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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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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.

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0 replies
jwelsh
Jul 1, 2025
0 replies
IMO ShortList 1998, geometry problem 5
nttu   34
N 4 minutes ago by ihatemath123
Source: IMO ShortList 1998, geometry problem 5
Let $ABC$ be a triangle, $H$ its orthocenter, $O$ its circumcenter, and $R$ its circumradius. Let $D$ be the reflection of the point $A$ across the line $BC$, let $E$ be the reflection of the point $B$ across the line $CA$, and let $F$ be the reflection of the point $C$ across the line $AB$. Prove that the points $D$, $E$ and $F$ are collinear if and only if $OH=2R$.
34 replies
+1 w
nttu
Oct 14, 2004
ihatemath123
4 minutes ago
IMO 2014 Problem 4
ipaper   174
N 5 minutes ago by reni_wee
Let $P$ and $Q$ be on segment $BC$ of an acute triangle $ABC$ such that $\angle PAB=\angle BCA$ and $\angle CAQ=\angle ABC$. Let $M$ and $N$ be the points on $AP$ and $AQ$, respectively, such that $P$ is the midpoint of $AM$ and $Q$ is the midpoint of $AN$. Prove that the intersection of $BM$ and $CN$ is on the circumference of triangle $ABC$.

Proposed by Giorgi Arabidze, Georgia.
174 replies
ipaper
Jul 9, 2014
reni_wee
5 minutes ago
IMO Genre Predictions
ohiorizzler1434   103
N 6 minutes ago by DangurKangur
Everybody, with IMO upcoming, what are you predictions for the problem genres?


Personally I predict: predict
103 replies
ohiorizzler1434
May 3, 2025
DangurKangur
6 minutes ago
exploding bombs on a board
gggzul   1
N 42 minutes ago by Razorrizelim
Integer $n\geq 2$ and a board $3n\times 3n$ are given. Evil Dan had chosen a number of squares on the board and placed bombs on them. After a minute, the bombs exploded and left smoke behind, which covered each square $3\times 3$ centered on a square that contained a bomb. Determine, in terms of $n$, the smallest number of bombs that were placed, such that after the explosion exactly $1$ square will not be covered by smoke.
1 reply
gggzul
2 hours ago
Razorrizelim
42 minutes ago
A^2+B^2=AB+BA
mathisreal   1
N Today at 10:50 AM by loup blanc
Source: OIMU 2023 #4
Determine all pairs of real matrices $(A,B)$ of size $2\times 2$ with $A\neq B$ such that
\[A^2+B^2=AB+BA=2I\]
1 reply
mathisreal
Today at 12:03 AM
loup blanc
Today at 10:50 AM
ISI UGB 2025 P3
SomeonecoolLovesMaths   14
N Today at 10:10 AM by Cats_on_a_computer
Source: ISI UGB 2025 P3
Suppose $f : [0,1] \longrightarrow \mathbb{R}$ is differentiable with $f(0) = 0$. If $|f'(x) | \leq f(x)$ for all $x \in [0,1]$, then show that $f(x) = 0$ for all $x$.
14 replies
SomeonecoolLovesMaths
May 11, 2025
Cats_on_a_computer
Today at 10:10 AM
ISI UGB 2025 P1
SomeonecoolLovesMaths   10
N Today at 8:03 AM by Cats_on_a_computer
Source: ISI UGB 2025 P1
Suppose $f \colon \mathbb{R} \longrightarrow \mathbb{R}$ is differentiable and $| f'(x)| < \frac{1}{2}$ for all $x \in \mathbb{R}$. Show that for some $x_0 \in \mathbb{R}$, $f \left( x_0 \right) = x_0$.
10 replies
SomeonecoolLovesMaths
May 11, 2025
Cats_on_a_computer
Today at 8:03 AM
2024 Putnam A1
KevinYang2.71   23
N Today at 12:05 AM by megahertz13
Determine all positive integers $n$ for which there exists positive integers $a$, $b$, and $c$ satisfying
\[
2a^n+3b^n=4c^n.
\]
23 replies
KevinYang2.71
Dec 10, 2024
megahertz13
Today at 12:05 AM
Matrices satisfy three conditions
ThE-dArK-lOrD   4
N Yesterday at 7:53 PM by Adustat
Source: IMC 2019 Day 1 P5
Determine whether there exist an odd positive integer $n$ and $n\times n$ matrices $A$ and $B$ with integer entries, that satisfy the following conditions:
[list=1]
[*]$\det (B)=1$;[/*]
[*]$AB=BA$;[/*]
[*]$A^4+4A^2B^2+16B^4=2019I$.[/*]
[/list]
(Here $I$ denotes the $n\times n$ identity matrix.)

Proposed by Orif Ibrogimov, ETH Zurich and National University of Uzbekistan
4 replies
ThE-dArK-lOrD
Jul 31, 2019
Adustat
Yesterday at 7:53 PM
Random points in the unit circle
math90   4
N Yesterday at 7:52 PM by Adustat
Source: IMC 2019 Day 2 P10
$2019$ points are chosen at random, independently, and distributed uniformly in the unit disc $\{(x,y)\in\mathbb R^2: x^2+y^2\le 1\}$. Let $C$ be the convex hull of the chosen points. Which probability is larger: that $C$ is a polygon with three vertices, or a polygon with four vertices?

Proposed by Fedor Petrov, St. Petersburg State University
4 replies
math90
Jul 31, 2019
Adustat
Yesterday at 7:52 PM
Spectrum of a function of permutations.
loup blanc   9
N Yesterday at 7:15 PM by loup blanc
Let $n$ be an even integer and $J_n$ be the $n\times n$ matrix of ones.
Let $S_n$ be the set of $n\times n$ permutation matrices, $f:P\in S_n\mapsto P^2+nP-J_n-I_n$
and $Z(P)=\{|\lambda|;\lambda\in spectrum(f(P))\}$.
Show that $\bigcap_{P\in S_n} Z(P)$ is constitued of 2 elements to be determined.
9 replies
loup blanc
Jun 7, 2025
loup blanc
Yesterday at 7:15 PM
Romania NMO 2023 Grade 11 P4
DanDumitrescu   6
N Yesterday at 6:58 PM by Filipjack
Source: Romania National Olympiad 2023
We consider a function $f:\mathbb{R} \rightarrow \mathbb{R}$ for which there exist a differentiable function $g : \mathbb{R} \rightarrow \mathbb{R}$ and exist a sequence $(a_n)_{n \geq 1}$ of real positive numbers, convergent to $0,$ such that

\[
    g'(x) = \lim_{n \to \infty} \frac{f(x + a_n) - f(x)}{a_n}, \forall x \in \mathbb{R}.
    \]
a) Give an example of such a function f that is not differentiable at any point $x \in \mathbb{R}.$

b) Show that if $f$ is continuous on $\mathbb{R}$, then $f$ is differentiable on $\mathbb{R}.$
6 replies
DanDumitrescu
Apr 14, 2023
Filipjack
Yesterday at 6:58 PM
The two-function hypothesis of the mean value theorem
fxandi   1
N Yesterday at 4:29 PM by fxandi
Given the constants $a$$,$ $b$$,$ $p$$,$ and $q$ with $p < a < b < q.$

Function $f : [a, b] \to \mathbb{R}$ satisfies the condition of the mean value theorem and function $g : [p, q] \to \mathbb{R}$ has the property $g(x) = f(x)$ on the interval $[a, b],$ but $g$ does not satisfy the condition of the mean value theorem on the interval $[p, q].$ Investigate whether there is a value of $c$ on the interval $(p, q)$ such that $g'(c) = \dfrac{g(q) - g(p)}{q - p}.$ Prove your answer$.$
1 reply
fxandi
Jul 6, 2025
fxandi
Yesterday at 4:29 PM
Nice series
Tricky123   6
N Yesterday at 1:45 PM by Riptide1901
Q) Does the series converges?
$\sum_{p\rightarrow prime}\frac{1}{p^{k}}$ where $k\in\mathbb{N}$

How to solve this question using basics analysis and number theory
6 replies
Tricky123
Jul 6, 2025
Riptide1901
Yesterday at 1:45 PM
A second final attempt to make a combinatorics problem
JARP091   2
N May 29, 2025 by JARP091
Source: At the time of writing this problem I do not know the source if any
Arthur Morgan is playing a game.

He has $n$ eggs, each with a hardness value $k_1, k_2, \dots, k_n$, where $\{k_1, k_2, \dots, k_n\}$ is a permutation of the set $\{1, 2, \dots, n\}$. He is throwing the eggs from an $m$-floor building.

When the $i$-th egg is dropped from the $j$-th floor, its new hardness becomes
\[
\left\lfloor \frac{k_i}{j+1} \right\rfloor.
\]If $\left\lfloor \frac{k_i}{j+1} \right\rfloor = 0$, then the egg breaks and cannot be used again.

Arthur can drop each egg from a particular floor at most once.
For which values of $n$ and $m$ can Arthur always determine the correct ordering of the eggs according to their initial hardness values?
Note: The problem might be wrong or too easy
2 replies
JARP091
May 25, 2025
JARP091
May 29, 2025
A second final attempt to make a combinatorics problem
G H J
Source: At the time of writing this problem I do not know the source if any
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JARP091
120 posts
#1
Y by
Arthur Morgan is playing a game.

He has $n$ eggs, each with a hardness value $k_1, k_2, \dots, k_n$, where $\{k_1, k_2, \dots, k_n\}$ is a permutation of the set $\{1, 2, \dots, n\}$. He is throwing the eggs from an $m$-floor building.

When the $i$-th egg is dropped from the $j$-th floor, its new hardness becomes
\[
\left\lfloor \frac{k_i}{j+1} \right\rfloor.
\]If $\left\lfloor \frac{k_i}{j+1} \right\rfloor = 0$, then the egg breaks and cannot be used again.

Arthur can drop each egg from a particular floor at most once.
For which values of $n$ and $m$ can Arthur always determine the correct ordering of the eggs according to their initial hardness values?
Note: The problem might be wrong or too easy
This post has been edited 1 time. Last edited by JARP091, May 25, 2025, 2:46 PM
Reason: Wrongly LaTeXted
Z K Y
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JARP091
120 posts
#2
Y by
Bump for this problem
Z K Y
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JARP091
120 posts
#3
Y by
Drop something from 1, if it breaks then its 0

if it doesnt break, it is currently 1, we have one more one, and we have a number we are trying to find out

we cant drop the number we are interested in from floor 1, otherwise we lose information

hence drop another egg from floor 2, if it doesnt break, it was 3, we started with a 2, and ther last egg is 1, if it breaks, then we either dropped a 1, or we dropped a 2, and so the possible outputs are (1,0,2) (1,0,3) (1,0,1), now we cant figure out what state the last egg is in, so it is impossible.

For n $>$ 3, n = 3 is a subproblem that cannot be solved.

Hence only possible solutions are:

i) n = 1, m $\geq$ 1

ii) n = 2, m $\geq$ 2
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