MYTS 2016 - Grade 7 - Round 2

A square and a regular pentagon are constructed outside an equilateral triangle. Find the marked angle. (Enter the number of degrees.)

Real numbers $x,y,z$ satisfy $|x|-3=|y|+4=10-|z|$. Find the maximum of $K=y(x+z)$.

An isosceles triangle $ABC$ has $AB=AC=8$ cm and median $BD=6$ cm. If $BC^2=x$ cm, find $x$.

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?

A tuple $K$ contains ten numbers, not necessarily distinct, and each number equals the sum of the squares of the other nine. Find every such tuple.

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?

Triangle $ABC$ has $\angle BAC=70^\circ$. The perpendicular bisector of $BC$ meets the external angle bisector at $A$ at point $T$. Given that $\angle BAT$ is obtuse and $\angle CAT$ is acute, find $\angle TBC$. (Enter the number of degrees.)

Five coconut trees are planted around a circular lake. The absolute height differences between consecutive trees are all equal. Prove that all five trees have the same height.