2009 xIMC keystage2-EMIC team

A $3\times60$ table has repeating rows: $1,2,3,4,5$; $1,2,3,4$; $1,2$. Add the three entries in each column. How many times does the most frequent sum occur?

All outer surfaces of the pictured T-shaped block are painted red. It has depth $5$ cm, a $10\times4$ cm top bar and a $3\times4$ cm stem, with overhangs $3$ and $4$ cm. Cut it into unit cubes. How many have no painted faces?

Kiran and Babu walk $100$ m from the same starting footprint. Their steps measure $0.8$ m and $0.6$ m. Count footprints up to and including $100$ m, counting exact overlaps only once.

Place the four marked $2\times1$ cards on the pictured board without overlaps and so no diagonals of different cards meet at a corner. Do not rotate or flip cards. Find every arrangement.

Water leaks out of a reservoir at a constant rate. Twenty identical pumps drain it in five hours; fifteen drain it in six hours. What is the fewest pumps to drain it within ten hours?

The integers fill rows $1$; $2,3$; $4,5,6$; $7,8,9,10$; and so on. If $a_{ij}=2009$, where $i$ is its row and $j$ its position in that row, find $i+j$.

Triangle $ABC$ has area $12$ cm$^2$. $DCFE$ is a parallelogram, $D$ on $AC$, $F$ beyond $C$ on line $BC$, and $BC=3CF$. Find the combined shaded triangles $ADE$ and $BDE$.

Semicircle $O$ has diameter $AB=12$ cm, trisected by $C,D$. Radius-$4$ arcs centred at $C,D$ meet at $M$, whose height above $AB$ is given as $3.464$ cm. Radius-$2$ semicircles centred at $C,D$ have endpoints $E,O$ and $O,F$. Find the shaded area shown, using $\pi=3.14$, to three decimal places.

In the pictured Galton board, a falling ball chooses left or right at each triangular block, eventually entering one of seven bottles. How many distinct complete paths are possible?

Place numbers $1$ through $7$ once each at vertices $A,B,C,D,E,F$ and centre $O$ of the regular hexagon shown. The rhombuses $ABOF,BCDO,DEFO$ must have equal vertex sums. Treat flipped or rotated solutions as identical. How many solutions?