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2006 Cyprus MO/Lyceum/Problems

Problem 1

A diary industry, in a quantity of milk with $4%$ (Error compiling LaTeX. Unknown error_msg) fat adds a quantity of milk with $1%$ (Error compiling LaTeX. Unknown error_msg) fat and produces $1200$kg of milk with $2%$ (Error compiling LaTeX. Unknown error_msg) fat. The quantity of milk with $1%$ (Error compiling LaTeX. Unknown error_msg) fat, that was added is (in kg)

A. $1000$

B. $600$

C. $800$

D. $120$

E. $480$

Solution

Problem 2

The operation $\alpha * \beta$ is defined by $\alpha * \beta = \alpha^2 - \beta^2$ $\forall \alpha , \beta \in R$. The value of the expression $K = \left[\left(1+\sqrt{3}\right) * 2\right]*\sqrt{2}$ is

A. $3$

B. $0$

C. $\sqrt{3}$

D. $9$

E. $1$

Solution

Problem 3

The domain of the function $f(x)=\sqrt{4+2x}$ is

A. $(-2,+\infty)$

B. $[0,+\infty)$

C. $[-2,+\infty)$

D. $[-2,0]$

E. $R$

Solution

Problem 4

Given the function $f(x)=\alpha x^2 +9x+ \frac{81}{4\alpha}$ , $\alpha \neq 0$ Which of the following is correct, about the graph of $f$?


A. intersects x-axis

B. touches y-axis

C. touches x-axis

D. has minimum point

E. has maximum point

Solution

Problem 5

If both integers $\alpha,\beta$ are bigger than 1 and satisfy $a^7=b^8$, then the minimum value of $\alpha+\beta$ is

A. $384$

B. $2$

C. $15$

D. $56$

E. $512$

Solution

Problem 6

The value of the expression $K=\sqrt{19+7\sqrt{3}}-\sqrt{7+4\sqrt{3}}$ is


A. $4$

B. $4\sqrt{3}$

C. $12+4\sqrt{3}$

D. $-2$

E. $2$

Solution

Problem 7

In the figure, $ABC$ is an equilateral triangle and $AD\perp BC$, $DE\perp AC$, $EZ\perp BC$. If $EZ=\sqrt{3}$, then the length of the side os the triangle ABC is


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

B. $8$

C. $4$

D. $3$

E. $9$

Solution

Problem 8

Solution

Problem 9

Solution

Problem 10

If $2^x=15$ and $15^y=256$, then the product $xy$ equals


A. $7$

B. $3$

C. $1$

D. $8$

E. $6$

Solution

Problem 11


Solution

Problem 12

Solution

Problem 13

Solution

Problem 14


Solution

Problem 15

Solution

Problem 16

Solution

Problem 17


Solution

Problem 18


Solution

Problem 19


Solution

Problem 20

Solution

Problem 21

Solution

Problem 22


Solution

Problem 23

Solution

Problem 24

Solution

Problem 25

Solution

Problem 26

Solution

Problem 27

Solution

Problem 28

Solution

Problem 29

Solution

Problem 30

Solution

See also

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