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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 9th (Webinar), 4:00pm PT/7:00pm ET, Learn about Video-based Summer Camps at the Virtual Campus
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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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a combinatorial geometry problem
xyz123456   5
N 2 minutes ago by Fishheadtailbody
$A_i\left(x_i{,}y_i\right){,}0\le x_i{,}y_i\le 1{,}1\le i\le 6.prove\ that:\exists \ 1\le i<j\le 6\ {,}\left|A_iA_j\right|\le \frac{\sqrt{13}}{6}$
5 replies
1 viewing
xyz123456
Mar 3, 2025
Fishheadtailbody
2 minutes ago
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problem....
Cobedangiu   2
N 6 minutes ago by sqing
$a,b,c>0$ and $a+b+c=1$
Prove that: $Q=ab+bc+ca-2abc\le\dfrac{7}{27}$
2 replies
Cobedangiu
3 hours ago
sqing
6 minutes ago
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Find all pairs (x; y)
sqing   5
N 21 minutes ago by EVKV
Source: 7th European Mathematical Cup , Junior Category, Q2
Find all pairs $ (x; y) $ of positive integers such that
$$xy | x^2 + 2y -1.$$
5 replies
sqing
Dec 25, 2018
EVKV
21 minutes ago
J
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Why is the old one deleted?
EeEeRUT   10
N 36 minutes ago by kamatadu
Source: EGMO 2025 P1
For a positive integer $N$, let $c_1 < c_2 < \cdots < c_m$ be all positive integers smaller than $N$ that are coprime to $N$. Find all $N \geqslant 3$ such that $$\gcd( N, c_i + c_{i+1}) \neq 1$$for all $1 \leqslant i \leqslant m-1$

Here $\gcd(a, b)$ is the largest positive integer that divides both $a$ and $b$. Integers $a$ and $b$ are coprime if $\gcd(a, b) = 1$.

Proposed by Paulius Aleknavičius, Lithuania
10 replies
EeEeRUT
Apr 16, 2025
kamatadu
36 minutes ago
J
No more topics!
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romanian concyclic, starting with <B=120^o, angle bisectors and 3 incenters
parmenides51   0
Sep 26, 2018
Source: IMAC Arhimede 2013 p3
Let $ABC$ be a triangle with $\angle ABC=120^o$ and triangle bisectors $(AA_1),(BB_1),(CC_1)$, respectively. $B_1F \perp A_1C_1$, where $F\in (A_1C_1)$. Let $R,I$ and $S$ be the centers of the circles which are inscribed in triangles $C_1B_1F,C_1B_1A_1, A_1B_1F$, and $B_1S\cap A_1C_1=\{Q\}$. Show that $R,I,S,Q$ are on the same circle.
0 replies
parmenides51
Sep 26, 2018
0 replies
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romanian concyclic, starting with <B=120^o, angle bisectors and 3 incenters
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Reveal topic tags
geometry
incenter
angle bisector
Concyclic
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Source: IMAC Arhimede 2013 p3
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parmenides51
30629 posts
parmenides51
#1 h  Sep 26, 2018, 9:59 PM • 1 Y
Y by Adventure10
Let $ABC$ be a triangle with $\angle ABC=120^o$ and triangle bisectors $(AA_1),(BB_1),(CC_1)$, respectively. $B_1F \perp A_1C_1$, where $F\in (A_1C_1)$. Let $R,I$ and $S$ be the centers of the circles which are inscribed in triangles $C_1B_1F,C_1B_1A_1, A_1B_1F$, and $B_1S\cap A_1C_1=\{Q\}$. Show that $R,I,S,Q$ are on the same circle.
This post has been edited 2 times. Last edited by parmenides51, Sep 26, 2018, 10:02 PM
Reason: name edit
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