Difference between revisions of "2007 Cyprus MO/Lyceum/Problems"

Revision as of 13:19, 6 May 2007

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

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$ABCD$ 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

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In the square ABCD the segment KB equals to the 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. $\sqrt{2}$ $-1$

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

Solution

Problem 15

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The non convex angles of the non convex octagon $ABCDEFGH$ measures $240^\circ$ each and the diagonals AE, GC are perpendicular, bisect each and are both equal to 2. Then the area of the ocatgon 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

Solution

Problem 20

Solution

Problem 21

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Solution

Problem 22

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Solution

Problem 23

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Solution

Problem 24

Solution

Problem 25

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Solution

Problem 26

Solution

Problem 27

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Solution

Problem 28

Solution

Problem 29

Solution

Problem 30

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Solution

@@ Line 263: / Line 263: @@
 == Problem 18 ==
+How many subsets are there for the set <math>A=\{1,2,3,4,5,6,7\}</math>?
+A. <math>7</math>
+B. <math>14</math>
+C. <math>49</math>
+D. <math>64</math>
+E. <math>128</math>
 [[2007 Cyprus MO/Lyceum/Problem 18|Solution]]
 == Problem 19 ==

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Difference between revisions of "2007 Cyprus MO/Lyceum/Problems"

Revision as of 13:19, 6 May 2007

Contents

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

Problem 14

Problem 15

Problem 16

Problem 17

Problem 18

Problem 19

Problem 20

Problem 21

Problem 22

Problem 23

Problem 24

Problem 25

Problem 26

Problem 27

Problem 28

Problem 29

Problem 30

See also