PIMSO 2025 - Mathematics - Secondary 4 - 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 $m/n$?

Let $f(x)=x-1$ and $g(x)=x^2$, if $f(g(x))g(f(x))=0$ find the sum of possible value of $x$.

The prime factorization of the product of factors of 15 is $3^2\times5^2$. If the prime factorization of the product of factors of 180 is $p_1^x\times p_2^y\times p_3^z$. What is $x+y+z$?

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

If $\{x\}$ denoted the fractional part of $x$, then $\left\{\frac{3^{2026}}8\right\}=$

The remainder obtained when 1! + 2! + 3! + ⋯ + 97! + 98! + 99! Is divided by 15, is

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

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

The diagonal length AB of a cube is $\sqrt{432}$ cm. What is the surface area of the cube?

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



Original choices: A. $2\sqrt2$ cm²; B. $2\sqrt3$ cm²; C. $3\sqrt2$ cm²; D. $3\sqrt3$ cm²

Editorial note: Explain whether the stated information determines a unique area. No automatic answer is assigned.

A sequence $x_n$ is defined by $x_1=3$ and $x_{n+1}=\frac{x_n}{1+nx_n}$ for all $n\geq1$. Then $x_{15}$ is

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?

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 $x+y\times z-s$?

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?

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

A cruise ship has cabins that can hold up to two passengers. Couples can share a cabin. Male passengers can only share cabins with other males, and female passengers can only share cabins with other females. Male staff can only share cabins with other males, and female staff can only share cabins with other females. The ship wants to be prepared to accommodate any group of 300 passengers and 8 staff under these rules. Staff and passengers cannot share cabins with each other. Additionally, one cabin must be reserved for the ship’s captain. How many cabins should the ship have in total to ensure everyone can be accommodated?

James plans to decorate his garden with flowers. A circular fountain has the equation $x^2+y^2-24x-32y+175=0$ and flowers are planted in a 3 units wide ring around the fountain. Find the ratio of the fountain area to flower area in term of $\pi$?