MathDB

Equation

Source: JBMO 2013 Shortlist

4/24/2016

\boxed{A1}

Find all ordered triplets of

(x,y,z)

real numbers that satisfy the following system of equation

x^3=\frac{z}{y}-\frac {2y}{z}

y^3=\frac{x}{z}-\frac{2z}{x}

z^3=\frac{y}{x}-\frac{2x}{y}

algebra

max no of different from {1,2,...,2013} so that no 2 have difference equal to 17

Source: JBMO Shortlist 2013 C1

4/24/2019

Find the maximum number of different integers that can be selected from the set

\{1,2,...,2013\}

so that no two exist that their difference equals to

17

.

number theorysets of integersSubsetDifference

2013 JBMO Shortlist G1

Source: 2013 JBMO Shortlist G1

10/8/2017

Let

{AB}

be a diameter of a circle

{\omega}

and center

{O}

,

{OC}

a radius of

{\omega}

perpendicular to

AB

,

{M}

be a point of the segment

\left( OC \right)

. Let

{N}

be the second intersection point of line

{AM}

with

{\omega}

and

{P}

the intersection point of the tangents of

{\omega}

at points

{N}

and

{B.}

Prove that points

{M,O,P,N}

are cocyclic.
(Albania)

geometryJBMO

JBMO 2013 Shortlist

Source: Secret

4/16/2015

\boxed{N1}

find all positive integers

n

for which

1^3+2^3+\cdots+{16}^3+{17}^n

is a perfect square.

number theory

1