Values for an angle

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0 replies

jlacosta

Apr 2, 2025

0 replies

Perpendicular if and only if Centre

shobber 3

N a minute ago by Tonne

Source: Pan African 2004

Let $ABCD$ be a cyclic quadrilateral such that $AB$ is a diameter of it's circumcircle. Suppose that $AB$ and $CD$ intersect at $I$ , $AD$ and $BC$ at $J$ , $AC$ and $BD$ at $K$ , and let $N$ be a point on $AB$ . Show that $IK$ is perpendicular to $JN$ if and only if $N$ is the midpoint of $AB$ .

3 replies

shobber

Oct 4, 2005

Tonne

a minute ago

$5^t + 3^x4^y = z^2$

Namisgood 0

7 minutes ago

Source: JBMO shortlist 2017

Solve in nonnegative integers the equation $5^t + 3^x4^y = z^2$

0 replies

Namisgood

7 minutes ago

0 replies

Concurrent lines

MathChallenger101 2

N 12 minutes ago by pigeon123

Let $A B C D$ be an inscribed quadrilateral. Circles of diameters $A B$ and $C D$ intersect at points $X_1$ and $Y_1$ , and circles of diameters $B C$ and $A D$ intersect at points $X_2$ and $Y_2$ . The circles of diameters $A C$ and $B D$ intersect in two points $X_3$ and $Y_3$ . Prove that the lines $X_1 Y_1, X_2 Y_2$ and $X_3 Y_3$ are concurrent.

2 replies

MathChallenger101

Feb 8, 2025

pigeon123

12 minutes ago

Find all natural numbers $n$

ItsBesi 7

N 17 minutes ago by justaguy_69

Source: Kosovo Math Olympiad 2025, Grade 9, Problem 2

Find all natural numbers $n$ such that $\frac{\sqrt{n}}{2}+\frac{10}{\sqrt{n}}$ is a natural number.

7 replies

ItsBesi

Nov 17, 2024

justaguy_69

17 minutes ago

No more topics!

Values for an angle

mavropnevma 3

N Oct 11, 2012 by mathbuzz

Source: Stars of Mathematics 2011 - Juniors - Problem 2

Let $ABC$ be an acute-angled, not equilateral triangle, where vertex $A$ lies on the perpendicular bisector of the segment $HO$ , joining the orthocentre $H$ to the circumcentre $O$ . Determine all possible values for the measure of angle $A$ .

(U.S.A. - 1989 IMO Shortlist)

3 replies

mavropnevma

Oct 10, 2012

mathbuzz

Oct 11, 2012

Values for an angle

G H J

Reveal topic tags

geometry

perpendicular bisector

IMO Shortlist

geometry proposed

G H BBookmark kLocked kLocked NReply

Source: Stars of Mathematics 2011 - Juniors - Problem 2

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mavropnevma

15142 posts

mavropnevma

#1 h Oct 10, 2012, 9:31 PM • 2 Y

Y by Adventure10, Mango247

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Faustus

1287 posts

Faustus

#2 h Oct 11, 2012, 12:45 AM • 2 Y

Y by Adventure10, Mango247

In the words, redirect here. Peace.
Faustus

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Lawasu

212 posts

Lawasu

#3 h Oct 11, 2012, 6:30 AM • 2 Y

Y by Adventure10, Mango247

This problem was also given at the Viitori Olimpici competition in Romania.

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mathbuzz

803 posts

mathbuzz

#4 h Oct 11, 2012, 12:01 PM • 2 Y

Y by Adventure10, Mango247

the midpoint of $OH$ is the nine-point centre $N$ . then $AN \perp OH$ .
we get that , $\Delta AON$ and $\Delta AHN$ are congruent. so , $AO=AH \implies R=2RcosA \implies cosA=1/2$ . so $A=60$ degrees.

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