# 2016 Grade 8 Math Challenge – Elimination Round with answer key – Part I

Below are the 2016 Grade 8 Math Challenge – Elimination Round questions and answers – Part I. More reviewers can be found on the Past Tests and All Posts pages.

1.) Simplify: $(a - b)^2 (a + b)^2 + 2a^2b^2$

2.) Simplify: $\left( \dfrac {125x^4y^3}{27x^{-2y^6}} \right)^\frac{1}{3}$

3.) Solve for x in the equation $x^4 - 5x^2 + 4 = 0$.

4.) In the arithmetic sequence $10 + 10\sqrt{3}, 11 + 9\sqrt{3}, 12 + 8\sqrt{3}, ... ,$ what term has no $\sqrt{3}$?

5.) If $x + y = 12$ and $xy = 50$, what is $x^2 + y^2$?

6.) What is the sum of the first ten terms of the geometric sequence 4, 8, 16, …?

7.) If the product of two consecutive odd integers is 783, what is the sum of theirs squares?

8.) If r and s are the roots of the equations $2x^2 - 3x + 4 = 0$, what is $4r^2 + 7rs + 4s^2$?

9.) A long wire is cut into three smaller pieces in the ratio of 7 : 3 : 2. If the shortest piece is 16 cm, what is the area of the largest rectangle that can be created using the longest piece?

10.) A boat takes $\dfrac{2}{3}$ as much to travel downstream as to its return. If the rate of the river’s current is 8 kph, what is the rate of the boat in still water?

11.) How many prime numbers between 40 and 240 ends with 4?

12.) Simplify: $x(1 - 6x) - (1 - 2x)(3x - 2)$

13.) What is the last digit of $7^{2016}$?

14.) If a = 3 and b = 7, what is $4a^3b + 6a^2b^2 + 4ab^3$?

15.) What is the median of the numbers $a + 1, a + 3, a - 2, a + 5 and a - 4$?

16.) A rectangle is formed by putting two squares side by side. If each square has perimeter 28 cm., what is the perimeter of the rectangle?

17.) Solve for x: $4(1 - 3x) - 2x(1 - 3x) + 5(1 - 3x) + 3x(1 - 3x) = 0$

18.) A jacket was worth Php 1,200. Hoping to gain more profit, the shop owner increased its price by 10%, but was later forced to reduce it by 15% since there were no takers. What was the final price of the jacket?

19.) Let $(a_n)$ be an arithmetic sequence. If $a_4 = 27$ and $a_9 = 67$, what is $a_1$?

20.) Triangle ABC is an isosceles with AB = AC. Let D be the foot of the altitude from A on BC, and let E be the point on side AC such that DE bisects $\angle ADC$. If $\angle DEC = 67^\circ$, what is $\angle BAC$?

21.) What is the greatest integer less than or equal to $(2 + \sqrt{3})^2$

22.) Each week, a pet owner buys m kilograms of bananas for its monkeys. If each monkey eats n kilograms of bananas each day, how many monkeys does the pet owner have? Express your answer in terms of m and n.

23.) If $8.07^3 = 525.557943$, what is $0.807^3$?

24.) Solve for x: $3 < |1 + 2x|$

25.) What is the value of $\dfrac {1}{3} + \dfrac {1}{15} + \dfrac{1}{35} + \dfrac {1}{63} + \dfrac {1}{99}$?

Answer key:

1.) $a^4 + b^4$
2.) $\dfrac{5x^2}{3y}$
3.) -2, -1, 1, 2
4.) 20 or 11th term
5.) 44
6.) 4092
7.) 1570
8.) 7
9.) $196 cm^2$
10.) 40 kph
11.) 0
12.) $-6x + 2$
13.) 1
14.) 7518
15.) a + 1
16.) 42 cm
17.) $x = -9$ or $x = \frac{1}{3}$
18.) Php 1,112
19.) 3
20.) $44^\circ$
21.) 13
22.) $\dfrac {m}{7n}$
23.) 0.525557943
24.) $x > 1$ or $x < -1$
25.) $\dfrac {5}{11}$

View Part II here: 2016 Grade 8 Math Challenge – Elimination Round with answer key – Part II