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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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Involved conditional geo
Assassino9931   1
N 4 minutes ago by hukilau17
Source: Balkan MO 2024 Shortlist G4
Let $ABC$ be an acute-angled triangle with $AB < AC$, orthocenter $H$, circumcircle $\Gamma$ and circumcentre $O$. Let $M$ be the midpoint of $BC$ and let $D$ be a point such that $ADOH$ is a parallellogram. Suppose that there exists a point $X$ on $\Gamma$ and on the opposite side of $DH$ to $A$ such that $\angle DXH + \angle DHA = 90^{\circ}$. Let $Y$ be the midpoint of $OX$. Prove that if $MY = OA$, then $OA = 2OH$.
1 reply
Assassino9931
39 minutes ago
hukilau17
4 minutes ago
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Inversion exercise
Assassino9931   2
N 10 minutes ago by awesomeming327.
Source: Balkan MO Shortlist 2024 G5
Let $ABC$ be an acute scalene triangle $ABC$, $D$ be the orthogonal projection of $A$ on $BC$, $M$ and $N$ are the midpoints of $AB$ and $AC$ respectively. Let $P$ and $Q$ are points on the minor arcs $\widehat{AB}$ and $\widehat{AC}$ of the circumcircle of triangle $ABC$ respectively such that $PQ \parallel BC$. Show that the circumcircles of triangles $DPQ$ and $DMN$ are tangent if and only if $M$ lies on $PQ$.
2 replies
Assassino9931
36 minutes ago
awesomeming327.
10 minutes ago
J
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all functions satisfying f(x+yf(x))+y = xy + f(x+y)
falantrng   26
N 24 minutes ago by awesomeming327.
Source: Balkan MO 2025 P3
Find all functions $f\colon \mathbb{R} \rightarrow \mathbb{R}$ such that for all $x,y \in \mathbb{R}$,
\[f(x+yf(x))+y = xy + f(x+y).\]
Proposed by Giannis Galamatis, Greece
26 replies
falantrng
Today at 11:52 AM
awesomeming327.
24 minutes ago
J
H
One more problem defined only with lines
Assassino9931   0
34 minutes ago
Source: Balkan MO 2024 Shortlist G6
Let $ABC$ be a triangle and the points $K$ and $L$ on $AB$, $M$ and $N$ on $BC$, and $P$ and $Q$ on $AC$ be such that $AK = LB < \frac{1}{2}AB, BM = NC < \frac{1}{2}BC$ and $CP = QA < \frac{1}{2}AC$. The intersections of $KN$ with $MQ$ and $LP$ are $R$ and $T$ respectively, and the intersections of $NP$ with $LM$ and $KQ$ are $D$ and $E$, respectively. Prove that the lines $DR, BE$ and $CT$ are concurrent.
0 replies
1 viewing
Assassino9931
34 minutes ago
0 replies
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No more topics!
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Perfect vertices in a tetrahedron
lifeisgood03   0
Nov 28, 2017
Source: 2000 Belarusian National Olympiad
A vertex of a tetrahedron is called perfect if the three edges at this vertex are sides of a certain triangle. How many perfect vertices can a tetrahedron have?
0 replies
lifeisgood03
Nov 28, 2017
0 replies
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Perfect vertices in a tetrahedron
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Reveal topic tags
geometry
3D geometry
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Source: 2000 Belarusian National Olympiad
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lifeisgood03
388 posts
lifeisgood03
#1 h  Nov 28, 2017, 10:17 PM • 2 Y
Y by Adventure10, Mango247
A vertex of a tetrahedron is called perfect if the three edges at this vertex are sides of a certain triangle. How many perfect vertices can a tetrahedron have?
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