We at Problem of the Week would like to wish you a wonderful, very restful holiday!

For those of you who were fearing what a week without Problem of the Week might entail, we have decided to release a holiday edition with problems for each week of the holiday, to give you something to think about if you tire of binge-watching on Netflix.

Happy problem solving, and as ever, do check the solutions page in September or email us at [email protected] to check / receive hints.

Problem 1

Given two positive integers $a \neq b$, let $f(a,b)$ be the smallest integer that divides exactly one of $a,b$, but not both. Determine the number of pairs of positive integers $(x,y)$, where

$x \neq y, 1 \leq x,y \leq 100$

and $gcd(f(x,y),gcd(x,y)) = 2$.

Problem 2

For how many integral values of $x$ can a triangle of positive area be formed having side lengths $log_2x, log_4x, 3$?

Problem 3

Graham the antisocial giant had 201 grandchildren. “Too many grandchildren!” he thought to himself glumly, although he was aware that this wasn’t the sort of thing you were really meant to say about your grandchildren. “Still, I suppose I’d better see them all soon, probably in the first 60 days of the year” he said.

Graham arranged to meet his first grandchild on the second day of the year, his second grandchild on the 4th day of the year, his nth grandchild on the $2n$th day of the year and so on up to and including day 60. That dealt with the first 30 of them. Then he arranged to meet his 31st grandchild on the 3rd day of the year, his 32nd grandchild on the 6th day of the year, his $(30+m)$th child on the $3m$th day of the year and so on. He continued this way, slotting in grandchildren on days numbered with multiples of 4, then of 5, all the way up to multiples of 60. Handily that took care of them all.

“Now” said Graham to himself, “on any day I that I have an even number of grandchildren visiting me it’s fine, because they can all pair off and talk to each other. But if I have an odd number of grandchildren visiting then I’ll have to be sociable. This seems as good a time as any to learn about Euler’s totient function”.

On how many of the 60 days will Graham have to be sociable?

Problem 4

The four points $A(−1, 2)$$B(3, −4)$$C(5, −6)$, and $D(−2, 8)$ lie in the coordinate plane. Compute the minimum possible value of $PA + PB + PC + PD$ over all points $P$.

Problem 5

Real numbers $x$, $y$, and $z$ are chosen from the interval $[-1,1]$ independently and uniformly at random. What is the probability that

$|x| + |y| + |z| + |x + y + z| = |x + y| + |y + z| + |z + x|$?

Problem 6

Planet Zog has exactly 60 seconds in a minute, 60 minutes in an hour, 24 hours in a day and 365 days in a year. It uses the same system of months, days and years that we do in a non-leap year (leap years don’t exist on planet Zog). Given a number $x$ with at least 10 digits, $x$-time on Zog is given from the first 10 digits of $x$ (ignoring any decimal places) as follows:

Day Month Year Year Year Year Sec Sec Min Hour

For example, since $π=3.1415926536….$$\pi$-time is the 3rd day of January, 4159, 26 seconds and five minutes past 3am (so no rounding).

For some numbers $x$ with at least 10 digits, $x$-time will not exist, e.g. for 9.012345678 (there is no month 0).

There are times which are not $x$-time for any value of $x$ (e.g. any time in October, or any time in the afternoon).

Find, in seconds, the time elapsed on Zog between $\phi$-time and $\textit{e}$-time, where $\phi$ is the golden ratio and $\textit{e} = 2.718281828...$ .

Problem 7

The diagram shows a square and a regular decagon that share an edge. One side of the square is extended to meet an extended side of the decagon. What is the value of $x$?

Problem 8

Students in a class take turns to practise their arithmetic skills. Initially a board contains the integers from 1 to 10 inclusive, each written ten times. On each turn a student first deletes two of the integers and then writes on the board the number that is one more than the sum of those two deleted integers. Turns are taken until there is only one number remaining on the board. Assuming no student makes a mistake, what is the remaining number?

Problem 9

The parabola $y = x^2 − x − 2$ and the line $y=x+1$ intersect in two points. What is the distance between these two points? Express your answer in simplest radical form.

Problem 10 (bonus!)

20 players are playing in a Super Smash Bros. Melee tournament. They are ranked 1 − 20, and player $n$ will always beat player $m$ if $n < m$. Out of all possible tournaments where each player plays 18 distinct other players exactly once, one is chosen uniformly at random. Find the expected number of pairs of players that win the same number of games.

Sources: PUMAC, AMC, MathsBombe, Purple Comet, HMMT, Ritangle, SMC, Senior Kangaroo, MathCounts