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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
J
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A Ball-Drawing problem
Vivacious_Owl   7
N 4 hours ago by Vivacious_Owl
Source: Inspired by a certain daily routine of mine
There are N identical black balls in a bag. I randomly take one ball out of the bag. If it is a black ball, I throw it away and put a white ball back into the bag instead. If it is a white ball, I simply throw it away and do not put anything back into the bag. The probability of getting any ball is the same.
Questions:
1. How many times will I need to reach into the bag to empty it?
2. What is the ratio of the expected maximum number of white balls in the bag to N in the limit as N goes to infinity?
7 replies
1 viewing
Vivacious_Owl
Apr 24, 2025
Vivacious_Owl
4 hours ago
J
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Putnam 2018 B3
62861   38
N Yesterday at 11:49 PM by Ilikeminecraft
Find all positive integers $n < 10^{100}$ for which simultaneously $n$ divides $2^n$, $n-1$ divides $2^n - 1$, and $n-2$ divides $2^n - 2$.
38 replies
62861
Dec 2, 2018
Ilikeminecraft
Yesterday at 11:49 PM
J
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D1020 : Special functional equation
Dattier   2
N Yesterday at 9:06 PM by alexheinis
Source: les dattes à Dattier
1) Are there any $(f,g) \in C(\mathbb R,\mathbb R_+)$ increasing with
$$\forall x \in \mathbb R, f(x)(\cos(x)+3/2)+g(x)(\sin(x)+3/2)=\exp(x)$$?

2) Are there any $(f,g) \in C(\mathbb R,\mathbb R_+)$ increasing with
$$\forall x \in \mathbb R, f(x)(\cos(x)+3/2)+g(x)(\sin(x)+3/2)=\exp(x/2)$$?
2 replies
Dattier
Thursday at 5:44 PM
alexheinis
Yesterday at 9:06 PM
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2022 Putnam A2
giginori   19
N Yesterday at 6:32 PM by Mathandski
Let $n$ be an integer with $n\geq 2.$ Over all real polynomials $p(x)$ of degree $n,$ what is the largest possible number of negative coefficients of $p(x)^2?$
19 replies
giginori
Dec 4, 2022
Mathandski
Yesterday at 6:32 PM
J
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J
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Determine the chromatic number of the Euclidean plane
paxtonw   1
N Apr 8, 2025 by solyaris
What is the smallest number of colors required to color every point in ℝ² so that no two points exactly 1 unit apart share the same color?
1 reply
paxtonw
Apr 7, 2025
solyaris
Apr 8, 2025
J
Determine the chromatic number of the Euclidean plane
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paxtonw
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paxtonw
#1 h  Apr 7, 2025, 3:03 PM
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What is the smallest number of colors required to color every point in ℝ² so that no two points exactly 1 unit apart share the same color?
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solyaris
632 posts
solyaris
#2 h  Apr 8, 2025, 7:33 AM
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This is known as the Hadwiger-Nelson problem. It is open. So far the best results give that four colours are not enough, whereas seven colours are enough.
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