Difference between revisions of "2007 Cyprus MO/Lyceum/Problems"
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I_like_pie (talk | contribs) (→Problem 14) |
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== Problem 14 == | == Problem 14 == | ||
{{image}} | {{image}} | ||
− | In the square ABCD the segment KB equals | + | In the square <math>ABCD</math> the segment <math>KB</math> equals a side of the square. The ratio of areas <math>\frac{S_1}{S_2}</math> is |
A. <math>\frac{1}{3}</math> | A. <math>\frac{1}{3}</math> | ||
Line 209: | Line 209: | ||
C. <math>\frac{1}{\sqrt{2}}</math> | C. <math>\frac{1}{\sqrt{2}}</math> | ||
− | D. <math>\sqrt{2} | + | D. <math>\sqrt{2}-1</math> |
E. <math>\frac{\sqrt{2}}{4}</math> | E. <math>\frac{\sqrt{2}}{4}</math> |
Revision as of 19:49, 6 May 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
If , then the value of the expression is
A.
B.
C.
D.
E.
Problem 2
Given the formula , then equals to
A.
B.
C.
D.
E.
Problem 3
A cyclist drives form town A to town B with velocity and comes back with velocity . The mean velocity in for the total distance is
A.
B.
C.
D.
E.
Problem 4
We define the operation , .
The value of is
A.
B.
C.
D.
E.
Problem 5
If the remainder of the division of with is , then the remainder of the division of with is
A.
B.
C.
D.
E.
Problem 6
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is a square of side length 2 and is an arc of the circle with centre the midpoint of the side and radius 2. The length of the segments is
A.
B.
C.
D.
E.
Problem 7
If a diagonal of a rectangle forms a angle with one of its sides, then the area of the rectangle is
A.
B.
C.
D.
E. None of these
Problem 8
If we subtract from 2 the inverse number of , we get the inverse of . Then the number equals to
A.
B.
C.
D.
E.
Problem 9
We consider the sequence of real numbers such that , and , . The value of the term is
A.
B.
C.
D.
E.
Problem 10
The volume of an orthogonal parallelepiped is and its dimensions are integers. The minimum sum of the dimensions is
A.
B.
C.
D.
E. None of these
Problem 11
If and , which of the following is correct?
A.
B.
C.
D.
E.
Problem 12
The function has the properties and , where is a constant. The value of is
A.
B.
C.
D.
E.
Problem 13
If are the roots of the equation and are the roots of the equation , then the expression equals to
A.
B.
C.
D.
E.
Problem 14
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In the square the segment equals a side of the square. The ratio of areas is
A.
B.
C.
D.
E.
Problem 15
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The reflex angles of the concave octagon measure each. Diagonals and are perpendicular, bisect each other, and are both equal to 2.
The area of the octagon is
A.
B.
C.
D.
E. None of these
Problem 16
The full time score of a football match was -. how many possible half time results can we have for this match?
A.
B.
C.
D.
E.
Problem 17
The last digit of the number is
A.
B.
C.
D.
E.
Problem 18
How many subsets are there for the set ?
A.
B.
C.
D.
E.
Problem 19
120 five-digit numbers can be written with the digits . If we place these numbers in increasing order, then the position of the number is
A.
B.
C.
D.
E. None of these
Problem 20
The mean value for 9 Math-tests that a student succeded was (in scale -). If we put the grades of these tests in incresing order, then the maximum grade of the test is
A.
B.
C.
D.
E.
Problem 21
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Problem 22
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Problem 23
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Problem 24
Costas sold two televisions for €198 each. From the sale of the first one he made a profit of 10% on its value and from the sale of the second one, he had a loss of 10% on its value. After the sale of the two televisions Costas had in total
A. profit €4
B. neither profit nor loss
C. loss €8
D. profit €8
E. loss €4
Problem 25
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Problem 26
The number of boys in a school is 3 times the number of girls and the number of girls is 9 times the number of teachers. Let us denote with , and , the number of boys, girls and teachers respectively. Then the total number of boys, girls and teachers equals to
A.
B.
C.
D.
E.
Problem 27
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Problem 28
The product of is an integer number whose last digits are zeros. How many zeros are there?
A.
B.
C.
D.
E.
Problem 29
The minimum value of a positive integer , for which the sum is a perfect square, is
A.
B.
C.
D.
E. None of these
Problem 30
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