SEAMO X 2020 — Paper D — Grade 7–8

Roy writes Pascal’s triangle, starting with row one containing one. The first five rows are shown. Find the sum of the numbers in row ten.

ABCD is a three-by-four rectangle with $AE=EF=FB=1$. Find the shaded area.

Nonzero real a,b,c satisfy $a+b+c=0$. Evaluate $x^{2020}-2020x$, where $x=\frac a{b+c}-\frac b{c+a}+\frac c{a+b}$.

An isosceles right triangle ABC encloses a semicircle of area $2\pi$. Its center O is on hypotenuse AB, and it is tangent to AC and BC. Find the triangle’s area.

How many odd integers between zero and one thousand have distinct digits?

Find the last two digits of $3^{2020}$.

Evaluate $\frac{2020^3-1}{2020\cdot2021+1}$.

Let $x=\cfrac1{1+\cfrac1{1+\cfrac1{1+\cdots}}}$. Find $x^2+x$.

If $\alpha,\beta$ are roots of $x^2+200x+20=0$, evaluate $\alpha/\beta+\beta/\alpha$.

Let $a=-\sqrt2+\sqrt3+\sqrt5$, $b=\sqrt2-\sqrt3+\sqrt5$, $c=\sqrt2+\sqrt3-\sqrt5$. Evaluate $a^2+b^2+c^2+ab+bc+ca$.

Cut an isosceles right triangle from each corner of a square of side two to form a regular octagon. Its area is $m+n\sqrt p$, where m,n,p are integers and p is minimum. Find m+n+p.

Choose positive integers a,b independently and uniformly from one through one hundred. If $N=2^a+3^b$ is divisible by five with probability m/n in lowest terms, find m+n.

The regular hexagon shown has area R. Its shaded region has area one. Find R.

Round one has the numbers one and two around a circle. In each next round, insert the sum of every adjacent pair between that pair, retaining all old numbers. Find the sum after round five.

Find the remainder when $1^{2020}+2^{2020}+\cdots+2020^{2020}$ is divided by ten.