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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)!

Be sure to mark your calendars for the following events:
[list][*]April 3rd (Webinar), 4pm PT/7:00pm ET, Learning with AoPS: Perspectives from a Parent, Math Camp Instructor, and University Professor
[*]April 8th (Math Jam), 4:30pm PT/7:30pm ET, 2025 MATHCOUNTS State Discussion
April 9th (Webinar), 4:00pm PT/7:00pm ET, Learn about Video-based Summer Camps at the Virtual Campus
[*]April 10th (Math Jam), 4:30pm PT/7:30pm ET, 2025 MathILy and MathILy-Er Math Jam: Multibackwards Numbers
[*]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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Break the stick!
BR1F1SZ   2
N 9 minutes ago by real_loser
Source: 2024 Argentina L2 P2
Ana and Beto play the following game with a stick of length $15$. Ana starts, and on her first turn, she cuts the stick into two pieces with integer lengths. Then, on each player's turn, they must cut one of the pieces, of their choice, into two new pieces with integer lengths. The player who, on their turn, leaves at least one piece with length equal to $1$ loses. Determine which of the two players has a winning strategy.
2 replies
BR1F1SZ
Nov 18, 2024
real_loser
9 minutes ago
J
H
all functions satisfying f(x+yf(x))+y = xy + f(x+y)
falantrng   29
N 12 minutes ago by Uzb_Math2010
Source: Balkan MO 2025 P3
Find all functions $f\colon \mathbb{R} \rightarrow \mathbb{R}$ such that for all $x,y \in \mathbb{R}$,
\[f(x+yf(x))+y = xy + f(x+y).\]
Proposed by Giannis Galamatis, Greece
29 replies
+2 w
falantrng
Apr 27, 2025
Uzb_Math2010
12 minutes ago
J
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2020 EGMO P4: n times the previous number of fresh permutations
alifenix-   22
N 12 minutes ago by math-olympiad-clown
Source: 2020 EGMO P4
A permutation of the integers $1, 2, \ldots, m$ is called fresh if there exists no positive integer $k < m$ such that the first $k$ numbers in the permutation are $1, 2, \ldots, k$ in some order. Let $f_m$ be the number of fresh permutations of the integers $1, 2, \ldots, m$.

Prove that $f_n \ge n \cdot f_{n - 1}$ for all $n \ge 3$.

For example, if $m = 4$, then the permutation $(3, 1, 4, 2)$ is fresh, whereas the permutation $(2, 3, 1, 4)$ is not.
22 replies
alifenix-
Apr 18, 2020
math-olympiad-clown
12 minutes ago
J
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Factorial Equation
Alidq   1
N 16 minutes ago by CHESSR1DER
Solve in $\mathbb{N}$ $$\frac{x!}{(x-y)!} = 10x+2y-29$$
1 reply
Alidq
an hour ago
CHESSR1DER
16 minutes ago
J
No more topics!
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J
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k QR = RT wanted, starting with 2 externally tangent circles
parmenides51   1
N Sep 26, 2020 by juckter
Source: 2017 Italy EGMO TST p1
Let $C_1$ and $C_2$ be two externally tangent circles at $S$ such that the radius of $C_2$ is three times the radius of $C_1$. Let us consider a tangent line to $C_1$ at $P \ne S$ and to $C_2$ in $Q \ne S$. Let $T$ be a point in $C_2$ such that $QT$ is a diameter of $C_2$. Finally the bisector of the angle $\angle SQT$ meets $ST$ in $R$. Prove that $QR = RT$.
1 reply
parmenides51
Sep 26, 2020
juckter
Sep 26, 2020
J
QR = RT wanted, starting with 2 externally tangent circles
G H J
Reveal topic tags
geometry
equal segments
tangent circles
L
G H BBookmark kLocked kLocked NReply
Source: 2017 Italy EGMO TST p1
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parmenides51
30650 posts
parmenides51
#1 h  Sep 26, 2020, 3:22 PM
Y by
Let $C_1$ and $C_2$ be two externally tangent circles at $S$ such that the radius of $C_2$ is three times the radius of $C_1$. Let us consider a tangent line to $C_1$ at $P \ne S$ and to $C_2$ in $Q \ne S$. Let $T$ be a point in $C_2$ such that $QT$ is a diameter of $C_2$. Finally the bisector of the angle $\angle SQT$ meets $ST$ in $R$. Prove that $QR = RT$.
Z Y
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juckter
323 posts
juckter
#2 h  Sep 26, 2020, 5:22 PM
Y by
Same as Mexico 2016 P1
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