Download „МАЭСТРО ПИФАГОР С ПЕРВОЙ ЕВРОПЕЙСКОЙ ГИМНАЗИЙ

„МАЭСТРО ПИФАГОР С ПЕРВОЙ ЕВРОПЕЙСКОЙ
ГИМНАЗИЙ ПЕТРА ВЕЛИКОГО”
8 КЛАСС
Problem 1. If 𝑎 < 𝑏 and 𝑎2 > 𝑏 2 then 𝑎 + 𝑏 equals:
А) a non-negative negative number B) a negative number
C) a positive number D) cannot be determined
Problem 2. The number of the solutions of the equation |−𝑥 2 + 3𝑥| = −𝑥 2 is:
А) 0
B) 1
C) 2
D) more than 2
Problem 3. It is well-known that the sum of the interior angles of a quadrilateral is 360 degrees.
If one of the angles of a given quadrilateral is equal to the arithmetic average of the other three
angles, then this angle is:
А) acute
B) right
C) obtuse
D) other
Problem 4. A polygon has more than 30 diagonals. The number of its sides is at least:
А) 9
B) 10
C) 11
D) 12
Problem 5. If 𝑎2 + 𝑏 2 + 𝑐 2 = 𝑎𝑏 + 𝑏𝑐 + 𝑐𝑎 , then 20𝑎 − 13𝑏 − 7𝑐 is identically equal to:
А) 𝑎
B) 𝑏
C) 𝑐
D) 0
Problem 6. The equality (𝑥 + 𝑎)(𝑥 2 + 𝑏𝑥 + 4) = 𝑥 3 − 𝑐𝑥 + 20
is identically true. Then, among the parameters a, b, and c, the one with lowest value is:
А) 𝑎
B) 𝑏
C) 𝑐
D) all values are equal
Problem 7. If ab>0 and a+b<0, then the value of (𝑎 − |𝑎|)(𝑏 − |𝑏|) is:
А) 2𝑎𝑏
B) 4𝑎𝑏
C) 𝑎𝑏
D) other answer
Problem 8. How many of the solutions of the equation (𝑥 − 1)(𝑥 − 2) = 0
are also solutions of the inequality (𝑥 − 1)(𝑥 2 + 2𝑥 + 2) < 0 ?
А) 0
B) 1
C) 2
D) 3
Problem 9. Suppose that in a given triangle two of the sides are 2 cm and 4 cm long. If the
median towards the third side is of length m cm, then it can be inferred that:
А) 𝑚 < 3
B) 2 < 𝑚 < 6
C) 2 < 𝑚 < 4
D) 0 < 𝑚 < 3
Problem 10. The smallest integer x for which 10 divides 22013 − 𝑥 with remainder 0 is:
А) 8
B) 6
C) 4
D) 2
Problem 11. A train travels at the speed of 20 m/s. How many kilometers will this train travel
for 1.5 hours including a 10 minutes waiting time?
А) 960
B) 108
C) 96
D) other
Problem 12. For certain values of the parameters a and b, the equationhas
(𝑎2 − 4)𝑥 = 𝑏 3 − 27 an infinite number of solutions. Then, the largest value of |𝑎 − 𝑏| is
А) 1
B) 2
C) 4
D) 5
Problem 13. The ten-digit numbers with sum of their digits equal to 2 are in total:
А) 2
B) 4
C) 9
D) 10
Problem 14. The point M lies inside square ABCD, so that ∢𝐶𝐷𝑀 = ∢𝐷𝐶𝑀 = 150 .
∢𝑀𝐴𝐵
Then, the ratio ∢𝐶𝐷𝑀 is equal to:
А) 2
B) 3
C) 4
D) 5
Problem 15. The factor decomposition of the binomial
𝑛4 + 4
consist two polynomials, one 𝑛2 + 2𝑛 + 2 of which isThe other is:
А) 𝑛2 + 2𝑛 − 2
B) 𝑛2 + 2𝑛 + 2
C) 𝑛2 − 2𝑛 + 2
D) −𝑛2 + 2𝑛 + 2
Problem 16. A given number is divisible by 2, 3, and 5, and has altogether 2013 divisors. The
smallest such number is
2𝑁 × 3𝑀 × 5𝑃
where 𝑀 + 𝑁 + 𝑃 =
Problem 17. A given rhombus has diagonals of lengths 8 cm and 6 cm. The quadrilateral with
vertices at the midpoints of the sides of the rhombus has area … sq.cm?
Problem 18. In a square 2,013 points are given. What is the greatest possible number of triangles
to which we can divide this square, if the triangles' vertices belong to the set of the given points
and the four vertices of the square.
Problem 19. A rectangular sheet of size 6 cm by 7 cm has been cut into the fewest number of
squares possible with integer sides in cm. How many are the squares?
Problem 20. The largest of 22 consecutive even natural numbers is four times bigger than the
smallest of them. Which is the fifth number in this sequence?