SEAMO 2018 — Paper D — Grade 7–8

Find the last two digits of $1+2+3+\cdots+2017+2018$.

It is now 3:15 PM. After how many minutes will the minute hand form an angle of $30^\circ$ with the hour hand?

For a positive integer n, find the remainder when $n^5-n$ is divided by five.

Find g in the shown multiplication $\overline{abcdef}\times f=\overline{gggggg}$.

p is prime and $p^2+2$ is also prime. Find p.

How many digits are in $4^{22}\times5^{40}$?

A box contains seven black, five white and four orange balls. Four are drawn at random. Find the probability that at least three are white.

Triangle ABC has area one. $BD:CD=AF:FD=CE:EF=1:2$, with the points placed as shown. Find the area of DEF.

Positive integers satisfy $1/x+1/y+1/z=5/6$. Which listed triple is possible?

Today is Saturday. What day is $3^{100}$ days later?

Given $a-b=3$ and $a^2+b^2=5$, find $1/a-1/b$.

Fifteen countries enter a World Cup. Each team plays exactly two matches with another team. A win gives three points, a loss zero, a draw one per team. What is the greatest possible points difference between champion and runner-up?

Positive integers $m\lt n\lt p$ satisfy $2^m+2^n+2^p=2336$. Find m.

How many odd digits are in $1111111111\times9999999999$?

Positive integer m gives a five-digit number $874m$ ending in ninety-two. Which of i: forty-eight, ii: fifty-eight, iii: sixty-eight, iv: one hundred eight are possible?

In rectangle ABCD, E and F are midpoints of AD and CE. Triangle BDF has area $7.5\text{ cm}^2$. Find the rectangle’s area.

Two thirds of the Lethbridge–Medicine Hat route is highway ending at St Paul; one third is country road. A van from Lethbridge travels at one hundred km/h on highway, sixty on country road. A car in the opposite direction travels at seventy km/h on country road, one hundred ten on highway. They meet forty-four km from St Paul. Find the full distance in km.

For positive integer n, $n^2+15n+36$ is a perfect square. Find n.

Chord AB of length $2\sqrt3$ is the side of an equilateral triangle in circle m and a regular hexagon in circle n. Find the area ratio m:n.

How many ways can MADAM be read in the grid, starting at any M and moving horizontally or vertically between adjacent cells?