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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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max n with n times n square are black
NicoN9   1
N 10 minutes ago by WallyWalrus
Source: Japan Junior MO Preliminary 2022 P5
Find the maximum positive integer $n$ such that for $45\times 45$ grid, no matter how you paint $2022$ unit squares black, there exists $n\times n$ square with all unit square painted black.
1 reply
NicoN9
Today at 9:19 AM
WallyWalrus
10 minutes ago
J
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Find triple
kalandar   2
N 15 minutes ago by mathlover1231
Find all triple ( x, y, z ) of natural numbers satisfying the equation $1+4^x+4^y=z^2$.
2 replies
kalandar
Apr 26, 2019
mathlover1231
15 minutes ago
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[SEIF Q1] FE on x^3+xy...( ͡° ͜ʖ ͡°)
EmilXM   17
N 17 minutes ago by jasperE3
Source: SEIF 2022
Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that any real numbers $x$ and $y$ satisfy
$$x^3+f(x)f(y)=f(f(x^3)+f(xy)).$$Proposed by EmilXM
17 replies
EmilXM
Mar 12, 2022
jasperE3
17 minutes ago
J
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IMO 2018 Problem 4
orthocentre   54
N 18 minutes ago by zuat.e
Source: IMO 2018
A site is any point $(x, y)$ in the plane such that $x$ and $y$ are both positive integers less than or equal to 20.

Initially, each of the 400 sites is unoccupied. Amy and Ben take turns placing stones with Amy going first. On her turn, Amy places a new red stone on an unoccupied site such that the distance between any two sites occupied by red stones is not equal to $\sqrt{5}$. On his turn, Ben places a new blue stone on any unoccupied site. (A site occupied by a blue stone is allowed to be at any distance from any other occupied site.) They stop as soon as a player cannot place a stone.

Find the greatest $K$ such that Amy can ensure that she places at least $K$ red stones, no matter how Ben places his blue stones.

Proposed by Gurgen Asatryan, Armenia
54 replies
orthocentre
Jul 10, 2018
zuat.e
18 minutes ago
J
No more topics!
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Disjoint discs tangent to 6 others
randomusername   2
N Jul 5, 2015 by Naysh
Source: Austrian-Polish 1978, Problem 6
We are given a family of discs in the plane, with pairwise disjoint interiors. Each disc is tangent to at least six other discs of the family. Show that the family is infinite.
2 replies
randomusername
Jul 5, 2015
Naysh
Jul 5, 2015
J
Disjoint discs tangent to 6 others
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Reveal topic tags
combinatorics
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G H BBookmark kLocked kLocked NReply
Source: Austrian-Polish 1978, Problem 6
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randomusername
1059 posts
randomusername
#1 h  Jul 5, 2015, 8:20 AM • 2 Y
Y by Adventure10, Mango247
We are given a family of discs in the plane, with pairwise disjoint interiors. Each disc is tangent to at least six other discs of the family. Show that the family is infinite.
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liberator
95 posts
liberator
#2 h  Jul 5, 2015, 9:09 PM • 2 Y
Y by Adventure10, Mango247
Suppose for contradiction that the family is finite. Construct a graph $G$ where the discs are vertices and there is an edge between two vertices iff their corresponding discs are tangent. Evidently, $G$ is planar, so it satisfies Euler's characteristic $V-E+F=2$, from which we may deduce $E \leq 3V - 6$. Hence the average degree $\tfrac{2E}{V}$ of the vertices of $G$ is less than $6$, which gives the required contradiction, since each disc is tangent to at least $6$ others.
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Naysh
2134 posts
Naysh
#3 h  Jul 5, 2015, 9:14 PM • 2 Y
Y by Adventure10, Mango247
Alternatively, if the family is finite, consider the minimal radius $r$ achieved. Among all discs of radius $r$ in the family, take the one who's center is furthest down and to the right. Then this one cannot be surrounded by six other discs, which is contradiction.
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