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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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Construct the orthocenter by drawing perpendicular bisectors
MarkBcc168   24
N 5 minutes ago by cj13609517288
Source: ELMO 2020 P3
Janabel has a device that, when given two distinct points $U$ and $V$ in the plane, draws the perpendicular bisector of $UV$. Show that if three lines forming a triangle are drawn, Janabel can mark the orthocenter of the triangle using this device, a pencil, and no other tools.

Proposed by Fedir Yudin.
24 replies
MarkBcc168
Jul 28, 2020
cj13609517288
5 minutes ago
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Problem involving Power of centroid
Mahdi_Mashayekhi   1
N 6 minutes ago by sami1618
Given is an triangle $ABC$ with centroid $G$. Let $p$ be the power of $G$ w.r.t circumcircle of $ABC$ and $q$ be the power of $G$ w.r.t incircle of $ABC$. prove that $\frac{a^2+b^2+c^2}{12} \le q-p < \frac{a^2+b^2+c^2}{3}$.
1 reply
Mahdi_Mashayekhi
33 minutes ago
sami1618
6 minutes ago
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Classical-looking inequality
Orestis_Lignos   8
N 24 minutes ago by Baimukh
Source: Greece National Olympiad 2022, Problem 3
The positive real numbers $a,b,c,d$ satisfy the equality
$$a+bc+cd+db+\frac{1}{ab^2c^2d^2}=18.$$Find the maximum possible value of $a$.
8 replies
Orestis_Lignos
Feb 26, 2022
Baimukh
24 minutes ago
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BMO 2024 SL C1
GreekIdiot   10
N 24 minutes ago by GreekIdiot
Let $n$, $k$ be positive integers. Julia and Florian play a game on a $2n \times 2n$ board. Julia
has secretly tiled the entire board with invisible dominos. Florian now chooses $k$ cells.
All dominos covering at least one of these cells then turn visible. Determine the minimal
value of $k$ such that Florian has a strategy to always deduce the entire tiling.
10 replies
GreekIdiot
Apr 27, 2025
GreekIdiot
24 minutes ago
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S_ {A_1B_1C_1} <= 1/4 S_ {ABC}, areas by cevians
parmenides51   0
Aug 18, 2021
Source: 2010 Armenia NMO X p3 XI p3
The points $A_1, B_1, C_1$ are taken on the sides $BC$, $CA$, $AB$ of triangle $ABC$ such that $AA_1$, $BB_1$, $CC_1$ intersect at the same point. Prove that $S_ {A_1B_1C_1} \le \frac14 S_ {ABC}$.

Note. Grade XI problem asked for the maximum value of area of triangle $A_1 B_1,C_1$ in terms of area of $ABC$.
0 replies
parmenides51
Aug 18, 2021
0 replies
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S_ {A_1B_1C_1} <= 1/4 S_ {ABC}, areas by cevians
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Source: 2010 Armenia NMO X p3 XI p3
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parmenides51
30650 posts
parmenides51
#1 h  Aug 18, 2021, 12:27 PM
Y by
The points $A_1, B_1, C_1$ are taken on the sides $BC$, $CA$, $AB$ of triangle $ABC$ such that $AA_1$, $BB_1$, $CC_1$ intersect at the same point. Prove that $S_ {A_1B_1C_1} \le \frac14 S_ {ABC}$.

Note. Grade XI problem asked for the maximum value of area of triangle $A_1 B_1,C_1$ in terms of area of $ABC$.
This post has been edited 4 times. Last edited by parmenides51, Aug 18, 2021, 4:57 PM
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