Parallelogram 31 • Level 3 • 4 Apr 2024In the art gallery

Each of points (0, 1), (1, 0), (0, -1) and (-1, 0) is 1 unit from the origin, but the point (1, 1) is at a distance of 2 units from the origin.

The person in position p is the only one who can see two of the others.

We are told that Caz can see Bea and Dan but everyone else can see just one other person or no-one. So it must be Caz who is at position P.

We can also see that Ali, who can see no-one, must be in the position shown. Bea and Dan must be in the other two positions, but the information we are given doesn't enable us to work out which is where.

Since half the 60 bulbs are yellow, 30 are yellow and 30 are not yellow. One third of the 30 bulbs that are not yellow are red, so 10 are red and 20 are neither yellow nor red. One quarter of the 20 that are neither yellow nor red, are pink. So 5 bulbs are pink.

This leaves 15 bulbs which are neither yellow, nor red nor pink. So there are 15 white bulbs. Therefore the fraction of bulbs that are white is 1560, that is, 14.

Jack won exactly 4 games. So Jill lost 4 games and gained 4 points from these losses. Jill gained 10 points altogether and so gained 10 − 4 = 6 points from her wins. Since there are 2 points for a win, Jill won 3 games. So Jack won 4 games, Jill won 3 games and altogether 4 + 3 = 7 games were played.

Crucially, the question said that x and y are positive integers.

The angles of a triangle add up to 180°. So 5x+3y+3x+20+10y+30=180. This gives 8x+13y+50=180. Therefore 8x+13y=130, and so

8x=130−13y=1310−y.

It follows that 8x is a multiple of 13 and hence, as 8 and 13 have no common factors, it must be that x is a multiple of 13. So x=13a, where a is a positive integer. It then follows that

813a=1310−y

and so, cancelling the factor 13

8a=10−y

and hence 10−y is a positive number which is a multiple of 8. The only positive value of y for which 10−y>0 and 10−y is a multiple of 8, is y=2, which gives a=1 and hence x=13. So x+y=13+2=15.