SEAMO 2019 — Paper E — Grade 9–10

Evaluate $\frac{4^8+4^8+4^8+4^8}{3^8+3^8+3^8}\times\frac{6^8+6^8+6^8+6^8+6^8+6^8}{2^8+2^8}$.

How many pairs of positive integers $(a,b)$ satisfy $a^2-b^2=361$?

Given $(62)_n=4(14)_n$, find $(36)_n$ in base ten.

How many integer pairs $(m,n)$ satisfy $m^3+6m^2+5m=21n^3+9n^2+3n+1$?

Find the units digit of $(2+1)(2^2+1)(2^4+1)(2^8+1)(2^{16}+1)(2^{32}+1)(2^{64}+1)$.

Given $\log_{10}2=m$ and $\log_{10}3=n$, express $\log_5 24$ in terms of $m,n$.

Given $0\le\theta\le\pi/2$ and $\sin2\theta=a$, find $\sin\theta+\cos\theta$.

In $\triangle ABC$, $AC=3AD$, $BC=3EC$, and $BG:GF:FD=3:1:2$ along $BD$. If $[\triangle ABC]=2700\,\mathrm{cm}^2$, find $[\triangle EFG]$.

How many paths spell “SEAMO”, starting from the central S and moving only up, down, left or right? A sample path is shaded in the figure.

Each of the eight corners of a rectangular prism is cut off. Two cuts are shown. The cutting planes do not intersect anywhere in or on the prism. How many edges does the resulting solid have?

A palindrome is a positive integer reading the same forwards and backwards, such as $12521$ or $270072$. A palindrome between $1000$ and $10000$ is chosen uniformly at random. Find the probability that it is divisible by $7$.

Evaluate $\sqrt{1+2019^2-\frac{2019^2}{2020^2}}-\frac1{2020}$.

For how many positive integers $n$ does $1+2+\cdots+n$ divide $12n$?

Square $ABCD$ has side $10\,\mathrm{cm}$. Points $E,F$ vary on $BC,CD$ with $BE=CF$. Find the smallest area of $\triangle AEF$ in $\mathrm{cm}^2$.

An eight-digit integer $m=\overline{2x192y20}$ is divisible by $41$. How many digit pairs $(x,y)$ are possible?

A badminton tournament has $100$ players. A loser is eliminated and the winner advances; there are no draws. In Round $1$, $28$ players receive a bye and the remaining $72$ are paired. Play continues until one player is unbeaten. How many matches are played in total?

An odometer displays the total kilometres a car has travelled. Leonard drives for a positive integer number of hours at average speed $55\,\mathrm{km/h}$. The reading changes from $\overline{abc}$ to $\overline{cba}$. Given $a\ge1$ and $a+b+c\le7$, find $a^3+b^3+c^3$.

In a triangle the sides $a,b,c$ are opposite angles $A,B,C$. Given $a^2=1$ and $b^2+c^2=2019$, find $\frac{\cot A}{\cot B+\cot C}$.

For $i=1,\ldots,100$, let $x_i$ be either $\sqrt3-\sqrt2$ or $\sqrt3+\sqrt2$. How many distinct positive integer values can $S=x_1x_2+x_3x_4+\cdots+x_{99}x_{100}$ take?

Thirteen identical circles of radius $1$ lie inside a larger circle as shown. Every intersection is a tangency. Find the shaded area.