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

A. $2$

B. $-2$

C. $1$

D. $-1$

E. $0$

Solution

Problem 2

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

A. $4$

B. $4^x$

C. $2\cdot4^x$

D. $4^{x+1}$

E. $3\cdot4^x$

Solution

Problem 3

A cyclist drives form town A to town B with velocity $40 \frac{\mathrm{km}}{\mathrm{h}}$ and comes back with velocity $60 \frac{\mathrm{km}}{\mathrm{h}}$ . The mean velocity in $\frac{\mathrm{km}}{\mathrm{h}}$ for the total distance is

A. $45$

B. $48$

C. $50$

D. $55$

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

A. $2$

B. $1$

C. $0$

D. $\frac{1}{2}$

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

A. $1$

B. $2$

C. $3$

D. $4$

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

A. $\frac{1}{4}$

B. $\frac{\sqrt{2}}{2}$

C. $2-\sqrt{3}$

D. $\sqrt{3}-1$

E. $\sqrt{2}$ $-1$

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

A. $\frac{d^2 \sqrt{3}}{4}$

B. $\frac{d^2}{2}$

C. $2d^2$

D. $d^2 \sqrt{2}$

E. None of these

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

A. $0$

B. $1$

C. $-1$

D. $3$

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

A. $0$

B. $-1$

C. $1$

D. $2$

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

A. $27\;\mathrm{ cm}$

B. $19\;\mathrm{ cm}$

C. $20\;\mathrm{ cm}$

D. $18\;\mathrm{ cm}$

E. None of these

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?

A. $X=2Y$

B. $Y=2X$

C. $X=Y$

D. $X=Y^2$

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

A. $1$

B. $-1$

C. $0$

D. $-2$

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

A. $a^2+b^2-2$

B. $a^2+b^2$

C. $\frac{a^2+b^2}{2}$

D. $a^2+b^2+1$

E. $a^2+b^2-4$

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

A. $\frac{1}{3}$

B. $\frac{2}{3}$

C. $\frac{1}{\sqrt{2}}$

D. $\sqrt2 -1$

E. $\frac{\sqrt{2}}{4}$

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

A. $\frac{6-2\sqrt{3}}{3}$

B. $8$

C. $1$

D. $\frac{6+2\sqrt{3}}{3}$

E. None of these

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?

A. $5$

B. $6$

C. $10$

D. $11$

E. $12$

Solution

Problem 17

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

A. $0$

B. $2$

C. $4$

D. $6$

E. $8$

Solution

Problem 18

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

A. $7$

B. $14$

C. $49$

D. $64$

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

A. $71^{st}$

B. $72^{nd}$

C. $73^{rd}$

D. $74^{th}$

E. None of these

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

A. $15$

B. $16$

C. $17$

D. $18$

E. $19$

Solution

Problem 21

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

Α. $(12+20\pi)$ m

Β. $(12 + \pi)$ m

C. $(12+ 4\pi)$ m

D. $\left(12+\frac\pi5\right)$

Ε. None of these

Solution

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

Α. $\frac{3a}{2}$

Β. $\frac{3a}{\sqrt{2}}$

C. $\frac{5a}{2}$

D. $\frac{3a}{\sqrt{3}}$

Ε. $2a$

Solution

Problem 23

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

Α. 5m

Β. 5,2m

C. 5,5m

D. 5,8m

Ε. 6m

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

A. profit €4

B. neither profit nor loss

C. loss €8

D. profit €8

E. loss €4

Solution

Problem 25

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