bourgondian cross (I think)

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0 replies

jlacosta

Apr 2, 2025

0 replies

PQ = r and 6 more conditions

avisioner 40

N 40 minutes ago by wu2481632

Source: 2023 ISL G2

Let $ABC$ be a triangle with $AC > BC,$ let $\omega$ be the circumcircle of $\triangle ABC,$ and let $r$ be its radius. Point $P$ is chosen on $\overline{AC}$ such taht $BC=CP,$ and point $S$ is the foot of the perpendicular from $P$ to $\overline{AB}$ . Ray $BP$ mets $\omega$ again at $D$ . Point $Q$ is chosen on line $SP$ such that $PQ = r$ and $S,P,Q$ lie on a line in that order. Finally, let $E$ be a point satisfying $\overline{AE} \perp \overline{CQ}$ and $\overline{BE} \perp \overline{DQ}$ . Prove that $E$ lies on $\omega$ .

40 replies

avisioner

Jul 17, 2024

wu2481632

40 minutes ago

Easy Quadric equation

VicKmath7 2

N an hour ago by RagvaloD

Source: Archimedes Junior 2015

Find all values of the real parameter $a$ , so that the equation $x^2+(a-2)x-(a-1)(2a-3)=0$ has two real roots, so that the one is the square of the other.

2 replies

1 viewing

VicKmath7

Mar 16, 2020

RagvaloD

an hour ago

Why is the old one deleted?

EeEeRUT 6

N an hour ago by Safal

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 $\gcd(a, b) = 1$ .

6 replies

EeEeRUT

Yesterday at 1:33 AM

Safal

an hour ago

Equation of exponents

shobber 4

N an hour ago by zhoujef000

Source: Canada 1998

Find all real numbers $x$ such that: $x = \sqrt{ x - \frac{1}{x} } + \sqrt{ 1 - \frac{1}{x} }$

4 replies

shobber

Mar 4, 2006

zhoujef000

an hour ago

No more topics!

bourgondian cross (I think)

Peter 3

N Aug 9, 2020 by parmenides51

Source: flanders '90

On the standard unit circle, draw 4 unit circles with centers [0,1],[1,0],[0,-1],[-1,0].
You get a figure as below, find the area of the colored part.

IMAGE

3 replies

Peter

Sep 27, 2005

parmenides51

Aug 9, 2020

bourgondian cross (I think)

G H J

Reveal topic tags

geometry

G H BBookmark kLocked kLocked NReply

Source: flanders '90

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Peter

3615 posts

Peter

#1 h Sep 27, 2005, 2:32 PM • 2 Y

Y by Adventure10, Mango247

On the standard unit circle, draw 4 unit circles with centers [0,1],[1,0],[0,-1],[-1,0].
You get a figure as below, find the area of the colored part.

Invalid image file

This post has been edited 3 times. Last edited by Peter, Sep 27, 2005, 7:37 PM

Z K Y

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aappaa

52 posts

aappaa

#2 h Sep 27, 2005, 5:18 PM • 2 Y

Y by Adventure10, Mango247

It's $3\sqrt{3}-\dfrac{4\pi}{3} )$

Click to reveal hidden text

Consider three circles with colinear centers. The first and the third intersect with the second one in four points. The line connecting the centers of the first and the third circle intersects with the second one in two points. Note that all these points are vertices of a regular hexagon. All we need to know is the area of the small circular segment as the difference between the areas of one sixth of the circle and one sixth of the hexagon.

$S=\dfrac{\pi}{6}-\dfrac{\sqrt{3}}{4}$

Now note that quarter of the 'cross' is simply $\dfrac{\pi}{6}-3\cdot S$ or $\dfrac{\sqrt{3}}{4}-2\cdot S$ and the result follows.

[edit:] The sketch helped a lot, thanks.

This post has been edited 1 time. Last edited by aappaa, Sep 27, 2005, 7:55 PM

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Peter

3615 posts

Peter

#3 h Sep 27, 2005, 5:48 PM • 2 Y

Y by Adventure10, Mango247

Edit: correct.

Z K Y

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parmenides51

30629 posts

parmenides51

#4 h Aug 9, 2020, 6:30 PM

Y by

On the standard unit circle, draw $4$ unit circles with centers $[0,1]$ , $[1,0]$ , $[0,-1]$ , $[-1,0]$ . You get a figure as below, find the area of the colored part.

https://cdn.artofproblemsolving.com/attachments/6/1/a479365778b8a1d10373a47a4f698150e0633f.png

This post has been edited 5 times. Last edited by parmenides51, Dec 24, 2022, 6:00 PM

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