MathDB

Maximum value

Source: JBMO 2013 Shortlist

5/15/2016

\boxed{\text{A2}}

Find the maximum value of

|\sqrt{x^2+4x+8}-\sqrt{x^2+8x+17}|

where

x

is a real number.

algebra

shots in a rectangle billiard, will the ball ever reach vertex B ?

Source: JBMO Shortlist 2013 C2

4/24/2019

In a billiard with shape of a rectangle

ABCD

with

AB=2013

and

AD=1000

, a ball is launched along the line of the bisector of

\angle BAD

. Supposing that the ball is reflected on the sides with the same angle at the impact point as the angle shot , examine if it shall ever reach at vertex B.

rectanglecombinatoricscombinatorial geometry

2013 JBMO Shortlist G2

Source: 2013 JBMO Shortlist G2

10/8/2017

Circles

{\omega_1}

,

{\omega_2}

are externally tangent at point M and tangent internally with circle

{\omega_3}

at points

{K}

and

L

respectively. Let

{A}

and

{B}

be the points that their common tangent at point

{M}

of circles

{\omega_1}

and

{\omega_2}

intersect with circle

{\omega_3.}

Prove that if

{\angle KAB=\angle LAB}

then the segment

{AB}

is diameter of circle

{\omega_3.}

Theoklitos Paragyiou (Cyprus)

geometryJBMO

20^x+13^y=2013^z

Source: JBMO Shortlist 2013 NT2

4/24/2019

Solve in integers

20^x+13^y=2013^z

.

number theoryDiophantine equationIntegers

2

Problems(4)