This is the 2018 MTAP Grade 10 Math Challenge questions 1 to 25. Questions 26 to 50 (including the pdf of Questions 1-50) can be found **here**. Solutions and answers will be posted later. More reviewers can be found on the **Past Tests** and **All Posts** pages.

**Questions**

1.) The average of a number and is . Find the number.

2.) Find if of is $latex 2.763%4 of .

3.) Simplify:

.

4.) If , find the integer closest to .

5.) Find the largest positive integer such that

6.) How many integers between 60 and 600(inclusive) are divisible by 2 or by 3?

7.) How many positive factors does the product of 3 different prime numbers have?

8.) A triangle has area 30 sq cm, and two sides are 5 cm and 12 cm long. How long is the third side?

9.) If 𝐴 = {all prime numbers between 1 and 100}, 𝐵 = {2,4,6,…,98,100}, and 𝐶 = {3,6,9, … , 96, 99}, how many elements does (𝐴 ∪ 𝐵) ∩ 𝐶 have?

10.) Find the perimeter of a regular hexagon inscribed inside a circle whose area is sq cm.

11.) Sets and are subsets of a set having elements. If set has elements, at least how many elements should set have to guarantee that and are not disjoint?

12.) If the 2nd and 5th terms of an arithmetic sequence are and , respectively, Find the 101st term.

13) Find the sum of the first 71 terms of the arithmetic sequence above.

14.) Find such that , , and are consecutive terms of a geometric sequence.

15.) Three numbers form an arithmetic sequence with common difference . If the first is doubled, the second is unchanged, and the third is increased by , a geometric sequence will be formed. Find the first number of the geometric sequence.

16.) The sum of an infinite geometric series have sum 14. If their squares have sum , find the sum of their cubes.

17.) A sequence has first two terms and . For every 𝑛 ≥ 3, is the sum of all the preceding terms of the sequence. Find

18.) Let and be the roots of . Find $latex

19.) If , find .

20.) Find the remainder when is divided by .

21.) Find the constant term of the quotient when is divided by .

22.) Find the constant if is a factor of .

23.) Find the largest real root of .

24.0 If is a 3rd degree polynomial, , and , find .

25.) The midpoint o f and is ; the midpoint of and is ; the midpoint of and is . Find the coordinates of , the midpoint of and .