2015 Grade 10 Math Challenge – Elimination Round with answer key – Part II

Below are the 2015 Grade 10 Math Challenge – Elimination Round questions and answers – Part II. More reviewers can be found on the Past Tests and All Posts pages.

26.) Solve the inequality: 3^{2x^2 + 3x - 2} > 1

27.) Two non-congruent circles have centers at C_1 and C_2. Diameter \overline {AB} of circle C_1 and diameter \overline {CD} of circle C_2 are perpendicular to \overline {{C_1}{C_2}} . If {C_1}{C_2} = 10, what is the area of the quadrilateral determined by A, B, C and D?

28.) Find the area of a triangle whose vertices have coordinates (2, 3), (-4, 2) and (10, 1).

29.) A jar contains only red and green balls. Ten red balls are added and the green balls now constitute 20% of the total. In addition, ten green balls are added, making the percentage of green balls equal to 40% of the total. How many balls were originally in the jar?

30.) If p + q = 22, what is the smallest value of p^2 + q^2?

31.) In the figure below, \triangle PAD is similar to which triangle?

32.) In the figure above, PA PA = 4, AC = 5 and PD = 20 (not drawn to scale). What is BD?

33.) Interestingly, it is true that (3!)(5!)(7!) is equal to n! What is n?

34.) For how many positive integers n Is 7n + 1 divisible by 3n + 5?

35.) If you randomly choose two different numbers from 1 to 10, what is the probability that one number is one more than the other?

36.) A class of 30 students took two quizzes. Sixteen passed the first test and 20 passed the second. If four students failed both quizzes, how many passed both?

37.) What is the radius of a circle whose equation is x^2 + y^2 - 4x + 8y - 16 = 0 .

38.) If (4^x)(5^x) = (2^x)(7^x), what is x?

39.) Suppose x, y and z are nonzero real numbers such that x + y + z = 0. What is the value of \dfrac {x}{y} + \dfrac {x}{z} + \dfrac {y}{z} + \dfrac {y}{x} + \dfrac {z}{x} + \dfrac {z}{y}?

40.) If x and y are positive real numbers satisfying x + y \leq xy , what is the smallest value of x + y?

41.) Simplify: \sqrt {23 + \sqrt {408}} - \sqrt {23 - \sqrt {408}}

42.) In how many ways can you write 7 as the sum of 1’s and 2’s?

43.) How many solutions does the equation log_x(4x - 3) = 3 have? Here it is understood that the logarithm function is defined only for positive real numbers.

44.) Rectangle ABCD is subdivided into four smaller rectangles by two lines, one parallel to AB and the other parallel to AD. If the areas of three smaller rectangles are 1, 3 and 5 square units, what are the possible values for the area of rectangle ABCD?

45.) Let x = 2^{100}, y = 3^{60} and z = 10^{30} . What is the smallest possible among the three?

46.) What is the minimum value of |x| + |x - 1| + |x + 2| ?

47.) Determine all real numbers x that satisfy |x^2 - x - 3| > |x^2 - x + 3|?

48.) How many real numbers satisfy x^6 + |x| = 7 ?

49.) Find all positive integers (x, y) that satisfy x^4 = y^2 + 97.

50.) Simplify: \dfrac {1}{2!} + \dfrac {2}{3!} + \dfrac {3}{4} + ... + \dfrac{11}{12!} .

Answer key:

26.) x < -2 or x > \dfrac{1}{2}
27.) 100 square units
28.) 10 square units
29.) 20
30.) 242
31.) \triangle PBC
32.) \dfrac{91}{5}
33.) 10
34.) 2
35.) \dfrac{1}{5}
36.) 10
37.) 6
38.) 0
39.) -3
40.) 4
41.) 2 \sqrt {6}
42.) 20
43.) 2
44.) 9 \dfrac {3}{5}, 10 \dfrac{2}{3} and 24 square units
45.) y
46.) 3
47.) 0 < x < 1
48.) 2
49.) x = 7, y = 48
50.) 1 - \dfrac {1}{12!} or \dfrac{12! - 1}{12!}

View Part I here: 2015 Grade 10 Math Challenge – Elimination Round with answer key – Part I

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