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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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BMO Shortlist 2021 A6
Lukaluce   12
N 10 minutes ago by jasperE3
Source: BMO Shortlist 2021
Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that
$$f(xy) = f(x)f(y) + f(f(x + y))$$holds for all $x, y \in \mathbb{R}$.
12 replies
Lukaluce
May 8, 2022
jasperE3
10 minutes ago
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Parallelograms and concyclicity
Lukaluce   19
N 27 minutes ago by cursed_tangent1434
Source: EGMO 2025 P4
Let $ABC$ be an acute triangle with incentre $I$ and $AB \neq AC$. Let lines $BI$ and $CI$ intersect the circumcircle of $ABC$ at $P \neq B$ and $Q \neq C$, respectively. Consider points $R$ and $S$ such that $AQRB$ and $ACSP$ are parallelograms (with $AQ \parallel RB, AB \parallel QR, AC \parallel SP$, and $AP \parallel CS$). Let $T$ be the point of intersection of lines $RB$ and $SC$. Prove that points $R, S, T$, and $I$ are concyclic.
19 replies
Lukaluce
Yesterday at 10:59 AM
cursed_tangent1434
27 minutes ago
J
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IMO 2014 Problem 5
codyj   72
N 41 minutes ago by math-olympiad-clown
For each positive integer $n$, the Bank of Cape Town issues coins of denomination $\frac1n$. Given a finite collection of such coins (of not necessarily different denominations) with total value at most most $99+\frac12$, prove that it is possible to split this collection into $100$ or fewer groups, such that each group has total value at most $1$.
72 replies
codyj
Jul 9, 2014
math-olympiad-clown
41 minutes ago
J
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Functional Equation with Surjectivity
spherical_charlie   2
N an hour ago by jasperE3
Can someone help me?
Find all surjective functions \( f : \mathbb{R} \to \mathbb{R} \) that satisfy the following equation for all real numbers \( x, y \):
\[ f\big(f(x - y)\big) = f(x) - f(y). \]
2 replies
spherical_charlie
3 hours ago
jasperE3
an hour ago
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Midlines of a Triangle
bluecarneal   1
N Aug 18, 2011 by oneplusone
Source: ToT - 2001 Spring Junior O-Level #2
One of the midlines of a triangle is longer than one of its medians. Prove that the triangle has an obtuse angle.
1 reply
bluecarneal
Aug 17, 2011
oneplusone
Aug 18, 2011
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Midlines of a Triangle
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Reveal topic tags
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geometry
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Source: ToT - 2001 Spring Junior O-Level #2
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bluecarneal
9294 posts
bluecarneal
#1 h  Aug 17, 2011, 12:41 PM • 1 Y
Y by Adventure10
One of the midlines of a triangle is longer than one of its medians. Prove that the triangle has an obtuse angle.
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oneplusone
1459 posts
oneplusone
#2 h  Aug 18, 2011, 1:06 PM • 3 Y
Y by Adventure10, Mango247, and 1 other user
We show that for any acute triangle, any side is less than twice of any median.

Given a triangle $ABC$ with median $AM$. Then $2AM>BC$ since $A$ lies outside the circle with diameter $BC$. So $2AM>AM+MB>AB$ and similarly $2AM>AC$ and we are done.
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