CYPRUS MATHEMATICAL SOCIETY
REGIONAL COMPETITION
NOVEMBER 2015
LYCEUM C’
Date: 07/11/2015
Time: 10:00 -12:00
INSTRUCTIONS
1. Solve all the problems by giving full answers.
2. Each problem is marked with 10 points.
3. Write with blue or black ink (Shapes can be drawn with pencil).
4. The use of corrective liquid (Tip-Ex) is not allowed.
5. The use of a calculator is not allowed.
PROBLEMS
Problem 1
Given the point Ρ(3,4) and the circle
.
Α) Find the equations of the tangents to the circle that pass through the point
Β) Find the angle formed by the two tangents
C) Find the distance between the point P and the circle’s chord formed by the contact points of the
tangents to the circle
Problem 2
Let
be any real number. Find the maximum and the minimum value of √
√
Problem 3
Given the expression
( )
Find the Range of
Problem 4
Let
be a scalene triangle with altitude
and median
From point of the
altitude
draw
and
perpendicular to
and
respectively ( and traces on
and
) Choose point H on the extention of
so that
and a point Θ on the
extension of AM so that
. If
and
intersect at , prove that
A) The
triangle is isosceles
B) The quadrilateral
can be inscribed in a circle.