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

Revision as of 13:45, 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

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

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Solution

Problem 22

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Solution

Problem 23

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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

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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 $b, g, t$ the number if boys, girls and the teachers respectively. Then the total number of boys, girls and teachers equals to

A. $31b$

B. $\frac{37b}{27}$

C. $13g$

D. $\frac{37g}{27}$

E. $\frac{37t}{27}$

Solution

Problem 27

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Solution

Problem 28

The product if 15^8 • 28^6 • 5^11 is an integer number whose last digits are zeores. How many are those zeroes?

A. $6$

B. $8$

C. $11$

D. $12$

E. $19$

Solution

Problem 29

The minimum value of the positive integer $k$ , for which the sum $S=k+(k+1)+(k+2)+...+(k+10)$ is a perfect square of an integer, is

A. $5$

B. $6$

C. $10$

D. $11$

E. None of these

Solution

Problem 30

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Solution

@@ Line 347: / Line 347: @@
 == 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, t the number if boys, girls and the teachers respectively. Then the total number of boys, girls and teachers equals to
+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 <math>b, g, t</math> the number if boys, girls and the teachers respectively. Then the total number of boys, girls and teachers equals to
-A. 31b
+A. <math>31b</math>
-B. \frac{37b}{27}
+B. <math>\frac{37b}{27}</math>
-C. 13g
+C. <math>13g</math>
-D. \frac{37g}{27}
+D. <math>\frac{37g}{27}</math>
-E. \frac{37t}{27}
+E. <math>\frac{37t}{27}</math>
 [[2007 Cyprus MO/Lyceum/Problem 26|Solution]]

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

Revision as of 13:45, 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