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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.

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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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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
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orl
Jul 16, 2008
eg4334
19 minutes ago
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parallelepiped is cut by a plane along a 6-gon
orl   3
N Feb 27, 2008 by greentreeroad
Source: Russia Olympiad 2004, problem 11.8
A parallelepiped is cut by a plane along a 6-gon. Supposed this 6-gon can be put into a certain rectangle $ \pi$ (which means one can put the rectangle $ \pi$ on the parallelepiped's plane such that the 6-gon is completely covered by the rectangle). Show that one also can put one of the parallelepiped' faces into the rectangle $ \pi.$
3 replies
orl
Apr 29, 2004
greentreeroad
Feb 27, 2008
J
parallelepiped is cut by a plane along a 6-gon
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Reveal topic tags
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Source: Russia Olympiad 2004, problem 11.8
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orl
3647 posts
orl
#1 h  Apr 29, 2004, 4:33 PM • 2 Y
Y by Adventure10, Mango247
A parallelepiped is cut by a plane along a 6-gon. Supposed this 6-gon can be put into a certain rectangle $ \pi$ (which means one can put the rectangle $ \pi$ on the parallelepiped's plane such that the 6-gon is completely covered by the rectangle). Show that one also can put one of the parallelepiped' faces into the rectangle $ \pi.$
This post has been edited 1 time. Last edited by orl, Sep 14, 2008, 11:25 AM
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iura
481 posts
iura
#2 h  May 1, 2004, 3:42 PM • 2 Y
Y by Adventure10, Mango247
I think the following lemma is correct:
If we have a hexagon with opposite parallel sides contained in a rectangle R,
there is a rectangle congruent to R containing it with one side parallel to one of hexagon's sides.
I'm 90% sure it's true, and if it is true, it solves the problem
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qxy
96 posts
qxy
#3 h  Jan 22, 2005, 4:52 AM • 2 Y
Y by Adventure10, Mango247
I've got a book, all the problems of ARMO 2004 are in it, but, orl gave a parallelepiped, in my book, that's not a parallelepiped but a cuboid. I don't know orl and my book which is right, could anyone give any advice? thank you!
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greentreeroad
484 posts
greentreeroad
#4 h  Feb 27, 2008, 8:17 PM • 2 Y
Y by Adventure10, Mango247
In my book is also a cube :wink:
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