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k a April Highlights and 2025 AoPS Online Class Information
jlacosta   0
Wednesday at 3:18 PM
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
Wednesday at 3:18 PM
0 replies
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Inspired by Polish 2025
sqing   0
2 minutes ago
Source: Own
Let $ a,b,c,d $ be reals such that $ a+b+c+d =0 $ and $ a^2+b^2+c^2+d^2=12.$ Prove that$$-3\leq abcd\leq 9$$Let $ a,b,c,d $ be reals such that $ a+b+c+d =0 $ and $ abcd=-3.$ Prove that$$a^2+b^2+c^2+d^2 \geq 12$$Let $ a,b,c,d $ be reals such that $ a+b+c+d =0 $ and $ abcd=9.$ Prove that$$a^2+b^2+c^2+d^2 \geq 12$$
0 replies
1 viewing
sqing
2 minutes ago
0 replies
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Problem 4
blug   0
2 minutes ago
Source: Polish Math Olympiad 2025 Finals P4
A positive integer $n\geq 2$ and a set $S$ consisting of $2n$ disting positive integers smaller than $n^2$ are given. Prove that there exists a positive integer $r\in \{1, 2, ..., n\}$ that can be written in the form $r=a-b$, for $a, b\in \mathbb{S}$ in at least $3$ different ways.
0 replies
blug
2 minutes ago
0 replies
J
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Problem 3
blug   0
6 minutes ago
Source: Polish Math Olympiad 2025 Finals P3
Positive integer $k$ and $k$ colors are given. We will say that a set of $2k$ points on a plane is $colorful$, if it contains exactly 2 points of each color and if lines connecting every two points of the same color are pairwise distinct. Find, in terms of $k$ the least integer $n\geq 2$ such that: in every set of $nk$ points of a plane, no three of which are collinear, consisting of $n$ points of every color there exists a $colorful$ subset.
0 replies
blug
6 minutes ago
0 replies
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D1010 : How it is possible ?
Dattier   15
N 12 minutes ago by maxal
Source: les dattes à Dattier
Is it true that$$\forall n \in \mathbb N^*, (24^n \times B \mod A) \mod 2 = 0 $$?

A=1728400904217815186787639216753921417860004366580219212750904
024377969478249664644267971025952530803647043121025959018172048
336953969062151534282052863307398281681465366665810775710867856
720572225880311472925624694183944650261079955759251769111321319
421445397848518597584590900951222557860592579005088853698315463
815905425095325508106272375728975

B=2275643401548081847207782760491442295266487354750527085289354
965376765188468052271190172787064418854789322484305145310707614
546573398182642923893780527037224143380886260467760991228567577
953725945090125797351518670892779468968705801340068681556238850
340398780828104506916965606659768601942798676554332768254089685
307970609932846902
15 replies
Dattier
Mar 10, 2025
maxal
12 minutes ago
J
No more topics!
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2 var inquality
sqing   0
Wednesday at 1:11 PM
Source: Own
Let $ a,b \ge 0 $ and $ a+b=2. $ Prove that
$$\sqrt{ a^2+b+6}+\sqrt{ b^2+a+6}\leq 8\sqrt{\frac{2- ab}{ab+1}} $$$$\sqrt{2a^2+b+1}+\sqrt{2b^2+a+1}\leq 4\sqrt{\frac{5-2ab}{ab+2}} $$$$\sqrt{2a^2+b}+\sqrt{2b^2+a}\leq 2\sqrt{\frac{3(5-2ab)}{ab+2}} $$
0 replies
sqing
Wednesday at 1:11 PM
0 replies
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2 var inquality
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Reveal topic tags
inequalities proposed
inequalities
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G H BBookmark kLocked kLocked NReply
Source: Own
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sqing
41409 posts
sqing
#1 h  Wednesday at 1:11 PM
Y by
Let $ a,b \ge 0 $ and $ a+b=2. $ Prove that
$$\sqrt{ a^2+b+6}+\sqrt{ b^2+a+6}\leq 8\sqrt{\frac{2- ab}{ab+1}} $$$$\sqrt{2a^2+b+1}+\sqrt{2b^2+a+1}\leq 4\sqrt{\frac{5-2ab}{ab+2}} $$$$\sqrt{2a^2+b}+\sqrt{2b^2+a}\leq 2\sqrt{\frac{3(5-2ab)}{ab+2}} $$
This post has been edited 2 times. Last edited by sqing, Wednesday at 1:25 PM
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