# 2007 Cyprus MO/Lyceum/Problems

## Problem 1

If $x-y=1$, then the value of the expression $K=x^2+x-2xy+y^2-y$ is

$\mathrm{(A)}\ 2\qquad\mathrm{(B)}\ -2\qquad\mathrm{(C)}\ 1\qquad\mathrm{(D)}\ -1\qquad\mathrm{(E)}\ 0$

## Problem 2

Given the formula $f(x) = 4^x$, then $f(x+1)-f(x)$ equals to

$\mathrm{(A)}\ 4\qquad\mathrm{(B)}\ 4^x\qquad\mathrm{(C)}\ 2\cdot4^x\qquad\mathrm{(D)}\ 4^{x+1}\qquad\mathrm{(E)}\ 3\cdot4^x$

## Problem 3

A cyclist drives form town A to town B with velocity $40 {}^{km}/{}_h$ and comes back with velocity $60 {}^{km}/{}_h$. The mean velocity in ${}^{km}/{}_h$ for the total distance is

$\mathrm{(A)}\ 45\qquad\mathrm{(B)}\ 48\qquad\mathrm{(C)}\ 50\qquad\mathrm{(D)}\ 55\qquad\mathrm{(E)}\ 100$

## Problem 4

We define the operation $a*b = \frac{1+a}{1+b^2}$, $\forall a,b \in \real$.

The value of $(2*0)*1$ is

$\mathrm{(A)}\ 2\qquad\mathrm{(B)}\ 1\qquad\mathrm{(C)}\ 0\qquad\mathrm{(D)}\ \frac{1}{2}\qquad\mathrm{(E)}\ \frac{5}{2}$

## Problem 5

If the remainder of the division of $a$ with $35$ is $23$, then the remainder of the division of $a$ with $7$ is

$\mathrm{(A)}\ 1\qquad\mathrm{(B)}\ 2\qquad\mathrm{(C)}\ 3\qquad\mathrm{(D)}\ 4\qquad\mathrm{(E)}\ 5$

## Problem 6

$ABCD$ is a square of side length 2 and $FG$ is an arc of the circle with centre the midpoint $K$ of the side $AB$ and radius 2. The length of the segments $FD=GC=x$ is

$\mathrm{(A)}\ \frac{1}{4}\qquad\mathrm{(B)}\ \frac{\sqrt{2}}{2}\qquad\mathrm{(C)}\ 2-\sqrt{3}\qquad\mathrm{(D)}\ \sqrt{3}-1\qquad\mathrm{(E)}\ \sqrt{2}-1$

## Problem 7

If a diagonal $d$ of a rectangle forms a $60^\circ$ angle with one of its sides, then the area of the rectangle is

$\mathrm{(A)}\ \frac{d^2 \sqrt{3}}{4}\qquad\mathrm{(B)}\ \frac{d^2}{2}\qquad\mathrm{(C)}\ 2d^2\qquad\mathrm{(D)}\ d^2 \sqrt{2}\qquad\mathrm{(E)}\ \mathrm{None\ of\ these}$

## Problem 8

If we subtract from 2 the inverse number of $x-1$, we get the inverse of $x-1$. Then the number $x+1$ equals to

$\mathrm{(A)}\ 0\qquad\mathrm{(B)}\ 1\qquad\mathrm{(C)}\ -1\qquad\mathrm{(D)}\ 3\qquad\mathrm{(E)}\ \frac{1}{2}$

## Problem 9

We consider the sequence of real numbers $a_1,a_2,a_3,...$ such that $a_1=0$, $a_2=1$ and $a_n=a_{n-1}-a_{n-2}$, $\forall n \in \{3,4,5,6,...\}$. The value of the term $a_{138}$ is

$\mathrm{(A)}\ 0\qquad\mathrm{(B)}\ -1\qquad\mathrm{(C)}\ 1\qquad\mathrm{(D)}\ 2\qquad\mathrm{(E)}\ -2$

## Problem 10

The volume of an orthogonal parallelepiped is $132\;\mathrm{cm}^3$ and its dimensions are integers. The minimum sum of the dimensions is

$\mathrm{(A)}\ 27\ \mathrm{cm}\qquad\mathrm{(B)}\ 19\ \mathrm{cm}\qquad\mathrm{(C)}\ 20\ \mathrm{cm}\qquad\mathrm{(D)}\ 18\ \mathrm{cm}\qquad\mathrm{(E)}\ \mathrm{None\ of\ these}$

## Problem 11

If $X=\frac{1}{2007 \sqrt{2006}+2006 \sqrt{2007}}$ and $Y=\frac{1}{\sqrt{2006}}-\frac{1}{\sqrt{2007}}$, which of the following is correct?

$\mathrm{(A)}\ X=2Y\qquad\mathrm{(B)}\ Y=2X\qquad\mathrm{(C)}\ X=Y\qquad\mathrm{(D)}\ X=Y^2\qquad\mathrm{(E)}\ Y=X^2$

## Problem 12

The function $f : \Re \rightarrow \Re$ has the properties $f(0) = -1$ and $f(xy)+f(x)+f(y)=x+y+xy+k$ $\forall x,y \in \Re$, where $k \in \Re$ is a constant. The value of $f(-1)$ is

$\mathrm{(A)}\ 1\qquad\mathrm{(B)}\ -1\qquad\mathrm{(C)}\ 0\qquad\mathrm{(D)}\ -2\qquad\mathrm{(E)}\ 3$

## Problem 13

If $x_1,x_2$ are the roots of the equation $x^2+ax+1=0$ and $x_3,x_4$ are the roots of the equation $x^2+bx+1=0$, then the expression $\frac{x_1}{x_2x_3x_4}+\frac{x_2}{x_1x_3x_4}+ \frac{x_3}{x_1x_2x_4}+\frac{x_4}{x_1x_2x_3}$equals to

$\mathrm{(A)}\ a^2+b^2-2\qquad\mathrm{(B)}\ a^2+b^2\qquad\mathrm{(C)}\ \frac{a^2+b^2}{2}\qquad\mathrm{(D)}\ a^2+b^2+1\qquad\mathrm{(E)}\ a^2+b^2-4$

## Problem 14

In the square $ABCD$ the segment $KB$ equals a side of the square. The ratio of areas $\frac{S_1}{S_2}$ is

$\mathrm{(A)}\ \frac{1}{3}\qquad\mathrm{(B)}\ \frac{2}{3}\qquad\mathrm{(C)}\ \frac{1}{\sqrt{2}}\qquad\mathrm{(D)}\ \sqrt{2}-1\qquad\mathrm{(E)}\ \frac{\sqrt{2}}{4}$

## Problem 15

The reflex angles of the concave octagon $ABCDEFGH$ measure $240^\circ$ each. Diagonals $AE$ and $GC$ are perpendicular, bisect each other, and are both equal to $2$.

The area of the octagon is

$\mathrm{(A)}\ \frac{6-2\sqrt{3}}{3}\qquad\mathrm{(B)}\ 8\qquad\mathrm{(C)}\ 1\qquad\mathrm{(D)}\ \frac{6+2\sqrt{3}}{3}\qquad\mathrm{(E)}\ \mathrm{None\ of\ these}$

## Problem 16

The full time score of a football match was $3$-$2$. How many possible half time results could there have been in this match?

$\mathrm{(A)}\ 5\qquad\mathrm{(B)}\ 6\qquad\mathrm{(C)}\ 10\qquad\mathrm{(D)}\ 11\qquad\mathrm{(E)}\ 12$

## Problem 17

The last digit of the number $a=2^{2007}+3^{2007}+5^{2007}+7^{2007}$ is

$\mathrm{(A)}\ 0\qquad\mathrm{(B)}\ 2\qquad\mathrm{(C)}\ 4\qquad\mathrm{(D)}\ 6\qquad\mathrm{(E)}\ 8$

## Problem 18

How many subsets are there for the set $A=\{1,2,3,4,5,6,7\}$?

$\mathrm{(A)}\ 7\qquad\mathrm{(B)}\ 14\qquad\mathrm{(C)}\ 49\qquad\mathrm{(D)}\ 64\qquad\mathrm{(E)}\ 128$

## Problem 19

120 five-digit numbers can be written with the digits $1,2,3,4,5$. If we place these numbers in increasing order, then the position of the number $41253$ is

$\mathrm{(A)}\ 71^{\mathrm{st}}\qquad\mathrm{(B)}\ 72^{\mathrm{nd}}\qquad\mathrm{(C)}\ 73^{\mathrm{rd}}\qquad\mathrm{(D)}\ 74^{\mathrm{th}}\qquad\mathrm{(E)}\ \mathrm{None\ of\ these}$

## Problem 20

The mean value for 9 Math-tests that a student succeded was $10$ (in scale $0$-$20$). If we put the grades of these tests in incresing order, then the maximum grade of the $5^{th}$ test is

$\mathrm{(A)}\ 15\qquad\mathrm{(B)}\ 16\qquad\mathrm{(C)}\ 17\qquad\mathrm{(D)}\ 18\qquad\mathrm{(E)}\ 19$

## Problem 21

In the following figure, three equal cycles of diameter $20\,\mathrm{ cm}$ represent pulleys, that are connected with a strap. If the distances between any two pulley center points are $AB=3\,\mathrm{m}$, $AC=4\,\mathrm{m}$ and $BC=5\,\mathrm{m}$, then the length of the strap is

$\mathrm{(A)}\ 12+20\pi)\ \mathrm{m}\qquad\mathrm{(B)}\ (12+\pi)\ \mathrm{m}\qquad\mathrm{(C)}\ (12+4\pi)\ \mathrm{m}\qquad\mathrm{(D)}\ \left(12+\frac{\pi}{5}\right)\ \mathrm{m}\qquad\mathrm{(E)}\ \mathrm{None\ of\ these}$

## Problem 22

In the following figure $ABCD$ is an orthogonal trapezium with $\ang A= \ang D=90^\circ$ (Error compiling LaTeX. Unknown error_msg) and bases $AB = a$ , $DC = 2a$ . If $AD = 3a$ and $M$ is the midpoint of the side $BC$, then $AM$ equals to

$\mathrm{(A)}\ \frac{3a}{2}\qquad\mathrm{(B)}\ \frac{3a}{\sqrt{2}}\qquad\mathrm{(C)}\ \frac{5a}{2}\qquad\mathrm{(D)}\ \frac{3a}{\sqrt{3}}\qquad\mathrm{(E)}\ 2a$

## Problem 23

In the figure above the right section of a parabolic tunnel is presented. Its maximum height is $OC=8\,\mathrm{m}$ and its maximum width is $AB=20\,\mathrm{m}$. If M is the midpoint of $OB$, then the height $MK$ of the tunnel at the point $M$ is

$\mathrm{(A)}\ 5\ \mathrm{m}\qquad\mathrm{(B)}\ 5.2\ \mathrm{m}\qquad\mathrm{(C)}\ 5.5\ \mathrm{m}\qquad\mathrm{(D)}\ 5.8\ \mathrm{m}\qquad\mathrm{(E)}\ 6\ \mathrm{m}$

## Problem 24

Costas sold two televisions for €198 each. From the sale of the first one he made a profit of 10% on its value and from the sale of the second one, he had a loss of 10% on its value. After the sale of the two televisions Costas had in total

$\mathrm{(A)}$ profit €4

$\mathrm{(B)}$ neither profit nor loss

$\qquad\mathrm{(C)}$ loss €8

$\qquad\mathrm{(D)}$ profit €8

$\qquad\mathrm{(E)}$ loss €4

## Problem 25

A jeweler makes crosses, according to the pattern shown above. The crosses are made from golden cyclical discs, with diameter of 1cm each. The height of a cross, which is made from 402 such discs is

$\mathrm{(A)}\ 198\ \mathrm{cm}\qquad\mathrm{(B)}\ 2\ \mathrm{m}\qquad\mathrm{(C)}\ 201\ \mathrm{cm}\qquad\mathrm{(D)}\ 202\ \mathrm{cm}\qquad\mathrm{(E)}\ 204\ \mathrm{cm}$

## Problem 26

The number of boys in a school is 3 times the number of girls and the number of girls is 9 times the number of teachers. Let us denote with $b$, $g$ and $t$, the number of boys, girls and teachers respectively. Then the total number of boys, girls and teachers equals to

$\mathrm{(A)}\ 31b\qquad\mathrm{(B)}\ \frac{37b}{27}\qquad\mathrm{(C)}\ 13g\qquad\mathrm{(D)}\ \frac{37g}{27}\qquad\mathrm{(E)}\ \frac{37t}{27}$

## Problem 27

In the following diagram, the light beam $\epsilon$ is reflected on the $x$-axis and the beam $d$, being reflected on a mirror parallel to the $y$-axis at distance 6, intersects the $y$-axis at point $B$.
The equation of line $f$ is given by

$\mathrm{(A)}\ x+y-11=0\qquad\mathrm{(B)}\ x+y+11=0\qquad\mathrm{(C)}\ x-y+11=0\qquad\mathrm{(D)}\ x-y-11=0\qquad\mathrm{(E)}\ y=-x+10$

## Problem 28

The product of $15^8\cdot28^6\cdot5^{11}$ is an integer number whose last digits are zeros. How many zeros are there?

$\mathrm{(A)}\ 6\qquad\mathrm{(B)}\ 8\qquad\mathrm{(C)}\ 11\qquad\mathrm{(D)}\ 12\qquad\mathrm{(E)}\ 19$

## Problem 29

The minimum value of a positive integer $k$, for which the sum $S=k+(k+1)+(k+2)+\ldots+(k+10)$ is a perfect square, is

$\mathrm{(A)}\ 5\qquad\mathrm{(B)}\ 6\qquad\mathrm{(C)}\ 10\qquad\mathrm{(D)}\ 11\qquad\mathrm{(E)}\ \mathrm{None\ of\ these}$

## Problem 30

A coin with a shape of a regular hexagon of side 1 is tangent to a square of side 6, as shown in the figure.

The coin rotates on the perimeter of the square, until it reaches its original position.

The length of the line which is being inscribed by the center of the hexagon is

$\mathrm{(A)}\ \frac{34\pi}{3}\qquad\mathrm{(B)}\ 24\qquad\mathrm{(C)}\ \frac{28\pi}{3}\qquad\mathrm{(D)}\ 6 \pi\sqrt{2}\qquad\mathrm{(E)}\ \mathrm{None\ of\ these}$