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k a April Highlights and 2025 AoPS Online Class Information
jlacosta   0
Yesterday at 3:18 PM
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
Yesterday at 3:18 PM
0 replies
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Sum of floors with primes p,q
WakeUp   6
N 7 minutes ago by L13832
Source: Baltic Way 2001
Let $p$ and $q$ be two different primes. Prove that
\[\left\lfloor\frac{p}{q}\right\rfloor+\left\lfloor\frac{2p}{q}\right\rfloor+\left\lfloor\frac{3p}{q}\right\rfloor+\ldots +\left\lfloor\frac{(q-1)p}{q}\right\rfloor=\frac{1}{2}(p-1)(q-1) \]
6 replies
WakeUp
Nov 17, 2010
L13832
7 minutes ago
J
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Floor double summation
CyclicISLscelesTrapezoid   51
N 7 minutes ago by cubres
Source: ISL 2021 A2
Which positive integers $n$ make the equation \[\sum_{i=1}^n \sum_{j=1}^n \left\lfloor \frac{ij}{n+1} \right\rfloor=\frac{n^2(n-1)}{4}\]true?
51 replies
CyclicISLscelesTrapezoid
Jul 12, 2022
cubres
7 minutes ago
J
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nice problem
hanzo.ei   3
N 8 minutes ago by X.Luser
Source: I forgot
Let triangle $ABC$ be inscribed in the circumcircle $(O)$ and circumscribed about the incircle $(I)$, with $AB < AC$. The incircle $(I)$ touches the sides $BC$, $CA$, and $AB$ at $D$, $E$, and $F$, respectively. A line through $I$, perpendicular to $AI$, intersects $BC$, $CA$, and $AB$ at $X$, $Y$, and $Z$, respectively. The line $AI$ meets $(O)$ at $M$ (distinct from $A$). The circumcircle of triangle $AYZ$ intersects $(O)$ at $N$ (distinct from $A$). Let $P$ be the midpoint of the arc $BAC$ of $(O)$. The line $AI$ cuts segments $DF$ and $DE$ at $K$ and $L$, respectively, and the tangents to the circle $(DKL)$ at $K$ and $L$ intersect at $T$. Prove that $AT \perp BC$.
3 replies
hanzo.ei
Mar 29, 2025
X.Luser
8 minutes ago
J
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Old problem :(
Drakkur   2
N 11 minutes ago by Drakkur
Let a, b, c be positive real numbers. Prove that
$$\dfrac{1}{\sqrt{a^2+bc}}+\dfrac{1}{\sqrt{b^2+ca}}+\dfrac{1}{\sqrt{c^2+ab}}\le \sqrt{2}\left(\dfrac{1}{a+b}+\dfrac{1}{b+c}+\dfrac{1}{c+a}\right)$$
2 replies
Drakkur
Yesterday at 3:38 PM
Drakkur
11 minutes ago
J
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Geometry problem
Raul_S_Baz   1
N 4 hours ago by sunken rock
IMAGE
1 reply
Raul_S_Baz
Yesterday at 8:49 PM
sunken rock
4 hours ago
J
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Prove that
abduqahhor_math   5
N 5 hours ago by krancky22
n^2+3n+5 is not divided to 121
5 replies
abduqahhor_math
Mar 31, 2025
krancky22
5 hours ago
J
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ez problem
Noname23   3
N 5 hours ago by KevinKV01
Find $x$
$7 + \sqrt7 + 2\sqrt{x + 4} + \sqrt{7x + 28} + \sqrt{14x + 7} + \sqrt{2x^2 + 9x + 4} - \sqrt{2x + 1} = 0$
3 replies
Noname23
Apr 1, 2025
KevinKV01
5 hours ago
J
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Solve the equetion
yt12   5
N 5 hours ago by KevinKV01
Solve the equetion:$\sin 2x+\tan x=2$
5 replies
yt12
Mar 31, 2025
KevinKV01
5 hours ago
J
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Inequalities
sqing   2
N 5 hours ago by sqing
Let $ a,b> 0 $ and $ a^2+ b^2+a+b= 2 . $ Prove that
$$ \frac{a^5}{a^5+ b^3}+ \frac{b^5}{b^5+ a^3}\leq 1$$
2 replies
sqing
5 hours ago
sqing
5 hours ago
J
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Real variables inequality
JK1603JK   1
N 6 hours ago by lbh_qys
Let a,b,c\in R then prove that \frac{15}{2}\cdot\frac{a^2+b^2+c^2}{(a+b+c)^2}+\frac{ab}{a^2+b^2}+\frac{bc}{b^2+c^2}+\frac{ca}{c^2+a^2}\ge 4
1 reply
JK1603JK
Today at 12:31 AM
lbh_qys
6 hours ago
J
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Inequalities
sqing   14
N 6 hours ago by sqing
Let $ a,b,c\geq 0 $ and $a+b+c=1$. Prove that$$a^3b+b^3c+c^3a+\frac{473}{256}abc\le\frac{27}{256}$$Equality holds when $ a=b=c=\frac{1}{3} $ or $ a=0,b=\frac{3}{4},c=\frac{1}{4} $ or $ a=\frac{1}{4} ,b=0,c=\frac{3}{4} $
or $ a=\frac{3}{4} ,b=\frac{1}{4},c=0. $
14 replies
sqing
Mar 22, 2025
sqing
6 hours ago
J
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Geo Mock #1
Bluesoul   2
N Today at 4:13 AM by jb2015007
Consider the rectangle $ABCD$ with $AB=4$. Point $E$ lies inside the rectangle such that $\triangle{ABE}$ is equilateral. Given that $C,E$ and the midpoint of $AD$ are on the same line, compute the length of $BC$.
2 replies
Bluesoul
Apr 1, 2025
jb2015007
Today at 4:13 AM
J
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pinkpig's Problem Collection - Signup
pinkpig   257
N Today at 4:13 AM by Yiyj1
Hello, all AoPS users!

I am very happy to release my Problem Collection. Here is the direct link to the forum for users interested in solving problems.

This problem collection will consist of various competition problems that I find very fun to solve. Some questions will be made by me, while others will be from competitions. There are Geometry, Intermediate Algebra, Precalculus, Number Theory, and Combinatorics questions. You may compete with other users in this forum. So, be competitive and active if you join!
Reviews
Sample Problems

Post \signup to join the fun!

Hope you enjoy the problems! :D
257 replies
pinkpig
Aug 16, 2021
Yiyj1
Today at 4:13 AM
J
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Easiest functional equation?
ZETA_in_olympiad   28
N Today at 4:07 AM by jkim0656
Here I want the users to post the functional equations that they think are the easiest. Everyone (including the one who posted the problem) are able to post solutions.
28 replies
ZETA_in_olympiad
Mar 19, 2022
jkim0656
Today at 4:07 AM
J
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J
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Gerretsen Inequality
dd00   2
N Dec 17, 2011 by palashahuja
For $\triangle ABC$ prove that
$s^2\leq 4R^2+4Rr+3r^2$
2 replies
dd00
Dec 15, 2011
palashahuja
Dec 17, 2011
J
Gerretsen Inequality
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Reveal topic tags
inequalities
geometry unsolved
geometry
L
G H BBookmark kLocked kLocked NReply
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dd00
15 posts
dd00
#1 h  Dec 15, 2011, 5:30 PM • 2 Y
Y by Adventure10, Mango247
For $\triangle ABC$ prove that
$s^2\leq 4R^2+4Rr+3r^2$
Z K Y
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Dr Sonnhard Graubner
16100 posts
Dr Sonnhard Graubner
#2 h  Dec 15, 2011, 5:37 PM • 2 Y
Y by Adventure10, Mango247
hello, see here
http://rgmia.org/papers/v6n3/wsh.pdf
Sonnhard.
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palashahuja
156 posts
palashahuja
#3 h  Dec 17, 2011, 2:16 AM • 2 Y
Y by Adventure10, Mango247
hey i have another proof see here
http://www.artofproblemsolving.com/Forum/viewtopic.php?f=51&t=439499
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