PIMSO 2021 - Mathematics - Grade 10 - National Round

What is the remainder when $\frac{81^{675}}{9^{45}}$ is divided by 28?

The highest power of 200 in 90! is n. The highest power of 9 in 30! is m. What is $\frac mn$?

Here “highest power” means its exponent.

What is the last two digits of $8^{581}\times71^{99}$?

How many numbers from 1 to 1500 are relatively prime to either 8 or 15 (but not both)?

There are n ordered pairs (a, b) that satisfy the equation ab=27404. If a and b are integers, what is n?

How many numbers that divides $3.24\times10^5$ are perfect square?

If $R_n=\sum_{k=1}^n\frac{2k+1}{k^4+2k^3+k^2}$, find $R_{10}$.

If $x=\sqrt{14+6\sqrt5}$, determine the value of $\frac{x^4-2x^3+3x^2+9x+10}{x^2-7x+12}$.

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

Simplify $(x^{2048}+1)(x^{1024}+1)(x^{512}+1)\ldots(x^4+1)(x^2+1)(x+1)$ and find its value when x=2. Express your answer in exponential form.

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

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

What is the numerical coefficient of the middle term of the binomial expansion $(2x+3y)^{10}$?

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

The diagonal length AB of a cube is $\sqrt{243}$ cm. What is the surface area of the cube?

What is the area of the biggest triangle that fits inside rectangle SPRT given that ST=10, SE=12, ER=13, and $TE=12\sqrt2$?

Editorial note: the printed data and diagram require manual review. Explain whether they determine a valid figure.

Angle TSA measures $60^\circ$. What is the area of quadrilateral SAOT if the radius of the circle is 2 cm?

Editorial note: justify whether the printed information determines a unique area. No unmarked symmetry may be assumed.

The radius of the circle A with the equation $x^2+y^2+4x+6y-12=0$ is the side of the equilateral triangle ABC. What is the area of triangle ABC?

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

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

Use the usual simplified-radical representation with positive integer coefficients and squarefree radicands.

The diagonal length AB of a prism is $\sqrt{299}$ cm. What is the volume of the box in Liters if the length and width are 13 and 7 respectively?

Use centimeters for all box dimensions, as in the diagonal measurement.

In a certain code system, ‘PIMSO’ is written as ‘OHLRN’. How can ‘ROBERT’ be written in that code?

A resort in Boracay has rooms that can accommodate up to two people. Couples can share a room. Male guests can only share with males and female guests only with females. How many rooms are needed to ensure that any group of 200 guests can be accommodated?