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 raised to

2.) Factor completely:

3.) The legs of a right triangle have lengths and . 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:

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 denote the sum of the first

*m*terms of an arithmetic series. If for some integer

*n*, find

5.) A sequence . . . satisfies the property that $ latex an+1 $is the average of the first

*n*terms if . If and find a = 2.

**Answer Key**

Easy

Easy

1.) 16

2.)

3.)

4.) 40

5.) 200°

6.)

7.) 160°

8.) 2028

9.) 56

10.) -5

Average

Average

1.) 1 or 4/3

2.) 1500

3.)

4.) 3/4

5.) 286

Difficult

Difficult

1.) 12

2.) -x +38

3.) 4 : 5

4.)

5.) 4035

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