# 2018 Grade 10 Math Challenge – National Level – Individual Finals

Below are the 2018 Grade 10 Math Challenge – National Level – Individual Finals questions with answers. More reviewers can be found on the Past Tests and All Posts pages.

Easy

1.) Find the value of $-\dfrac{1}{32}$ raised to $-\dfrac{4}{5}$

2.) Factor completely: $4x^3 + 9x^2 - 4x - 9.$

3.) The legs of a right triangle have lengths $\sqrt{5}cm$ and $\sqrt{7}cm$. How long (in cm) is the median to the hypotenuse?

4.) Find the sum of the numerator and the denominator when the repeating decimal 0.212 121 . . . is written as a fraction in lowest terms?

5.) Let A and B be two distinct points on a circle. If major arc AB is 40° more than minor arc AB, find the

6.) Set A has 4 elements and is a subset of B which has 2022 elements. How many sets S are possible if A is a subset of S, and S is a subset of B?

7.) Find the measure in degrees of the smaller angle formed by the hands of a clock at 9:20.

8.) Find the 4th term of an arithmetic sequence whose first 3 terms are x + 3, 2018, and x + 13.

9.) An ant on the plane is traveling from (0, 0) to (5, 3) and in each move it can only go 1 unit up or 1 unit to the right, at each time. How many distinct paths of 8 moves can the ant possibly take?

10.) The sum of two numbers is 1 and their product is 2. Find the sum of their cubes.

Average

1.) The 3rd, 6th, and 10th terms of an arithmetic sequence form a geometric sequence. Find the common ratio.

2.) Find the number of positive integers less than 500 000 which contain the block 678, with the 3 digits appearing consecutively and in this order.

3.) Solve for x in the equation: $x^2 + \dfrac{4}{x^2} = 9 + \dfrac{4}{9}$

4.) Find the cosine of the smallest acute angle of the triangle whose sides have lengths 4, 5, and 6.

5.) How many ordered quadruples (a, b, c, d) of nonnegative integers are there such that a + b + c + d = 10?

Difficult

1.) Find the sum of two positive integers if their quotient, sum, and product are in the ratio 1 ∶ 6 ∶ 16

2.) A polynomial has remainder 20 when divided by x − 18, and remainder 18 when divided by x − 20. Find the remainder when divided by (x − 18)(x − 20).

3.) Triangle ABC has sides AB = 6, BC = 9, and AC = 9. Point P is chosen on side BC such that AP = 6. Find the ratio of BP to CP.

4.) Let $S_m$ denote the sum of the first m terms of an arithmetic series. If $\dfrac{S_{2n}}{S_n} = 3$ for some integer n, find $\dfrac{S_{5n}}{S_{3n}}$

5.) A sequence $a_1, a_{2}, a_3$ . . . satisfies the property that $latex an+1$is the average of the first n terms if $n \geq 2$. If $a_1 = 1$ and $a_{2018},$ find a = 2.

Answer Key

Easy

1.) 16
2.) $(4x + 9)(x + 1)(x - 1)$
3.) $\sqrt{3} cm$
4.) 40
5.) 200°
6.) $2^{2018}$
7.) 160°
8.) 2028
9.) 56
10.) -5

Average

1.) 1 or 4/3
2.) 1500
3.) $\pm 3, \pm \dfrac{2}{3}$
4.) 3/4
5.) 286

Difficult

1.) 12
2.) -x +38
3.) 4 : 5
4.) $\dfrac{5}{2}$
5.) 4035

View Team Orals here: Math Challenge – National Level – Team Orals

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