Plan ahead for the next school year. Schedule your class today!

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k a July Highlights and 2025 AoPS Online Class Information
jwelsh   0
Jul 1, 2025
We are halfway through summer, so be sure to carve out some time to keep your skills sharp and explore challenging topics at AoPS Online and our AoPS Academies (including the Virtual Campus)!

[list][*]Over 60 summer classes are starting at the Virtual Campus on July 7th - check out the math and language arts options for middle through high school levels.
[*]At AoPS Online, we have accelerated sections where you can complete a course in half the time by meeting twice/week instead of once/week, starting on July 8th:
[list][*]MATHCOUNTS/AMC 8 Basics
[*]MATHCOUNTS/AMC 8 Advanced
[*]AMC Problem Series[/list]
[*]Plus, AoPS Online has a special seminar July 14 - 17 that is outside the standard fare: Paradoxes and Infinity
[*]We are expanding our in-person AoPS Academy locations - are you looking for a strong community of problem solvers, exemplary instruction, and math and language arts options? Look to see if we have a location near you and enroll in summer camps or academic year classes today! New locations include campuses in California, Georgia, New York, Illinois, and Oregon and more coming soon![/list]

MOP (Math Olympiad Summer Program) just ended and the IMO (International Mathematical Olympiad) is right around the corner! This year’s IMO will be held in Australia, July 10th - 20th. Congratulations to all the MOP students for reaching this incredible level and best of luck to all selected to represent their countries at this year’s IMO! Did you know that, in the last 10 years, 59 USA International Math Olympiad team members have medaled and have taken over 360 AoPS Online courses. Take advantage of our Worldwide Online Olympiad Training (WOOT) courses
and train with the best! Please note that early bird pricing ends August 19th!
Are you tired of the heat and thinking about Fall? You can plan your Fall schedule now with classes at either AoPS Online, AoPS Academy Virtual Campus, or one of our AoPS Academies around the US.

Our full course list for upcoming classes is below:
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0 replies
jwelsh
Jul 1, 2025
0 replies
2^n away from additivity
MarkBcc168   4
N 11 minutes ago by PHSH
Source: IMO Shortlist 2024 A4
Let \(\mathbb{Z}_{>0}\) be the set of all positive integers. Determine all subsets \(\mathcal{S}\) of \(\{2^{0},2^{1},2^{2},\ldots\}\) for which there exists a function \(f\colon\mathbb{Z}_{>0}\to\mathbb{Z}_{>0}\) such that
\[\mathcal{S}=\{f(a+b)-f(a)-f(b)\mid a,b\in\mathbb{Z}_{>0}\}.\]Pitchayut Saengrungkongka, Thailand
4 replies
MarkBcc168
Jul 16, 2025
PHSH
11 minutes ago
original problem with double angle identities
ACalculationError   0
21 minutes ago
Problem Statement: Evaluate
$$\frac{1}{\sin^2 \theta} + \frac{1}{\cos^2 \theta}$$given that
$$\tan(2\theta) = 4, \quad 0 < \theta < \frac{\pi}{4}$$Answer Confirmation
Solution
0 replies
+1 w
ACalculationError
21 minutes ago
0 replies
functional equation
COCBSGGCTG3   7
N 28 minutes ago by JARP091
Source: Azerbaijan Senior Math Olympiad Training TST 2025 P2
Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that the following equality holds for any real numbers $x$ and $y$.
$f(f(x) + xf(y)) = xf(y + 1)$
7 replies
COCBSGGCTG3
Today at 4:41 AM
JARP091
28 minutes ago
Functional xf(x+f(y))=(y-x)f(f(x)) for all reals x,y
cretanman   65
N 31 minutes ago by youochange
Source: BMO 2023 Problem 1
Find all functions $f\colon \mathbb{R} \rightarrow \mathbb{R}$ such that for all $x,y \in \mathbb{R}$,
\[xf(x+f(y))=(y-x)f(f(x)).\]
Proposed by Nikola Velov, Macedonia
65 replies
cretanman
May 10, 2023
youochange
31 minutes ago
No more topics!
another n x n table problem.
pohoatza   3
N May 16, 2025 by reni_wee
Source: Romanian JBTST III 2007, problem 3
Consider a $n$x$n$ table such that the unit squares are colored arbitrary in black and white, such that exactly three of the squares placed in the corners of the table are white, and the other one is black. Prove that there exists a $2$x$2$ square which contains an odd number of unit squares white colored.
3 replies
pohoatza
May 13, 2007
reni_wee
May 16, 2025
another n x n table problem.
G H J
Source: Romanian JBTST III 2007, problem 3
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pohoatza
1145 posts
#1 • 3 Y
Y by Adventure10, Mango247, and 1 other user
Consider a $n$x$n$ table such that the unit squares are colored arbitrary in black and white, such that exactly three of the squares placed in the corners of the table are white, and the other one is black. Prove that there exists a $2$x$2$ square which contains an odd number of unit squares white colored.
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PhilAndrew
207 posts
#2 • 3 Y
Y by Anajar, Adventure10, Mango247
My solution from the contest
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Sashsiam_2
81 posts
#3 • 3 Y
Y by Anajar, Adventure10, Mango247
Let us try to color the table in such a way that all the 2 x 2 squares have an even number of white unit squares. We start coloring the first two rows. If the first column has the two squares of the same color, then so has the second, and easy induction shows that it holds for the entire "2-row". If the first column has the two squares of different colors, then so have the second and the others.

Take the first square and the last square of the bottom row. Assume that they are of the same color. The previous discussion shows that the first square and the last square of the second row must be both black or both white. The same then applies to the third column an so on. We see that we must have an even number of white corners. With a similar reasoning, if the first square and last square of the bottom row were of different colors then so would be the first square and last square of the second row and so on. Again the number of white corners is even, what finishes the problem.

Remark: this proof works for an m x n table.
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reni_wee
94 posts
#4
Y by
Solved with cursed_tangent1434. Consider the set of all $2\times 2$ patches in our $n \times n$ grid. Assume for the sake of contradiction that each $2\times 2$ grid contains an even number of white cells. Let a $2\times 2$ grid be known as a square in what follows.

Each of the corner cells is contained within exactly one square, each non-corner cell along the sides of the grid in exactly two squares and every other cell in exactly four squares. Summing the number of occurrences of white cells in a square with multiplicity by assumption we have that this is even. However this is also $3+2x+4y$ for some non-negative integers $x$ and $y$ which is clearly odd. Hence our assumption must have been false and the result follows.
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