Square and equilateral triangle

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]
Our full course list for upcoming classes is below:
All classes run 7:30pm-8:45pm ET/4:30pm - 5:45pm PT unless otherwise noted.

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0 replies

jlacosta

Apr 2, 2025

0 replies

GCD of a sequence

oVlad 6

N 3 minutes ago by Rohit-2006

Source: Romania EGMO TST 2017 Day 1 P2

Determine all pairs $(a,b)$ of positive integers with the following property: all of the terms of the sequence $(a^n+b^n+1)_{n\geqslant 1}$ have a greatest common divisor $d>1.$

6 replies

+1 w

oVlad

5 hours ago

Rohit-2006

3 minutes ago

Maximum with the condition $x^2+y^2+z^2=1$

hlminh 1

N 4 minutes ago by rchokler

Let $x,y,z$ be real numbers such that $x^2+y^2+z^2=1,$ find the largest value of $E=|x-2y|+|y-2z|+|z-2x|.$

1 reply

hlminh

Today at 9:20 AM

rchokler

4 minutes ago

Mock 22nd Thailand TMO P10

korncrazy 2

N 4 minutes ago by korncrazy

Source: own

Prove that there exists infinitely many triples of positive integers $(a,b,c)$ such that $a>b>c,\,\gcd(a,b,c)=1$ and $a^2-b^2,a^2-c^2,b^2-c^2$ are all perfect square.

2 replies

korncrazy

Apr 13, 2025

korncrazy

4 minutes ago

IMO Shortlist 2014 N6

hajimbrak 26

N 5 minutes ago by ihategeo_1969

Let $a_1 < a_2 < \cdots <a_n$ be pairwise coprime positive integers with $a_1$ being prime and $a_1 \ge n + 2$ . On the segment $I = [0, a_1 a_2 \cdots a_n ]$ of the real line, mark all integers that are divisible by at least one of the numbers $a_1 , \ldots , a_n$ . These points split $I$ into a number of smaller segments. Prove that the sum of the squares of the lengths of these segments is divisible by $a_1$ .

Proposed by Serbia

26 replies