# 2018 Grade 10 Math Challenge Questions 1-25

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 $\sqrt{7} + \sqrt{5}$ is $\sqrt{7} - \sqrt{5}$. Find the number.

2.) Find $n$ if$27.63%$ of $349$ is $latex 2.763%4 of $n$. 3.) Simplify: $\dfrac{101!}{99!}$. 4.) If $3^n = 90$, find the integer closest to $n$. 5.) Find the largest positive integer $n$ such that $(n - 18)^{4036} \leq 99^{2018}.$ 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 $49 \pi$ sq cm. 11.) Sets $A$ and $B$ are subsets of a set having $20$ elements. If set $A$ has $134$ elements, at least how many elements should set $B$ have to guarantee that $A$ and $B$ are not disjoint? 12.) If the 2nd and 5th terms of an arithmetic sequence are $-2$ and $7$, respectively, Find the 101st term. 13) Find the sum of the first 71 terms of the arithmetic sequence above. 14.) Find $k$ such that $k - 2$, $2k + 2$, and $10k + 2$ are consecutive terms of a geometric sequence. 15.) Three numbers form an arithmetic sequence with common difference $15$. If the first is doubled, the second is unchanged, and the third is increased by $21$, 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 $\frac{196}{3}$ , find the sum of their cubes. 17.) A sequence $a_n$ has first two terms $a_1 = 3$ and $a_2 = 2$. For every 𝑛 ≥ 3, $a_n$ is the sum of all the preceding terms of the sequence. Find $a_{12}$ 18.) Let $r$ and $s$ be the roots of $x^2 - 9x + 7$. Find$latex $(r+1)(s + 1)$

19.) If $P(x + 2) = 3x^2 + 5x + 4$, find $P(x)$.

20.) Find the remainder when$P(x) = 27x^3 + 81x^2 + 5x + 20$ is divided by $x + 3$.

21.) Find the constant term of the quotient when $p(x) = 6x^4 - 8x^3 + x^2 + 9x + 7$ is divided by $x + \frac{2}{3}$.

22.) Find the constant $k$ if $x + 3$ is a factor of $f(x) = 2x^4 - 8x^3 + (k^2 + 1)x^2 + kx + 15$.

23.) Find the largest real root $x$ of $2x^3 - 5x^2 - 2x + 2 = 0$.

24.0 If  $p(x)$ is a 3rd degree polynomial,  $p(\frac{5}{3}) = p(4) = p(-5) = 0$, and $p(0) = -200$, find $p(2)$.

25.) The midpoint o f $P(1,2)$ and $A$ is $Q$; the midpoint of $P$ and $C$ is $S$; the midpoint of $Q$ and $S$ is $R(5, 1.5)$. Find the coordinates of $B$, the midpoint of $A$ and $C$.