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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.

WOOT early bird pricing is in effect, don’t miss out! If you took MathWOOT Level 2 last year, no worries, it is all new problems this year! Our Worldwide Online Olympiad Training program is for high school level competitors. AoPS designed these courses to help our top students get the deep focus they need to succeed in their specific competition goals. Check out the details at this link for all our WOOT programs in math, computer science, chemistry, and physics.

Looking for summer camps in math and language arts? Be sure to check out the video-based summer camps offered at the Virtual Campus that are 2- to 4-weeks in duration. There are middle and high school competition math camps as well as Math Beasts camps that review key topics coupled with fun explorations covering areas such as graph theory (Math Beasts Camp 6), cryptography (Math Beasts Camp 7-8), and topology (Math Beasts Camp 8-9)!

Be sure to mark your calendars for the following events:
[list][*]April 3rd (Webinar), 4pm PT/7:00pm ET, Learning with AoPS: Perspectives from a Parent, Math Camp Instructor, and University Professor
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April 9th (Webinar), 4:00pm PT/7:00pm ET, Learn about Video-based Summer Camps at the Virtual Campus
[*]April 10th (Math Jam), 4:30pm PT/7:30pm ET, 2025 MathILy and MathILy-Er Math Jam: Multibackwards Numbers
[*]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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Functional Equation
Keith50   1
N 25 minutes ago by jasperE3
Source: Own
Find all functions $f:\mathbb{R} \rightarrow \mathbb{R}$ such that \[f(x+f(x)+2f(y))+f(2f(x)-y)=4x+f(y)\]holds for all reals $x$ and $y$.
1 reply
Keith50
Jun 24, 2021
jasperE3
25 minutes ago
J
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Cursed F.E. #3
EmilXM   6
N 26 minutes ago by jasperE3
Source: Own
Find all $f: \mathbb{R} \rightarrow \mathbb{R}$, such that:
$$f(2f(x)y) = f(xy) + xf(y)$$$\forall x,y \in \mathbb{R}$, and if $\exists t \in \mathbb{R}$, such that $f(2^{2021} + 1 -t)= 2^{2021} + \frac{t-1}{2}$, then $t=1$.
Note
$\rule{100cm}{0.1px}$
[center]<Previous $\hspace{3.14in}$Next>[/center]
6 replies
EmilXM
Jun 3, 2021
jasperE3
26 minutes ago
J
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Functional equation in R^2
EmilXM   2
N 27 minutes ago by jasperE3
Source: Own
Find all functions $f:\mathbb{R}^2 \longrightarrow \mathbb{R}$, such that: \begin{align*} f(f(x,y),y^2) = f(x^2,0) + 2yf(x,y) \end{align*}For all $x,y \in \mathbb{R}$.

@below fixed.
2 replies
EmilXM
Oct 14, 2020
jasperE3
27 minutes ago
J
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Inequality with a,b,c
GeoMorocco   0
an hour ago
Source: Morocco Training 2025
Let $ a,b,c $ be positive real numbers such that : $ ab+bc+ca=3 $ . Prove that : $$\frac{a\sqrt{3+bc}}{b+c}+\frac{b\sqrt{3+ca}}{c+a}+\frac{c\sqrt{3+ab}}{a+b}\ge a+b+c $$
0 replies
GeoMorocco
an hour ago
0 replies
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geometric inequality with sum of principal diagonals in convex 101-gon
parmenides51   0
Dec 28, 2020
Source: 2014 Euler Olympiad P8, a Junior Olympiad from Russia
A diagonal of a convex $101$-gon will be called principal if there are $50$ vertices on one side of it and $49$ vertices on the other. Several principal diagonals have been chosen that have no common ends. Prove that the sum of the lengths of these diagonals is less than the sum of the lengths of the remaining principal diagonals.
0 replies
parmenides51
Dec 28, 2020
0 replies
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geometric inequality with sum of principal diagonals in convex 101-gon
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Source: 2014 Euler Olympiad P8, a Junior Olympiad from Russia
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parmenides51
30630 posts
parmenides51
#1 h  Dec 28, 2020, 9:04 AM
Y by
A diagonal of a convex $101$-gon will be called principal if there are $50$ vertices on one side of it and $49$ vertices on the other. Several principal diagonals have been chosen that have no common ends. Prove that the sum of the lengths of these diagonals is less than the sum of the lengths of the remaining principal diagonals.
This post has been edited 1 time. Last edited by parmenides51, Dec 28, 2020, 9:10 AM
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