SEAMO X 2023 — Paper C — Grade 5–6

For coprime positive integers $m,n$, $\left(\frac12+\cdots+\frac1{72}\right)\left(1+\frac12+\cdots+\frac1{71}\right)-\left(1+\frac12+\cdots+\frac1{72}\right)\left(\frac12+\cdots+\frac1{71}\right)=m/n$. Find $m+n$.

Nuggets are sold only in boxes of five or seven. What is the largest number of nuggets that cannot be bought exactly?

Find the remainder when $3979^{5678}$ is divided by $39$.

Choose a positive integer at most $200$ uniformly. Its probability of being divisible by three or five is $m/n$ in lowest terms. Find $m+n$.

Evaluate $\left(10-\frac49\times5\right)+\left(9-\frac49\times6\right)+\cdots+\left(5-\frac49\times10\right)$.

The pattern starts $1^3=1$, $1^3+2^3=9$, $1^3+2^3+3^3=36$, $1^3+2^3+3^3+4^3=100$. Find $1^3+2^3+\cdots+10^3$.

Find the units digit of $1^4+2^4+\cdots+2023^4$.

Find the smallest integral value of $k$ for which $3^{83}+k$ is divisible by thirteen.

Square $ABCD$ circumscribes square $PQRS$ as shown. The inner square has area $128$ square units. Find the outer perimeter $k$.

Given $x+y=4$ and $xy=-12$, find $(x-y)^2$.

Which is largest: $2^{485}$, $3^{291}$ or $5^{194}$?

In trapezium $ABCD$, $AB=\frac23CD$. An interior point is joined to the four vertices. The unshaded triangles on bases $AB,CD$ have areas ten and twelve respectively. Find the shaded area.

Container A has fifteen litres of pure spirit; B has twenty litres of water. Some water is transferred from B to A and mixed. Then $k$ litres of this mixture are transferred back to B. The final spirit concentrations are $75\%$ in A and $30\%$ in B. Find $k$.

Everyone plays exactly one of soccer, rugby or tennis. Twenty percent play soccer, $a/7$ play rugby, and ninety-one play tennis. Find the number of people.

An integer $x$ satisfies that $x-45$ and $x+44$ are perfect squares. Find $x$.