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k a May Highlights and 2025 AoPS Online Class Information
jlacosta   0
Thursday at 11:16 PM
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.

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[*]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
Thursday at 11:16 PM
0 replies
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Random modulos
m4thbl3nd3r   6
N a few seconds ago by GreekIdiot
Find all pair of integers $(x,y)$ s.t $x^2+3=y^7$
6 replies
m4thbl3nd3r
Apr 7, 2025
GreekIdiot
a few seconds ago
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Concurrency in Parallelogram
amuthup   89
N a minute ago by happypi31415
Source: 2021 ISL G1
Let $ABCD$ be a parallelogram with $AC=BC.$ A point $P$ is chosen on the extension of ray $AB$ past $B.$ The circumcircle of $ACD$ meets the segment $PD$ again at $Q.$ The circumcircle of triangle $APQ$ meets the segment $PC$ at $R.$ Prove that lines $CD,AQ,BR$ are concurrent.
89 replies
amuthup
Jul 12, 2022
happypi31415
a minute ago
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deleting multiple or divisor in pairs from 2-50 on a blackboard
parmenides51   1
N an hour ago by TheBaiano
Source: 2023 May Olympiad L2 p3
The $49$ numbers $2,3,4,...,49,50$ are written on the blackboard . An allowed operation consists of choosing two different numbers $a$ and $b$ of the blackboard such that $a$ is a multiple of $b$ and delete exactly one of the two. María performs a sequence of permitted operations until she observes that it is no longer possible to perform any more. Determine the minimum number of numbers that can remain on the board at that moment.
1 reply
parmenides51
Mar 24, 2024
TheBaiano
an hour ago
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at everystep a, b, c are replaced by a+\gcd(b,c), b+\gcd(a,c), c+\gcd(a,b)
NJAX   9
N an hour ago by atdaotlohbh
Source: 2nd Al-Khwarizmi International Junior Mathematical Olympiad 2024, Day2, Problem 8
Three positive integers are written on the board. In every minute, instead of the numbers $a, b, c$, Elbek writes $a+\gcd(b,c), b+\gcd(a,c), c+\gcd(a,b)$ . Prove that there will be two numbers on the board after some minutes, such that one is divisible by the other.
Note. $\gcd(x,y)$ - Greatest common divisor of numbers $x$ and $y$

Proposed by Sergey Berlov, Russia
9 replies
NJAX
May 31, 2024
atdaotlohbh
an hour ago
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trigonometric functions
VivaanKam   11
N 3 hours ago by aok
Hi could someone explain the basic trigonometric functions to me like sin, cos, tan etc.
Thank you!
11 replies
VivaanKam
Apr 29, 2025
aok
3 hours ago
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1201 divides sum of powers
V0305   1
N 4 hours ago by vincentwant
(Source: me) Prove that for all positive integers $n$, $1201 \mid 2^{2^n} + 59^{2^n} + 61^{2^n}$.
1 reply
V0305
4 hours ago
vincentwant
4 hours ago
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Interesting geometry
polarLines   5
N 5 hours ago by Mathworld314
Let $ABC$ be an equilateral triangle of side length $2$. Point $A'$ is chosen on side $BC$ such that the length of $A'B$ is $k<1$. Likewise points $B'$ and $C'$ are chosen on sides $CA$ and $AB$. with $CB'=AC'=k$. Line segments are drawn from points $A',B',C'$ to their corresponding opposite vertices. The intersections of these line segments form a triangle, labeled $PQR$. Prove that $\Delta PQR$ is an equilateral triangle with side length ${4(1-k) \over \sqrt{k^2-2k+4}}$.
5 replies
polarLines
May 20, 2018
Mathworld314
5 hours ago
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Showing that certain number is divisible by 13
BBNoDollar   3
N 5 hours ago by Shan3t
Show that 3^(n+2) + 9^(n+1) + 4^(2n+1) + 4^(4n+1) is divisible by 13 for every n natural number.
3 replies
BBNoDollar
Today at 2:54 PM
Shan3t
5 hours ago
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Inequality
tom-nowy   0
6 hours ago
Let $0<a,b,c,<1$. Show that
$$ \frac{3(a+b+c)}{a+b+c+3abc} > \frac{1}{1+a} + \frac{1}{1+b} + \frac{1}{1+c} .$$
0 replies
tom-nowy
6 hours ago
0 replies
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Logarithm of a product
axsolers_24   2
N Today at 2:59 PM by axsolers_24
Let $x_1=97 ,$ $x_2=\frac{2}{x_1} ,$ $x_3=\frac{3}{x_2} ,$$... , $ $x_8=\frac{8}{x_7}$
then
$ \log_{3\sqrt{2}} \left(\prod_{i=1}^8 x_i-60\right)$
2 replies
axsolers_24
Today at 10:42 AM
axsolers_24
Today at 2:59 PM
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Inequalities
sqing   1
N Today at 2:48 PM by sqing
Let $ a,b>0 , a^2 + 2b^2 = a + 2b $. Prove that $$\sqrt{\frac{a}{b( a+2)}} + \sqrt{\frac{b}{a(2b+1)}} \geq \frac {2}{\sqrt{3}} $$Let $ a,b>0 , a^3 + 2b^3 = a + 2b $. Prove that $$\sqrt[3]{\frac{a}{b( a+2)}} + \sqrt[3]{\frac{b}{a(2b+1)}} \geq \frac {2}{\sqrt[3]{3}} $$
1 reply
sqing
Today at 2:27 PM
sqing
Today at 2:48 PM
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Coprime sequence
Ecrin_eren   4
N Today at 2:37 PM by Pal702004
"Let N be a natural number. Show that any two numbers from the following sequence are coprime:

2^1 + 1, 2^2 + 1, 2^4+ 1,2^8+1 ..., 2^(2^N )+ 1."
4 replies
Ecrin_eren
May 1, 2025
Pal702004
Today at 2:37 PM
J
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Hard Inequality
William_Mai   0
Today at 2:13 PM
Given $a, b, c \in \mathbb{R}$ such that $a^2 + b^2 + c^2 = 1$.
Find the minimum value of $P = ab + 2bc + 3ca$.

Source: Pham Le Van
0 replies
William_Mai
Today at 2:13 PM
0 replies
J
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Find the minimum
Ecrin_eren   5
N Today at 2:05 PM by Jackson0423
The polynomial is given by P(x) = x^4 + ax^3 + bx^2 + cx + d, and its roots are x1, x2, x3, x4. Additionally, it is stated that d ≥ 5.Find the minimum value of the product:

(x1^2 + 1)(x2^2 + 1)(x3^2 + 1)(x4^2 + 1).

5 replies
Ecrin_eren
May 1, 2025
Jackson0423
Today at 2:05 PM
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J
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Combinatorial Sum
P162008   1
N Today at 12:56 PM by Mathzeus1024
Evaluate $\sum_{n=0}^{\infty} \frac{2^n + 1}{(2n + 1) \binom{2n}{n}}$
1 reply
P162008
Apr 24, 2025
Mathzeus1024
Today at 12:56 PM
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Combinatorial Sum
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P162008
175 posts
P162008
#1 h  Apr 24, 2025, 1:02 AM
Y by
Evaluate $\sum_{n=0}^{\infty} \frac{2^n + 1}{(2n + 1) \binom{2n}{n}}$
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Mathzeus1024
862 posts
Mathzeus1024
#2 h  Today at 12:56 PM
Y by
$\sum_{n=0}^{\infty} \frac{2^n + 1}{(2n + 1) \binom{2n}{n}} = \sum_{n=0}^{\infty} \frac{2^n + 1}{\binom{2n}{n}}\cdot \int_{0}^{1} x^{2n} dx = \int_{0}^{1} \sum_{n=0}^{\infty} \frac{(\sqrt{2}x)^{2n} + x^{2n}}{\binom{2n}{n}} dx$;

or $\int_{0}^{1} \frac{4x \cdot \arcsin(x/2) +4\sqrt{4-x^2}}{(4-x^2)^{3/2}} + \frac{2x \cdot \arcsin(x/\sqrt{2})+2\sqrt{2-x^2}}{(2-x^2)^{3/2}} dx$;

or $\frac{4\arcsin(x/2)}{\sqrt{4-x^2}} + \frac{2\arcsin(x/\sqrt{2})}{\sqrt{2-x^2}}|_{0}^{1} = \textcolor{red}{\frac{(9+4\sqrt{3})\pi}{18}}$.
This post has been edited 5 times. Last edited by Mathzeus1024, Today at 1:10 PM
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