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

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(→‎Problem 9)
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We consider the sequence of real numbers <math>a_1,a_2,a_3,...</math> such that <math>a_1=0</math>, <math>a_2=1</math> and <math>a_n=a_{n-1}-a_{n-2}</math>, <math>\forall n \in \{3,4,5,6,...\}</math>. The value of the term <math>a_{138}</math> is
 
We consider the sequence of real numbers <math>a_1,a_2,a_3,...</math> such that <math>a_1=0</math>, <math>a_2=1</math> and <math>a_n=a_{n-1}-a_{n-2}</math>, <math>\forall n \in \{3,4,5,6,...\}</math>. The value of the term <math>a_{138}</math> is
  
−
A. 0
+
A. <math>0</math>
  
−
B. -1
+
B. <math>-1</math>
  
−
C. 1
+
C. <math>1</math>
  
−
D. 2
+
D. <math>2</math>
  
−
E. -2
+
E. <math>-2</math>
  
 
[[2007 Cyprus MO/Lyceum/Problem 9|Solution]]
 
[[2007 Cyprus MO/Lyceum/Problem 9|Solution]]

Revision as of 11:44, 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

An image is supposed to go here. You can help us out by creating one and editing it in. Thanks.

$ABCD$ 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

Solution

Problem 11

Solution

Problem 12

Solution

Problem 13

Solution

Problem 14

Solution

Problem 15

Solution

See also

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