SEAMO 2020 — Paper E — Grade 9–10

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

Positive real numbers $a,b$ satisfy $(a+b)^2=400$ and $ab=50$. Find $a^3+b^3$.

Following the directions of the arrows in the figure, how many paths lead from $A$ to $B$?

In $\triangle ABC$, cevians $BM,CN$ meet at $P$. Given $[CMP]=3$, $[BNP]=4$, $[BCP]=5$, find $[ABC]$.

A semicircle is inscribed in an equilateral triangle, which is inscribed in a larger semicircle as shown. Find the ratio of the large radius to the small radius.

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

Positive real numbers $a,b,c$ satisfy $abc(a+b+c)=100$. Find the minimum 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 of $f(x)=\frac{9x^2\sin^2x+4}{x\sin x}$ for $0\lt x\lt \pi$.

The monic cubic $p(x)=x^3+ax^2+bx+c$ has real roots $x_1,x_2,x_3$. Given $a+c=10$, $b=5$, find $(x_1^2-1)(x_2^2-1)(x_3^2-1)$.

Two people independently go to the gym at uniformly random arrival times within the same three-hour interval. Each stays for $1.5$ hours. What is the probability that they meet?

The figure arranges $1$ through $9$ row by row in a $3\times3$ square, with diagonal sum $15$. Construct the analogous array of $1$ through $100$. What is its diagonal sum?

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

Amos, Bryan and Catherine play one-on-one chess games. Two play while the third watches; the winner stays to face the watcher and the loser sits out. Amos plays $7$ games, Bryan $10$, Catherine $13$. Who loses the second game?

Five connected circles are shown. Using four distinct colours, how many colourings make every pair of connected circles different in colour?

Circles $C_1,C_2$ of different sizes are internally tangent. Given $AB=18$ and $CD=10$ in the figure, find the shaded area.

Point $P$ lies inside equilateral $\triangle ABC$, with $\angle APB=120^\circ$ and $\angle BPC=130^\circ$. A new triangle has side lengths $AP,BP,CP$. Find its largest angle.

Real $a,b$ satisfy $ab\ne1$, $3a^2-4a-5=0$, $5b^2+4b-3=0$. Evaluate $\frac{(a+1)(b+1)}b$.

Twenty-five marketing specialists sit around a round table. A team of three is chosen uniformly at random. The probability that at least one pair in the team sits next to each other is $m/n$ in lowest terms. Find $m+n$.