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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
Nordic 2025 P3
anirbanbz   8
N 4 minutes ago by lksb
Source: Nordic 2025
Let $ABC$ be an acute triangle with orthocenter $H$ and circumcenter $O$. Let $E$ and $F$ be points on the line segments $AC$ and $AB$ respectively such that $AEHF$ is a parallelogram. Prove that $\vert OE \vert = \vert OF \vert$.
8 replies
anirbanbz
Mar 25, 2025
lksb
4 minutes ago
another functional inequality?
Scilyse   32
N 13 minutes ago by ihategeo_1969
Source: 2023 ISL A4
Let $\mathbb R_{>0}$ be the set of positive real numbers. Determine all functions $f \colon \mathbb R_{>0} \to \mathbb R_{>0}$ such that \[x \big(f(x) + f(y)\big) \geqslant \big(f(f(x)) + y\big) f(y)\]for every $x, y \in \mathbb R_{>0}$.
32 replies
+2 w
Scilyse
Jul 17, 2024
ihategeo_1969
13 minutes ago
Mount Inequality erupts in all directions!
BR1F1SZ   1
N 17 minutes ago by sami1618
Source: Austria National MO Part 1 Problem 1
Let $a$, $b$ and $c$ be pairwise distinct nonnegative real numbers. Prove that
\[
(a + b + c) \left( \frac{a}{(b - c)^2} + \frac{b}{(c - a)^2} + \frac{c}{(a - b)^2} \right) > 4.
\](Karl Czakler)
1 reply
BR1F1SZ
an hour ago
sami1618
17 minutes ago
Division involving difference of squares
BR1F1SZ   1
N 35 minutes ago by grupyorum
Source: Austria National MO Part 1 Problem 4
Determine all integers $n$ that can be written in the form
\[
n = \frac{a^2 - b^2}{b},
\]where $a$ and $b$ are positive integers.

(Walther Janous)
1 reply
BR1F1SZ
an hour ago
grupyorum
35 minutes ago
Sequence with GCD involved
mathematics2004   3
N 5 hours ago by anudeep
Source: 2021 Simon Marais, A2
Define the sequence of integers $a_1, a_2, a_3, \ldots$ by $a_1 = 1$, and
\[ a_{n+1} = \left(n+1-\gcd(a_n,n) \right) \times a_n \]for all integers $n \ge 1$.
Prove that $\frac{a_{n+1}}{a_n}=n$ if and only if $n$ is prime or $n=1$.
Here $\gcd(s,t)$ denotes the greatest common divisor of $s$ and $t$.
3 replies
mathematics2004
Nov 2, 2021
anudeep
5 hours ago
Putnam 2000 B2
ahaanomegas   20
N 6 hours ago by reni_wee
Prove that the expression \[ \dfrac {\text {gcd}(m, n)}{n} \dbinom {n}{m} \] is an integer for all pairs of integers $ n \ge m \ge 1 $.
20 replies
ahaanomegas
Sep 6, 2011
reni_wee
6 hours ago
f(x)f'(x)≥cos, f(∞)=undef. if f is bounded
jasperE3   2
N Today at 4:21 PM by Rohit-2006
Source: VJIMC 2013 1.1
Let $f:[0,\infty)\to\mathbb R$ be a differentiable function with $|f(x)|\le M$ and $f(x)f'(x)\ge\cos x$ for $x\in[0,\infty)$, where $M>0$. Prove that $f(x)$ does not have a limit as $x\to\infty$.
2 replies
jasperE3
May 30, 2021
Rohit-2006
Today at 4:21 PM
Trace with minimal polynomial x^n + x - 1
Assassino9931   5
N Today at 2:42 PM by loup blanc
Source: Vojtech Jarnik IMC 2025, Category I, P4
Let $A$ be an $n\times n$ real matrix with minimal polynomial $x^n + x - 1$. Prove that the trace of $(nA^{n-1} + I)^{-1}A^{n-2}$ is zero.
5 replies
Assassino9931
May 2, 2025
loup blanc
Today at 2:42 PM
Alternating series and integral
jestrada   5
N Today at 2:33 PM by jestrada
Source: own
Prove that for all $\alpha\in\mathbb{R}, \alpha>-1$, we have
$$ \frac{1}{\alpha+1}-\frac{1}{\alpha+2}+\frac{1}{\alpha+3}-\frac{1}{\alpha+4}+\cdots=\int_0^1 \frac{x^{\alpha}}{x+1}  \,dx. $$
5 replies
jestrada
Saturday at 10:56 PM
jestrada
Today at 2:33 PM
Rolles theorem
sasu1ke   5
N Today at 1:46 PM by GentlePanda24

Let \( f: \mathbb{R} \to \mathbb{R} \) be a twice differentiable function such that
\[
f(0) = 2, \quad f'(0) = -2, \quad \text{and} \quad f(1) = 1.
\]Prove that there exists a point \( \xi \in (0, 1) \) such that
\[
f(\xi) \cdot f'(\xi) + f''(\xi) = 0.
\]

5 replies
sasu1ke
May 3, 2025
GentlePanda24
Today at 1:46 PM
Convergent series with weight becomes divergent
P_Fazioli   3
N Today at 1:42 PM by solyaris
Initially, my problem was : is it true that if we fix $(b_n)$ positive such that $b_n\underset{{n}\longrightarrow{+\infty}}\longrightarrow +\infty$, then there exists $(a_n)$ positive such that $\displaystyle\sum_{n\geq 0}a_n$ converges and $\displaystyle\sum_{n\geq 0}a_nb_n$ diverges ?

Thinking about the continuous case : if $g:\mathbb{R}_+\longrightarrow\mathbb{R}$ is continuous, positive with $g(x)\underset{{x}\longrightarrow{+\infty}}\longrightarrow +\infty$, does $f$ continuous and positive exist on $\mathbb{R}_+$ such that $\displaystyle\int_0^{+\infty}f(x)\text{d}x$ converges and $\displaystyle\int_0^{+\infty}f(x)g(x)\text{d}x$ diverges ?

To the last question, the answer seems to be yes if $g$ is in the $\mathcal{C}^1$ class, increasing : I chose $f=\dfrac{g'}{g^2}$. With this idea, I had the idea to define $a_n=\dfrac{b_{n+1}-b_n}{b_n^2}$ but it is not clear that it is ok, even if $(b_n)$ is increasing.

Now I have some questions !

1) The main problem : is it true that if we fix $(b_n)$ positive such that $b_n\underset{{n}\longrightarrow{+\infty}}\longrightarrow +\infty$, then there exists $(a_n)$ positive such that $\displaystyle\sum_{n\geq 0}a_n$ converges and $\displaystyle\sum_{n\geq 0}a_nb_n$ diverges ? And if $(b_n)$ is increasing ?
2) is it true that if we fix $(b_n)$ positive increasing such that $b_n\underset{{n}\longrightarrow{+\infty}}\longrightarrow +\infty$
and $\frac{b_{n+1}}{b_n}\underset{{n}\longrightarrow{+\infty}}\longrightarrow 1$, then $\displaystyle\sum_{n\geq 0}\left(\frac{b_{n+1}}{b_n}-1\right)$ diverges ?
3) is it true that if we fix $(b_n)$ positive increasing such that $b_n\underset{{n}\longrightarrow{+\infty}}\longrightarrow +\infty$
and $\frac{b_{n+1}}{b_n}\underset{{n}\longrightarrow{+\infty}}\longrightarrow 1$, then $\displaystyle\sum_{n\geq 0}\frac{b_{n+1}-b_n}{b_n^2}$ converges ?
4) if $(b_n)$ is positive increasing and such that $b_n\underset{{n}\longrightarrow{+\infty}}\longrightarrow +\infty$
and $\frac{b_{n+1}}{b_n}$ does not converge to $1$, can $\displaystyle\sum_{n\geq 0}\frac{b_{n+1}-b_n}{b_n^2}$ diverge ?
5) for the continuous case, is it true if we suppose $g$ only to be continuous ?

3 replies
P_Fazioli
Today at 5:37 AM
solyaris
Today at 1:42 PM
Putnam 1954 B1
sqrtX   6
N Today at 12:48 PM by AshAuktober
Source: Putnam 1954
Show that the equation $x^2 -y^2 =a^3$ has always integral solutions for $x$ and $y$ whenever $a$ is a positive integer.
6 replies
sqrtX
Jul 17, 2022
AshAuktober
Today at 12:48 PM
Cube Colouring Problems
Saucepan_man02   1
N Today at 10:43 AM by removablesingularity
Could anyone kindly post some problems (and hopefully along the solution thread/final answer) related to combinatorial colouring of cube?
1 reply
Saucepan_man02
May 3, 2025
removablesingularity
Today at 10:43 AM
D1026 : An equivalent
Dattier   4
N Today at 4:56 AM by 3ch03s
Source: les dattes à Dattier
Let $u_0=1$ and $\forall n \in \mathbb N, u_{2n+1}=\ln(1+u_{2n}), u_{2n+2}=\sin(u_{2n+1})$.

Find an equivalent of $u_n$.
4 replies
Dattier
May 3, 2025
3ch03s
Today at 4:56 AM
Radius of circle tangent to two equal circles and a common line
rilarfer   1
N Apr 19, 2025 by Lankou
Source: ASJTNic 2005
Two circles of radius 2 are tangent to each other and to a straight line. A third circle is placed so that it is tangent to both of the other circles and also tangent to the same straight line.

What is the radius of the third circle?

IMAGE
1 reply
rilarfer
Apr 19, 2025
Lankou
Apr 19, 2025
Radius of circle tangent to two equal circles and a common line
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Source: ASJTNic 2005
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rilarfer
26 posts
#1
Y by
Two circles of radius 2 are tangent to each other and to a straight line. A third circle is placed so that it is tangent to both of the other circles and also tangent to the same straight line.

What is the radius of the third circle?

[asy]
size(150);
draw((-1,0)--(3,0)); // ground line
draw(circle((0,1),1)); // left big circle
draw(circle((2,1),1)); // right big circle
draw(circle((1,0.25),0.25)); // small circle in between
[/asy]
Z K Y
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Lankou
1396 posts
#2 • 1 Y
Y by teomihai
We have the relation $(2+r)^2=(2-r)^2+2^2$; $r=\frac{1}{2}$
This post has been edited 1 time. Last edited by Lankou, Apr 19, 2025, 7:08 PM
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