Difference between revisions of "2006 Cyprus MO/Lyceum/Problems"
(→Problem 19) |
(→Problem 22) |
||
Line 360: | Line 360: | ||
</div> | </div> | ||
+ | <math>ABCD</math> is rectangular and the points <math>K,L,M,N</math> lie on the sides <math>AB, BC, CD, DA</math> respectively so that <math>\frac{AK}{KB}=\frac{BL}{LC}=\frac{CM}{MD}=\frac{DN}{NA}=2</math>. If <math>E_1</math> is the area of <math>KLMN</math> and <math>E_2</math> is the area of the rectangle <math>ABCD</math>, the ratio <math>\frac{E_1}{E_2}</math> equals | ||
+ | |||
+ | A. <math>\frac{5}{9}</math> | ||
+ | |||
+ | B. <math>\frac{1}{3}</math> | ||
+ | |||
+ | C. <math>\frac{9}{5}</math> | ||
+ | |||
+ | D. <math>\frac{3}{5}</math> | ||
+ | |||
+ | E. None of these | ||
[[2006 Cyprus MO/Lyceum/Problem 22|Solution]] | [[2006 Cyprus MO/Lyceum/Problem 22|Solution]] |
Revision as of 15:12, 5 July 2007
Contents
- 1 Problem 1
- 2 Problem 2
- 3 Problem 3
- 4 Problem 4
- 5 Problem 5
- 6 Problem 6
- 7 Problem 7
- 8 Problem 8
- 9 Problem 9
- 10 Problem 10
- 11 Problem 11
- 12 Problem 12
- 13 Problem 13
- 14 Problem 14
- 15 Problem 15
- 16 Problem 16
- 17 Problem 17
- 18 Problem 18
- 19 Problem 19
- 20 Problem 20
- 21 Problem 21
- 22 Problem 22
- 23 Problem 23
- 24 Problem 24
- 25 Problem 25
- 26 Problem 26
- 27 Problem 27
- 28 Problem 28
- 29 Problem 29
- 30 Problem 30
- 31 See also
Problem 1
A diary industry, in a quantity of milk with $4%$ (Error compiling LaTeX. Unknown error_msg) fat adds a quantity of milk with $1%$ (Error compiling LaTeX. Unknown error_msg) fat and produces kg of milk with $2%$ (Error compiling LaTeX. Unknown error_msg) fat.
The quantity of milk with $1%$ (Error compiling LaTeX. Unknown error_msg) fat, that was added is (in kg)
A.
B.
C.
D.
E.
Problem 2
The operation is defined by
.
The value of the expression
is
A.
B.
C.
D.
E.
Problem 3
The domain of the function is
A.
B.
C.
D.
E.
Problem 4
Given the function ,
Which of the following is correct, about the graph of
?
A. intersects x-axis
B. touches y-axis
C. touches x-axis
D. has minimum point
E. has maximum point
Problem 5
If both integers are bigger than 1 and satisfy
, then the minimum value of
is
A.
B.
C.
D.
E.
Problem 6
The value of the expression is
A.
B.
C.
D.
E.
Problem 7
In the figure, is an equilateral triangle and
,
,
. If
, then the length of the side os the triangle ABC is
A.
B.
C.
D.
E.
Problem 8
In the figure is a regular 5-sided polygon and
,
,
,
,
are the points of intersections of the extensions of the sides.
If the area of the "star"
is 1, then the area of the shaded quadrilateral
is
A.
B.
C.
D.
E. None of these
Problem 9
If and
, then which of the following is correct
A.
B.
C.
D.
E. None of these
Problem 10
If and
, then the product
equals
A.
B.
C.
D.
E.
Problem 11
The lines and
intersect at the point
. If the line
intersects the axes
and
to the points
and
respectively, then the ratio of the area of the triangle
to the area of the triangle
equals
A.
B.
C.
D.
E.
Problem 12
If
then equals
A.
B.
C.
D.
E.
Problem 13
The sum of the digits of the number is
A.
B.
C.
D.
E. None of these
Problem 14
The rectangle is a small garden divided to the rectangle
and to the square
, so that
and the shaded area of the triangle
is
. The area of the whole garden is
A.
B.
C.
D.
E.
Problem 15
The expression : equals
A.
B.
C.
D.
E.
Problem 16
If are the roots of the equation
, then
are the roots of the equation
A.
B.
C.
D.
E.
Problem 17
is equilateral triangle of side
and
. The measure of the angle $\ang CPE$ (Error compiling LaTeX. Unknown error_msg) is
A.
B.
C.
D.
E.
Problem 18
is the minimum point of the parabola and the parabola intersects the y-axis at the point
.
If the area if the rectangle
is
, then the equation of the parabola is
A.
B.
C.
D.
E.
Problem 19
In the figure is isosceles triangle with
and $\ang A=45^\circ$ (Error compiling LaTeX. Unknown error_msg). If
is altitude of the triangle and the sector
belongs to the circle
, the area of the shaded region is
A.
B.
C.
D.
E. None of these
Problem 20
The sequence satisfies
.
Given that
, then
equals
A.
B.
C.
D.
E.
Problem 21
A convex polygon has sides and
diagonals. Then
equals
A.
B.
C.
D.
E. None of these
Problem 22
is rectangular and the points
lie on the sides
respectively so that
. If
is the area of
and
is the area of the rectangle
, the ratio
equals
A.
B.
C.
D.
E. None of these
Problem 23
Of students taking Mathematics, Physics and Chemistry, no student takes one subject only. The number of students taking Mathematics and Chemistry only, equals to four times the number taking Mathematics and Physics only. If the number of students taking Physics and Chemistry only equals to three times the number of students taking all three subjects, then the number of students taking all three subjects is
A.
B.
C.
D.
E.
Problem 24
The number of divisors of the number is
A.
B.
C.
D.
E.