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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
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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.

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0 replies
jwelsh
Jul 1, 2025
0 replies
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A cyclic inequality
Nguyenhuyen_AG   3
N 6 minutes ago by Victoria_Discalceata1
Let $a, \, b, \,c$ be positive real numbers. Prove that
\[(5a^3-b^3)(a-b)^2+(5b^3-c^3)(b-c)^2 + (5c^3-a^3)(c-a)^2 \geqslant 0.\]
3 replies
+1 w
Nguyenhuyen_AG
Jul 11, 2025
Victoria_Discalceata1
6 minutes ago
J
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d(2025^{a_i}-1) divides a_{n+1}
navi_09220114   3
N 7 minutes ago by ihategeo_1969
Source: TASIMO 2025 Day 2 Problem 5
Let $a_n$ be a strictly increasing sequence of positive integers such that for all positive integers $n\ge 1$
\[d(2025^{a_n}-1)|a_{n+1}.\]Show that for any positive real number $c$ there is a positive integers $N_c$ such that $a_n>n^c$ for all $n\geq N_c$.

Note. Here $d(m)$ denotes the number of positive divisors of the positive integer $m$.
3 replies
navi_09220114
May 19, 2025
ihategeo_1969
7 minutes ago
J
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A question
boylearnmath   0
16 minutes ago
Source: somewhere else
Can someone explain this solution in the link below, I don't understand and can someone give another solution.
https://artofproblemsolving.com/community/c4h2905032_find_a_b_are_positive_integers_such_that_ale2018_and_a5b7vdots_2018
0 replies
boylearnmath
16 minutes ago
0 replies
J
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Forgotten coaxiality lemma and more
Vivouaf   0
17 minutes ago
Source: self discovered
Let $\omega_1$, $\omega_2$ be two circles. We define for a point $X$ in the plane $\mathbb{P}_{\omega}(X)$ to be the power of $X$ wrt to circle $\omega$.
- Prove that the locus of points $X$ such that $\frac{\mathbb{P}_{\omega_1}(X)}{\mathbb{P}_{\omega_2}(X)} = K$ for some real constant $K$ is a circle coaxial to $\omega_1$ and $\omega_2$, which will be denoted by $S_K$.
- Prove that the center $O_K$ of $S_K$, $O_1$ of $\omega_1$, and $O_2$ of $\omega_2$ are such that $\frac{\overline{O_KO_1}}{\overline{O_KO_2}} = K$
0 replies
Vivouaf
17 minutes ago
0 replies
J
No more topics!
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J
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one variable function
youochange   1
N May 17, 2025 by Fishheadtailbody
$f:\mathbb R-\{0,1\} \to \mathbb R$


$f(x)+f(\frac{1}{1-x})=2x$
1 reply
youochange
May 17, 2025
Fishheadtailbody
May 17, 2025
J
one variable function
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Reveal topic tags
function
functional equation
algebra
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G H BBookmark kLocked kLocked NReply
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youochange
194 posts
youochange
#1 h  May 17, 2025, 2:49 PM • 1 Y
Y by PikaPika999
$f:\mathbb R-\{0,1\} \to \mathbb R$


$f(x)+f(\frac{1}{1-x})=2x$
Z K Y
The post below has been deleted. Click to close.
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Fishheadtailbody
9 posts
Fishheadtailbody
#2 h  May 17, 2025, 3:13 PM • 1 Y
Y by youochange
Notice that if $g(x) = \frac{1}{1 - x}$, then $g^3(x) = g(x)$.
Hence we can consider the following, which is simply a system of equations with three variables,
\begin{align*} f(x) + f(g(x)) = 2x, \\ f(g(x)) + f(g(g(x)) = 2g(x), \\ f(g^2(x)) + f(x) = 2g^2(x). \end{align*}Thus,
$f(x) = x - g(x) + g^2(x) = x - \frac{1}{1 - x} + \frac{x - 1}{x}$.
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