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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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Medium geometry with AH diameter circle
v_Enhance   94
N 20 minutes ago by alexanderchew
Source: USA TSTST 2016 Problem 2, by Evan Chen
Let $ABC$ be a scalene triangle with orthocenter $H$ and circumcenter $O$. Denote by $M$, $N$ the midpoints of $\overline{AH}$, $\overline{BC}$. Suppose the circle $\gamma$ with diameter $\overline{AH}$ meets the circumcircle of $ABC$ at $G \neq A$, and meets line $AN$ at a point $Q \neq A$. The tangent to $\gamma$ at $G$ meets line $OM$ at $P$. Show that the circumcircles of $\triangle GNQ$ and $\triangle MBC$ intersect at a point $T$ on $\overline{PN}$.

Proposed by Evan Chen
94 replies
v_Enhance
Jun 28, 2016
alexanderchew
20 minutes ago
J
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problem interesting
Cobedangiu   7
N 21 minutes ago by vincentwant
Let $a=3k^2+3k+1 (a,k \in N)$
$i)$ Prove that: $a^2$ is the sum of $3$ square numbers
$ii)$ Let $b \vdots a$ and $b$ is the sum of $3$ square numbers. Prove that: $b^n$ is the sum of $3$ square numbers
7 replies
Cobedangiu
Yesterday at 5:06 AM
vincentwant
21 minutes ago
J
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another problem
kjhgyuio   1
N 42 minutes ago by lpieleanu
........
1 reply
kjhgyuio
an hour ago
lpieleanu
42 minutes ago
J
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2^x+3^x = yx^2
truongphatt2668   9
N 42 minutes ago by Jackson0423
Prove that the following equation has infinite integer solutions:
$$2^x+3^x = yx^2$$
9 replies
truongphatt2668
Apr 22, 2025
Jackson0423
42 minutes ago
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min circumradius is (a-b)/2 as angle <XOY varies
parmenides51   0
Oct 10, 2021
Source: 2002 IberoAmerican Shortlist G9
An angle $\angle XOY = \alpha$ and points $ A$ and $ B$ on $OY$ are given such that $OA = a$ and $OB = b$ ($a> b$). A circle passes through points $ A$ and $ B$ and is tangent to $OX$.
a) Calculate the radius of this circle in terms of $a, b$ and $\alpha$.
b) If $a$ and $b$ are constant and $\alpha$ varies, show that the minimum value of the radius of the circumference is $\frac{a-b}{2}$
0 replies
parmenides51
Oct 10, 2021
0 replies
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min circumradius is (a-b)/2 as angle <XOY varies
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Source: 2002 IberoAmerican Shortlist G9
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parmenides51
30650 posts
parmenides51
#1 h  Oct 10, 2021, 12:13 AM
Y by
An angle $\angle XOY = \alpha$ and points $ A$ and $ B$ on $OY$ are given such that $OA = a$ and $OB = b$ ($a> b$). A circle passes through points $ A$ and $ B$ and is tangent to $OX$.
a) Calculate the radius of this circle in terms of $a, b$ and $\alpha$.
b) If $a$ and $b$ are constant and $\alpha$ varies, show that the minimum value of the radius of the circumference is $\frac{a-b}{2}$
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