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

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(→‎Problem 10)
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== Problem 10 ==
 
== Problem 10 ==
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The volume of an orthogonal parallelepiped is <math>132 {cm}^3 </math> and its dimensions are integer numbres. The minimum sum of the dimensions is
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A. <math>27cm</math>
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B. <math>19cm</math>
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C. <math>20cm</math>
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D. <math>18cm</math>
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E. None of these
  
 
[[2007 Cyprus MO/Lyceum/Problem 10|Solution]]
 
[[2007 Cyprus MO/Lyceum/Problem 10|Solution]]

Revision as of 12:48, 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 2 and $FG$ is an arc of the circle with centre the midpoint $K$ os 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 the angle of the diagonal d of a rectangle forms an angle $60^\circ$ with one of its sides, then the area of the recangle 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 substract 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 $132 {cm}^3$ and its dimensions are integer numbres. The minimum sum of the dimensions is

A. $27cm$

B. $19cm$

C. $20cm$

D. $18cm$

E. None of these

Solution

Problem 11

Solution

Problem 12

Solution

Problem 13

Solution

Problem 14

Solution

Problem 15

Solution

See also

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