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Difference between revisions of "2006 Cyprus MO/Lyceum/Problems"

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In the figure, <math>ABC</math> is an equilateral triangle and <math>AD\perp BC</math>, <math>DE\perp AC</math>, <math>EZ\perp BC</math>. If <math>EZ=\sqrt{3}</math>, then the length of the side os the triangle ABC is
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A. <math>\frac{3\sqrt{3}}{2}</math>
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B. <math>8</math>
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C.  <math>4</math>
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D. <math>3</math>
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E. <math>9</math>
  
 
[[2006 Cyprus MO/Lyceum/Problem 7|Solution]]
 
[[2006 Cyprus MO/Lyceum/Problem 7|Solution]]

Revision as of 09:41, 30 June 2007

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

2006 CyMO-7.PNG

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

2006 CyMO-8.PNG

Solution

Problem 9

Solution

Problem 10

Solution

Problem 11

2006 CyMO-11.PNG


Solution

Problem 12

Solution

Problem 13

Solution

Problem 14

2006 CyMO-17.PNG


Solution

Problem 15

Solution

Problem 16

Solution

Problem 17

2006 CyMO-17.PNG


Solution

Problem 18

2006 CyMO-18.PNG


Solution

Problem 19

2006 CyMO-19.PNG


Solution

Problem 20

Solution

Problem 21

Solution

Problem 22

2006 CyMO-22.PNG


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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