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

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(→‎Problem 7)
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== Problem 7 ==
 
== Problem 7 ==
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If the angle of the diagonal d of a rectangle forms an angle <math>60^\circ</math> with one of its sides, then the area of the recangle is
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A. <math>\frac{d^2 \sqrt{3}}{4}</math>
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 +
B. <math>\frac{d^2}{2}</math>
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C. <math>2d^2</math>
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 +
D. <math>d^2 \sqrt{2}</math>
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 +
E. None of these
  
 
[[2007 Cyprus MO/Lyceum/Problem 7|Solution]]
 
[[2007 Cyprus MO/Lyceum/Problem 7|Solution]]

Revision as of 11:31, 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 $40 \frac{km}{h}$ and comes back with velocity $60 \frac{km}{h}$. The mean valocity in $\frac{km}{h}$ 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

Solution

Problem 9

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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