Isosceles into Isosceleses

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0 replies

jlacosta

Apr 2, 2025

0 replies

functional equation interesting

skellyrah 9

N 8 minutes ago by Blackbeam999

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

9 replies

skellyrah

Apr 24, 2025

Blackbeam999

8 minutes ago

Line through orthocenter

juckter 14

N 21 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

21 minutes ago

Geometry with parallel lines.

falantrng 33

N 23 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

23 minutes ago

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

N.T.TUAN 9

N 34 minutes 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

34 minutes ago

No more topics!

Isosceles into Isosceleses

mira74 4

N Apr 16, 2020 by Green4Applez

Source: 2018-2019 Fall SDPC #1

An isosceles triangle $T$ has the following property: it is possible to draw a line through one of the three vertices of $T$ that splits it into two smaller isosceles triangles $R$ and $S$ , neither of which are similar to $T$ . Find all possible values of the vertex (apex) angle of $T$ .

4 replies

mira74

Apr 13, 2020

Green4Applez

Apr 16, 2020

Isosceles into Isosceleses

G H J

Reveal topic tags

algebra

G H BBookmark kLocked kLocked NReply

Source: 2018-2019 Fall SDPC #1

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mira74

1010 posts

mira74

#1 h Apr 13, 2020, 10:48 PM

Y by

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dictator2002

146 posts

dictator2002

#2 h Apr 14, 2020, 10:09 PM

Y by

What is SDPC?

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brainiacmaniac31

2170 posts

brainiacmaniac31

#3 h Apr 14, 2020, 10:35 PM • 3 Y

Y by Mango247, Mango247, Mango247

dictator2002 wrote:

What is SDPC?

San Diego Power Contest.

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dictator2002

146 posts

dictator2002

#4 h Apr 15, 2020, 10:19 PM

Y by

brainiacmaniac31 wrote:

dictator2002 wrote:

What is SDPC?

San Diego Power Contest.

Got it, thanks

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Green4Applez

125 posts

Green4Applez

#5 h Apr 16, 2020, 12:57 AM

Y by

One of the Possibilities

Suppose isosceles triangle $T$ has vertices $A, B,$ and $C$ where $A$ is the vertex angle. For the first case, suppose a line is drawn from $B$ to intersect $AC$ at $D$ such that $BD = BC$ and $BD=AD$ . From this, we can suppose that $m \angle A = x$ and, by the Triangle Exterior Angle Theorem, we know that $m \angle ACM=m \angle ABC=2x$ . So, since the sum of all of the angles in a triangle sum to $180^{\circ}$ , $x+2x+2x=5x=180 \implies \boxed{x=36^{\circ}}.$ [\hide]

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