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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.

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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
May 1, 2025
0 replies
D1026 : An equivalent
Dattier   1
N 2 hours ago by Hello_Kitty
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 $u_n$.
1 reply
Dattier
Yesterday at 1:39 PM
Hello_Kitty
2 hours ago
A problem in point set topology
tobylong   2
N 2 hours ago by cosmicgenius
Source: Basic Topology, Armstrong
Let $f:X\to Y$ be a closed map with the property that the inverse image of each point in $Y$ is a compact subset of $X$. Prove that $f^{-1}(K)$ is compact whenever $K$ is compact in $Y$.
2 replies
tobylong
Yesterday at 3:14 AM
cosmicgenius
2 hours ago
Alternating series and integral
jestrada   2
N 2 hours ago by jestrada
Source: own
Prove that for all $\alpha\in\mathbb{R}, \alpha>-1$, we have
$$ \frac{1}{\alpha+1}-\frac{1}{\alpha+2}+\frac{1}{\alpha+3}-\frac{1}{\alpha+4}+\cdots=\int_0^1 \frac{x^{\alpha}}{x+1}  \,dx. $$
2 replies
jestrada
3 hours ago
jestrada
2 hours ago
Equivalent condition of the uniformly continuous fo a function
Alphaamss   1
N 3 hours ago by alexheinis
Source: Personal
Let $f_{a,b}(x)=x^a\cos(x^b),x\in(0,\infty)$. Get all the $(a,b)\in\mathbb R^2$ such that $f_{a,b}$ is uniformly continuous on $(0,\infty)$.
1 reply
Alphaamss
Yesterday at 7:35 AM
alexheinis
3 hours ago
Sintetic geometry problem
ICE_CNME_4   0
5 hours ago
Let there be the triangle ABC and the points E ∈ (AC), F ∈ (AB), such that BE and CF are concurrent in O.
If {L} = AO ∩ EF and K ∈ BC, such that LK ⊥ BC, show that EKL = FKL.
0 replies
ICE_CNME_4
5 hours ago
0 replies
Values of x
Ecrin_eren   5
N 6 hours ago by Math-lover1
Given 0 ≤ x < 2π, what is the difference between the largest and the smallest of the values of x
that satisfy the equation 5cosx + 2sin2x = 4 in radians?
5 replies
Ecrin_eren
Friday at 6:42 PM
Math-lover1
6 hours ago
Algebra problem
Deomad123   0
Yesterday at 7:29 PM
Let $n$ be a positive integer.Prove that there is a polynomial $P$ with integer coefficients so that $a+b+c=0$,then$$a^{2n+1}+b^{2n+1}+c^{2n+1}=abc[P(a,b)+P(b,c)+P(a,c)]$$.
0 replies
Deomad123
Yesterday at 7:29 PM
0 replies
9 Rate my profile
Evanlovemath   2
N Yesterday at 7:15 PM by Evanlovemath
https://artofproblemsolving.com/alcumus/profile/613246 Not doing Alcumus right now, to many classes :(
2 replies
Evanlovemath
Yesterday at 7:13 PM
Evanlovemath
Yesterday at 7:15 PM
trigonometric functions
VivaanKam   11
N Yesterday at 5:50 PM 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
Yesterday at 5:50 PM
1201 divides sum of powers
V0305   1
N Yesterday at 5:09 PM 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
Yesterday at 4:36 PM
vincentwant
Yesterday at 5:09 PM
Interesting geometry
polarLines   5
N Yesterday at 4:25 PM 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
Yesterday at 4:25 PM
Showing that certain number is divisible by 13
BBNoDollar   3
N Yesterday at 4:24 PM 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
Yesterday at 2:54 PM
Shan3t
Yesterday at 4:24 PM
Inequality
tom-nowy   0
Yesterday at 3:07 PM
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
Yesterday at 3:07 PM
0 replies
Logarithm of a product
axsolers_24   2
N Yesterday at 2:59 PM by axsolers_24
Let $x_1=97 ,$ $x_2=\frac{2}{x_1} ,$ $x_3=\frac{3}{x_2} ,$$... , $ $x_8=\frac{8}{x_7}$
then
$ \log_{3\sqrt{2}} \left(\prod_{i=1}^8 x_i-60\right)$
2 replies
axsolers_24
Yesterday at 10:42 AM
axsolers_24
Yesterday at 2:59 PM
Integral involving hypergeometric function
Yimself   1
N May 27, 2018 by pprime
Greetings, show that $$\int_{0}^{1} \, _2F_1\left(\frac{1} {2}, 1;\frac{3} {2} ;-x^2 \right)\,dx=G$$Where G is the Catalan constant: https://en.wikipedia.org/wiki/Catalan%27s_constant
1 reply
Yimself
May 27, 2018
pprime
May 27, 2018
Integral involving hypergeometric function
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Yimself
1309 posts
#1 • 1 Y
Y by Adventure10
Greetings, show that $$\int_{0}^{1} \, _2F_1\left(\frac{1} {2}, 1;\frac{3} {2} ;-x^2 \right)\,dx=G$$Where G is the Catalan constant: https://en.wikipedia.org/wiki/Catalan%27s_constant
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pprime
600 posts
#2 • 2 Y
Y by Adventure10, Mango247
we know that ${}_{2}{{F}_{1}}\left( a,b;c;z \right)=\sum\limits_{n=0}^{+\infty }{\frac{{{\left( a \right)}_{n}}{{\left( b \right)}_{n}}}{{{\left( c \right)}_{n}}}\cdot \frac{{{z}^{n}}}{n!}}$ for $\left| z \right|<1$
where ${{\left( q \right)}_{n}}=\left\{ \begin{matrix}
   1\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ n=0  \\
   q\left( q+1 \right)\cdot \cdot \cdot \left( q+n-1 \right)\ \ \ n>0  \\
\end{matrix} \right.$

then
$\int\limits_{0}^{1}{{}_{2}{{F}_{1}}\left( \frac{1}{2},1;\frac{3}{2};-{{x}^{2}} \right)dx}=\int\limits_{0}^{1}{\sum\limits_{n=0}^{+\infty }{\frac{{{\left( \frac{1}{2} \right)}_{n}}\cdot {{\left( 1 \right)}_{n}}}{{{\left( \frac{3}{2} \right)}_{n}}}\cdot \frac{{{\left( -{{x}^{2}} \right)}^{n}}}{n!}dx}}=\int\limits_{0}^{1}{\left( 1+\sum\limits_{n=1}^{+\infty }{\frac{{{\left( \frac{1}{2} \right)}_{n}}\cdot {{\left( 1 \right)}_{n}}}{{{\left( \frac{3}{2} \right)}_{n}}}\cdot \frac{{{\left( -{{x}^{2}} \right)}^{n}}}{n!}} \right)dx}$

$=\int\limits_{0}^{1}{\left( 1+\sum\limits_{n=1}^{+\infty }{\frac{\left( \frac{1}{2}\left( \frac{1}{2}+1 \right)\left( \frac{1}{2}+2 \right)\cdot \cdot \cdot \left( \frac{1}{2}+n-1 \right) \right)\cdot \left( 1\cdot 2\cdot 3\cdot \cdot \cdot \left( n-1 \right)n \right)}{\left( \frac{3}{2}\left( \frac{3}{2}+1 \right)\left( \frac{3}{2}+2 \right)\cdot \cdot \cdot \left( \frac{3}{2}+n-1 \right) \right)}\cdot \frac{{{\left( -{{x}^{2}} \right)}^{n}}}{n!}} \right)dx}$

$=\int\limits_{0}^{1}{\left( 1+\sum\limits_{n=1}^{+\infty }{\frac{\frac{1\cdot 3\cdot 5\cdot \cdot \cdot \left( 2n-1 \right)}{{{2}^{n-1}}}}{\frac{3\cdot 5\cdot 7\cdot \cdot \cdot \left( 2n+1 \right)}{{{2}^{n-1}}}}\cdot {{\left( -{{x}^{2}} \right)}^{n}}} \right)dx}=\int\limits_{0}^{1}{\left( 1+\sum\limits_{n=1}^{+\infty }{\frac{{{\left( -{{x}^{2}} \right)}^{n}}}{2n+1}} \right)dx}=\int\limits_{0}^{1}{\left( \sum\limits_{n=0}^{+\infty }{\frac{{{\left( -{{x}^{2}} \right)}^{n}}}{2n+1}} \right)dx}=\sum\limits_{n=0}^{+\infty }{\frac{{{\left( -1 \right)}^{n}}}{{{\left( 2n+1 \right)}^{2}}}}=G$

:-D
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