SEAMO 2016 — Upper Secondary — Paper E

Given $a$, $b$ and $c$ are three consecutive even numbers and $b^3=1728$. Find the value of $ac$.

Find the simplest expression for $\dfrac{3^{40}}{9^{20}}$.

Find the possible value(s) of $m$ in $$\frac1{m-2}+\frac1{m+1}-\frac12=0.$$

Find the value of $12345^2-12344^2$.

Given $\overline{aabb}$ is a square number, find the value of $(a+b)$.

$ABC$ is an equilateral triangle of side $6$ cm. It first rotates about $B$, then $C_1$, without sliding. As a result, Vertex $A$ travels from $A$ to $A_1$, then comes to rest at $A_2$. Find the length, in cm, of the path travelled by vertex $A$, in terms of $\pi$.

The figure shows a can of height $8$ cm. The circumference of its base is $44$ cm. Find the shortest distance from $A$ to $B$, without cutting through the can.

In triangle $ABC$, $\angle ABC=90^\circ$, $AB=7$ cm and $BC=24$ cm. $P$ is a point inside the triangle such that its shortest distance to each side of the triangle is the same. What is this distance?

$M$ and $N$ are midpoints of $AD$ and $AB$, respectively, in rectangle $ABCD$. Lines $BM$ and $DN$ intersect at point $P$. Find $\angle MPD$, given that $\angle MBC=\angle BCN+31^\circ$.

Suppose $m^2+1=3m$ and $n^2+1=3n$, then the value of $\dfrac1{m^2}+\dfrac1{n^2}$ is _____.

Find the value of $$\sum_{i=1}^{5}3i-\sum_{i=1}^{5}(2i-3).$$

In the figure below, $BA$ and $BC$ are tangents to the circle with centre $O$. Suppose $EC=3$ cm and $DA=2$ cm, find $BC$ in cm.

Find the remainder when $55^{2015}+17$ is divided by $8$.

The sum of a positive integer and 100 is a square number. The sum of same positive integer and 168 is also a square number. Find the integer.

The area of rectangle $ABCD$ is $35\text{ cm}^2$. Points $E$ and $F$ divide side $AB$ into $3$ equal segments. $G$ is the midpoint of side $CD$. Find the area of green triangle $GHI$ in $\text{cm}^2$.

Given that $m+\dfrac1m=4$, find the value of $m^4+\dfrac1{m^4}$.

Find the integer part of
$$\frac{1}{\frac1{2010}+\frac1{2011}+\frac1{2012}+\frac1{2013}+\frac1{2014}+\frac1{2015}+\frac1{2016}}.$$

An integer is chosen from the set $\{1,2,3,\ldots,499,500\}$. The probability that this integer is divisible by $9$ or $11$ is $\dfrac mn$, in its lowest terms. Find the value of $m+n$.

It is given $x^2+2x=3$. Find the value of
$$x^4+7x^3+8x^2-13x+12.$$

The circumference of a circular lake is $30\text{ km}$. Paul and Mary cycle around the lake in opposite directions, both starting from the same point.



Paul's speed is $250\text{ m/min}$, but he rests for $5$ minutes after every $55$ minutes of cycling. Mary's speed is $200\text{ m/min}$, but she rests for $10$ minutes after every $1\text{ h}$ of cycling. Find the time taken, in hours, for them to meet for the first time.