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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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All classes run 7:30pm-8:45pm ET/4:30pm - 5:45pm PT unless otherwise noted.

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0 replies
jlacosta
Apr 2, 2025
0 replies
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Straight-edge is a social construct
anantmudgal09   27
N 4 minutes ago by cj13609517288
Source: INMO 2023 P6
Euclid has a tool called cyclos which allows him to do the following:
[list]
[*] Given three non-collinear marked points, draw the circle passing through them.
[*] Given two marked points, draw the circle with them as endpoints of a diameter.
[*] Mark any intersection points of two drawn circles or mark a new point on a drawn circle.
[/list]

Show that given two marked points, Euclid can draw a circle centered at one of them and passing through the other, using only the cyclos.

Proposed by Rohan Goyal, Anant Mudgal, and Daniel Hu
27 replies
anantmudgal09
Jan 15, 2023
cj13609517288
4 minutes ago
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Reducibility of 2x^2 cyclotomic
vincentwant   2
N 16 minutes ago by vincentwant
Let $S$ denote the set of all positive integers less than $1020$ that are relatively prime to $1020$. Let $\omega=\cos\frac{\pi}{510}+i\sin\frac{\pi}{510}$. Is the polynomial $$\prod_{n\in S}(2x^2-\omega^n)$$reducible over the rational numbers, given that it has integer coefficients?
2 replies
vincentwant
2 hours ago
vincentwant
16 minutes ago
J
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Weighted Blocks
ilovemath04   51
N 22 minutes ago by Maximilian113
Source: ISL 2019 C2
You are given a set of $n$ blocks, each weighing at least $1$; their total weight is $2n$. Prove that for every real number $r$ with $0 \leq r \leq 2n-2$ you can choose a subset of the blocks whose total weight is at least $r$ but at most $r + 2$.
51 replies
ilovemath04
Sep 22, 2020
Maximilian113
22 minutes ago
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Very easy NT
GreekIdiot   7
N 23 minutes ago by Primeniyazidayi
Prove that there exists no natural number $n>1$ such that $n \mid 2^n-1$.
7 replies
GreekIdiot
3 hours ago
Primeniyazidayi
23 minutes ago
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No more topics!
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5 spheres in a cone, 4 equal ones
parmenides51   0
Jan 24, 2021
Source: 2015 SPbU finals, grades 10-11 p6 v6 - Saint Petersburg State University School Olympiad
Four identical balls are placed in a cone whose base height and radius are $7$. Each of them touches the other two (externally), the base and the lateral surface of the cone. The fifth ball touches the lateral surface of the cone and all identical balls (externally). Find the radius of the fifth ball.

result
0 replies
parmenides51
Jan 24, 2021
0 replies
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5 spheres in a cone, 4 equal ones
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Reveal topic tags
geometry
3D geometry
sphere
cone
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Source: 2015 SPbU finals, grades 10-11 p6 v6 - Saint Petersburg State University School Olympiad
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parmenides51
30650 posts
parmenides51
#1 h  Jan 24, 2021, 10:16 PM
Y by
Four identical balls are placed in a cone whose base height and radius are $7$. Each of them touches the other two (externally), the base and the lateral surface of the cone. The fifth ball touches the lateral surface of the cone and all identical balls (externally). Find the radius of the fifth ball.

result
This post has been edited 4 times. Last edited by parmenides51, Jan 24, 2021, 10:25 PM
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