SEAMO X 2021 — Paper D — Grade 7–8

Find the remainder when $2020^{2020^{2020}}$ is divided by two thousand twenty-one.

How many positive perfect cubes divide $5\cdot5!\cdot8\cdot8!$?

Positive a,b,c satisfy $(a+b)(a+b+c)=18$, $(b+c)(a+b+c)=24$, $(c+a)(a+b+c)=30$. Evaluate $ab/c+bc/a+ca/b$.

How many positive integers n have $1+2+\cdots+n$ dividing $100n$?

A bag contains five red and three blue marbles. Remove them randomly without replacement until all marbles of one color have been removed. Find the probability that the final removed marble is blue.

Walking on a moving escalator, Terry reaches the next floor in fifty seconds. Walking at twice his speed takes forty seconds. How many seconds are needed if he stands still?

For what p is the sum of the squares of the roots of $x^2-px=100-p$ least?

Square ABCD has side eight. E is on AB with $AE=6$, and M varies on BD. Find the minimum of MA+ME.

In how many ways can SINGAPORE be divided into contiguous parts, each containing at least one vowel?

A positive integer n becomes its cube root when its three rightmost digits are removed. Find the sum of its digits.

Five distinct married couples sit around a circle, each couple together. How many arrangements are there?

Evaluate $\frac{1^4+99^4+100^4}{1^2+99^2+100^2}$.

A quadratic f satisfies $f(10)=193$ and $x^2+4x+5\le f(x)\le2x^2+8x+9$ for every real x. Find f(16).

In triangle ABC, $\angle B=2\angle C$. AD bisects angle BAC and meets BC at D. If AB=CD, find angle BAC.

Positive integers a,b,c satisfy $4abc+2ab+2bc+2ca+a+b+c=136$. Find a+b+c.