SEAMO 2017 — Paper E — Grade 9–10

Evaluate $\left(1-\frac1{2^2}\right)\left(1-\frac1{3^2}\right)\cdots\left(1-\frac1{2006^2}\right)\left(1-\frac1{2007^2}\right)$.

Given $y=(x+11)(x-13)(x-11)(x+13)$, find the minimum value of $y$.

When $1245,1794,2160$ are divided by a positive integer $x$, their remainders are all equal to another positive integer $y$. Find $x+y$.

The circle with centre $O_1$ touches a $120^\circ$ sector at $C,D,E$. Find its circumference, given that arc $\widehat{AB}$ has length $1$.

Four primes are written in ascending order. The product of the first three is $385$ and that of the last three is $1001$. Find the first number.

A bag has $b$ black and $a$ white marbles. The probability of picking white is $\frac3{10}$. After adding $10$ white marbles it becomes $\frac13$. Find $a+b$.

Neha skates from $A$ at $8\,\mathrm{m/s}$ along a ray making $60^\circ$ with $AB$. Harsha starts from $B$ simultaneously at $7\,\mathrm{m/s}$. In the figure $AB=100\,\mathrm m$. Find the shortest distance Harsha travels before meeting Neha.

It is known that $\cos Q=\frac{m^2-n^2}{m^2+n^2}$. Find $\sin Q$.

Simplify $\sqrt{21+12\sqrt3}-\sqrt{21-12\sqrt3}$.

Find the largest number $n$ such that exactly one integer $k$ satisfies $\frac8{15}\lt \frac n{n+k}\lt \frac7{13}$.

Given $m=\frac{\sqrt3-\sqrt2}{\sqrt3+\sqrt2}$ and $n=\frac{\sqrt3+\sqrt2}{\sqrt3-\sqrt2}$, find $\frac n{m^2}+\frac m{n^2}$.

For the first part of a journey, Group B walks while A takes the bus; for the second, A walks while B takes the bus. The bus speed is $40\,\mathrm{km/h}$ when loaded and $50\,\mathrm{km/h}$ when empty. Walking speed is $4\,\mathrm{km/h}$. Both groups arrive together. Find walking distance as a fraction of bus distance.

In a sequence of positive integers, every term after the first two is the mean of all preceding terms. If the $2559$th term is $2017$, how many different sequences are possible?

Four identical circles are circumscribed by a square of area $64\,\mathrm{cm}^2$. A smaller circle touches all four circles as shown. Find the smaller circle’s diameter.

Find the remainder when $1^5+2^5+\cdots+99^5+100^5$ is divided by $4$.

Evaluate $\frac1{a-b}+\frac1{a+b}+\frac{2a}{a^2+b^2}+\frac{4a^3}{a^4+b^4}$.

Let $n_1,n_2,\ldots,n_{2017}$ be positive integers and $x=(n_1+\cdots+n_{2016})(n_2+\cdots+n_{2017})$, $y=(n_1+\cdots+n_{2017})(n_2+\cdots+n_{2016})$. Which statement is true?

A small circle with radius $r$ and centre $D$ is tangent to three identical semicircles with centres $A,B,C$ and radius $R$ as shown. Find $R:r$.

$\alpha,\beta$ are roots of $x^2+mx+n=0$, with $\alpha\gt 1$, $-1\lt \beta\lt 1$, and integers $m,n$. Find $m,n$ when $\alpha^3+\beta^3\le4$.

Find $N$ in $\frac1{2!15!}+\frac1{3!14!}+\frac1{4!13!}+\frac1{5!12!}=\frac N{1!16!}$.