Summer and Fall classes are open for enrollment. Schedule today!

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k a July Highlights and 2025 AoPS Online Class Information
jwelsh   0
Jul 1, 2025
We are halfway through summer, so be sure to carve out some time to keep your skills sharp and explore challenging topics at AoPS Online and our AoPS Academies (including the Virtual Campus)!

[list][*]Over 60 summer classes are starting at the Virtual Campus on July 7th - check out the math and language arts options for middle through high school levels.
[*]At AoPS Online, we have accelerated sections where you can complete a course in half the time by meeting twice/week instead of once/week, starting on July 8th:
[list][*]MATHCOUNTS/AMC 8 Basics
[*]MATHCOUNTS/AMC 8 Advanced
[*]AMC Problem Series[/list]
[*]Plus, AoPS Online has a special seminar July 14 - 17 that is outside the standard fare: Paradoxes and Infinity
[*]We are expanding our in-person AoPS Academy locations - are you looking for a strong community of problem solvers, exemplary instruction, and math and language arts options? Look to see if we have a location near you and enroll in summer camps or academic year classes today! New locations include campuses in California, Georgia, New York, Illinois, and Oregon and more coming soon![/list]

MOP (Math Olympiad Summer Program) just ended and the IMO (International Mathematical Olympiad) is right around the corner! This year’s IMO will be held in Australia, July 10th - 20th. Congratulations to all the MOP students for reaching this incredible level and best of luck to all selected to represent their countries at this year’s IMO! Did you know that, in the last 10 years, 59 USA International Math Olympiad team members have medaled and have taken over 360 AoPS Online courses. Take advantage of our Worldwide Online Olympiad Training (WOOT) courses
and train with the best! Please note that early bird pricing ends August 19th!
Are you tired of the heat and thinking about Fall? You can plan your Fall schedule now with classes at either AoPS Online, AoPS Academy Virtual Campus, or one of our AoPS Academies around the US.

Our full course list for upcoming classes is below:
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0 replies
jwelsh
Jul 1, 2025
0 replies
functional equation
cipher703516247   1
N 2 minutes ago by cipher703516247
$f: \mathbb{R}^+ \to \mathbb{R}^+$
$f(x)f(x+f(y))=f(x)^2+2x^3f(y)+f(y)^2f(x)$


This is an enhanced version of the above problem, where you are asked to find the minimum m.


$f: \mathbb{R}^+ \to \mathbb{R}^+$
$f(x)^2f(x+f(y))=f(x)^3+mx^5f(y)+f(x)^2f(y^2)$

Can anyone expand on this functional equation?
1 reply
cipher703516247
Today at 1:13 AM
cipher703516247
2 minutes ago
Sum of lengths of each pair of opposite sides of q is equal
Amir Hossein   26
N 7 minutes ago by QueenArwen
The feet of the perpendiculars from the intersection point of the diagonals of a convex cyclic quadrilateral to the sides form a quadrilateral $q$. Show that the sum of the lengths of each pair of opposite sides of $q$ is equal.
26 replies
Amir Hossein
Oct 4, 2011
QueenArwen
7 minutes ago
JBMO 2024 SL N5
MuradSafarli   5
N 10 minutes ago by Assassino9931
Source: JBMO 2024 Shortlist
Does there exist a positive integer $k$ such that the number of quadruples $(a, b, c, n)$ of positive integers satisfying
\[
a^5 + b^6 + c^{15} - n! = k
\]is finite?
5 replies
MuradSafarli
Jun 26, 2025
Assassino9931
10 minutes ago
max sum 1/a_i if adjacent gcd's increase
v_Enhance   10
N 25 minutes ago by math90
Source: USA TSTST 2025/4
Let $n\ge 2$ be a positive integer. Let $a_1$, $a_2$, $\dots$, $a_n$ be a sequence of positive integers such that \[\operatorname{gcd}(a_1,a_2),\,\operatorname{gcd}(a_2,a_3),\,\dots,\,\operatorname{gcd}(a_{n-1},a_n)\]is a strictly increasing sequence. Find, in terms of $n$, the maximum possible value of \[\frac{1}{a_1}+\frac{1}{a_2}+\dots+\frac{1}{a_n}\]over all such sequences.

Maxim Li
10 replies
+1 w
v_Enhance
Jul 1, 2025
math90
25 minutes ago
Challenge: Make as many positive integers from 2 zeros
Biglion   14
N 3 hours ago by Biglion
How many positive integers can you make from at most 2 zeros, any math operation and cocatination?
New Rule: The successor function can only be used at most 3 times per number
Starting from 0, 0=0
14 replies
Biglion
Yesterday at 4:48 PM
Biglion
3 hours ago
Given $d,e\in\Bbb Z^{+}.$ Show that the two arithmetic progressions $a,a+d,a+2d,
Vulch   1
N Today at 10:52 AM by Mathzeus1024
Given $d,e\in\Bbb Z^{+}.$ Show that the two arithmetic progressions $a,a+d,a+2d,\cdots;$ and $b,b+e,b+2e,\cdots$ have a number in common if and only if $\gcd(d,e)|b - a$
1 reply
Vulch
Jun 20, 2022
Mathzeus1024
Today at 10:52 AM
2016 preRMO p10, [M], M is max of (6x-3y-8z) when 2x^2+3y^2+4z^2 = 1
parmenides51   5
N Today at 10:49 AM by fathersayno
Let $M$ be the maximum value of $(6x-3y-8z)$, subject to $2x^2+3y^2+4z^2 = 1$. Find $[M]$.
5 replies
parmenides51
Aug 9, 2019
fathersayno
Today at 10:49 AM
Find all n
Darealzolt   1
N Today at 10:02 AM by HAL9000sk
Find the number of natural numbers \(n\), such that \(3^n-4n\) is a perfect square.
1 reply
Darealzolt
Today at 7:09 AM
HAL9000sk
Today at 10:02 AM
Trigonometry
Maaaaaaath   2
N Today at 9:30 AM by Mathzeus1024
In a geometric progression we have $ a_1 = \sin x $ , $a_2 = \cos x $ and $a_3 = \tan x$ .Which term of this progression is equal to $1+\cos x$ ?
2 replies
Maaaaaaath
Yesterday at 6:39 PM
Mathzeus1024
Today at 9:30 AM
Find how much sides in the n-gon
Darealzolt   0
Today at 9:28 AM
Let there be a regular \(n-gon\), with \(n\geq4\). Let points \(A,B,C,D\) be points on the \(n-gon\) with \(AB\) being perpendicular with the angle bisector of \( \angle BDC\). Hence find \(n\).
0 replies
Darealzolt
Today at 9:28 AM
0 replies
Simple Angle Chasing Problem Stumping Me
WheatNeat   0
Today at 8:54 AM
Isoceles triangle $ABC$ with $AC = BC$, has angle $CAB$ as $80$ degrees. Let points $E$ and $D$ be on sides $BC$ and $AC$ respectively, such that angle $CBD$ is 20 degrees and angle $CAE$ is 10 degrees. Find angle $DEA$.
0 replies
WheatNeat
Today at 8:54 AM
0 replies
2018 preRMO p11, teacups in the kitchen, some wiht handles
parmenides51   3
N Today at 8:24 AM by cortex_classes
There are several teacups in the kitchen, some with handles and the others without handles. The number of ways of selecting two cups without a handle and three with a handle is exactly $1200$. What is the maximum possible number of cups in the kitchen?
3 replies
parmenides51
Aug 8, 2019
cortex_classes
Today at 8:24 AM
2018 preRMO p10, (BC^2 + CA^2 + AB^2)/100, perp. medians
parmenides51   2
N Today at 8:23 AM by cortex_classes
In a triangle $ABC$, the median from $B$ to $CA$ is perpendicular to the median from $C$ to $AB$.
If the median from $A$ to $BC$ is $30$, determine $\frac{BC^2 + CA^2 + AB^2}{100}$.
2 replies
parmenides51
Aug 8, 2019
cortex_classes
Today at 8:23 AM
2018 preRMO p9, a+b is root of x^2 +ax+b = 0, a,b integers
parmenides51   9
N Today at 8:23 AM by cortex_classes
Suppose $a, b$ are integers and $a+b$ is a root of $x^2 +ax+b = 0$. What is the maximum possible value of $b^2$?
9 replies
parmenides51
Aug 8, 2019
cortex_classes
Today at 8:23 AM
Inequality with x+y+z=1.
FrancoGiosefAG   4
N May 21, 2025 by sqing
Let $x,y,z$ be positive real numbers such that $x+y+z=1$. Show that
\[ \frac{x^2-yz}{x^2+x}+\frac{y^2-zx}{y^2+y}+\frac{z^2-xy}{z^2+z}\leq 0. \]
4 replies
FrancoGiosefAG
May 20, 2025
sqing
May 21, 2025
Inequality with x+y+z=1.
G H J
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FrancoGiosefAG
121 posts
#1
Y by
Let $x,y,z$ be positive real numbers such that $x+y+z=1$. Show that
\[ \frac{x^2-yz}{x^2+x}+\frac{y^2-zx}{y^2+y}+\frac{z^2-xy}{z^2+z}\leq 0. \]
Z K Y
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Blackbeam999
52 posts
#2
Y by
Use this identity
2(x^2-yz)=(x-y)(x+z)+(x+y)(x-z) and try sos
Z K Y
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PurpleJam
7 posts
#3
Y by
Simple solution using Titu's Lemma:
\[ \sum \limits_{cyc}^{}\dfrac{yz+x}{x^2+x} = \sum \limits_{cyc}^{} \left ( \dfrac{(yz)^2}{x^2yz+xyz} + \dfrac{1}{x+1} \right ) \geq \dfrac{(xy+xz+yz)^2}{xyz(3+x+y+z)} + \dfrac{9}{x+y+z+3} \geq \dfrac{3xyz(x+y+z)}{4xyz}+ \dfrac{9}{4}=3\]
This post has been edited 1 time. Last edited by PurpleJam, May 21, 2025, 5:20 AM
Z K Y
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JARP091
113 posts
#4
Y by
PurpleJam wrote:
Simple solution using Titu's Lemma:
\[ \sum \limits_{cyc}^{}\dfrac{yz+x}{x^2+x} = \sum \limits_{cyc}^{} \left ( \dfrac{yz}{x^2+x} + \dfrac{1}{x+1} \right ) \geq \dfrac{(xy+xz+yz)^2}{xyz(3+x+y+z)} + \dfrac{9}{x+y+z+3} \geq \dfrac{3xyz(x+y+z)}{4xyz}+ \dfrac{9}{4}=3\]

I dont think this is correct
Z K Y
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sqing
43018 posts
#5
Y by
FrancoGiosefAG wrote:
Let $x,y,z$ be positive real numbers such that $x+y+z=1$. Show that
\[ \frac{x^2-yz}{x^2+x}+\frac{y^2-zx}{y^2+y}+\frac{z^2-xy}{z^2+z}\leq 0. \]
https://artofproblemsolving.com/community/c6h299899p1644375
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