PIMSO 2021 - Mathematics - Grade 8 - National Round Mock Test

What is the units digit of the simplest form of $\frac{27^{120}}{3^{36}}$?

What is the highest power of 200 in 90!?

What is the two rightmost digits of $8^{581}$?

How many numbers from 1 to 1000 are relatively prime to either 6 or 35 (but not both)?

There are m ordered pairs (a, b) that satisfy the equation ab=728. If a and b are integers, what is m?

How many numbers that divides 81,000 are perfect square?

X, Y, and Z are prime numbers and $X\lt Y\lt Z$. What is the value of Y in 3X + 2Y + 5Z = 143?

If $x=\sqrt{14+6\sqrt5}$, determine the value of $\frac{x^2+5x}{x^2+2x-15}$.

Simplify $(x^{64}+1)(x^{32}+1)(x^{16}+1)\cdots(x^4+1)(x^2+1)(x+1)$ and find its value when x=2. Express your answer in exponential form.

If x is a real number, so that $2x^2-x-2=0$, find the value of $x^8+\frac1{x^8}$.

If $(a+b)^4+(a-b)^4=4177$ and $a^2-b^2=24$, where a and b are both real numbers, find the value of $(a^2+b^2)$.

Given that the minimum value of $f(x)=3x^2-12x+19$ is F and the maximum value of $g(x)=-4x^2-40x-115$ is G, find the value of F-G.

The equation $(3-x)^3+(4-x)^3=(7-2x)^3$ has roots $x_1,x_2,x_3$. What is the value of $(x_1+x_2+x_3)$?

RHST is a rectangle with length SH = 15 m, and PHIL is a square with side 12 m. The vertex S of the rectangle lies on side LI of the square the vertex P of the square lies on side TR of the rectangle. Find the area of the rectangle.

Find the value of $\frac{\tan35^\circ\tan70^\circ\tan75^\circ}{\tan35^\circ+\tan70^\circ+\tan75^\circ}$.

In rectangle PHIL, R is the midpoint of PH and S is the midpoint of LR. The area of triangle PSI is 30 cm². Find the area of the rectangle.

The quadrilateral PHIL below is a rectangle. B is the midpoint of side PH. The areas of triangles HAG and CID are 60 cm² and 20 cm², respectively. The areas of quadrilaterals FECH and HEDL are 420 cm² and 480 cm², respectively. Find the area of triangle BFG.

Find the maximum value of $4\cos(x+\frac{7\pi}{24})+5\sin(x+\frac\pi8)$.

In the figure shown below, lines PI, PL, and PS and straight lines. HO ∥ IL and MO ∥ SL. The area of triangles MOP and POH are 10 cm² and 20 cm², respectively. PH = 4cm and HI = 3cm. Find the sum of the areas of quadrilaterals SMOL and HILO. Express your answer in three decimal places.

Three rhombus parquets are used to decorate a hall. These are glued edge to edge to form different shapes as in the given figure. How many distinct shapes can be formed? (Shapes that look a reflection or rotation of certain shape are not considered distinct.)