SEAMO 2026 Paper E SET 2 – Self-practice

Evaluate $\sqrt[3]{8+3\sqrt{21}}+\sqrt[3]{8-3\sqrt{21}}$.

Suppose a and b are positive real numbers such that $(a+b)^2=400$ and $ab=50$. Find the value of $a^3+b^3$.

Following the direction of the arrows in the figure below, how many ways are there to get from A to B?

In $\triangle ABC$, BM and CN intersect at P. Given the areas $[CMP]=3$, $[BNP]=4$ and $[BCP]=5$, find $[ABC]$.

In the figure below, a semicircle is inscribed in an equilateral triangle, which is inscribed in a larger semicircle. Find the ratio of the radius of the larger semicircle to that of the smaller semicircle.

Evaluate $\frac{\cos^4 15^\circ+\sin^4 15^\circ+2\sin^2 15^\circ\cos^2 15^\circ}{\cos^6 15^\circ+\sin^6 15^\circ+3\sin^2 15^\circ\cos^2 15^\circ}$.

Let p, q, and r be the roots of the equation $2x^3-5x^2-6x+2=0$. Evaluate $\frac1p+\frac1q+\frac1r$.

Suppose a, b and c are positive real numbers such that $abc(a+b+c)=100$. Find the minimum value of $(a+b)(b+c)$.

Evaluate $\frac{2020}{\sqrt1+\sqrt5}+\frac{2020}{\sqrt5+\sqrt9}+\frac{2020}{\sqrt9+\sqrt{13}}+\cdots+\frac{2020}{\sqrt{2021}+\sqrt{2025}}$.

Find the minimum value of $f(x)=\frac{9x^2\sin^2x+4}{x\sin x}$, $0<x<\pi$.

Let $p(x)=x^3+ax^2+bx+c$ be a third-degree polynomial with three roots $x_1,x_2,x_3$. Suppose c+a=10 and b=5. Find $(x_1^2-1)(x_2^2-1)(x_3^2-1)$.

In the same 3-hour interval, two people went to the gym at random times. Given that they each spent 1.5 hours working out there, what is the probability that they met?

Consider the following array, in which the integers 1 to 9 are placed as shown below. The diagonal sum of integers is 15. If we construct a similar array with integers from 1 to 100, what is the diagonal sum of integers?

Find the remainder when $2020!\left(1+\frac12+\frac13+\cdots+\frac1{2020}\right)$ is divided by 2021.

Amos, Bryan, and Catherine play a series of 1-on-1 chess games. At any one time, 2 people play while 1 watches. The winner continues playing against the next competitor while the loser switches out. Amos played 7 games in total. Bryan played 10 games in total. Catherine played 13 games in total. Who lost the second game?

The figure below shows 5 connected circles. Given 4 distinct colours, how many different ways are there to colour the circles such that no two connected circles are of the same colour?

Two circles C 1 and C 2 of different sizes are internally tangential. Given that AB=18 and CD=10, find the area of the shaded region.

Point P lies in equilateral triangle ABC. Suppose $\angle APB=120^\circ$ and $\angle BPC=130^\circ$. If segments AP, BP and CP were to form a new triangle, what would be its largest angle?

Given that a and b are real numbers, where $ab\ne1$, satisfying $3a^2-4a-5=0$ and $5b^2+4b-3=0$, evaluate $\frac{(a+1)(b+1)}b$.

During a round-table meeting, the CEO wanted to select a team of 3 out of 25 marketing specialists to handle a new project. If the probability that 2 of the selected specialists were sitting next to each other is $m/n$, where m and n are relatively prime, find m+n.