VTMO 2014 — Grade 5 — final

From the three illustrated views of the same die, list its three opposite-face pairs.

Evaluate $\frac1{1\times2\times3}+\frac1{2\times3\times4}+\frac1{3\times4\times5}+\frac1{4\times5\times6}$.

How many triangles of area $4 \,\mathrm{cm}^{2}$ have vertices among the pictured points? Consecutive points on a row or column are $1 \,\mathrm{cm}$ apart.

Each shape is a distinct value. The four row sums are $26 , 24 , 27 , 33$ as pictured. Find square and heart.

Using digits $3 , 5 , 7 , 0$ with repetition allowed, how many four-digit palindromes can be formed?

A closed $15 \times 15 \times 15 \,\mathrm{cm}$ cube has two small holes $5 \,\mathrm{cm}$ from the marked edges as shown. What is the maximum volume of water it can hold?

There are $280$ students, $40 \%$ girls. How many girls must join to make girls $60 \%$?

Teams $A$ and $B$ each have $10$ workers. A builds the road in $14$ days and bridge $25$; $B$ needs $18$ and $15$. Team $C$ combines $5$ from each and builds road then bridge. How many days are required? Assume equal, constant individual productivity within each team.