2007 Cyprus MO/Lyceum/Problems

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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

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

Solution

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

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

Solution

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

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

Solution

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

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

Solution

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

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

Solution

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

$\mathrm{(A)}\ \frac{d^2 \sqrt{3}}{4}\qquad\mathrm{(B)}\ \frac{d^2}{2}\qquad\mathrm{(C)}\ 2d^2\qquad\mathrm{(D)}\ d^2 \sqrt{2}...$

Solution

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

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

Solution

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

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

Solution

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

$\mathrm{(A)}\ 27\ \mathrm{cm}\qquad\mathrm{(B)}\ 19\ \mathrm{cm}\qquad\mathrm{(C)}\ 20\ \mathrm{cm}\qquad\mathrm{(D)}\ 18\ \m...$

Solution

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?

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

Solution

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

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

Solution

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

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

Solution

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

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

Solution

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

$\mathrm{(A)}\ \frac{6-2\sqrt{3}}{3}\qquad\mathrm{(B)}\ 8\qquad\mathrm{(C)}\ 1\qquad\mathrm{(D)}\ \frac{6+2\sqrt{3}}{3}\qquad\...$

Solution

Problem 16

The full time score of a football match was $3$ - $2$ . how many possible half time results can we have for this match?

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

Solution

Problem 17

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

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

Solution

Problem 18

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

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

Solution

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

$\mathrm{(A)}\ 71^{\mathrm{st}}\qquad\mathrm{(B)}\ 72^{\mathrm{nd}}\qquad\mathrm{(C)}\ 73^{\mathrm{rd}}\qquad\mathrm{(D)}\ 74^...$

Solution

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

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

Solution

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

$\mathrm{(A)}\ 12+20\pi)\ \mathrm{m}\qquad\mathrm{(B)}\ (12+\pi)\ \mathrm{m}\qquad\mathrm{(C)}\ (12+4\pi)\ \mathrm{m}\qquad\ma...$

Solution

Problem 22

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

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

Solution

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

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

Solution

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

Solution

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

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

Solution

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

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

Solution

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