Difference between revisions of "2006 Cyprus MO/Lyceum/Problems"
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== Problem 2 == | == Problem 2 == | ||
− | The operation <math>\alpha*\beta</math> is defined by <math>\alpha*\beta=\alpha^2-\beta^2\forall\alpha,\beta\in\mathbb{R}</math>. | + | The operation <math>\alpha*\beta</math> is defined by <math>\alpha*\beta=\alpha^2-\beta^2\ \forall\alpha,\beta\in\mathbb{R}</math>. |
The value of the expression <math>K=\left[\left(1+\sqrt{3}\right)*2\right]*\sqrt{2}</math> is | The value of the expression <math>K=\left[\left(1+\sqrt{3}\right)*2\right]*\sqrt{2}</math> is | ||
Latest revision as of 10:56, 27 April 2008
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 fat adds a quantity of milk with
fat and produces
kg of milk with
fat.
The quantity of milk with
fat, that was added is (in kg)
Problem 2
The operation is defined by
.
The value of the expression
is
Problem 3
The domain of the function is
Problem 4
Given the function ,
Which of the following is correct, about the graph of
?
Problem 5
If both integers are bigger than 1 and satisfy
, then the minimum value of
is
Problem 6
The value of the expression is
Problem 7
In the figure, is an equilateral triangle and
,
,
. If
, then the length of the side of the triangle
is
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
Problem 9
If and
, then which of the following is correct?
Problem 10
If and
, then the product
equals
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
Problem 12
If
then equals
Problem 13
The sum of the digits of the number is
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
Problem 15
The expression equals
Problem 16
If are the roots of the equation
, then
are the roots of the equation
Problem 17
is equilateral triangle of side
and
. The measure of the angle $\ang \Gamma PE$ (Error compiling LaTeX. Unknown error_msg) is
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
Problem 19
In the figure, is an isosceles triangle with
and $\ang A=45^\circ$ (Error compiling LaTeX. Unknown error_msg). If
is an altitude of the triangle and the sector
belongs to the circle
, the area of the shaded region is
Problem 20
The sequence satisfies
.
Given that
, then
equals
Problem 21
A convex polygon has sides and
diagonals. Then
equals
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
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
Problem 24
The number of divisors of the number is
Problem 25
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Problem 26
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Problem 27
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Problem 28
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Problem 29
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Problem 30
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