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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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[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 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!
H
J
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angle bisector wanted, circumcircle related
parmenides51   1
N Jun 29, 2020 by Math-48
Source: All-Siberian Open School Olympiad 2011-12 9.3
The line tangent to the circumcircle of the triangle $ABC$ at point $A$, intersects the line $BC$ at point $K$. On the line $BC$ from point $K$ to the points $B$ and $C$, a segment $KM$ is laid out, the length of which is equal to the length $AK$. Prove that $AM$ is the bisector of the angle of $BAC$.
1 reply
parmenides51
Jun 29, 2020
Math-48
Jun 29, 2020
J
angle bisector wanted, circumcircle related
G H J
Reveal topic tags
geometry
angle bisector
circumcircle
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G H BBookmark kLocked kLocked NReply
Source: All-Siberian Open School Olympiad 2011-12 9.3
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parmenides51
30629 posts
parmenides51
#1 h  Jun 29, 2020, 3:01 PM • 1 Y
Y by Muaaz.SY
The line tangent to the circumcircle of the triangle $ABC$ at point $A$, intersects the line $BC$ at point $K$. On the line $BC$ from point $K$ to the points $B$ and $C$, a segment $KM$ is laid out, the length of which is equal to the length $AK$. Prove that $AM$ is the bisector of the angle of $BAC$.
This post has been edited 1 time. Last edited by parmenides51, Jun 29, 2020, 3:02 PM
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Math-48
44 posts
Math-48
#3 h  Jun 29, 2020, 4:07 PM • 4 Y
Y by Bst, Moustafa, Muaaz.SY, mariam10
Here's my solution :-D
$\angle BAM=\angle KAM-\angle KAB$
$\angle BAM=\angle KMA-\angle KAB$
$\angle BAM=180^o-\angle ABM-\angle BAM-\angle KAB$
$\implies 2\angle BAM=\angle BAC$ done.
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