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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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primes,exponentials,factorials
skellyrah   3
N 8 minutes ago by skellyrah
find all primes p,q such that $$ \frac{p^q+q^p-p-q}{p!-q!} $$is a prime number
3 replies
skellyrah
3 hours ago
skellyrah
8 minutes ago
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af(a)+bf(b)+2ab=x^2 for all natural a, b - show that f(a)=a
shoki   26
N 13 minutes ago by MathLuis
Source: Iran TST 2011 - Day 4 - Problem 3
Suppose that $f : \mathbb{N} \rightarrow \mathbb{N}$ is a function for which the expression $af(a)+bf(b)+2ab$ for all $a,b \in \mathbb{N}$ is always a perfect square. Prove that $f(a)=a$ for all $a \in \mathbb{N}$.
26 replies
shoki
May 14, 2011
MathLuis
13 minutes ago
J
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Very easy NT
GreekIdiot   8
N 14 minutes ago by vsamc
Prove that there exists no natural number $n>1$ such that $n \mid 2^n-1$.
8 replies
GreekIdiot
Today at 2:49 PM
vsamc
14 minutes ago
J
H
Another quadrilateral in a circle
v_Enhance   110
N 18 minutes ago by Marco22
Source: APMO 2013, Problem 5
Let $ABCD$ be a quadrilateral inscribed in a circle $\omega$, and let $P$ be a point on the extension of $AC$ such that $PB$ and $PD$ are tangent to $\omega$. The tangent at $C$ intersects $PD$ at $Q$ and the line $AD$ at $R$. Let $E$ be the second point of intersection between $AQ$ and $\omega$. Prove that $B$, $E$, $R$ are collinear.
110 replies
v_Enhance
May 3, 2013
Marco22
18 minutes ago
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No more topics!
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AG // BC wantnted, <EFC = 2 < ECD, circle through A tangent to BC
parmenides51   0
Oct 31, 2020
Source: 2011 Thailand TST 6.2
A circle $\Omega$ through the vertex $A$ of $\vartriangle ABC$ is tangent to segment $BC$ at $D$. Let $E$ be the projection from $D$ onto $AC$, and let $F$ be a point on $\Omega$ lying on the opposite side of $D$ with respect to line $EC$ so that $\angle EFC = 2 \angle ECD$. Suppose that $CF$ meets $\Omega$ again at $G$. Prove that $AG$ is parallel to $BC$.
0 replies
parmenides51
Oct 31, 2020
0 replies
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AG // BC wantnted, <EFC = 2 < ECD, circle through A tangent to BC
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Reveal topic tags
equal angles
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geometry
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Source: 2011 Thailand TST 6.2
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parmenides51
30650 posts
parmenides51
#1 h  Oct 31, 2020, 6:03 AM
Y by
A circle $\Omega$ through the vertex $A$ of $\vartriangle ABC$ is tangent to segment $BC$ at $D$. Let $E$ be the projection from $D$ onto $AC$, and let $F$ be a point on $\Omega$ lying on the opposite side of $D$ with respect to line $EC$ so that $\angle EFC = 2 \angle ECD$. Suppose that $CF$ meets $\Omega$ again at $G$. Prove that $AG$ is parallel to $BC$.
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