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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
and train with the best! Please note that early bird pricing ends August 19th!
Are you tired of the heat and thinking about Fall? You can plan your Fall schedule now with classes at either AoPS Online, AoPS Academy Virtual Campus, or one of our AoPS Academies around the US.

Our full course list for upcoming classes is below:
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0 replies
jwelsh
Jul 1, 2025
0 replies
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
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
again ineq)
Maksat_B   38
N 21 minutes ago by n-k-p
Source: JBMO 2025 pr1
For all positive real numbers \( a, b, c \), prove that
\[
\frac{(a^2 + bc)^2}{b + c} + \frac{(b^2 + ca)^2}{c + a} + \frac{(c^2 + ab)^2}{a + b} \geq \frac{2abc(a + b + c)^2}{ab + bc + ca}.
\]
Proposed by Hakan Karakuş, Türkiye
38 replies
Maksat_B
Jun 26, 2025
n-k-p
21 minutes ago
continuous f(xyf(x+y)) = f(x)+f(y)
jasperE3   1
N 28 minutes ago by Mathzeus1024
Source: mse
Find all continuous functions $f:\mathbb R^+\to\mathbb R^+$ such that:
$$f(xyf(x+y))=f(x)+f(y)$$for all $x,y>0$.
1 reply
jasperE3
4 hours ago
Mathzeus1024
28 minutes ago
Ratio of magical to non-magical numbers
Timta27   0
29 minutes ago
Source: own
Let us call a natural number $n$ magical if there exists a divisor $k$ of $n$ ($k \neq 1$ and $k \neq n$), such that $k^{k} \equiv k \hspace{0.5em} (mod \hspace{0.5em} n)$.

Is it true that the ratio of magical to non-magical numbers in the interval $[1; t]$ trends to some limit as $t \to \infty$?
0 replies
Timta27
29 minutes ago
0 replies
IMO 2025 P2
sarjinius   26
N 30 minutes ago by Royal_mhyasd
Source: 2025 IMO P2
Let $\Omega$ and $\Gamma$ be circles with centres $M$ and $N$, respectively, such that the radius of $\Omega$ is less than the radius of $\Gamma$. Suppose $\Omega$ and $\Gamma$ intersect at two distinct points $A$ and $B$. Line $MN$ intersects $\Omega$ at $C$ and $\Gamma$ at $D$, so that $C, M, N, D$ lie on $MN$ in that order. Let $P$ be the circumcentre of triangle $ACD$. Line $AP$ meets $\Omega$ again at $E\neq A$ and meets $\Gamma$ again at $F\neq A$. Let $H$ be the orthocentre of triangle $PMN$.

Prove that the line through $H$ parallel to $AP$ is tangent to the circumcircle of triangle $BEF$.
26 replies
+3 w
sarjinius
6 hours ago
Royal_mhyasd
30 minutes ago
Bonza functions
KevinYang2.71   23
N 38 minutes ago by ravengsd
Source: 2025 IMO P3
Let $\mathbb{N}$ denote the set of positive integers. A function $f\colon\mathbb{N}\to\mathbb{N}$ is said to be bonza if
\[
f(a)~~\text{divides}~~b^a-f(b)^{f(a)}
\]for all positive integers $a$ and $b$.

Determine the smallest real constant $c$ such that $f(n)\leqslant cn$ for all bonza functions $f$ and all positive integers $n$.
23 replies
+3 w
KevinYang2.71
6 hours ago
ravengsd
38 minutes ago
Prove two circles intersect on line BC
62861   78
N an hour ago by ezpotd
Source: USA Winter TST for IMO 2019, Problem 1 and TST for EGMO 2019, Problem 2, by Merlijn Staps
Let $ABC$ be a triangle and let $M$ and $N$ denote the midpoints of $\overline{AB}$ and $\overline{AC}$, respectively. Let $X$ be a point such that $\overline{AX}$ is tangent to the circumcircle of triangle $ABC$. Denote by $\omega_B$ the circle through $M$ and $B$ tangent to $\overline{MX}$, and by $\omega_C$ the circle through $N$ and $C$ tangent to $\overline{NX}$. Show that $\omega_B$ and $\omega_C$ intersect on line $BC$.

Merlijn Staps
78 replies
62861
Dec 10, 2018
ezpotd
an hour ago
Polish MO Finals 2014, Problem 5
j___d   15
N an hour ago by de-Kirschbaum
Source: Polish MO Finals 2014
Find all pairs $(x,y)$ of positive integers that satisfy
$$2^x+17=y^4$$.
15 replies
j___d
Jul 27, 2016
de-Kirschbaum
an hour ago
number theory or algebra ?
tabel   2
N an hour ago by math90
Source: Romanian NMO 2025 9th grade shortlist
Let $n \geq 1$ be an integer and let $x_1, x_2, \ldots, x_n$ be positive integers (i.e., $x_i \in \mathbb{N}^*$ for all $i$), such that
\[
x_k \leq k \quad \text{for all } k = 1, 2, \ldots, n,
\]and
\[
x_1 + x_2 + \cdots + x_n \text{ is an odd integer}.
\]Prove that there exist $\varepsilon_1, \varepsilon_2, \ldots, \varepsilon_n \in \{1, -1\}$ such that
\[
\varepsilon_1 x_1 + \varepsilon_2 x_2 + \cdots + \varepsilon_n x_n = 1.
\]
2 replies
tabel
Jul 12, 2025
math90
an hour ago
GCD is a perfect square
Timta27   0
an hour ago
Source: own
Let $s(n)$ be the sum of all natural numbers $k$ ($1 \leq k \leq n$) such that $GCD(k,n)$ is a perfect square.

For example, $s(9) = 1+2+4+5+7+8+9 = 36$.

Prove that for every natural number $x$: $s(x^{2})$ is divisible by $x^{2}$.
0 replies
2 viewing
Timta27
an hour ago
0 replies
Combinatorics
BQK   0
an hour ago
I have some problems, can anyone help me please.
0 replies
BQK
an hour ago
0 replies
Problem 2
delegat   153
N 2 hours ago by Konigsberg
Source: 0
Let $n\ge 3$ be an integer, and let $a_2,a_3,\ldots ,a_n$ be positive real numbers such that $a_{2}a_{3}\cdots a_{n}=1$. Prove that
\[(1 + a_2)^2 (1 + a_3)^3 \dotsm (1 + a_n)^n > n^n.\]

Proposed by Angelo Di Pasquale, Australia
153 replies
delegat
Jul 10, 2012
Konigsberg
2 hours ago
Sunny lines
sarjinius   17
N 2 hours ago by thdwlgh1229
Source: 2025 IMO P1
A line in the plane is called $sunny$ if it is not parallel to any of the $x$axis, the $y$axis, or the line $x+y=0$.

Let $n\ge3$ be a given integer. Determine all nonnegative integers $k$ such that there exist $n$ distinct lines in the plane satisfying both of the following:
[list]
[*] for all positive integers $a$ and $b$ with $a+b\le n+1$, the point $(a,b)$ lies on at least one of the lines; and
[*] exactly $k$ of the $n$ lines are sunny.
[/list]
17 replies
+1 w
sarjinius
6 hours ago
thdwlgh1229
2 hours ago
Basic 3-variable inequality
Iveela   11
N 2 hours ago by math-olympiad-clown
Source: 2025 IRN-MNG Friendly Competition
Let $x, y, z$ be three positive reals such that $x + y + z = 3$. Prove that
\[\frac{1}{x + 2} + \frac{1}{y + 2} + \frac{1}{z + 2} + \frac{xyz}{4} \leq \frac{5}{4}.\]
11 replies
Iveela
Jun 8, 2025
math-olympiad-clown
2 hours ago
Geometry finale: radical axis bisects D-altitude
v_Enhance   53
N 2 hours ago by mira74
Source: USA TSTST 2016 Problem 6, by Danielle Wang
Let $ABC$ be a triangle with incenter $I$, and whose incircle is tangent to $\overline{BC}$, $\overline{CA}$, $\overline{AB}$ at $D$, $E$, $F$, respectively. Let $K$ be the foot of the altitude from $D$ to $\overline{EF}$. Suppose that the circumcircle of $\triangle AIB$ meets the incircle at two distinct points $C_1$ and $C_2$, while the circumcircle of $\triangle AIC$ meets the incircle at two distinct points $B_1$ and $B_2$. Prove that the radical axis of the circumcircles of $\triangle BB_1B_2$ and $\triangle CC_1C_2$ passes through the midpoint $M$ of $\overline{DK}$.

Proposed by Danielle Wang
53 replies
v_Enhance
Jun 29, 2016
mira74
2 hours ago
Eventually constant sequence with condition
PerfectPlayer   4
N Jun 1, 2025 by kujyi
Source: Turkey TST 2025 Day 3 P8
A positive real number sequence $a_1, a_2, a_3,\dots $ and a positive integer \(s\) is given.
Let $f_n(0) = \frac{a_n+\dots+a_1}{n}$ and for each $0<k<n$
\[f_n(k)=\frac{a_n+\dots+a_{k+1}}{n-k}-\frac{a_k+\dots+a_1}{k}\]Then for every integer $n\geq s,$ the condition
\[a_{n+1}=\max_{0\leq k<n}(f_n(k))\]is satisfied. Prove that this sequence must be eventually constant.
4 replies
PerfectPlayer
Mar 18, 2025
kujyi
Jun 1, 2025
Eventually constant sequence with condition
G H J
G H BBookmark kLocked kLocked NReply
Source: Turkey TST 2025 Day 3 P8
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PerfectPlayer
15 posts
#1 • 1 Y
Y by sami1618
A positive real number sequence $a_1, a_2, a_3,\dots $ and a positive integer \(s\) is given.
Let $f_n(0) = \frac{a_n+\dots+a_1}{n}$ and for each $0<k<n$
\[f_n(k)=\frac{a_n+\dots+a_{k+1}}{n-k}-\frac{a_k+\dots+a_1}{k}\]Then for every integer $n\geq s,$ the condition
\[a_{n+1}=\max_{0\leq k<n}(f_n(k))\]is satisfied. Prove that this sequence must be eventually constant.
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ehuseyinyigit
868 posts
#2 • 2 Y
Y by sami1618, MS_asdfgzxcvb
For $a_{n+1}=f_n(p)$, prove $a_{n+2}=f_{n+1}(p)$.
This post has been edited 1 time. Last edited by ehuseyinyigit, Mar 23, 2025, 9:24 AM
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egxa
215 posts
#3 • 8 Y
Y by swynca, bin_sherlo, PerfectPlayer, hakN, D.C., AlperenINAN, farhad.fritl, Sadigly
ehuseyinyigit wrote:
Obviously $a_2=a_1$. Also $a_3=max(a_1,a_2-a_1)=a_1$. We have $a_1=a_2=a_3$. We will proceed by using induction. Suppose that $a_1=a_2=\cdots=a_p$ holds, we will prove $a_{p+1}=a_p$. On the other hand, for all $k=1,2,\cdots,p-1$
$$f_p(k)=\dfrac{\sum_{i=k+1}^{p}{a_i}}{n-k}+\dfrac{\sum_{i=1}^{k}{a_i}}{k}=a_k-a_1=0$$Thus, we obtain the following
$$a_{p+1}=max(f_p(0),f_p(1),\cdots,f_p(p-1))=max(a_1,0,0,\cdots,0)=a_1$$implying that the sequence $(a)_1^n$ must be eventually constant as desired.

adam olimpiyati bitirmiş
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egxa
215 posts
#4
Y by
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This post has been edited 1 time. Last edited by egxa, Mar 23, 2025, 8:05 AM
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kujyi
2 posts
#5
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any answers in detail?
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