MYTS 2016 - Grade 6 - Round 2

In the figure, parallelogram $ABCD$ has area 174 cm². Point $E$ is inside it, and triangle $ECD$ has area 28 cm². If triangle $ABE$ has area $x$ cm², find $x$.

Some brothers discover a treasure of gold and silver. They divide it so that each receives exactly 120 kg of gold and silver combined. The eldest receives 30 kg of gold, more gold than each other brother, and also receives $\frac18$ of all the silver. How many kilograms of gold are in the treasure? (Enter the number of kilograms.)

Find the sum of the digits of $A=234\times\underbrace{555\ldots555}_{66\text{ digits}}$, where the second factor consists of 66 copies of the digit 5.

A box contains one ball labelled 1, two balls labelled 2, ..., and 100 balls labelled 100. Balls are drawn without looking. What is the minimum number that must be drawn to guarantee ten balls with the same label?

Find a six-digit number $\overline{abcdef}$ such that moving its units digit $f$ to the far left gives a number five times the original: $\overline{fabcde}=5\times\overline{abcdef}$.

A football is sewn from 32 leather panels. Each black pentagonal panel is joined to five white panels, and each white hexagonal panel is joined to three black panels, as shown. How many white panels are there?

A room contains 26 people, each either always truthful or always lying. The first says, “There are no truthful people in this room.” The second says, “There is at most one truthful person in this room.” The third says, “There are at most two truthful people in this room.” This pattern continues, with the last saying, “There are at most 25 truthful people in this room.” How many truthful people are there?

Magic Wood has only three animal species: 12 snakes, 23 mice and 31 cats. When a snake eats a cat, it turns into a mouse. When a cat eats a mouse, it turns into a snake. When a snake eats a mouse, it turns into a cat. What is the greatest possible number of animals remaining when no animal can eat another?