PIMSO 2025 - Mathematics - Secondary 2 - National Round

What is the unit digit of the simplest form of $\frac{27^{2025}}{3^{2026}}$?

What is the highest power of 20 in 90!?

What is the sum of roots of $x^2-ax+b=0$, where a and b are prime numbers and the roots are positive, and distinct and are elements of the set of integers?

The LCM of $(16^4)^2$, $(6^7)^4$, and m is $36^{16}$. How many possible values of m are there?

Take m to be a positive integer.

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 < Y < Z and X, Y and Z are consecutive prime number. What is the value of Z - X in 2X + 3Y + 4Z = 153?

If $x=\sqrt{9+4\sqrt5}$, determine the value of $\frac{x^2+2x}{x^2+x-6}$. Hint! Simplify x in the form $\sqrt a+\sqrt b$.

The sum of the digits of the number $100^{2025}-2025$, written in decimal notation is

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 $(x_1+x_2+x_3)$? Hint! $a^3+b^3=(a+b)^3-3ab(a+b)$.

Find the sum of all real numbers satisfying $6^x+2^{2x}+24^x-36^x-16^x=1$.

If $(a+b)^4+(a-b)^4=337$ and $a^2-b^2=8$, where a and b are both real numbers, find the value of $(a^2+b^2)$.

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.

The number of value of k for which the equation $x^3-3x+k=0$ has two distinct roots lying in the interval (0, 1) are

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

The In the circular sector below, PH and HL are straight lines that intersect at H on the circumference of circle O. Find ∠PHL.

A video game character must climb 10 levels to reach the top of a tower. At each move, the character can jump up either 1 level or 2 levels. In how many different ways can the character reach the top of the tower?

The lengths of the sides of triangle ABC are 10, 15, x, and the triangle is not isosceles. How many possible values of x?

The source omits a domain for x. For the printed finite-choice task, assume integer side lengths.

3 distinguishable balls are distributed to 3 boxes at random. Number of ways in which we set at most empty box is

The source omits “one”. Read “at most one empty box”; the boxes are distinct.