2004 xIMC keystage2-EMIC team

A grandfather, father and son have total age $144$. The grandfather is under $100$, his age is even, reversing its digits gives the father’s age, and adding those digits gives the son’s age. How old is the grandfather?

Three cubes of volumes $1,8,27\text{ cm}^3$ are glued together at their faces. Find the smallest possible exposed surface area.

From a $324$ m by $141$ m rectangle, cut off squares of side $141$ m. Repeat on the remaining rectangle, always using its shorter side as the square side, until only squares remain. Find the side of the smallest square.

Different letters represent different whole numbers; a word’s value is the product of its letters. For example F=$5$, O=$3$, X=$2$ gives FOX=$30$. Given TEEN=$52$, TILT=$77$, TALL=$363$, find TATTLE.

$A=1\times2+2\times3+\cdots+98\times99$ and $B=1^2+2^2+\cdots+98^2$. Find $A+B$.

Nine chairs in a row seat six students and three distinct professors Alpha, Beta, Gamma. The professors choose first, so each professor will have a student immediately on both sides. How many ways can the professors choose their chairs?

Compute $\frac31+\frac3{1+2}+\frac3{1+2+3}+\cdots+\frac3{1+2+\cdots+100}$.

How many different three-digit numbers satisfy the multiplication diagram? A three-digit number is multiplied by a two-digit number whose tens digit is $9$, giving a four-digit product whose hundreds digit is $2$.

Sixteen containers, each of capacity $5$ litres, form the original $4\times4$ array. Numbers outside give row, column and diagonal totals. Redistribute water taken from only one container so all these totals become equal. Fill the final array.

Place digits $1$ to $9$, each once, in the nine closed regions of the five-circle figure, one digit per region. Every circle must have the same digit sum. How many arrangements are possible? Draw all of them.