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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)!

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[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 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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Three variables inequality
Headhunter   2
N 36 minutes ago by Headhunter
$\forall a\in R$ ,$~\forall b\in R$ ,$~\forall c \in R$
Prove that at least one of $(a-b)^{2}$, $(b-c)^{2}$, $(c-a)^{2}$ is not greater than $\frac{a^{2}+b^{2}+c^{2}}{2}$.

I assume that all are greater than it, but can't go more.
2 replies
Headhunter
3 hours ago
Headhunter
36 minutes ago
J
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Inequalities
sqing   0
40 minutes ago
Let $ x,y\geq 0 $ such that $ 2(x+y)=4+xy. $ Prove that$$x+y+\frac{1}{x}+\frac{1}{y}\geq5 $$Let $ x,y\geq 0 $ such that $ 3(x+y)=8+xy. $ Prove that$$x+y+\frac{1}{x}+\frac{1}{y}\geq5 $$Let $ x,y\geq 0 $ such that $ 3(x+y)=9+xy. $ Prove that$$x+y+\frac{1}{x}+\frac{1}{y}\geq \frac{20}{3} $$Let $ x,y\geq 0 $ such that $ 3(x+y)=6+xy. $ Prove that$$x+y+\frac{1}{x}+\frac{1}{y}\geq7-\frac{5}{\sqrt 3} $$
0 replies
1 viewing
sqing
40 minutes ago
0 replies
J
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Very Easy Combinatorics Problem
zeta1   0
an hour ago
Ali and Veli goes to hunting. The probability that each will successfully hit a duck is $1/2$ on any given shot. During the hunt, Ali shoots $12$ times, and Veli shoots $13$ times. What is the probability that Veli hits more ducks than Ali?

$ \textbf{(A)}\ \dfrac 12 \qquad\textbf{(B)}\ \dfrac{13}{25} \qquad\textbf{(C)}\ \dfrac{13}{24} \qquad\textbf{(D)}\ \dfrac{7}{13} \qquad\textbf{(E)}\ \dfrac{3}{4} $
0 replies
zeta1
an hour ago
0 replies
J
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Inequalities
sqing   11
N 2 hours ago by sqing
Let $ a,b,c> 0 $ and $ \frac{a}{a^2+ab+c}+\frac{b}{b^2+bc+a}+\frac{c}{c^2+ca+b} \geq 1$. Prove that
$$ a+b+c\leq 3 $$
11 replies
sqing
Apr 4, 2025
sqing
2 hours ago
J
No more topics!
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J
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Quadratic Function with Hidden Symmetry
synxxx   0
Apr 14, 2025
It's a multiple choice question where the answer choices are between 120, 146, 176, and 180. Hopefully, a sufficiently good bound should solve the question because solving for the exact value doesn't seem feasible on first glance. Could someone help with that then? Or if there's a way to solve for the exact value by hand and it's efficient then that would be nice too.
0 replies
synxxx
Apr 14, 2025
0 replies
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Quadratic Function with Hidden Symmetry
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synxxx
#1 h  Apr 14, 2025, 7:09 AM
Y by
It's a multiple choice question where the answer choices are between 120, 146, 176, and 180. Hopefully, a sufficiently good bound should solve the question because solving for the exact value doesn't seem feasible on first glance. Could someone help with that then? Or if there's a way to solve for the exact value by hand and it's efficient then that would be nice too.
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