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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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Common tangent to diameter circles
Stuttgarden   1
N a few seconds ago by jrpartty
Source: Spain MO 2025 P2
The cyclic quadrilateral $ABCD$, inscribed in the circle $\Gamma$, satisfies $AB=BC$ and $CD=DA$, and $E$ is the intersection point of the diagonals $AC$ and $BD$. The circle with center $A$ and radius $AE$ intersects $\Gamma$ in two points $F$ and $G$. Prove that the line $FG$ is tangent to the circles with diameters $BE$ and $DE$.
1 reply
Stuttgarden
Mar 31, 2025
jrpartty
a few seconds ago
J
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Two Functional Inequalities
Mathdreams   2
N a minute ago by kokcio
Source: 2025 Nepal Mock TST Day 2 Problem 2
Determine all functions $f : \mathbb{R} \rightarrow \mathbb{R}$ such that $$f(x) \le x^3$$and $$f(x + y) \le f(x) + f(y) + 3xy(x + y)$$for any real numbers $x$ and $y$.

(Miroslav Marinov, Bulgaria)
2 replies
Mathdreams
13 minutes ago
kokcio
a minute ago
J
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Beautiful problem
luutrongphuc   15
N 2 minutes ago by r7di048hd3wwd3o3w58q
Let triangle $ABC$ be circumscribed about circle $(I)$, and let $H$ be the orthocenter of $\triangle ABC$. The circle $(I)$ touches line $BC$ at $D$. The tangent to the circle $(BHC)$ at $H$ meets $BC$ at $S$. Let $J$ be the midpoint of $HI$, and let the line $DJ$ meet $(I)$ again at $X$. The tangent to $(I)$ parallel to $BC$ meets the line $AX$ at $T$. Prove that $ST$ is tangent to $(I)$.
15 replies
luutrongphuc
Apr 4, 2025
r7di048hd3wwd3o3w58q
2 minutes ago
J
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FE with a lot of terms
MrHeccMcHecc   0
3 minutes ago
Determine all functions $f : \mathbb{R} \rightarrow \mathbb{R}$ such that for all $x,y \in \mathbb{R}$ $$f(x)f(y)+f(x+y)=xf(y)+yf(x)+f(xy)+x+y+1$$
0 replies
MrHeccMcHecc
3 minutes ago
0 replies
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No more topics!
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AB^2+CD^2+EF^2=BC^2+DE^2+FA^2 for intersecting equal equilaterals
parmenides51   1
N Aug 18, 2021 by parmenides51
Source: 2009 Armenia NMO IX p6 X p4
Two congruent equilateral triangles intersect at the points $A,B,C,D,E,F$ (as shown in the figure). Prove that $$AB^2+CD^2+EF^2=BC^2+DE^2+FA^2$$IMAGE
1 reply
parmenides51
Aug 18, 2021
parmenides51
Aug 18, 2021
J
AB^2+CD^2+EF^2=BC^2+DE^2+FA^2 for intersecting equal equilaterals
G H J
Reveal topic tags
geometry
equilaterals
Equilateral
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G H BBookmark kLocked kLocked NReply
Source: 2009 Armenia NMO IX p6 X p4
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parmenides51
30629 posts
parmenides51
#1 h  Aug 18, 2021, 12:11 AM
Y by
Two congruent equilateral triangles intersect at the points $A,B,C,D,E,F$ (as shown in the figure). Prove that $$AB^2+CD^2+EF^2=BC^2+DE^2+FA^2$$https://cdn.artofproblemsolving.com/attachments/8/0/dc0c3d9000ff908b78d9addc8e3dcf1210e580.png
This post has been edited 3 times. Last edited by parmenides51, Aug 18, 2021, 12:16 AM
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parmenides51
30629 posts
parmenides51
#2 h  Aug 18, 2021, 12:12 AM
Y by
posted for the image links
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This post has been edited 2 times. Last edited by parmenides51, Aug 18, 2021, 12:56 AM
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