PIMSO 2021 - Mathematics - Grade 9 - National Round

What is the remainder when $\frac{8^{5^{43}}}{2^{36}}$ is divided by 9?

The highest power of 8 in 90! is n. The highest power of 25 in 90! is m. What is $\frac mn$?

What is the last two digits of $41^{126}\times81^{99}$?

How many numbers from 1 to 1500 are relatively prime to both 8 and 15?

There are n ordered pairs (a, b) that satisfy the equation ab=6851. If a and b are integers, what is n?

How many divisors of 10,125 are perfect cubes?

Count positive divisors.

How many different measures of sides are there in a square whose area is between $5^6$ and $8^{10}$?

Editorial caveat: the source does not restrict side lengths to integers. Explain the resulting ambiguity; manual grading.

If A = {p, i, m, s, o} and B = {1, 2, 3, 4, 5, 6, 7, 8, 9} then D = A × B = {p, i, m, s, o} × {1, 2, 3, 4, 5, 6, 7, 8, 9}. How many subsets of D are there? Express your answer in exponential form.

Input format: use ^ for powers, sqrt(...) for square roots, * for multiplication, and / for division.

If $b\lt a\lt0$, $c>0$, and $|c|>|b|>|a|$, then simplify $2|a+b|+|c+b|+|c+a|$?

Input format: use ^ for powers, sqrt(...) for square roots, * for multiplication, and / for division.

If (153x + 765y) is a square number and both x and y are positive integers, what is the minimum value of (x + y)?

In the equation 15x + 16y + 17z = 208, where x, y, and z are integers, find the value of 4(x + y + z).

Editorial caveat: the stated integer condition may not determine one value. Explain; manual grading.

Find the sum of the coefficients of the polynomial $(4x^3-7x+12)(11+5x-10x^2)^2(5x-2)^3$.

What is the remainder when $(x+5)^{2019}+(x+4)^{2021}$ is divided by $(x^2+9x+20)$?

Input format: use ^ for powers, sqrt(...) for square roots, * for multiplication, and / for division.

Let $f(x)=x+1$ and $g(x)=x^2$. Find $f(g(x))g(f(x))$.

Input format: use ^ for powers, sqrt(...) for square roots, * for multiplication, and / for division.

Find the sum of the numbers in the series below.

Source defect: the series is absent from the supplied PDF. No terms have been reconstructed. Manual editorial review is required.

Simplify $S=\frac1{10}+\frac2{10^2}+\frac3{10^3}+\frac4{10^4}+\frac5{10^5}$.

The area of triangle SPE is 20 cm², and the measures of segments SH, HE, PH, and TH are 5 cm, 3 cm, 4 cm, and 2 cm, respectively. What is the area of triangle THE?



Editorial caveat: the lengths and stated area are inconsistent. Discuss the inconsistency and any conditional calculation; manual grading.

The diagonal length AB of a cube is $\sqrt{300}$ cm. How many liters is the volume of the cube?

The perimeter of a triangle formed by points A(1,2), B(3,-4), and C(-2,-3) is in the form $x\sqrt{5y}+\sqrt{17y}+\sqrt{2z}$. What is x + y + z?

Convention: match the reduced positive-radical lengths AB, AC, BC in the displayed order, using positive integer parameters.

Angle TSA measures 60°. What is $(\sin\theta+\cos\theta)^2$?



Input format: use ^ for powers, sqrt(...) for square roots, * for multiplication, and / for division.