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k a April Highlights and 2025 AoPS Online Class Information
jlacosta   0
Wednesday at 3:18 PM
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
Wednesday at 3:18 PM
0 replies
J
H
Geo Mock #6
Bluesoul   1
N 26 minutes ago by vanstraelen
Consider triangle $ABC$ with $AB=5, BC=8, AC=7$, denote the incenter of the triangle as $I$. Extend $BI$ to meet the circumcircle of $\triangle{AIC}$ at $Q\neq I$, find the length of $QC$.
1 reply
Bluesoul
Apr 1, 2025
vanstraelen
26 minutes ago
J
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Congruence
Ecrin_eren   1
N an hour ago by Ecrin_eren
Find the number of integer pairs (x, y) satisfying the congruence equation:

3y² + 3x²y + y³ ≡ 3x² (mod 41)

for 0 ≤ x, y < 41.

1 reply
Ecrin_eren
Yesterday at 10:34 AM
Ecrin_eren
an hour ago
J
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Probability
Ecrin_eren   1
N an hour ago by Ecrin_eren
In a board, James randomly writes A , B or C in each cell. What is the probability that, for every row and every column, the number of A 's modulo 3 is equal to the number of B's modulo 3?

1 reply
Ecrin_eren
Yesterday at 11:21 AM
Ecrin_eren
an hour ago
J
H
Excalibur Identity
jjsunpu   9
N 2 hours ago by fruitmonster97
proof is below
9 replies
jjsunpu
Yesterday at 3:27 PM
fruitmonster97
2 hours ago
J
No more topics!
H
J
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Prove that \( S \) contains all integers.
nhathhuyyp5c   1
N Mar 30, 2025 by GreenTea2593
Let \( S \) be a set of integers satisfying the following property: For every positive integer \( n \) and every set of coefficients \( a_0, a_1, \dots, a_n \in S \), all integer roots of the polynomial $P(x) = a_0 + a_1 x + \dots + a_n x^n $ are also elements of \( S \). It is given that \( S \) contains all numbers of the form \( 2^a - 2^b \) where \( a, b \) are positive integers. Prove that \( S \) contains all integers.









1 reply
nhathhuyyp5c
Mar 29, 2025
GreenTea2593
Mar 30, 2025
J
Prove that \( S \) contains all integers.
G H J
Reveal topic tags
number theory
algebra
polynomial
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nhathhuyyp5c
57 posts
nhathhuyyp5c
#1 h  Mar 29, 2025, 3:53 PM
Y by
Let \( S \) be a set of integers satisfying the following property: For every positive integer \( n \) and every set of coefficients \( a_0, a_1, \dots, a_n \in S \), all integer roots of the polynomial $P(x) = a_0 + a_1 x + \dots + a_n x^n $ are also elements of \( S \). It is given that \( S \) contains all numbers of the form \( 2^a - 2^b \) where \( a, b \) are positive integers. Prove that \( S \) contains all integers.
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GreenTea2593
230 posts
GreenTea2593
#2 h  Mar 30, 2025, 3:19 AM • 1 Y
Y by nhathhuyyp5c
Wrong forum, this is also 2019 IMO Shortlist N3
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