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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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Movie Collections of Students
Eray   9
N 7 minutes ago by complex2math
Source: Turkey TST 2016 P2
In a class with $23$ students, each pair of students have watched a movie together. Let the set of movies watched by a student be his movie collection. If every student has watched every movie at most once, at least how many different movie collections can these students have?
9 replies
Eray
Apr 10, 2016
complex2math
7 minutes ago
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Something nice
KhuongTrang   26
N 7 minutes ago by KhuongTrang
Source: own
Problem. Given $a,b,c$ be non-negative real numbers such that $ab+bc+ca=1.$ Prove that

$$\sqrt{a+1}+\sqrt{b+1}+\sqrt{c+1}\le 1+2\sqrt{a+b+c+abc}.$$
26 replies
KhuongTrang
Nov 1, 2023
KhuongTrang
7 minutes ago
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Inspired by BMO 2024 SL A4
sqing   0
19 minutes ago
Source: Own
Let \(a \geq b \geq c \geq 0\) and \(ab + bc + ca = 3\). Prove that
$$2 + \left(2 - \frac{2}{\sqrt{3}}\right) \cdot \frac{(b-c)^2}{b+(\sqrt{2}-1)c} \leq a+b+c$$$$3+ 2\left(2 - \sqrt{3}\right) \cdot \frac{(b-c)^2}{a+b+2(\sqrt{3}-1)c} \leq a+b+c$$
0 replies
1 viewing
sqing
19 minutes ago
0 replies
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Balkan MO Shortlist official booklet
guptaamitu1   9
N 20 minutes ago by envision2017
These days I was trying to find the official booklet of Balkan MO Shortlist. But apparently, there's no big list of all Balkan shortlists for previous years. Through some sources, I have been able to find the official booklet for the following years. So if people have it for other years too, can they please put it on this thread, so that everything is in one place.
[list]
[*] 2021
[*] 2020
[*] 2019
[*] 2018
[*] 2017
[*] 2016
[/list]
9 replies
guptaamitu1
Jun 19, 2022
envision2017
20 minutes ago
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sum of inradii independent of triangulation on cyclic n-gon
parmenides51   0
Jun 25, 2021
Source: 2011 Olympiad of Rusanovsky Lyceum in Kyiv, Ukraine , IX-X Team p3
The cyclic $n$-gon is divided by disjoint diagonals into ($n - 2$) triangles. Let ${{r} _ {1}}$, ${{r} _ {2}}$ , $...$, ${{r} _ {n-2}} $ be the radii inscribed in these triangles are circles. Prove that the sum $\left ({{r} _ {1}} + {{r} _ {2}} + \ldots + {{r} _ {n-2}} \right)$ does not depend on the way of dividing the $n$-gon into triangles.

(Sangaku)
0 replies
parmenides51
Jun 25, 2021
0 replies
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sum of inradii independent of triangulation on cyclic n-gon
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Reveal topic tags
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Source: 2011 Olympiad of Rusanovsky Lyceum in Kyiv, Ukraine , IX-X Team p3
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parmenides51
30650 posts
parmenides51
#1 h  Jun 25, 2021, 5:50 PM
Y by
The cyclic $n$-gon is divided by disjoint diagonals into ($n - 2$) triangles. Let ${{r} _ {1}}$, ${{r} _ {2}}$ , $...$, ${{r} _ {n-2}} $ be the radii inscribed in these triangles are circles. Prove that the sum $\left ({{r} _ {1}} + {{r} _ {2}} + \ldots + {{r} _ {n-2}} \right)$ does not depend on the way of dividing the $n$-gon into triangles.

(Sangaku)
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