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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)!

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[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 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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A=b
k2c901_1   84
N 15 minutes ago by sharknavy75
Source: Taiwan 1st TST 2006, 1st day, problem 3
Let $a$, $b$ be positive integers such that $b^n+n$ is a multiple of $a^n+n$ for all positive integers $n$. Prove that $a=b$.

Proposed by Mohsen Jamali, Iran
84 replies
k2c901_1
Mar 29, 2006
sharknavy75
15 minutes ago
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IMO ShortList 2001, combinatorics problem 4
orl   12
N 27 minutes ago by Maximilian113
Source: IMO ShortList 2001, combinatorics problem 4
A set of three nonnegative integers $\{x,y,z\}$ with $x < y < z$ is called historic if $\{z-y,y-x\} = \{1776,2001\}$. Show that the set of all nonnegative integers can be written as the union of pairwise disjoint historic sets.
12 replies
orl
Sep 30, 2004
Maximilian113
27 minutes ago
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2007 SL A4
the.sceth   90
N 34 minutes ago by bin_sherlo
Source: ISL 2007, A4, Ukrainian TST 2008 Problem 5
Find all functions $ f: \mathbb{R}^{ + }\to\mathbb{R}^{ + }$ satisfying $ f\left(x + f\left(y\right)\right) = f\left(x + y\right) + f\left(y\right)$ for all pairs of positive reals $ x$ and $ y$. Here, $ \mathbb{R}^{ + }$ denotes the set of all positive reals.

Proposed by Paisan Nakmahachalasint, Thailand
90 replies
the.sceth
Jun 24, 2008
bin_sherlo
34 minutes ago
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Inequality with a,b,c
GeoMorocco   4
N an hour ago by arqady
Source: Morocco Training 2025
Let $ a,b,c $ be positive real numbers such that : $ ab+bc+ca=3 $ . Prove that : $$\frac{a\sqrt{3+bc}}{b+c}+\frac{b\sqrt{3+ca}}{c+a}+\frac{c\sqrt{3+ab}}{a+b}\ge a+b+c $$
4 replies
GeoMorocco
Yesterday at 9:51 PM
arqady
an hour ago
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Iran(3rd round)2009
shoki   1
N Sep 13, 2009 by Luis González
Source: Problem 3 Geometry
3-There is given a trapezoid $ ABCD$ in the plane with $ BC||AD$.We know that the angle bisectors of the angles of the trapezoid are concurrent at $ O$.Let $ T$ be the intersection of the diagonals $ AC,BD$.Let $ Q$ be on $ CD$ such that $ \angle OQD = 90^\circ$.Prove that if the circumcircle of the triangle $ OTQ$ intersects $ CD$ again at $ P$ then $ TP||AD$.
1 reply
shoki
Sep 13, 2009
Luis González
Sep 13, 2009
J
Iran(3rd round)2009
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Reveal topic tags
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trapezoid
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Source: Problem 3 Geometry
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shoki
843 posts
shoki
#1 h  Sep 13, 2009, 12:28 PM • 2 Y
Y by Adventure10, Mango247
3-There is given a trapezoid $ ABCD$ in the plane with $ BC||AD$.We know that the angle bisectors of the angles of the trapezoid are concurrent at $ O$.Let $ T$ be the intersection of the diagonals $ AC,BD$.Let $ Q$ be on $ CD$ such that $ \angle OQD = 90^\circ$.Prove that if the circumcircle of the triangle $ OTQ$ intersects $ CD$ again at $ P$ then $ TP||AD$.
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Luis González
4147 posts
Luis González
#2 h  Sep 13, 2009, 5:54 PM • 2 Y
Y by AlastorMoody, Adventure10
The trapezoid has incircle $ (O)$ and $ X,Y,Q$ are the tangency point of $ (O)$ with $ BC,AD,CD,$ respectively. By Newton's theorem, the diagonal intersection $ T$ lies on $ XY,$ which is perpendicular to its bases. Since the quadrilateral $ PQTO$ is cyclic, it follows that $ \angle OQP = \angle PTO = 90^{\circ}$ $\Longrightarrow$ $ TP \perp XY$ $ \Longrightarrow$ $ TP \parallel AD, $ as desired.
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