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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
J
H
Dou Fang Geometry in Taiwan TST
Li4   0
3 minutes ago
Source: 2025 Taiwan TST Round 3 Mock P2
Let $\omega$ and $\Omega$ be the incircle and circumcircle of the acute triangle $ABC$, respectively. Draw a square $WXYZ$ so that all of its sides are tangent to $\omega$, and $X$, $Y$ are both on $BC$. Extend $AW$ and $AZ$, intersecting $\Omega$ at $P$ and $Q$, respectively. Prove that $PX$ and $QY$ intersects on $\Omega$.

Proposed by kyou46, Li4, Revolilol.
0 replies
Li4
3 minutes ago
0 replies
J
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[EGMO3] GCD Bounds
socrates   18
N 24 minutes ago by Ilikeminecraft
Source: EGMO 2015, Problem 3
Let $n, m$ be integers greater than $1$, and let $a_1, a_2, \dots, a_m$ be positive integers not greater than $n^m$. Prove that there exist positive integers $b_1, b_2, \dots, b_m$ not greater than $n$, such that \[ \gcd(a_1 + b_1, a_2 + b_2, \dots, a_m + b_m) < n, \] where $\gcd(x_1, x_2, \dots, x_m)$ denotes the greatest common divisor of $x_1, x_2, \dots, x_m$.
18 replies
socrates
Apr 16, 2015
Ilikeminecraft
24 minutes ago
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a_n = c x \phi (a_{n-1}) , is bounded
parmenides51   4
N 31 minutes ago by Ilikeminecraft
Source: (2021-) 2022 XV 15th D&uuml;rer Math Competition Finals Day 1 E+1
Let $c \ge 2$ be a fixed integer. Let $a_1 = c$ and for all $n \ge 2$ let $a_n = c \cdot \phi (a_{n-1})$. What are the numbers $c$ for which sequence $(a_n)$ will be bounded?

$\phi$ denotes Euler’s Phi Function, meaning that $\phi (n)$ gives the number of integers within the set $\{1, 2, . . . , n\}$ that are relative primes to $n$. We call a sequence $(x_n)$ bounded if there exist a constant $D$ such that $|x_n| < D$ for all positive integers $n$.
4 replies
parmenides51
Nov 29, 2022
Ilikeminecraft
31 minutes ago
J
H
Polynomial mapping set of divisors to set of divisors
eduD_looC   11
N 32 minutes ago by Ilikeminecraft
Source: CMO 2023 P4
Let $f(x)$ be a non-constant polynomial with integer coefficients such that $f(1) \neq 1$. For a positive integer $n$, define $\text{divs}(n)$ to be the set of positive divisors of $n$.

A positive integer $m$ is $f$-cool if there exists a positive integer $n$ for which $$f[\text{divs}(m)]=\text{divs}(n).$$Prove that for any such $f$, there are finitely many $f$-cool integers.

(The notation $f[S]$ for some set $S$ denotes the set $\{f(s):s \in S\}$.)
11 replies
eduD_looC
Mar 11, 2023
Ilikeminecraft
32 minutes ago
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Playing cards 1
prof.   1
N Apr 17, 2025 by prof.
In how many ways can a deck of 52 cards be divided among 13 players, each with 4 cards, so that one player has all 4 suits and the others have one suit?
1 reply
prof.
Apr 15, 2025
prof.
Apr 17, 2025
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Playing cards 1
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#1 h  Apr 15, 2025, 5:59 AM
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In how many ways can a deck of 52 cards be divided among 13 players, each with 4 cards, so that one player has all 4 suits and the others have one suit?
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#2 h  Apr 17, 2025, 12:40 PM
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Reminder :blush:
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