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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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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
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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 w
xyz123456
Mar 3, 2025
Fishheadtailbody
2 minutes ago
J
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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
J
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Find all pairs (x; y)
sqing   5
N 20 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
20 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!
H
J
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concentric circumcircles wanted, starting with isosceles trapezoid
parmenides51   1
N Jan 25, 2022 by qwertyboyfromalotoftime
Source: Iranian Geometry Olympiad 2021 IGO Elementary p4
In isosceles trapezoid $ABCD$ ($AB \parallel CD$) points $E$ and $F$ lie on the segment $CD$ in such a way that $D, E, F$ and $C$ are in that order and $DE = CF$. Let $X$ and $Y$ be the reflection of $E$ and $C$ with respect to $AD$ and $AF$. Prove that circumcircles of triangles $ADF$ and $BXY$ are concentric.

Proposed by Iman Maghsoudi - Iran
1 reply
parmenides51
Jan 25, 2022
qwertyboyfromalotoftime
Jan 25, 2022
J
concentric circumcircles wanted, starting with isosceles trapezoid
G H J
Reveal topic tags
geometry
reflection
trapezoid
concentric circles
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G H BBookmark kLocked kLocked NReply
Source: Iranian Geometry Olympiad 2021 IGO Elementary p4
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parmenides51
30629 posts
parmenides51
#1 h  Jan 25, 2022, 1:56 PM
Y by
In isosceles trapezoid $ABCD$ ($AB \parallel CD$) points $E$ and $F$ lie on the segment $CD$ in such a way that $D, E, F$ and $C$ are in that order and $DE = CF$. Let $X$ and $Y$ be the reflection of $E$ and $C$ with respect to $AD$ and $AF$. Prove that circumcircles of triangles $ADF$ and $BXY$ are concentric.

Proposed by Iman Maghsoudi - Iran
Z K Y
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qwertyboyfromalotoftime
156 posts
qwertyboyfromalotoftime
#2 h  Jan 25, 2022, 3:51 PM • 1 Y
Y by JUSTemom
parmenides51 wrote:
In isosceles trapezoid $ABCD$ ($AB \parallel CD$) points $E$ and $F$ lie on the segment $CD$ in such a way that $D, E, F$ and $C$ are in that order and $DE = CF$. Let $X$ and $Y$ be the reflection of $E$ and $C$ with respect to $AD$ and $AF$. Prove that circumcircles of triangles $ADF$ and $BXY$ are concentric.

Proposed by Iman Maghsoudi - Iran
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This post has been edited 3 times. Last edited by qwertyboyfromalotoftime, Jan 25, 2022, 4:08 PM
Reason: typo
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