# 2006 Cyprus MO/Lyceum/Problems

## Contents

- 1 Problem 1
- 2 Problem 2
- 3 Problem 3
- 4 Problem 4
- 5 Problem 5
- 6 Problem 6
- 7 Problem 7
- 8 Problem 8
- 9 Problem 9
- 10 Problem 10
- 11 Problem 11
- 12 Problem 12
- 13 Problem 13
- 14 Problem 14
- 15 Problem 15
- 16 Problem 16
- 17 Problem 17
- 18 Problem 18
- 19 Problem 19
- 20 Problem 20
- 21 Problem 21
- 22 Problem 22
- 23 Problem 23
- 24 Problem 24
- 25 Problem 25
- 26 Problem 26
- 27 Problem 27
- 28 Problem 28
- 29 Problem 29
- 30 Problem 30
- 31 See also

## Problem 1

A diary industry, in a quantity of milk with fat adds a quantity of milk with fat and produces kg of milk with fat. The quantity of milk with fat, that was added is (in kg)

A.

B.

C.

D.

E.

## Problem 2

The operation is defined by . The value of the expression is

A.

B.

C.

D.

E.

## Problem 3

The domain of the function is

A.

B.

C.

D.

E.

## Problem 4

Given the function , Which of the following is correct, about the graph of ?

A. intersects x-axis

B. touches y-axis

C. touches x-axis

D. has minimum point

E. has maximum point

## Problem 5

If both integers are bigger than 1 and satisfy , then the minimum value of is

A.

B.

C.

D.

E.

## Problem 6

The value of the expression is

A.

B.

C.

D.

E.

## Problem 7

In the figure, is an equilateral triangle and , , . If , then the length of the side of the triangle is

A.

B.

C.

D.

E.

## Problem 8

In the figure is a regular 5-sided polygon and , , , , are the points of intersections of the extensions of the sides. If the area of the "star" is 1, then the area of the shaded quadrilateral is

A.

B.

C.

D.

E. None of these

## Problem 9

If and , then which of the following is correct?

A.

B.

C.

D.

E. None of these

## Problem 10

If and , then the product equals

A.

B.

C.

D.

E.

## Problem 11

The lines and intersect at the point . If the line intersects the axes and to the points and respectively, then the ratio of the area of the triangle to the area of the triangle equals

A.

B.

C.

D.

E.

## Problem 12

If

then equals

A.

B.

C.

D.

E.

## Problem 13

The sum of the digits of the number is

A.

B.

C.

D.

E. None of these

## Problem 14

The rectangle is a small garden divided to the rectangle and to the square , so that and the shaded area of the triangle is . The area of the whole garden is

A.

B.

C.

D.

E.

## Problem 15

The expression : equals

A.

B.

C.

D.

E.

## Problem 16

If are the roots of the equation , then are the roots of the equation

A.

B.

C.

D.

E.

## Problem 17

is equilateral triangle of side and . The measure of the angle $\ang \Gamma PE$ (Error compiling LaTeX. ! Undefined control sequence.) is

A.

B.

C.

D.

E.

## Problem 18

is the minimum point of the parabola and the parabola intersects the y-axis at the point . If the area if the rectangle is , then the equation of the parabola is

A.

B.

C.

D.

E.

## Problem 19

In the figure, is an isosceles triangle with and $\ang A=45^\circ$ (Error compiling LaTeX. ! Undefined control sequence.). If is altitude of the triangle and the sector belongs to the circle , the area of the shaded region is

A.

B.

C.

D.

E. None of these

## Problem 20

The sequence satisfies . Given that , then equals

A.

B.

C.

D.

E.

## Problem 21

A convex polygon has sides and diagonals. Then equals

A.

B.

C.

D.

E. None of these

## Problem 22

is rectangular and the points lie on the sides respectively so that . If is the area of and is the area of the rectangle , the ratio equals

A.

B.

C.

D.

E. None of these

## Problem 23

Of students taking Mathematics, Physics and Chemistry, no student takes one subject only. The number of students taking Mathematics and Chemistry only, equals to four times the number taking Mathematics and Physics only. If the number of students taking Physics and Chemistry only equals to three times the number of students taking all three subjects, then the number of students taking all three subjects is

A.

B.

C.

D.

E.

## Problem 24

The number of divisors of the number is

A.

B.

C.

D.

E.