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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
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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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IMO ShortList 1998, geometry problem 5
nttu   32
N 18 minutes ago by lpieleanu
Source: IMO ShortList 1998, geometry problem 5
Let $ABC$ be a triangle, $H$ its orthocenter, $O$ its circumcenter, and $R$ its circumradius. Let $D$ be the reflection of the point $A$ across the line $BC$, let $E$ be the reflection of the point $B$ across the line $CA$, and let $F$ be the reflection of the point $C$ across the line $AB$. Prove that the points $D$, $E$ and $F$ are collinear if and only if $OH=2R$.
32 replies
nttu
Oct 14, 2004
lpieleanu
18 minutes ago
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IMO 2010 Problem 5
mavropnevma   53
N 26 minutes ago by shanelin-sigma
Each of the six boxes $B_1$, $B_2$, $B_3$, $B_4$, $B_5$, $B_6$ initially contains one coin. The following operations are allowed

Type 1) Choose a non-empty box $B_j$, $1\leq j \leq 5$, remove one coin from $B_j$ and add two coins to $B_{j+1}$;

Type 2) Choose a non-empty box $B_k$, $1\leq k \leq 4$, remove one coin from $B_k$ and swap the contents (maybe empty) of the boxes $B_{k+1}$ and $B_{k+2}$.

Determine if there exists a finite sequence of operations of the allowed types, such that the five boxes $B_1$, $B_2$, $B_3$, $B_4$, $B_5$ become empty, while box $B_6$ contains exactly $2010^{2010^{2010}}$ coins.

Proposed by Hans Zantema, Netherlands
53 replies
mavropnevma
Jul 8, 2010
shanelin-sigma
26 minutes ago
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a_n < b_n for large n
tastymath75025   11
N an hour ago by torch
Source: 2017 ELMO Shortlist A1
Let $0<k<\frac{1}{2}$ be a real number and let $a_0, b_0$ be arbitrary real numbers in $(0,1)$. The sequences $(a_n)_{n\ge 0}$ and $(b_n)_{n\ge 0}$ are then defined recursively by

$$a_{n+1} = \dfrac{a_n+1}{2} \text{ and } b_{n+1} = b_n^k$$
for $n\ge 0$. Prove that $a_n<b_n$ for all sufficiently large $n$.

Proposed by Michael Ma
11 replies
tastymath75025
Jul 3, 2017
torch
an hour ago
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primes,exponentials,factorials
skellyrah   4
N an hour ago by aaravdodhia
find all primes p,q such that $$ \frac{p^q+q^p-p-q}{p!-q!} $$is a prime number
4 replies
skellyrah
Yesterday at 6:31 PM
aaravdodhia
an hour ago
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No more topics!
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collinear wanted, 2 parallelograms related
parmenides51   0
Sep 6, 2020
Source: Champions Tournament (Ukraine) - Турнір чемпіонів - 2010 Seniors p3
On the sides $AB$ and $BC$ arbitrarily mark points $M$ and $N$, respectively. Let $P$ be the point of intersection of segments $AN$ and $BM$. In addition, we note the points $Q$ and $R$ such that quadrilaterals $MCNQ$ and $ACBR$ are parallelograms. Prove that the points $P,Q$ and $R$ lie on one line.
0 replies
parmenides51
Sep 6, 2020
0 replies
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collinear wanted, 2 parallelograms related
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Reveal topic tags
geometry
parallelogram
collinear
Champions Tournament
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Source: Champions Tournament (Ukraine) - Турнір чемпіонів - 2010 Seniors p3
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parmenides51
30650 posts
parmenides51
#1 h  Sep 6, 2020, 6:29 AM
Y by
On the sides $AB$ and $BC$ arbitrarily mark points $M$ and $N$, respectively. Let $P$ be the point of intersection of segments $AN$ and $BM$. In addition, we note the points $Q$ and $R$ such that quadrilaterals $MCNQ$ and $ACBR$ are parallelograms. Prove that the points $P,Q$ and $R$ lie on one line.
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