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k a July Highlights and 2025 AoPS Online Class Information
jwelsh   0
Jul 1, 2025
We are halfway through summer, so be sure to carve out some time to keep your skills sharp and explore challenging topics at AoPS Online and our AoPS Academies (including the Virtual Campus)!

[list][*]Over 60 summer classes are starting at the Virtual Campus on July 7th - check out the math and language arts options for middle through high school levels.
[*]At AoPS Online, we have accelerated sections where you can complete a course in half the time by meeting twice/week instead of once/week, starting on July 8th:
[list][*]MATHCOUNTS/AMC 8 Basics
[*]MATHCOUNTS/AMC 8 Advanced
[*]AMC Problem Series[/list]
[*]Plus, AoPS Online has a special seminar July 14 - 17 that is outside the standard fare: Paradoxes and Infinity
[*]We are expanding our in-person AoPS Academy locations - are you looking for a strong community of problem solvers, exemplary instruction, and math and language arts options? Look to see if we have a location near you and enroll in summer camps or academic year classes today! New locations include campuses in California, Georgia, New York, Illinois, and Oregon and more coming soon![/list]

MOP (Math Olympiad Summer Program) just ended and the IMO (International Mathematical Olympiad) is right around the corner! This year’s IMO will be held in Australia, July 10th - 20th. Congratulations to all the MOP students for reaching this incredible level and best of luck to all selected to represent their countries at this year’s IMO! Did you know that, in the last 10 years, 59 USA International Math Olympiad team members have medaled and have taken over 360 AoPS Online courses. Take advantage of our Worldwide Online Olympiad Training (WOOT) courses
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0 replies
jwelsh
Jul 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
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
J
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Nice Inequality
MathRook7817   0
4 minutes ago
Nice question I found:
0 replies
MathRook7817
4 minutes ago
0 replies
J
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Polynomial NT with analytic flavour?
Aiden-1089   3
N 8 minutes ago by EeEeRUT
Source: APMO 2025 Problem 3
Let $P(x)$ be a non-constant polynomial with integer coefficients such that $P(0) \neq 0$. Let $a_1, a_2, a_3, \dots$ be an infinite sequence of integers such that $P(i - j)$ divides $a_i-a_j$ for all distinct positive integers $i,j$. Prove that the sequence $a_1, a_2, a_3, \dots$ must be constant, that is, $a_n$ equals a constant $c$ for all positive integers $n$.
3 replies
Aiden-1089
2 hours ago
EeEeRUT
8 minutes ago
J
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Inequality
SunnyEvan   8
N 15 minutes ago by arqady
Source: Own
Let $ a,b,c>0 ,$ such that: $ abc=1 .$ Prove that :
$$ \sqrt{64a^2+225}+\sqrt{64b^2+225}+\sqrt{64c^2+225} \leq (7\sqrt3-4)(a+b+c)+21(3-\sqrt3) $$
8 replies
SunnyEvan
Tuesday at 11:45 PM
arqady
15 minutes ago
J
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winning strategy, writing words on a blackboard
parmenides51   1
N 16 minutes ago by LeYohan
Source: 1st Mathematics Regional Olympiad of Mexico Northwest 2018 P2
Alicia and Bob take turns writing words on a blackboard.
The rules are as follows:
a) Any word that has been written cannot be rewritten.
b) A player can only write a permutation of the previous word, or can simply simply remove one letter (whatever you want) from the previous word.
c) The first person who cannot write another word loses.
If Alice starts by typing the word ''Olympics" and Bob's next turn, who, do you think, has a winning strategy and what is it?
1 reply
parmenides51
Sep 6, 2022
LeYohan
16 minutes ago
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Challenge: Make as many positive integers from 2 zeros
Biglion   48
N 26 minutes ago by huajun78
How many positive integers can you make from at most 2 zeros, any math operation and cocatination?
New Rule: The successor function can only be used at most 3 times per number
Starting from 0, 0=0
48 replies
Biglion
Jul 2, 2025
huajun78
26 minutes ago
J
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Arithmetic Sequence of Products
GrantStar   20
N 31 minutes ago by BS2012
Source: IMO Shortlist 2023 N4
Let $a_1, \dots, a_n, b_1, \dots, b_n$ be $2n$ positive integers such that the $n+1$ products
\[a_1 a_2 a_3 \cdots a_n, b_1 a_2 a_3 \cdots a_n, b_1 b_2 a_3 \cdots a_n, \dots, b_1 b_2 b_3 \cdots b_n\]form a strictly increasing arithmetic progression in that order. Determine the smallest possible integer that could be the common difference of such an arithmetic progression.
20 replies
GrantStar
Jul 17, 2024
BS2012
31 minutes ago
J
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A combinatorics stranger.
Imanamiri   1
N an hour ago by Imanamiri
Source: Oral 2004
Let \( a_1, a_2, \ldots, a_n \) be natural numbers such that the residues of \( a_i \mod (n+10) \) include 20 distinct values. Prove that there exists a non-empty subset of these numbers whose sum is divisible by \( n+10 \).
1 reply
Imanamiri
Jul 29, 2025
Imanamiri
an hour ago
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Geo seems familiar?
Aiden-1089   3
N an hour ago by wassupevery1
Source: APMO 2025 Problem 1
Let $ABC$ be an acute triangle inscribed in a circle $\Gamma$. Let $A_1$ be the orthogonal projection of $A$ onto $BC$ so that $AA_1$ is an altitude. Let $B_1$ and $C_1$ be the orthogonal projections of $A_1$ onto $AB$ and $AC$, respectively. Point $P$ is such that quadrilateral $AB_1PC_1$ is convex and has the same area as triangle $ABC$. Is it possible that $P$ strictly lies in the interior of circle $\Gamma$? Justify your answer.
3 replies
Aiden-1089
2 hours ago
wassupevery1
an hour ago
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If OAB and OAC share equal angles and sides, why aren't they congruent?
Merkane   1
N an hour ago by nudinhtien

Problem 1.39 (CGMO 2012/5). Let ABC be a triangle. The incircle of ABC is tangent
to AB and AC at D and E respectively. Let O denote the circumcenter of BCI .
Prove that ∠ODB = ∠OEC. Hints: 643 89 Sol: p.241

While I have solved the problem, I encountered a step that seems logically sound but leads to a contradiction, and I would like help identifying the flaw.

Here is the reasoning I followed:

The quadrilateral ABOC is cyclic.

OB = OC.

∠OAB = ∠OCB.
Similarly, ∠OAC = ∠OBC.

From symmetry and the above, it seems that ∠OAB = ∠OAC.

Since OA is a shared side, I concluded that triangle OAB ≅ triangle OAC.


But clearly, OAB and OAC are not congruent.
Where exactly is the logical error in this argument?
1 reply
Merkane
Today at 4:33 AM
nudinhtien
an hour ago
J
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Functional Equation with non-linear solutions
MrHeccMcHecc   0
an hour ago
Determine all functions $f : \mathbb{Q} \rightarrow \mathbb{Q}$ that satisfy $$2f \left( 2xy+ \frac 12 \right) + 2f(x-y) - 8f(x)f(y)=1$$for all choices of rational numbers $x,y$.
0 replies
MrHeccMcHecc
an hour ago
0 replies
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Maybe unoriginal?
GreekIdiot   5
N an hour ago by Feita
Source: Couldnt find it anywhere
Find all $f:\mathbb R_+ \to \mathbb R_+$ such that for all $x$, $y$ in $\mathbb R_+$ we have $f(x+4f(y))=f(x+3y)+f(y)$.
5 replies
GreekIdiot
2 hours ago
Feita
an hour ago
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Travelling through Mictlán
Justpassingby   2
N an hour ago by LeYohan
Source: 2021 Mexico Center Zone Regional Olympiad, problem 2
The Mictlán is an $n\times n$ board and each border of each $1\times 1$ cell is painted either purple or orange. Initially, a catrina lies inside a $1\times 1$ cell and may move in four directions (up, down, left, right) into another cell with the condition that she may move from one cell to another only if the border that joins them is painted orange. We know that no matter which cell the catrina chooses as a starting point, she may reach all other cells through a sequence of valid movements and she may not abandon the Mictlán (she may not leave the board). What is the maximum number of borders that could have been colored purple?

Proposed by CDMX
2 replies
Justpassingby
Jan 17, 2022
LeYohan
an hour ago
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T belongs on the Newton-Gauss Line
TigerOnion   4
N an hour ago by TigerOnion
Source: Own
Hi guys!
It has been quite a long while ago since I first came across this interesting configuration regarding a property of cyclic quadrilaterals.. Maybe this is a known lemma, but I cannot find any relevant sources regarding it.

Let $ABCD$ be a non-trapezium cyclic quadrilateral. Let $E$ and $F$ be the respective points of intersection of $AD, BC$ and $AB, CD$. The respective angle bisectors $\angle CED$. and $\angle AFD$ meet at point $T$. Prove that $T$ lies on the Newton-Gauss Line of the complete quadrilateral $DAEBF$.
4 replies
TigerOnion
Jul 26, 2025
TigerOnion
an hour ago
J
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Beautiful geo
justalonelyguy   4
N 2 hours ago by geombi
Let \( \triangle ABC \) be an acute triangle with \( AB < AC \). The internal angle bisectors of angles \( \angle ABC \) and \( \angle ACB \) intersect at point \( I \). Let \( D \) be the feet of the perpendicular from \( I \) to \( BC \). Let points \( K \) and \( L \) lie on segments \( IB \) and \( IC \), respectively, such that \( \angle KDL = 90^\circ \). Prove that \[ \angle KAL = \frac{1}{2} \angle BAC. \]
4 replies
justalonelyguy
Jul 28, 2025
geombi
2 hours ago
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Function
AOSP2405   1
N Jun 13, 2025 by Mathzeus1024
If f:R->R, f(x) =-x+{x}(a+2)-{ax} find f(f(x)), where {x} is the fractional part of x.
1 reply
AOSP2405
Feb 17, 2019
Mathzeus1024
Jun 13, 2025
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Reveal topic tags
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algebra
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AOSP2405
235 posts
AOSP2405
#1 h  Feb 17, 2019, 11:44 PM • 1 Y
Y by Adventure10
If f:R->R, f(x) =-x+{x}(a+2)-{ax} find f(f(x)), where {x} is the fractional part of x.
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Mathzeus1024
1081 posts
Mathzeus1024
#2 h  Jun 13, 2025, 9:30 AM
Y by
We can rewrite $f$ as the following (knowing that $\{x\} =x-\lfloor x \rfloor$):

$f(x) = -x+(a+2)(x-\lfloor x \rfloor) - (ax - \lfloor ax \rfloor) = x-(a+2)\lfloor x \rfloor +\lfloor ax \rfloor$;

which leads to:

$\textcolor{red}{f(f(x)) = [x-(a+2)\lfloor x \rfloor +\lfloor ax \rfloor]-(a+2)\lfloor x-(a+2)\lfloor x \rfloor +\lfloor ax \rfloor\rfloor + \lfloor a[ x-(a+2)\lfloor x \rfloor +\lfloor ax \rfloor]\rfloor}$;
This post has been edited 3 times. Last edited by Mathzeus1024, Jun 13, 2025, 10:30 AM
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