SEAMO 2018 — Paper E — Grade 9–10

Arrange $4^{392},6^{294},5^{196}$ in ascending order.

Georgetown and Greenville are $120\,\mathrm{km}$ apart. An SUV and a sedan leave Georgetown for Greenville simultaneously. After one hour, the SUV is $24\,\mathrm{km}$ ahead. The sedan arrives $25$ minutes after the SUV. Find the sedan’s speed.

Given $x^2-8x+1=0$, find $x^4+\frac1{x^4}$.

Find the last two digits of $\underbrace{18^{2018}+18^{2018}+\cdots+18^{2018}}_{8\text{ terms}}$.

$P,Q$ are the centres of the circles shown. Chord $AB$ is tangent to the circle centred at $P$, $PQ\parallel AB$ and $AB=24\,\mathrm{cm}$. Find the shaded area in $\mathrm{cm}^2$.

Neha adds the page numbers $1,2,3,\ldots,n$ of a book and obtains $2018$. She then notices one number was added twice. Find that number.

Find a four-digit perfect square $\overline{aabb}$, where $a\ne b$.

Goldbach’s conjecture suggests every even integer greater than $2$ is a sum of two primes. Write $186$ as such a sum. What is the greatest possible product of the two primes?

Evaluate $\frac56+\frac{23}{24}+\frac{59}{60}+\frac{119}{120}+\frac{209}{210}$.

In trapezium $ABCD$, $AD\parallel BC$, $AB\perp BC$, $AB=BC$ and $AE=AD$. Point $E$ lies on $AB$ and $\angle BEC=75^\circ$. Find $\angle DCE$.

Find the remainder when $2^{2018}$ is divided by $33$.

Find $x$ if $\log\frac{13}{19}+\log\frac{95}{91}-\log\frac57+\log5=\log x$.

$A,B,C,D$ are concyclic. $AC$ is a diameter, $AD=DC$, and $DE\perp AB$ as shown. The area of $ABCD$ is $20\,\mathrm{cm}^2$. Find $DE$.

The groups are $(1),(2,3),(4,5,6),(7,8,9,10),\ldots$. Find the sum of the numbers in the $n$th group.

Given $a+2b+c=4$ for $a,b,c\in\mathbb R$, find the maximum of $ab+bc+ca$.

In $\triangle ABC$, $9a^2+9b^2=19c^2$. Find $\frac{\cot C}{\cot A+\cot B}$.

Barrel $A$ contains $11$ litres of alcohol and barrel $B$ contains $15$ litres of water. Some alcohol is poured from $A$ to $B$, then some well-mixed liquid is poured back from $B$ to $A$. The final alcohol percentages are $62.5\%$ in $A$ and $25\%$ in $B$. How many litres are poured back?

How many real solutions does the system $x+y=2$, $xy-z^2=1$ have?

Two fair dice are thrown. Which statements about the absolute difference are true? (i) Probability of $2$ is $\frac29$. (ii) Probability of $3$ is $\frac16$. (iii) Probability of at least $2$ is $\frac13$. (iv) Probability of at most $4$ is $\frac79$.

If $3x^2+x=1$, find $6x^3-x^2-3x+2018$.