Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

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

m (→‎Problem 8)
m (all problems)
Line 208: Line 208:
  
 
[[2007 Cyprus MO/Lyceum/Problem 15|Solution]]
 
[[2007 Cyprus MO/Lyceum/Problem 15|Solution]]
 +
 +
== Problem 16 ==
 +
 +
[[2007 Cyprus MO/Lyceum/Problem 16|Solution]]
 +
 +
 +
== Problem 17 ==
 +
 +
[[2007 Cyprus MO/Lyceum/Problem 17|Solution]]
 +
 +
 +
== Problem 18 ==
 +
 +
[[2007 Cyprus MO/Lyceum/Problem 18|Solution]]
 +
 +
 +
== Problem 19 ==
 +
 +
[[2007 Cyprus MO/Lyceum/Problem 19|Solution]]
 +
 +
 +
== Problem 20 ==
 +
 +
[[2007 Cyprus MO/Lyceum/Problem 20|Solution]]
 +
 +
 +
== Problem 21 ==
 +
{{image}}
 +
 +
[[2007 Cyprus MO/Lyceum/Problem 21|Solution]]
 +
 +
 +
== Problem 22 ==
 +
{{image}}
 +
 +
[[2007 Cyprus MO/Lyceum/Problem 22|Solution]]
 +
 +
 +
== Problem 23 ==
 +
{{image}}
 +
 +
[[2007 Cyprus MO/Lyceum/Problem 23|Solution]]
 +
 +
 +
== Problem 24 ==
 +
 +
[[2007 Cyprus MO/Lyceum/Problem 24|Solution]]
 +
 +
 +
== Problem 25 ==
 +
{{image}}
 +
 +
[[2007 Cyprus MO/Lyceum/Problem 25|Solution]]
 +
 +
 +
== Problem 26 ==
 +
 +
[[2007 Cyprus MO/Lyceum/Problem 26|Solution]]
 +
 +
 +
== Problem 27 ==
 +
{{image}}
 +
 +
[[2007 Cyprus MO/Lyceum/Problem 27|Solution]]
 +
 +
 +
== Problem 28 ==
 +
 +
[[2007 Cyprus MO/Lyceum/Problem 28|Solution]]
 +
 +
 +
== Problem 29 ==
 +
 +
[[2007 Cyprus MO/Lyceum/Problem 29|Solution]]
 +
 +
 +
== Problem 30 ==
 +
{{image}}
 +
 +
[[2007 Cyprus MO/Lyceum/Problem 30|Solution]]
 +
  
 
== See also ==
 
== See also ==

Revision as of 13:43, 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 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 $132 {cm}^3$ 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

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

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

Solution

Problem 15

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

Solution

Problem 16

Solution


Problem 17

Solution


Problem 18

Solution


Problem 19

Solution


Problem 20

Solution


Problem 21

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

Solution


Problem 22

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

Solution


Problem 23

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

Solution


Problem 24

Solution


Problem 25

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

Solution


Problem 26

Solution


Problem 27

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

Solution


Problem 28

Solution


Problem 29

Solution


Problem 30

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

Solution


See also

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2007_Cyprus_MO/Lyceum/Problems&oldid=14879"