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
AIMEification
ethanhansummerfun   9
N a minute ago by ethanhansummerfun
(moved from MSM with modifications, if this is too hard go to MSM for an easier ver.)

(pls don’t flame me if this is too hard, I thought of it in the shower).

people this came up in the shower I was considering it and got a loosely defined statement on paper and copied it up here, please if there are errors don’t be mad. yall chilllll

To “AIMEify” a number $\frac{m}{n}$ is to convert it into $m+n$, given $gcd(m,n) = 1$ . To AIMEify a number $m\sqrt[k]{n}$ is to convert it into $m+k+n$, where $n$ is power free (i.e. not divisible by the power of any prime). Define a function $f(x)$ to AIMEify any square-root radical and any rational number such that it must always output a 3-digit number from $000$ to $999$, inclusive.

Find all $k$ such that there exists a real number $x$ such that $f(x^2) = f(x)^2 + k$, or prove none exist.

(I haven’t even solven this :rotfl: so feel free to cook)
9 replies
ethanhansummerfun
24 minutes ago
ethanhansummerfun
a minute ago
Diophantine equation (easy?)
molikpagaria   7
N 2 hours ago by molikpagaria
\[
x^4 + y^4 + z^4 - 2x^2y^2 - 2y^2z^2 - 2x^2z^2 = 24
\]
7 replies
molikpagaria
Yesterday at 4:40 PM
molikpagaria
2 hours ago
General math problems
wimpykid   2
N 2 hours ago by whwlqkd

$\textbf{Problem 1}$ In an equilateral triangle of $\frac{n(n + 1)}{2}$ pennies, with $n$ pennies along each side of the triangle, all but one penny shows heads. A $move$ consists of choosing two adjacent pennies with centers $A$ and $B$ and flipping every penny on line $AB$. Determine all initial arrangements - the value of $n$ and the position of the coin initially showing tails - from which one can make all the coins show tails after finitely many moves.

$\textbf{Problem 2}$ Find all functions $f:\mathbb{N} \rightarrow \mathbb{N}$ such that
$$f(f(f(n))) + f(f(n)) + f(n) = 3n$$for all $n\in \mathbb{N}$.
2 replies
wimpykid
4 hours ago
whwlqkd
2 hours ago
Find the minimum area
cataquino2007   1
N 4 hours ago by Mathzeus1024
Let ABC be an equilateral triangle with AC = 12. Points M and K lie on sides AB and BC, respectively. A new triangle is formed with sides equal to AK, CM, and KM. What is the minimum possible area of this triangle, given that MK = 4√6?
1 reply
cataquino2007
Jul 28, 2025
Mathzeus1024
4 hours ago
Bijection on set of maps
enter16180   1
N 4 hours ago by Tintarn
Source: IMC 2025, Problem 5
For a positive integer $n$, let $[n]=\{1,2, \ldots, n\}$. Denote by $S_n$ the set of all bijections from $[n]$ to $[n]$, and let $T_n$ be the set of all maps from $[n]$ to $[n]$. Define the order $\operatorname{ord}(\tau)$ of a map $\tau \in T_n$ as the number of distinct maps in the set $\{\tau, \tau \circ \tau, \tau \circ \tau \circ \tau, \ldots\}$ where o denotes composition. Finally, let
$$
f(n)=\max _{\tau \in S_n} \operatorname{ord}(\tau) \quad \text { and } \quad g(n)=\max _{\tau \in T_n} \operatorname{ord}(\tau) .
$$Prove that $g(n)<f(n)+n^{0.501}$ for sufficiently large $n$.
1 reply
enter16180
Today at 11:23 AM
Tintarn
4 hours ago
Polynomial
enter16180   4
N 4 hours ago by grupyorum
Source: IMC 2025, Problem 1
Let $P \in \mathbb{R}[x]$ be a polynomial with real coefficients, and suppose $\operatorname{deg}(P) \geq 2$. For every $x \in \mathbb{R}$, let $\ell_x \subset \mathbb{R}^2$ denote the line tangent to the graph of $P$ at the point ( $x, P(x)$ ).
a) Suppose that the degree of $P$ is odd. Show that $\bigcup_{x \in \mathbb{R}} \ell_x=\mathbb{R}^2$.
b) Does there exist a polynomial of even degree for which the above equality still holds?
4 replies
enter16180
Today at 11:12 AM
grupyorum
4 hours ago
Twice continuously differntiable function
enter16180   5
N 4 hours ago by bsf714
Source: IMC 2025, Problem 2
Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a twice continuously differentiable function, and suppose that $\int_{-1}^1 f(x) \mathrm{d} x=0$ and $f(1)=f(-1)=1$. Prove that
$$
\int_{-1}^1\left(f^{\prime \prime}(x)\right)^2 \mathrm{~d} x \geq 15
$$and find all such functions for which equality holds.
5 replies
enter16180
Today at 11:14 AM
bsf714
4 hours ago
Real symmetric matrix of rank 1
enter16180   3
N 4 hours ago by grupyorum
Source: IMC 2025, Problem 3
Denote by $\mathcal{S}$ the set of all real symmetric $2025 \times 2025$ matrices of rank $1$ whose entries take values $-1$ or $+1$ . Let $A, B \in \mathcal{S}$ be matrices chosen independently uniformly at random. Find the probability that $A$ and $B$ commute, i.e. $A B=B A$.
3 replies
enter16180
Today at 11:16 AM
grupyorum
4 hours ago
Trigonometric functional equation
Eul12   0
Today at 11:18 AM
Source: Dubikajtis
Find all derivates functions f : IR---->IR such that
2*(f(x))^2 + f(pi/2 - 2*x) = 1
for all real x.
Wecome for any ideas
0 replies
Eul12
Today at 11:18 AM
0 replies
Wallis limit extension
P0tat0b0y   0
Today at 10:13 AM
Source: Own
Extension of Wallis limit: $\underset{n\to \infty }{\mathop{\lim }}\,\left( \prod\limits_{k=1}^{n}{{{\left( \frac{2k}{2k-1} \right)}^{2}}}-n\pi  \right)=\frac{\pi }{4}$
0 replies
P0tat0b0y
Today at 10:13 AM
0 replies
Putnam 1972 B6
Kunihiko_Chikaya   4
N Today at 7:52 AM by smileapple
Let $ n_1<n_2<n_3<\cdots <n_k$ be a set of positive integers. Prove that the polynomial $ 1+z^{n_1}+z^{n_2}+\cdots +z^{n_k}$ has no roots inside the circle $ |z|<\frac{\sqrt{5}-1}{2}$.
4 replies
Kunihiko_Chikaya
Jun 5, 2008
smileapple
Today at 7:52 AM
?k/nk is an irrational number
fxandi   2
N Today at 6:13 AM by fxandi
If \(\dfrac{n_{k+1}}{n_k} \geq Q > 1\) for \(k \in \mathbb{N}\). Prove that \(\displaystyle\sum \dfrac{\varepsilon_k}{n_k}\) is an irrational number.
In the problem above \(\{\varepsilon_k\}\) is arbitrary sequence with elements \(+1\) and \(-1\).
2 replies
fxandi
Jul 28, 2025
fxandi
Today at 6:13 AM
Frankl Theorem?
EthanWYX2009   1
N Today at 4:34 AM by Twoisaprime
Source: 2024 February 谜之竞赛-3
Let \( n \) be a positive integer, \( f(n) \) denote the maximum possible size of a family \(\mathcal{F}\) of subsets of \(\{1, 2, \cdots, n\}\) such that for any two subsets \( X, Y \in \mathcal{F} \), \( |X \cap Y| \) is a perfect square.

Show that $2^{\lfloor \sqrt{n} \rfloor} \leq f(n) \leq 100^{\sqrt{n} \ln n}.$

Proposed by Qianhong Cheng, Guangdong Experimental High School
1 reply
EthanWYX2009
Jul 26, 2025
Twoisaprime
Today at 4:34 AM
an integral
Svyatoslav   2
N Today at 2:19 AM by Alphaamss
How do we prove analytically that
$$\int_0^{\pi/2}\frac{\ln(1+\cos x)-x}{\sqrt{\sin x}}\,dx=0\quad?$$The sourse: Quora

Numeric evaluation
2 replies
Svyatoslav
Jul 23, 2025
Alphaamss
Today at 2:19 AM
Maximum value of function (with two variables)
Saucepan_man02   1
N May 22, 2025 by Saucepan_man02
If $f(\theta) = \min(|2x-7|+|x-4|+|x-2 -\sin \theta|)$, where $x, \theta \in \mathbb R$, then maximum value of $f(\theta)$.
1 reply
Saucepan_man02
May 22, 2025
Saucepan_man02
May 22, 2025
Maximum value of function (with two variables)
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If $f(\theta) = \min(|2x-7|+|x-4|+|x-2 -\sin \theta|)$, where $x, \theta \in \mathbb R$, then maximum value of $f(\theta)$.
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