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k a August Highlights and 2025 AoPS Online Class Information
jwelsh   0
Friday at 2:14 PM
CONGRATULATIONS to all the competitors at this year’s International Mathematical Olympiad (IMO)! The US Team took second place with 5 gold medals and 1 silver - we are proud to say that each member of the 2025 IMO team has participated in an AoPS WOOT (Worldwide Online Olympiad Training) class!

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0 replies
jwelsh
Friday at 2:14 PM
0 replies
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Geometry Problem
Hopeooooo   14
N 21 minutes ago by Royal_mhyasd
Source: SRMC 2022 P1
Convex quadrilateral $ABCD$ is inscribed in circle $w.$Rays $AB$ and $DC$ intersect at $K.\ L$ is chosen on the diagonal $BD$ so that $\angle BAC= \angle DAL.\ M$ is chosen on the segment $KL$ so that $CM \mid\mid BD.$ Prove that line $BM$ touches $w.$
(Kungozhin M.)
14 replies
1 viewing
Hopeooooo
May 23, 2022
Royal_mhyasd
21 minutes ago
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Police officers catching thief
JustPostNorthKoreaTST   0
41 minutes ago
Source: 2016 North Korea TST P6
There are $n$ stations in the sky, and any two stations are connected by exactly one bridge; every bridge has the same length and does not intersect with any other. $n$ police officers are standing at these stations (there may be more than one police officer at a station), and a thief is standing on a bridge. During a patrol, each police officer can choose to cross a bridge to move to the next station or remain in place, but at least one police officer must move. The thief can choose to move from one bridge to another if he is willing to. Assume that each police officer moves at the same speed and the thief moves at five times the speed of the police officers. If a thief and a police officer are in the same location at a certain moment, then the thief will be caught.
Given a positive integer $k$, show that for any initial position of the police officers, the thief has an initial position such that for any $k$ patrols, the thief has a strategy to avoid being caught during the $k$ patrols.
0 replies
JustPostNorthKoreaTST
41 minutes ago
0 replies
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IMO ShortList 1998, combinatorics theory problem 7
orl   10
N an hour ago by legogubbe
Source: IMO ShortList 1998, combinatorics theory problem 7
A solitaire game is played on an $m\times n$ rectangular board, using $mn$ markers which are white on one side and black on the other. Initially, each square of the board contains a marker with its white side up, except for one corner square, which contains a marker with its black side up. In each move, one may take away one marker with its black side up, but must then turn over all markers which are in squares having an edge in common with the square of the removed marker. Determine all pairs $(m,n)$ of positive integers such that all markers can be removed from the board.
10 replies
orl
Oct 22, 2004
legogubbe
an hour ago
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hard 3 vars symetric
perfect_square   0
an hour ago
Let $a,b,c \ge 0$ which satisfy:
$ \begin{cases} a+b+c=4 \\ a^4+b^4+c^4 =18 \end{cases} $
Prove that: $ab+bc+ca \le 5$
0 replies
perfect_square
an hour ago
0 replies
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Limit of expression
enter16180   8
N Today at 5:13 AM by YaoAOPS
Source: IMC 2025, Problem 10
For any positive integer $N$, let $S_N$ be the number of pairs of integers $1 \leq a, b \leq N$ such that the number $\left(a^2+a\right)\left(b^2+b\right)$ is a perfect square. Prove that the limit
$$ \lim _{N \rightarrow \infty} \frac{S_N}{N} $$exists and find its value.
8 replies
enter16180
Jul 31, 2025
YaoAOPS
Today at 5:13 AM
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expected value of maximum of random process
enter16180   4
N Today at 12:01 AM by Agsh2005
Source: IMC 2025, Problem 9
Let $n$ be a positive integer. Consider the following random process which produces $n$ sequence of $n$ distinct positive integers $X_1, X_2 \ldots, X_n$.
First, $X_1$ is chosen randomly with $\mathbb{P}\left(X_1=i\right)=2^{-i}$ for every positive integer $i$. For $1 \leq j \leq n-1$. having chosen $X_1, \ldots, X_j$, arrange the remaining positive integers in increasing order as $n_1<n_2<$ $\cdots$, and choose $X_{j+1}$ randomly with $\mathbb{P}\left(X_{j+1}=n_i\right)=2^{-i}$ for every positive integer $i$.
Let $Y_n=\max \left\{X_1, \ldots, X_n\right\}$. Show that
$$ \mathbb{E}\left[Y_n\right]=\sum_{i=1}^n \frac{2^i}{2^i-1} $$where $\mathbb{E}\left[Y_n\right]$ is the expected value of $Y_n$.
4 replies
enter16180
Jul 31, 2025
Agsh2005
Today at 12:01 AM
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Fourier Series
EthanWYX2009   0
Yesterday at 11:35 PM
Source: 2025 Spring NSTE(2)-3
Let \( x_1, x_2, \cdots, x_n \) be real numbers. Define \(\|x\| = \min_{n \in \mathbb{Z}} |x - n|\). Prove that:
\[ \sum_{1 \leq i, j \leq n} 2^{\|x_i - x_j\|} \leq \sum_{1 \leq i, j \leq n} 2^{\|x_i - x_j + \frac{1}{2}\|}. \]Proposed by Site Mu
0 replies
EthanWYX2009
Yesterday at 11:35 PM
0 replies
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Putnam 2016 A5
Kent Merryfield   10
N Yesterday at 9:29 PM by ransun
Suppose that $G$ is a finite group generated by the two elements $g$ and $h,$ where the order of $g$ is odd. Show that every element of $G$ can be written in the form
\[g^{m_1}h^{n_1}g^{m_2}h^{n_2}\cdots g^{m_r}h^{n_r}\]with $1\le r\le |G|$ and $m_n,n_1,m_2,n_2,\dots,m_r,n_r\in\{1,-1\}.$ (Here $|G|$ is the number of elements of $G.$)
10 replies
Kent Merryfield
Dec 4, 2016
ransun
Yesterday at 9:29 PM
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Rotation of matrix and eignavalues
enter16180   2
N Yesterday at 9:05 PM by ZNatox
Source: IMC 2025, Problem 8
For an $n \times n$ real matrix $A \in M_n(\mathbb{R})$, denote by $A^{\mathbb{R}}$ its counter-clockwise $90^{\circ}$ rotation.
(10 points) For example,
$$ \left[\begin{array}{lll} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{array}\right]^R=\left[\begin{array}{lll} 3 & 6 & 9 \\ 2 & 5 & 8 \\ 1 & 4 & 7 \end{array}\right] $$Prove that if $A=A^R$ then for any eigenvalue $\lambda$ of $A$, we have $\operatorname{Re} \lambda=0$ or $\operatorname{Im} \lambda=0$.
2 replies
enter16180
Jul 31, 2025
ZNatox
Yesterday at 9:05 PM
J
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Easy Limit problem
Fermat_Fanatic108   2
N Yesterday at 3:12 PM by Fermat_Fanatic108
Evaluate
\[ \lim_{x \to 0^+} \left\{ \lim_{n \to \infty} \left( \frac{\left\lfloor 1^2 (\sin x)^x \right\rfloor + \left\lfloor 2^2 (\sin x)^x \right\rfloor + \cdots + \left\lfloor n^2 (\sin x)^x \right\rfloor}{n^3} \right) \right\}, \]where $\left\lfloor \cdot \right\rfloor$ denotes the floor function
2 replies
Fermat_Fanatic108
Jul 31, 2025
Fermat_Fanatic108
Yesterday at 3:12 PM
J
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2024 Putnam A5
KevinYang2.71   10
N Yesterday at 8:21 AM by ray66
Consider the circle $\Omega$ with radius $9$ and center at the origin $(0,\,0)$, and a disk $\Delta$ with radius $1$ and center at $(r,\,0)$, where $0\leq r\leq 8$. Two points $P$ and $Q$ are chosen independently and uniformly at random on $\Omega$. Which value(s) of $r$ minimize the probability that the chord $\overline{PQ}$ intersects $\Delta$?
10 replies
1 viewing
KevinYang2.71
Dec 10, 2024
ray66
Yesterday at 8:21 AM
J
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An Integral
Saucepan_man02   1
N Yesterday at 7:50 AM by Calcul8er
$\int_0^1\min_{n\ \in Z^+}\left|nx-1\right|$
1 reply
Saucepan_man02
Friday at 1:53 PM
Calcul8er
Yesterday at 7:50 AM
J
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2024 Putnam A2
KevinYang2.71   10
N Yesterday at 7:46 AM by ray66
For which real polynomials $p$ is there a real polynomial $q$ such that
\[ p(p(x))-x=(p(x)-x)^2q(x) \]for all real $x$?
10 replies
KevinYang2.71
Dec 10, 2024
ray66
Yesterday at 7:46 AM
J
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2024 Putnam A1
KevinYang2.71   25
N Yesterday at 7:04 AM by ray66
Determine all positive integers $n$ for which there exists positive integers $a$, $b$, and $c$ satisfying
\[ 2a^n+3b^n=4c^n. \]
25 replies
KevinYang2.71
Dec 10, 2024
ray66
Yesterday at 7:04 AM
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J
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An algorithm for discovering prime numbers?
Lukaluce   4
N May 30, 2025 by alexanderhamilton124
Source: 2025 Junior Macedonian Mathematical Olympiad P3
Is there an infinite sequence of prime numbers $p_1, p_2, ..., p_n, ...,$ such that for every $i \in \mathbb{N}, p_{i + 1} \in \{2p_i - 1, 2p_i + 1\}$ is satisfied? Explain the answer.
4 replies
Lukaluce
May 18, 2025
alexanderhamilton124
May 30, 2025
J
An algorithm for discovering prime numbers?
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Reveal topic tags
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Source: 2025 Junior Macedonian Mathematical Olympiad P3
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Lukaluce
286 posts
Lukaluce
#1 h  May 18, 2025, 3:31 PM
Y by
Is there an infinite sequence of prime numbers $p_1, p_2, ..., p_n, ...,$ such that for every $i \in \mathbb{N}, p_{i + 1} \in \{2p_i - 1, 2p_i + 1\}$ is satisfied? Explain the answer.
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grupyorum
1469 posts
grupyorum
#2 h  May 18, 2025, 6:31 PM
Y by
We first show that there is an $n_0$ and $\epsilon\in\{-1,1\}$ such that for every $n\ge n_0$, $p_{n+1} = 2p_n+\epsilon$.

To see this, suppose $p_1>3$. If $p_1\equiv 1\pmod{3}$ then $p_{n+1}=2p_n-1$ must hold necessarily (otherwise $3\mid 2p_n+1$ but $p_n>3$). Likewise if $p_1\equiv -1\pmod{3}$ then $p_{n+1}\equiv 2p_n+1$ must hold. If $p_1\le 3$, then $p_j>3$ for some $j>1$, so the same argument carries through. Shifting if necessary, we will analyze the sequence $p_{n+1} =2p_n-1$ and $p_{n+1}=2p_n+1$ for $p_1>3$.

Case 1. Let $p_{n+1} = 2p_n-1$ for $n\ge 1$. Set $b_n:=p_n-1$ to obtain $b_{n+1} = 2b_n$. Iterating, we find $b_n = 2^{n-1}b_1$. Consequently, $p_n = 2^{n-1}(p_1-1)+1$. Taking $n=k(p_1-1)+1$ for suitably large $k$, Fermat's theorem asserts $2^{n-1}\equiv 1\pmod{p_1}$. So, $p_1\mid p_n$ but $p_n>p_1$, hence $p_n$ cannot be a prime.

Case 2. Let $p_{n+1}=2p_n+1$ for $n\ge 1$. Set $b_n:=p_n+1$ to obtain $p_n = 2^{n-1}(p_1+1)-1$. The same choice of $n$ ensures $p_1\mid p_n$, a contradiction.

So, no such infinite sequence exists.

Remark. This is an old Bulgarian problem (between 2003-2010 I think), though I don't remember the exact year.
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Assassino9931
1553 posts
Assassino9931
#3 h  May 29, 2025, 4:17 PM
Y by
@above Hm, haven't seen this in Bulgaria, but it is popular from Baltic Way 2004.
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TopGbulliedU
24 posts
TopGbulliedU
#4 h  May 29, 2025, 5:15 PM • 1 Y
Y by alexanderhamilton124
hahaha I was in the comp,after i got out I told everyone that nobody could solve this after the results came it was only me :-D
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alexanderhamilton124
407 posts
alexanderhamilton124
#5 h  May 30, 2025, 2:47 PM
Y by
TopGbulliedU wrote:
hahaha I was in the comp,after i got out I told everyone that nobody could solve this after the results came it was only me :-D

orz gj man
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