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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
J
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n-variables inequality
JK1603JK   0
8 minutes ago
Source: unknown?
Given $a_i>0,\ \forall i=1,2,\cdots n$ satisfying $\prod_{i=1}^n a_i\le p^n$ where $p\ge n-1.$ Prove that $$\sum_{i=1}^n \frac{1}{1+a_i^n}\ge \frac{n}{1+p^n}.$$
0 replies
JK1603JK
8 minutes ago
0 replies
J
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A cyclic inequality
KhuongTrang   1
N 14 minutes ago by KhuongTrang
Source: own-CRUX
IMAGE
Link
1 reply
KhuongTrang
Apr 2, 2025
KhuongTrang
14 minutes ago
J
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Something nice
KhuongTrang   25
N 15 minutes ago by KhuongTrang
Source: own
Problem. Given $a,b,c$ be non-negative real numbers such that $ab+bc+ca=1.$ Prove that

$$\sqrt{a+1}+\sqrt{b+1}+\sqrt{c+1}\le 1+2\sqrt{a+b+c+abc}.$$
25 replies
KhuongTrang
Nov 1, 2023
KhuongTrang
15 minutes ago
J
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IMO 2008, Question 1
orl   154
N 19 minutes ago by eg4334
Source: IMO Shortlist 2008, G1
Let $ H$ be the orthocenter of an acute-angled triangle $ ABC$. The circle $ \Gamma_{A}$ centered at the midpoint of $ BC$ and passing through $ H$ intersects the sideline $ BC$ at points $ A_{1}$ and $ A_{2}$. Similarly, define the points $ B_{1}$, $ B_{2}$, $ C_{1}$ and $ C_{2}$.

Prove that the six points $ A_{1}$, $ A_{2}$, $ B_{1}$, $ B_{2}$, $ C_{1}$ and $ C_{2}$ are concyclic.

Author: Andrey Gavrilyuk, Russia
154 replies
1 viewing
orl
Jul 16, 2008
eg4334
19 minutes ago
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No more topics!
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J
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A cyclic quad
Rushil   3
N Oct 31, 2024 by namanrobin08
Source: Indian RMO 2002 Problem 5
The circumference of a circle is divided into eight arcs by a convex quadrilateral $ABCD$ with four arcs lying inside the quadrilateral and the remaining four lying outside it. The lengths of the arcs lying inside the quadrilateral are denoted by $p,q,r,s$ in counter-clockwise direction. Suppose $p+r = q+s$. Prove that $ABCD$ is cyclic.
3 replies
Rushil
Oct 27, 2005
namanrobin08
Oct 31, 2024
J
A cyclic quad
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Reveal topic tags
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G H BBookmark kLocked kLocked NReply
Source: Indian RMO 2002 Problem 5
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Rushil
1592 posts
Rushil
#1 h  Oct 27, 2005, 2:32 PM • 2 Y
Y by Adventure10, Mango247
The circumference of a circle is divided into eight arcs by a convex quadrilateral $ABCD$ with four arcs lying inside the quadrilateral and the remaining four lying outside it. The lengths of the arcs lying inside the quadrilateral are denoted by $p,q,r,s$ in counter-clockwise direction. Suppose $p+r = q+s$. Prove that $ABCD$ is cyclic.
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stergiu
1648 posts
stergiu
#2 h  Oct 28, 2005, 1:47 PM • 2 Y
Y by Adventure10, Mango247
I think ,I have seen this also as a problem in

Bundeswettberwerb - Mathematik, Germany
babis
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10000th User
3049 posts
10000th User
#3 h  Jun 23, 2008, 5:22 PM • 2 Y
Y by Adventure10, Mango247
For purposes of the Resources section where this problem is featured, here is a link to a solution:

http://www.mathlinks.ro/viewtopic.php?t=209261
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namanrobin08
99 posts
namanrobin08
#4 h  Oct 31, 2024, 11:18 AM
Y by
Link to official solution:
https://www.isical.ac.in/~rmo/papers/rmo/rmo-2002.pdf
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