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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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6 variable inequality
ChuongTk17   4
N 13 minutes ago by arqady
Source: Own
Given real numbers a,b,c,d,e,f in the interval [-1;1] and positive x,y,z,t such that $$2xya+2xzb+2xtc+2yzd+2yte+2ztf=x^2+y^2+z^2+t^2$$. Prove that: $$a+b+c+d+e+f \leq 2$$
4 replies
ChuongTk17
Nov 29, 2024
arqady
13 minutes ago
J
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APMO 2015 P1
aditya21   62
N an hour ago by Tonne
Source: APMO 2015
Let $ABC$ be a triangle, and let $D$ be a point on side $BC$. A line through $D$ intersects side $AB$ at $X$ and ray $AC$ at $Y$ . The circumcircle of triangle $BXD$ intersects the circumcircle $\omega$ of triangle $ABC$ again at point $Z$ distinct from point $B$. The lines $ZD$ and $ZY$ intersect $\omega$ again at $V$ and $W$ respectively.
Prove that $AB = V W$

Proposed by Warut Suksompong, Thailand
62 replies
aditya21
Mar 30, 2015
Tonne
an hour ago
J
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Or statement function
ItzsleepyXD   2
N an hour ago by cursed_tangent1434
Source: Own , Mock Thailand Mathematic Olympiad P2
Find all $f: \mathbb{R} \to \mathbb{Z^+}$ such that $$f(x+f(y))=f(x)+f(y)+1\quad\text{ or }\quad f(x)+f(y)-1$$for all real number $x$ and $y$
2 replies
ItzsleepyXD
Yesterday at 9:07 AM
cursed_tangent1434
an hour ago
J
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Trivial fun Equilateral
ItzsleepyXD   4
N 2 hours ago by cursed_tangent1434
Source: Own , Mock Thailand Mathematic Olympiad P1
Let $ABC$ be a scalene triangle with point $P$ and $Q$ on the plane such that $\triangle BPC , \triangle CQB$ is an equilateral . Let $AB$ intersect $CP$ and $CQ$ at $X$ and $Z$ respectively and $AC$ intersect $BP$ and $BQ$ at $Y$ and $W$ respectively .
Prove that $XY\parallel ZW$
4 replies
ItzsleepyXD
Yesterday at 9:05 AM
cursed_tangent1434
2 hours ago
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concurrency of 4 altitudes in tetrahedron criterion
parmenides51   0
Apr 19, 2021
Source: 1978 British Mathematical Olympiad p4 BrMO
An altitude of a tetrahedron is a line through a vertex perpendicular to the opposite face.
Prove that the four altitudes of a tetrahedron are concurrent if and only if each edge of the tetrahedron is perpendicular to its opposite edge.
0 replies
parmenides51
Apr 19, 2021
0 replies
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concurrency of 4 altitudes in tetrahedron criterion
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Reveal topic tags
geometry
3D geometry
tetrahedron
concurrency
concurrent
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Source: 1978 British Mathematical Olympiad p4 BrMO
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parmenides51
30650 posts
parmenides51
#1 h  Apr 19, 2021, 7:42 PM • 1 Y
Y by FaThEr-SqUiRrEl
An altitude of a tetrahedron is a line through a vertex perpendicular to the opposite face.
Prove that the four altitudes of a tetrahedron are concurrent if and only if each edge of the tetrahedron is perpendicular to its opposite edge.
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