MYTS 2016 - Grade 8 - Round 2

Let $ABCDEF$ be a regular hexagon, $G$ the midpoint of $FA$, and $I$ a point on diagonal $CF$ such that $IC=\frac13IF$. Prove that triangle $IGE$ is equilateral.

Let $\alpha$ be the unique root of $P(x)=x^3+9x-5$, and $\beta$ the unique real root of $Q(x)=x^3-15x^2+84x-165$. Prove that $\alpha+\beta=5$.

Mr Du writes the integers from 1 to $n$ on a board but accidentally omits one number. The mean of the numbers written is $\frac{599}{17}$. Which number did he omit?

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?

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.

In triangle $ABC$, let $D,E,F$ be the midpoints of $BC,CA,AB$, respectively. A line through $A$ meets $DE,DF$ at $M,N$, respectively. Let $P$ be the intersection of $BM$ and $CN$. Prove that $E,F,P$ are collinear.

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.

Find all positive integers $n$ such that $\frac{7n-12}{2^n}+\frac{2n-14}{3^n}+\frac{24n}{6^n}=1$.