# 2019 Grade 10 Math Challenge – Division Level with answer key

Below are the 2019 MTAP Grade 10 Math Challenge Division level questions and answers. Solutions will be posted later. More reviewers can be found on the Past Tests and All Posts pages.

Easy

1.) Find the solution set of the inequality $x(x + 1) < x.$

2.) A circle, centered at the origin, passes through the points (-2, 3) and (3, k). What is k?

3.) The 4th and 10th terms of an arithmetic sequence are 48 and 68. What is the 1st term?

4.) If -1 is a root of the equation $2x^4 + 2x^3 = 3x^2 + kx + 2$, what is k?

5.) A 3 cm chord is 3 cm away from the center of the circle. What is the exact area of the circle?

6.) In an ordinary deck of playing cards, how many five-card hands of consecutive ranks in one suit are possible?

7.) A square is inscribed in a right triangle with two of its sides lying on the legs of the triangle. if the legs of t he triangle are 6 and 12 cm long, what is the area of the square?

8.) In rolling two dice at random, what is the probability of obtaining a sum that is a perfect square?

9.) The sides of an isosceles triangle are 5, 5 and 6 cm long. How long is the altitude from one base vertex to the opposite leg?

10.) Find the sum of all even positive integers less than 1000.

11.) What is the minimum value of the function f(x) = $2x^2 - 4x + 5$?

Average

1.) How many different code words can be formed from the letters DIVISION such that S and N are next to each other?

2.) In the sequence 25, a, b, c, $\frac {25}{49}$, the 1st, 3rd, and 5th terms form a harmonic sequence and the last 3 terms form a geometric sequence. What is c?

3.) Solve for x in $3x^4 + 4x^3 = 5x^2 + 2x$.

4.) The dimensions of a wooden rectangular prism are 5, 7, and 8 units. Its faces are painted blue, and then the prism is cut into unit cubes. If two cubes are selected at random, what is the probability that one has exactly one blue face and the other has exactly two blue faces?

5.) A circle passes through the points (1, 3), (2, -2), and (6, 4). What is its radius?

6.) The point (k, 7) lies on the perpendicular bisector of the segment with endpoints (-1, 2) and (2, 9). What is k?

Difficult

1.) The numbers 3, a, b, c, 23 328 for a geometric sequence. What is $\sqrt {abcd}$?

2.) If $23 + 7x - 5x^2 - 2x^3 = a + b(x + 2) + c(x + 2)^2 + d(x + 2)^3$ is an identity, what is c?

3.) The lines $3y = 2x + 3$ and $2x = 3y + 3$ are parallel. Find the distance between those lines?

4.) An arithmetic sequence ${a_n}$ has ${a_4} = 6$ and ${a_7} = 4$. What is $n so that {a_1} + {a_2} + ... + {a_n} = 42$ ?

5.) The hypotenuse AC of right triangle ABC is trisected at P and Q. If $BP^2 + BQ^2 = 10cm^2$, how long is AC?

6.) The lengths (in cm) of the sides of a triangle are the roots of the equations $x^3 + 84x = 16x^2 + 144$. Find the area of the triangle.

Tiebreaker

1.) If $\sqrt {x}$ is between 6 and 7, between what two consecutive integers is $\sqrt {x}$?

2.) How many sides does a convex polygon have if it has 10 times as many diagonals as sides?

3.) Find all points with integral coordinates that are 5 units away from the point (-3, -2).

Do-or-Die

1.) The zeroes of a polynomial $P(x) = ax^4 + bx^3 + cx^2 + e$ are $-\frac {3}{5}, \frac {1}{3},$ 2, and 3. If $P(1) = 32,$ what is c?

Easy
1.) $\varnothing$
2.) $\pm 2$
3.) 38
4.) 5
5.) $\dfrac {45 \pi}{4} cm^2$
6.) 36
7.) $16 cm^2$
8.) $\dfrac {7}{36}$
9.) $\dfrac {24}{5} cm$
10.) 249 500
11.) 3

Average
1.) 1680
2.) $\pm \dfrac {5}{7}$
3.) $-2, -\dfrac {1}{3}, 0, 1$
4.) $\dfrac {28}{155}$
5.) $\sqrt {13}$ units
6.) -3

Difficult
1.) $108 \sqrt {6}$
2.) 7
3.) $\dfrac {6\sqrt {13}}{3}$ units
4.) 7, 18
5.) $3 \sqrt {2} cm$
6.) $8 \sqrt {2} cm^2$ units

Tiebreaker
1.) 3 and 4
2.) 23
3.) (-8, -2), (-7, -5), (-7, 1), (-6, -6), (-6, 2), (-3, -7), (-3, 3), (0, -6), (0, 2), (2, -2), (1, -5), (1, 1)

Do-or-Die
1.) 104