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k a April Highlights and 2025 AoPS Online Class Information
jlacosta   0
Apr 2, 2025
Spring is in full swing and summer is right around the corner, what are your plans? At AoPS Online our schedule has new classes starting now through July, so be sure to keep your skills sharp and be prepared for the Fall school year! Check out the schedule of upcoming classes below.

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[*]April 22nd (Webinar), 4:00pm PT/7:00pm ET, Competitive Programming at AoPS (USACO).[/list]
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0 replies
jlacosta
Apr 2, 2025
0 replies
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Composite sum
rohitsingh0812   39
N 4 minutes ago by YaoAOPS
Source: INDIA IMOTC-2006 TST1 PROBLEM-2; IMO Shortlist 2005 problem N3
Let $ a$, $ b$, $ c$, $ d$, $ e$, $ f$ be positive integers and let $ S = a+b+c+d+e+f$.
Suppose that the number $ S$ divides $ abc+def$ and $ ab+bc+ca-de-ef-df$. Prove that $ S$ is composite.
39 replies
rohitsingh0812
Jun 3, 2006
YaoAOPS
4 minutes ago
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Problem 1
SpectralS   145
N 6 minutes ago by IndexLibrorumProhibitorum
Given triangle $ABC$ the point $J$ is the centre of the excircle opposite the vertex $A.$ This excircle is tangent to the side $BC$ at $M$, and to the lines $AB$ and $AC$ at $K$ and $L$, respectively. The lines $LM$ and $BJ$ meet at $F$, and the lines $KM$ and $CJ$ meet at $G.$ Let $S$ be the point of intersection of the lines $AF$ and $BC$, and let $T$ be the point of intersection of the lines $AG$ and $BC.$ Prove that $M$ is the midpoint of $ST.$

(The excircle of $ABC$ opposite the vertex $A$ is the circle that is tangent to the line segment $BC$, to the ray $AB$ beyond $B$, and to the ray $AC$ beyond $C$.)

Proposed by Evangelos Psychas, Greece
145 replies
SpectralS
Jul 10, 2012
IndexLibrorumProhibitorum
6 minutes ago
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Help me please
sealight2107   0
7 minutes ago
Let $m,n,p,q$ be positive reals such that $m+n+p+q+\frac{1}{mnpq} = 18$. Find the minimum and maximum value of $m,n,p,q$
0 replies
sealight2107
7 minutes ago
0 replies
J
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Rectangular line segments in russia
egxa   2
N 10 minutes ago by mohsen
Source: All Russian 2025 9.1
Several line segments parallel to the sides of a rectangular sheet of paper were drawn on it. These segments divided the sheet into several rectangles, inside of which there are no drawn lines. Petya wants to draw one diagonal in each of the rectangles, dividing it into two triangles, and color each triangle either black or white. Is it always possible to do this in such a way that no two triangles of the same color share a segment of their boundary?
2 replies
egxa
Apr 18, 2025
mohsen
10 minutes ago
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Integrable function: + and - on every subinterval.
SPQ   3
N Today at 7:06 AM by solyaris
Provide a function integrable on [a, b] such that f takes on positive and negative values on every subinterval (c, d) of [a, b]. Prove your function satisfies both conditions.
3 replies
SPQ
Today at 2:40 AM
solyaris
Today at 7:06 AM
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Putnam 1999 A4
djmathman   7
N Today at 7:05 AM by P162008
Sum the series \[\sum_{m=1}^\infty\sum_{n=1}^\infty\dfrac{m^2n}{3^m(n3^m+m3^n)}.\]
7 replies
djmathman
Dec 22, 2012
P162008
Today at 7:05 AM
J
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Find the greatest possible value of the expression
BEHZOD_UZ   1
N Today at 6:34 AM by alexheinis
Source: Yandex Uzbekistan Coding and Math Contest 2025
Let $a, b, c, d$ be complex numbers with $|a| \le 1, |b| \le 1, |c| \le 1, |d| \le 1$. Find the greatest possible value of the expression $$|ac+ad+bc-bd|.$$
1 reply
BEHZOD_UZ
Yesterday at 11:56 AM
alexheinis
Today at 6:34 AM
J
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Problem vith lcm
snowhite   2
N Today at 6:21 AM by snowhite
Prove that $\underset{n\to \infty }{\mathop{\lim }}\,\sqrt[n]{lcm(1,2,3,...,n)}=e$
Please help me! Thank you!
2 replies
snowhite
Today at 5:19 AM
snowhite
Today at 6:21 AM
J
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combinatorics
Hello_Kitty   2
N Yesterday at 10:23 PM by Hello_Kitty
How many $100$ digit numbers are there
- not including the sequence $123$ ?
- not including the sequences $123$ and $231$ ?
2 replies
Hello_Kitty
Apr 17, 2025
Hello_Kitty
Yesterday at 10:23 PM
J
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Sequence of functions
Squeeze   2
N Yesterday at 10:22 PM by Hello_Kitty
Q) let $f_n:[-1,1)\to\mathbb{R}$ and $f_n(x)=x^{n}$ then is this uniformly convergence on $(0,1)$ comment on uniformly convergence on $[0,1]$ where in general it is should be uniformly convergence.

My I am trying with some contradicton method like chose $\epsilon=1$ and trying to solve$|f_n(a)-f(a)|<\epsilon=1$
Next take a in (0,1) and chose a= 2^1/N but not solution
How to solve like this way help.
2 replies
Squeeze
Apr 18, 2025
Hello_Kitty
Yesterday at 10:22 PM
J
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A in M2(prime), A=B^2 and det(B)=p^2
jasperE3   1
N Yesterday at 9:59 PM by KAME06
Source: VJIMC 2012 1.2
Determine all $2\times2$ integer matrices $A$ having the following properties:

$1.$ the entries of $A$ are (positive) prime numbers,
$2.$ there exists a $2\times2$ integer matrix $B$ such that $A=B^2$ and the determinant of $B$ is the square of a prime number.
1 reply
jasperE3
May 31, 2021
KAME06
Yesterday at 9:59 PM
J
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Equation over a finite field
loup blanc   1
N Yesterday at 9:30 PM by alexheinis
Find the set of $x\in\mathbb{F}_{5^5}$ such that the equation in the unknown $y\in \mathbb{F}_{5^5}$:
$x^3y+y^3+x=0$ admits $3$ roots: $a,a,b$ s.t. $a\not=b$.
1 reply
loup blanc
Yesterday at 6:08 PM
alexheinis
Yesterday at 9:30 PM
J
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Integration Bee Kaizo
Calcul8er   51
N Yesterday at 7:41 PM by BaidenMan
Hey integration fans. I decided to collate some of my favourite and most evil integrals I've written into one big integration bee problem set. I've been entering integration bees since 2017 and I've been really getting hands on with the writing side of things over the last couple of years. I hope you'll enjoy!
51 replies
Calcul8er
Mar 2, 2025
BaidenMan
Yesterday at 7:41 PM
J
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interesting integral
Martin.s   1
N Yesterday at 2:46 PM by ysharifi
$$\int_0^\infty \frac{\sinh(t)}{t \cosh^3(t)} dt$$
1 reply
Martin.s
Monday at 3:12 PM
ysharifi
Yesterday at 2:46 PM
J
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J
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Let $a,b,c$ be nonnegative real numbers. Prove that
khanhsy   0
Mar 29, 2025
Let $a,b,c,$ be nonnegative real numbers such that $a+b+c=3$. Prove that:
$$\dfrac{1}{\sqrt{a^3+3bc}}+\dfrac{1}{\sqrt{b^3+3ca}}+\dfrac{1}{\sqrt{c^3+3ab}}\ge \dfrac{1}{\sqrt{a^3+b^3+c^3+abc}}+\dfrac{2\sqrt{2}}{\sqrt{(a+b)(b+c)(c+a)}}.$$
0 replies
khanhsy
Mar 29, 2025
0 replies
J
Let $a,b,c$ be nonnegative real numbers. Prove that
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khanhsy
129 posts
khanhsy
#1 h  Mar 29, 2025, 9:21 AM
Y by
Let $a,b,c,$ be nonnegative real numbers such that $a+b+c=3$. Prove that:
$$\dfrac{1}{\sqrt{a^3+3bc}}+\dfrac{1}{\sqrt{b^3+3ca}}+\dfrac{1}{\sqrt{c^3+3ab}}\ge \dfrac{1}{\sqrt{a^3+b^3+c^3+abc}}+\dfrac{2\sqrt{2}}{\sqrt{(a+b)(b+c)(c+a)}}.$$
This post has been edited 3 times. Last edited by khanhsy, Mar 29, 2025, 9:23 AM
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