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k a May Highlights and 2025 AoPS Online Class Information
jlacosta   0
May 1, 2025
May is an exciting month! National MATHCOUNTS is the second week of May in Washington D.C. and our Founder, Richard Rusczyk will be presenting a seminar, Preparing Strong Math Students for College and Careers, on May 11th.

Are you interested in working towards MATHCOUNTS and don’t know where to start? We have you covered! If you have taken Prealgebra, then you are ready for MATHCOUNTS/AMC 8 Basics. Already aiming for State or National MATHCOUNTS and harder AMC 8 problems? Then our MATHCOUNTS/AMC 8 Advanced course is for you.

Summer camps are starting next month at the Virtual Campus in math and language arts that are 2 - to 4 - weeks in duration. Spaces are still available - don’t miss your chance to have an enriching summer experience. 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 upcoming events:
[list][*]May 9th, 4:30pm PT/7:30pm ET, Casework 2: Overwhelming Evidence — A Text Adventure, a game where participants will work together to navigate the map, solve puzzles, and win! All are welcome.
[*]May 19th, 4:30pm PT/7:30pm ET, What's Next After Beast Academy?, designed for students finishing Beast Academy and ready for Prealgebra 1.
[*]May 20th, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 1 Math Jam, Problems 1 to 4, join the Canada/USA Mathcamp staff for this exciting Math Jam, where they discuss solutions to Problems 1 to 4 of the 2025 Mathcamp Qualifying Quiz!
[*]May 21st, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 2 Math Jam, Problems 5 and 6, Canada/USA Mathcamp staff will discuss solutions to Problems 5 and 6 of the 2025 Mathcamp Qualifying Quiz![/list]
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0 replies
jlacosta
May 1, 2025
0 replies
Serbian selection contest for the IMO 2025 - P6
OgnjenTesic   15
N 5 minutes ago by math90
Source: Serbian selection contest for the IMO 2025
For an $n \times n$ table filled with natural numbers, we say it is a divisor table if:
- the numbers in the $i$-th row are exactly all the divisors of some natural number $r_i$,
- the numbers in the $j$-th column are exactly all the divisors of some natural number $c_j$,
- $r_i \ne r_j$ for every $i \ne j$.

A prime number $p$ is given. Determine the smallest natural number $n$, divisible by $p$, such that there exists an $n \times n$ divisor table, or prove that such $n$ does not exist.

Proposed by Pavle Martinović
15 replies
OgnjenTesic
May 22, 2025
math90
5 minutes ago
Easy Number Theory
math_comb01   39
N 21 minutes ago by Adywastaken
Source: INMO 2024/3
Let $p$ be an odd prime and $a,b,c$ be integers so that the integers $$a^{2023}+b^{2023},\quad b^{2024}+c^{2024},\quad a^{2025}+c^{2025}$$are divisible by $p$.
Prove that $p$ divides each of $a,b,c$.
$\quad$
Proposed by Navilarekallu Tejaswi
39 replies
math_comb01
Jan 21, 2024
Adywastaken
21 minutes ago
Painting Beads on Necklace
amuthup   46
N 28 minutes ago by quantam13
Source: 2021 ISL C2
Let $n\ge 3$ be a fixed integer. There are $m\ge n+1$ beads on a circular necklace. You wish to paint the beads using $n$ colors, such that among any $n+1$ consecutive beads every color appears at least once. Find the largest value of $m$ for which this task is $\emph{not}$ possible.

Carl Schildkraut, USA
46 replies
amuthup
Jul 12, 2022
quantam13
28 minutes ago
Iran geometry
Dadgarnia   38
N 36 minutes ago by cursed_tangent1434
Source: Iranian TST 2018, first exam day 2, problem 4
Let $ABC$ be a triangle ($\angle A\neq 90^\circ$). $BE,CF$ are the altitudes of the triangle. The bisector of $\angle A$ intersects $EF,BC$ at $M,N$. Let $P$ be a point such that $MP\perp EF$ and $NP\perp BC$. Prove that $AP$ passes through the midpoint of $BC$.

Proposed by Iman Maghsoudi, Hooman Fattahi
38 replies
Dadgarnia
Apr 8, 2018
cursed_tangent1434
36 minutes ago
Quadruple Binomial Coefficient Sum
P162008   4
N Yesterday at 8:40 PM by vmene
Source: Self made by my Elder brother
$\sum_{p=0}^{\infty} \sum_{r=0}^{\infty} \sum_{q=1}^{\infty} \sum_{s=0}^{p+q - 1} \frac{((-1)^{p+r+s+1})(2^{p+q-1}) \binom{p + q - s - 1}{p + q - 2s - 1}}{4^s(2p^2q + 2pqr + pq + qr)(2p + 2q + 2r + 3)}.$
4 replies
P162008
Thursday at 8:04 PM
vmene
Yesterday at 8:40 PM
IMC 1994 D1 P5
j___d   5
N Yesterday at 5:39 PM by krigger
a) Let $f\in C[0,b]$, $g\in C(\mathbb R)$ and let $g$ be periodic with period $b$. Prove that $\int_0^b f(x) g(nx)\,\mathrm dx$ has a limit as $n\to\infty$ and
$$\lim_{n\to\infty}\int_0^b f(x)g(nx)\,\mathrm dx=\frac 1b \int_0^b f(x)\,\mathrm dx\cdot\int_0^b g(x)\,\mathrm dx$$
b) Find
$$\lim_{n\to\infty}\int_0^\pi \frac{\sin x}{1+3\cos^2nx}\,\mathrm dx$$
5 replies
j___d
Mar 6, 2017
krigger
Yesterday at 5:39 PM
2023 Putnam A2
giginori   21
N Yesterday at 3:32 PM by pie854
Let $n$ be an even positive integer. Let $p$ be a monic, real polynomial of degree $2 n$; that is to say, $p(x)=$ $x^{2 n}+a_{2 n-1} x^{2 n-1}+\cdots+a_1 x+a_0$ for some real coefficients $a_0, \ldots, a_{2 n-1}$. Suppose that $p(1 / k)=k^2$ for all integers $k$ such that $1 \leq|k| \leq n$. Find all other real numbers $x$ for which $p(1 / x)=x^2$.
21 replies
giginori
Dec 3, 2023
pie854
Yesterday at 3:32 PM
Putnam 2019 A1
awesomemathlete   33
N Yesterday at 3:25 PM by cursed_tangent1434
Source: 2019 William Lowell Putnam Competition
Determine all possible values of $A^3+B^3+C^3-3ABC$ where $A$, $B$, and $C$ are nonnegative integers.
33 replies
awesomemathlete
Dec 10, 2019
cursed_tangent1434
Yesterday at 3:25 PM
IMC 1994 D1 P2
j___d   5
N Yesterday at 3:11 PM by krigger
Let $f\in C^1(a,b)$, $\lim_{x\to a^+}f(x)=\infty$, $\lim_{x\to b^-}f(x)=-\infty$ and $f'(x)+f^2(x)\geq -1$ for $x\in (a,b)$. Prove that $b-a\geq\pi$ and give an example where $b-a=\pi$.
5 replies
j___d
Mar 6, 2017
krigger
Yesterday at 3:11 PM
A Construction in Multivariable Analysis
MrOrange   0
Yesterday at 2:11 PM
Source: Garling's A COURSE IN MATHEMATICAL ANALYSIS
Construct a continuous real valued function \( f \) on \( \mathbb{R}^2 \) for which
\[
\lim_{R \to \infty} \int_{\|x\|_2 \leq R} f(x) \, dx = 0
\]and for which
\[
\lim_{R \to \infty} \int_{\|x\|_\infty \leq R} f(x) \, dx \text{ does not exist.}
\]
0 replies
MrOrange
Yesterday at 2:11 PM
0 replies
Possible values of determinant of 0-1 matrices
mathematics2004   4
N Yesterday at 1:56 PM by loup blanc
Source: 2021 Simon Marais, A3
Let $\mathcal{M}$ be the set of all $2021 \times 2021$ matrices with at most two entries in each row equal to $1$ and all other entries equal to $0$.
Determine the size of the set $\{ \det A : A \in M \}$.
Here $\det A$ denotes the determinant of the matrix $A$.
4 replies
mathematics2004
Nov 2, 2021
loup blanc
Yesterday at 1:56 PM
ISI UGB 2025
Entrepreneur   1
N Yesterday at 1:49 PM by Knight2E4
Source: ISI UGB 2025
1.)
Suppose $f:\mathbb R\to\mathbb R$ is differentiable and $|f'(x)|<\frac 12\;\forall\;x\in\mathbb R.$ Show that for some $x_0\in\mathbb R,f(x_0)=x_0.$

3.)
Suppose $f:[0,1]\to\mathbb R$ is differentiable with $f(0)=0.$ If $|f'(x)|\le f(x)\;\forall\;x\in[0,1],$ then show that $f(x)=0\;\forall\;x.$

4.)
Let $S^1=\{z\in\mathbb C:|z|=1\}$ be the unit circle in the complex plane. Let $f:S^1\to S^1$ be the map given by $f(z)=z^2.$ We define $f^{(1)}:=f$ and $f^{(k+1)}=f\circ f^{(k)}$ for $k\ge 1.$ The smallest positive integer $n$ such that $f^n(z)=z$ is called period of $z.$ Determine the total number of points $S^1$ of period $2025.$

6.)
Let $\mathbb N$ denote the set of natural numbers, and let $(a_i,b_i), 1\le i\le 9,$ be nine distinct tuples in $\mathbb N\times\mathbb N.$ Show that there are $3$ distinct elements in the set $\{2^{a_i}3^{b_i}:1\le i\le 9\}$ whose product is a perfect cube.

8.)
Let $n\ge 2$ and let $a_1\le a_2\le\cdots\le a_n$ be positive integers such that $$\sum_{i=1}^n a_i=\prod_{i=1}^n a_i.$$Prove that $$\sum_{i=1}^n a_i\le 2n$$and determine when equality holds.
1 reply
Entrepreneur
May 27, 2025
Knight2E4
Yesterday at 1:49 PM
Recurrence trouble
SomeonecoolLovesMaths   3
N Yesterday at 1:44 PM by Knight2E4
Let $0 < x_0 < y_0$ be real numbers. Define $x_{n+1} = \frac{x_n + y_n}{2}$ and $y_{n+1} = \sqrt{x_{n+1}y_n}$.
Prove that $\lim_{n \to \infty} x_n = \lim_{n \to \infty} y_n$ and hence find the limit.
3 replies
SomeonecoolLovesMaths
May 28, 2025
Knight2E4
Yesterday at 1:44 PM
Trigo or Complex no.?
hzbrl   5
N Yesterday at 9:20 AM by GreenKeeper
(a) Let $y=\cos \phi+\cos 2 \phi$, where $\phi=\frac{2 \pi}{5}$. Verify by direct substitution that $y$ satisfies the quadratic equation $2 y^2=3 y+2$ and deduce that the value of $y$ is $-\frac{1}{2}$.
(b) Let $\theta=\frac{2 \pi}{17}$. Show that $\sum_{k=0}^{16} \cos k \theta=0$
(c) If $z=\cos \theta+\cos 2 \theta+\cos 4 \theta+\cos 8 \theta$, show that the value of $z$ is $-(1-\sqrt{17}) / 4$.



I could solve (a) and (b). Can anyone help me with the 3rd part please?
5 replies
hzbrl
May 27, 2025
GreenKeeper
Yesterday at 9:20 AM
3D geometry theorem
KAME06   1
N Apr 22, 2025 by mathuz
Let $M$ a point in the space and $G$ the centroid of a tetrahedron $ABCD$. Prove that:
$$\frac{1}{4}(AB^2+AC^2+AD^2+BC^2+BD^2+CD^2)+4MG^2=MA^2+MB^2+MC^2+MD^2$$
1 reply
KAME06
Apr 21, 2025
mathuz
Apr 22, 2025
3D geometry theorem
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KAME06
161 posts
#1
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Let $M$ a point in the space and $G$ the centroid of a tetrahedron $ABCD$. Prove that:
$$\frac{1}{4}(AB^2+AC^2+AD^2+BC^2+BD^2+CD^2)+4MG^2=MA^2+MB^2+MC^2+MD^2$$
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mathuz
1525 posts
#2 • 1 Y
Y by KAME06
Just use the vectors (like in Leibniz's formula for a triangle).
Or, see this: https://vixra.org/pdf/1003.0187v1.pdf
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