2011 xIMC keystage2-EMIC team

Bag $k$ contains $k^2$ candies, for $k=1,2,\ldots,18$. Divide the eighteen bags into three piles of six bags with equal candy totals. Give one division.

Eight positive integers form a row, each from the third equal to the sum of the preceding two. The eighth is $2011$. What is the largest possible first number?

A light beam reflects around a circle: at each boundary point its angles with the radius before and after reflection are equal. Starting at $A_0$, it hits $A_1,A_2,\ldots$. Point $A_{95}$ first coincides with $A_0$. How many choices of $A_1$ are possible?

The capacities of one large pipe and four identical small pipes, in cubic metres per hour, are positive integers. The large pipe delivers $6$ more than one small pipe. The four small pipes fill a pool two hours faster than the large pipe. Find the maximum pool volume.

The white Key Stage II soccer team is positioned against the black team as shown. Pass in any of eight row, column or diagonal directions to the first teammate on that line, without passing through an opponent. Starting at the white goalkeeper on the right, let every white player touch the ball once, then shoot into the left goal.

A palindrome reads the same backwards. How many unordered pairs of four-digit palindromes have a five-digit palindrome as their sum? For example $2882+9339=12221$.

Place $1$–$7$ in the empty arrow boxes, keeping $0$ and $8$ fixed. The arrows of the box with $k$ must point along a row or column to the box with $k+1$, possibly past another box. Give the completed diagram.

Digits use the six-panel patterns shown. Only one panel of each digit in the three-digit subtraction remains visible. What is the largest possible three-digit difference?

Six villages lie equally spaced along a road, with a one-hour bicycle ride between neighbours. A mailman can order the six delivery packets as he likes; a van drops him at the first village and collects him after the last. He earns $20000$ rupiahs per hour cycling. What is his greatest daily pay?

In how many ways is $90$ a sum of at least two consecutive positive integers?