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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
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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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IMO 2014 Problem 4
ipaper   168
N 9 minutes ago by Bonime
Let $P$ and $Q$ be on segment $BC$ of an acute triangle $ABC$ such that $\angle PAB=\angle BCA$ and $\angle CAQ=\angle ABC$. Let $M$ and $N$ be the points on $AP$ and $AQ$, respectively, such that $P$ is the midpoint of $AM$ and $Q$ is the midpoint of $AN$. Prove that the intersection of $BM$ and $CN$ is on the circumference of triangle $ABC$.

Proposed by Giorgi Arabidze, Georgia.
168 replies
+1 w
ipaper
Jul 9, 2014
Bonime
9 minutes ago
J
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function Z to Z..
Jackson0423   3
N 21 minutes ago by jasperE3
Let \( f : \mathbb{Z} \to \mathbb{Z} \) be a function satisfying
\[ f(f(x)) = x^2 - 6x + 6 \quad \text{for all} \quad x \in \mathbb{Z}. \]Given that
\[ f(i) < f(i+1) \quad \text{for} \quad i = 0, 1, 2, 3, 4, 5, \]find the value of
\[ f(0) + f(1) + f(2) + \cdots + f(6). \]
3 replies
Jackson0423
Yesterday at 2:49 PM
jasperE3
21 minutes ago
J
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Functional equation
tenplusten   9
N an hour ago by Jakjjdm
Source: Bulgarian NMO 2015 P4
Find all functions $f:\mathbb{R^+}\to\mathbb {R^+} $ such that for all $x,y\in R^+$ the followings hold:
$i) $ $f (x+y)\ge f (x)+y $
$ii) $ $f (f (x))\le x $
9 replies
1 viewing
tenplusten
Apr 29, 2018
Jakjjdm
an hour ago
J
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Prove DK and BC are perpendicular.
yunxiu   61
N an hour ago by Avron
Source: 2012 European Girls’ Mathematical Olympiad P1
Let $ABC$ be a triangle with circumcentre $O$. The points $D,E,F$ lie in the interiors of the sides $BC,CA,AB$ respectively, such that $DE$ is perpendicular to $CO$ and $DF$ is perpendicular to $BO$. (By interior we mean, for example, that the point $D$ lies on the line $BC$ and $D$ is between $B$ and $C$ on that line.)
Let $K$ be the circumcentre of triangle $AFE$. Prove that the lines $DK$ and $BC$ are perpendicular.

Netherlands (Merlijn Staps)
61 replies
yunxiu
Apr 13, 2012
Avron
an hour ago
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No more topics!
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convex polyhedron without a quadr. or pentagonal face has >=4 triang. faces
parmenides51   0
Oct 28, 2020
Source: 2014 Thailand TST 8.3.1
Prove that every convex polyhedron without a quadrilateral or pentagonal face must have at least $4$ triangular faces.
0 replies
parmenides51
Oct 28, 2020
0 replies
J
convex polyhedron without a quadr. or pentagonal face has >=4 triang. faces
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Reveal topic tags
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Source: 2014 Thailand TST 8.3.1
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parmenides51
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parmenides51
#1 h  Oct 28, 2020, 7:57 PM
Y by
Prove that every convex polyhedron without a quadrilateral or pentagonal face must have at least $4$ triangular faces.
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