SEAMO X 2019 — Paper D — Grade 7–8

Which is largest: $6^{291},4^{388},15^{194}$?

The six-digit number $\overline{m2019n}$ is divisible by seventy-two. Find it.

A,B,C,D lie on a circle, with equal minor arcs AB,BC,CD. BA and CD extended meet at E, where $\angle BEC=38^\circ$. Find $\angle BAC+\angle ACD$.

Real a,b,c satisfy $a^2+b^2+c^2-6a-10b-12c+70=0$. Find $a^2+b^2+c^2$.

Twenty students passed at least one subject, seventeen at least two, fifteen at least three and ten at least four. No student passed more than four subjects. How many passes were there in total?

Solve $\frac{x-15}{x-16}-\frac{x-16}{x-17}=\frac{x-18}{x-19}-\frac{x-19}{x-20}$.

Positive integers m,n satisfy $2/3\lt m/n\lt 5/7$. Choose the smallest possible n and find $m^2+n^2$.

Find the coefficient of x in $(1-x)-(1-x)^2+\cdots+(1-x)^{2017}-(1-x)^{2018}$.

Twelve matchsticks are removed in successive turns, taking one, two or three per turn. In how many different orders of group sizes can all be removed?

In rectangle ABCD, AB is eight cm. Semicircles with diameters AB and CD meet at P,Q. The circle with diameter PQ is tangent to AB and CD. Find the shaded area inside this circle and outside the overlap of the semicircles.

ABC is equilateral with side two thousand nineteen. Through an interior point P draw DE parallel to BC, FG parallel to CA and HI parallel to AB, with endpoints on the triangle. If $DE:FG:HI=3:4:5$, find $DE+FG+HI$.

Nonnegative a,b,c satisfy $a+b=15$ and $b+c=13$. If M and N are the greatest and least values of $a+c$, find M+N.

Real a,b are roots of $x^2+3mx+3m=3$. Find the least value of $a^2+b^2$.

Triangle ABC has $AB=35,BC=40,CA=45$. P may be any point. Find the minimum of $3PA+5PB+8PC$.

Real a,b satisfy $ab\lt 0$, $1/a^2+1/b^2=25$ and $a+b=1/12$. Find $1/a^4+1/b^4$.