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k a May Highlights and 2025 AoPS Online Class Information
jlacosta   0
May 1, 2025
May is an exciting month! National MATHCOUNTS is the second week of May in Washington D.C. and our Founder, Richard Rusczyk will be presenting a seminar, Preparing Strong Math Students for College and Careers, on May 11th.

Are you interested in working towards MATHCOUNTS and don’t know where to start? We have you covered! If you have taken Prealgebra, then you are ready for MATHCOUNTS/AMC 8 Basics. Already aiming for State or National MATHCOUNTS and harder AMC 8 problems? Then our MATHCOUNTS/AMC 8 Advanced course is for you.

Summer camps are starting next month at the Virtual Campus in math and language arts that are 2 - to 4 - weeks in duration. Spaces are still available - don’t miss your chance to have an enriching summer experience. 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][*]May 9th, 4:30pm PT/7:30pm ET, Casework 2: Overwhelming Evidence — A Text Adventure, a game where participants will work together to navigate the map, solve puzzles, and win! All are welcome.
[*]May 19th, 4:30pm PT/7:30pm ET, What's Next After Beast Academy?, designed for students finishing Beast Academy and ready for Prealgebra 1.
[*]May 20th, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 1 Math Jam, Problems 1 to 4, join the Canada/USA Mathcamp staff for this exciting Math Jam, where they discuss solutions to Problems 1 to 4 of the 2025 Mathcamp Qualifying Quiz!
[*]May 21st, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 2 Math Jam, Problems 5 and 6, Canada/USA Mathcamp staff will discuss solutions to Problems 5 and 6 of the 2025 Mathcamp Qualifying Quiz![/list]
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0 replies
jlacosta
May 1, 2025
0 replies
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IMO Shortlist 2014 N4
hajimbrak   73
N a minute ago by Orzify
Let $n > 1$ be a given integer. Prove that infinitely many terms of the sequence $(a_k )_{k\ge 1}$, defined by \[a_k=\left\lfloor\frac{n^k}{k}\right\rfloor,\] are odd. (For a real number $x$, $\lfloor x\rfloor$ denotes the largest integer not exceeding $x$.)

Proposed by Hong Kong
73 replies
3 viewing
hajimbrak
Jul 11, 2015
Orzify
a minute ago
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Inspired by lbh_qys
sqing   0
5 minutes ago
Source: Own
Let $ a, b $ be real numbers such that $ (a-3)(b-3)(a-b)\neq 0 $ and $ a + b =6 . $ Prove that
$$ \left( \frac{a+k-1}{b - 3} + \frac{b+k-1}{3 - a} + \frac{k+2}{a - b} \right)^2 + 2(a^2 + b^2 )\geq6(k+8)$$Where $ k\in N^+.$
0 replies
sqing
5 minutes ago
0 replies
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Prove the inequality
Butterfly   1
N 6 minutes ago by Mathzeus1024

Let $a,b,c,d$ be positive real numbers. Prove $$a^2+b^2+c^2+d^2+abcd-3(a+b+c+d)+7\ge 0.$$
1 reply
+1 w
Butterfly
an hour ago
Mathzeus1024
6 minutes ago
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Inspired by lbh_qys
sqing   3
N 18 minutes ago by sqing
Source: Own
Let $ a, b $ be real numbers such that $ (a-3)(b-3)(a-b)\neq 0 $ and $ a + b =6 . $ Prove that
$$ \left( \frac{a}{b - 3} + \frac{b}{3 - a} + \frac{3}{a - b} \right)^2 + 2(a^2 + b^2 )\geq54$$Equality holds when $ (a,b)=\left(\frac{3}{2},\frac{9}{2}\right)$
$$\left( \frac{a+1}{b - 3} + \frac{b+1}{3 - a} + \frac{4}{a - b} \right)^2 + 2(a^2 + b^2 )\geq 60$$Equality holds when $ (a,b)=\left(3-\sqrt 3,3+\sqrt 3\right)$
$$ \left( \frac{a+3}{b - 3} + \frac{b+3}{3 - a} + \frac{6}{a - b} \right)^2 + 2\left(a^2 + b^2\right)\geq 72$$Equality holds when $ (a,b)=\left(3-\frac{3}{\sqrt 2},3+\frac{3}{\sqrt 2}\right)$
3 replies
1 viewing
sqing
4 hours ago
sqing
18 minutes ago
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Combinatorial Double Sum
P162008   0
Apr 24, 2025
Compute $\sum_{r=0}^{n} \sum_{k=0}^{r} (-1)^k (k + 1)(k + 2) \binom {n + 5}{r - k}$
0 replies
P162008
Apr 24, 2025
0 replies
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Combinatorial Double Sum
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P162008
#1 h  Apr 24, 2025, 1:03 AM
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Compute $\sum_{r=0}^{n} \sum_{k=0}^{r} (-1)^k (k + 1)(k + 2) \binom {n + 5}{r - k}$
This post has been edited 1 time. Last edited by P162008, Apr 25, 2025, 3:25 AM
Reason: Typo
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