VTMO 2015 — Grade 5 — final

Define ⊛$5 = 5 + 4 + 3 + 2 + 1$. Find ⊛$15$−⊛$10$.

Write $1 , \ldots , 10$. Each step replaces two numbers by their sum plus $1$. Repeat until one remains. Find it.

Two equal rulers overlap $10 \,\mathrm{cm}$ and span $50 \,\mathrm{cm}$. What overlap gives $a = 56 \,\mathrm{cm}$ span?

$P , K$ are distinct digits.$39 P \times K 3 = 32951$. Find $P + K$.

$IQ = 5 D$ has $21$ students with mean $8.0$; $IQ = 5 P$ has $14$ with mean $7.0$. Find the combined mean.

A two-digit $N$ satisfies $N - 4$ divisible by $4 , N - 5$ by $5 , N - 6$ by $6$. Find $N$.

A square cloth has one corner folded as shown. Its original area and visible area after folding are consecutive even natural numbers. Find the original area.

Transfer from container $1$ to $2$ an amount equal to $2$’s contents, then from $2$ to $1$ an amount equal to $1$’s current contents. Both end with $32$ litres. Find initial contents.

No two girls share a birth weekday, and no two boys share a birth month. Adding any one girl or boy would violate the property. How many students are there?

There are $6000$ ballots for two candidates. After $90 \%$ are counted, $A$ has $53 \%$ of counted ballots. How many remaining votes does $B$ need at minimum to win?

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.