r
#3 h  Apr 4, 2025, 5:17 AM • 1 Y
Y by fAaAtDoOoG
No, this is just a step in a geometry problem I was solving.
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