angle bisectors and radii

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0 replies

jlacosta

Apr 2, 2025

0 replies

functional equation interesting

skellyrah 10

N 12 minutes ago by jasperE3

find all functions IR->IR such that $xf(x+yf(xy)) + f(f(x)) = f(xf(y))^2 + (x+1)f(x)$

10 replies

skellyrah

Apr 24, 2025

jasperE3

12 minutes ago

Line through orthocenter

juckter 14

N 36 minutes ago by lpieleanu

Source: Mexico National Olympiad 2011 Problem 2

Let $ABC$ be an acute triangle and $\Gamma$ its circumcircle. Let $l$ be the line tangent to $\Gamma$ at $A$ . Let $D$ and $E$ be the intersections of the circumference with center $B$ and radius $AB$ with lines $l$ and $AC$ , respectively. Prove the orthocenter of $ABC$ lies on line $DE$ .

14 replies

juckter

Jun 22, 2014

lpieleanu

36 minutes ago

Geometry with parallel lines.

falantrng 33

N 37 minutes ago by joshualiu315

Source: RMM 2018,D1 P1

Let $ABCD$ be a cyclic quadrilateral an let $P$ be a point on the side $AB.$ The diagonals $AC$ meets the segments $DP$ at $Q.$ The line through $P$ parallel to $CD$ mmets the extension of the side $CB$ beyond $B$ at $K.$ The line through $Q$ parallel to $BD$ meets the extension of the side $CB$ beyond $B$ at $L.$ Prove that the circumcircles of the triangles $BKP$ and $CLQ$ are tangent .

33 replies

falantrng

Feb 24, 2018

joshualiu315

37 minutes ago

subsets of {1,2,...,mn}

N.T.TUAN 9

N an hour ago by AshAuktober

Source: USA TST 2005, Problem 1

Let $n$ be an integer greater than $1$ . For a positive integer $m$ , let $S_{m}= \{ 1,2,\ldots, mn\}$ . Suppose that there exists a $2n$ -element set $T$ such that
(a) each element of $T$ is an $m$ -element subset of $S_{m}$ ;
(b) each pair of elements of $T$ shares at most one common element;
and
(c) each element of $S_{m}$ is contained in exactly two elements of $T$ .

Determine the maximum possible value of $m$ in terms of $n$ .

9 replies

1 viewing

N.T.TUAN

May 14, 2007

AshAuktober

an hour ago

No more topics!

angle bisectors and radii

rogue 2

N May 11, 2010 by moldovan

Let $O$ be the circumcenter and $I$ be the incenter of triangle $ABC.$ Prove that if $AI\perp OB$ and $BI\perp OC$ then $CI\parallel OA$ .

2 replies

rogue

May 8, 2010

moldovan

May 11, 2010

angle bisectors and radii

G H J

Reveal topic tags

Kyiv mathematical festival

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rogue

554 posts

rogue

#1 h May 8, 2010, 11:37 PM • 2 Y

Y by Adventure10, Mango247

Let $O$ be the circumcenter and $I$ be the incenter of triangle $ABC.$ Prove that if $AI\perp OB$ and $BI\perp OC$ then $CI\parallel OA$ .

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Batominovski

1608 posts

Batominovski

#2 h May 9, 2010, 3:22 AM • 3 Y

Y by Adventure10, Mango247, and 1 other user

Angle chasing to get
$\angle{A}=\frac{2\pi}{7}\,,\;\angle{B}=\frac{4\pi}{7}\,,\text{ and }\angle{C}=\frac{\pi}{7}\,.$
The remaining is obvious.

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moldovan

1311 posts

moldovan

#3 h May 11, 2010, 7:52 PM • 2 Y

Y by Adventure10, Mango247

$AI \perp OB \Rightarrow \frac{A}{2}+\frac{\pi-2C}{2}=\frac{\pi}{2} \Rightarrow C=\frac{A}{2}$

$BI \perp OC \Rightarrow \frac{B}{2}+\frac{\pi-2A}{2}=\frac{\pi}{2} \Rightarrow A=\frac{B}{2}$

$\Rightarrow A=\frac{2 \pi}{7}, B=\frac{4 \pi}{7} , C=\frac{\pi}{7}$

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