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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
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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
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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
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No more topics!
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B'C' tangent to incircle iff (s -b)^2+(s-c)^2=(s-a)^2, touchpoints of excircls
parmenides51   1
N Dec 26, 2021 by rafaello
Source: Romanian Mathematical Magazine SP 310 https://artofproblemsolving.com/community/c2691230_
In $\vartriangle ABC$, $B' \in (AC)$ the contact point of the external circumscription circle of side $AC$ and $C'$ the contact point of the external circumscription circle of side $AB$. Prove that $B'C'$ is tangent of the inscribed circle in $ABC$ if and only if $(s - b)^2 + (s - c)^2 = (s - a)^2$

by Marian Ursarescu - Romania

My note: $B'$ is touchpoint of $B$-excircle with $AC$, $C'$ is touchpoint of $C$-excircle with $AB$, $s$ is the semiperimeter
1 reply
parmenides51
Dec 13, 2021
rafaello
Dec 26, 2021
J
B'C' tangent to incircle iff (s -b)^2+(s-c)^2=(s-a)^2, touchpoints of excircls
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Reveal topic tags
geometry
incircle
excircle
tangent
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G H BBookmark kLocked kLocked NReply
Source: Romanian Mathematical Magazine SP 310 https://artofproblemsolving.com/community/c2691230_
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parmenides51
30629 posts
parmenides51
#1 h  Dec 13, 2021, 3:12 PM
Y by
In $\vartriangle ABC$, $B' \in (AC)$ the contact point of the external circumscription circle of side $AC$ and $C'$ the contact point of the external circumscription circle of side $AB$. Prove that $B'C'$ is tangent of the inscribed circle in $ABC$ if and only if $(s - b)^2 + (s - c)^2 = (s - a)^2$

by Marian Ursarescu - Romania

My note: $B'$ is touchpoint of $B$-excircle with $AC$, $C'$ is touchpoint of $C$-excircle with $AB$, $s$ is the semiperimeter
Z K Y
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rafaello
1079 posts
rafaello
#2 h  Dec 26, 2021, 12:05 AM
Y by
Let $x=s-a$, $y=s-b$ and $z=s-c$. We have $$2\cos{\angle A}=\frac{b^2+c^2-a^2}{bc}=\frac{(x+y)^2+(x+z)^2-(y+z)^2}{(x+y)(x+z)}=\frac{2x^2+2xy+2xz-2yz}{x^2+xy+xz+yz}.$$By Pitot's theorem and Law of Cosines, we have \begin{align*} (s-b)^2+(s-c)^2-2(s-b)(s-c)\cos{\angle A}&=(2(s-a)-(s-b)-(s-c))^2\Longleftrightarrow\\ y^2+z^2-\frac{2yz(x^2+xy+xz-yz)}{x^2+xy+xz+yz}&=(2x-y-z)^2\Longleftrightarrow\\ \frac{-x^2yz-xy^2z-xyz^2+y^2z^2}{x^2+xy+xz+yz}&=2x^2+yz-2xy-2xz\Longleftrightarrow\\ 2x^2(x^2-y^2-z^2)&=0, \end{align*}as desired. $\blacksquare$
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