DK-GEORGMOHR 2018

A blackboard contains $2018$ instances of the digit $1$ separated by spaces. Georg and his mother play a game where they take turns filling in one of the spaces between the digits with either a $+$ or a $\times$ . Georg begins, and the game ends when all spaces have been filled. Georg wins if the value of the expression is even, and his mother wins if it is odd. Which player may prepare a strategy which secures him/her victory?

The figure shows a large circle with radius $2$ m and four small circles with radii $1$ m. It is to be painted using the three shown colours. What is the cost of painting the figure?

The positive integers $a, b$ and $c$ satisfy that the three fractions $\frac{b}{a}$ , $\frac{c + 100}{b}$ and $\frac{a + b + 169}{2c + 200}$ are all integers. Determine all possible values of $a$ .

A sequence $a_1, a_2, a_3, . . . , a_{100}$ of $100$ (not necessarily distinct) positive numbers satisfy that the $99$ fractions $$\frac{a_1}{a_2},\frac{a_2}{a_3},\frac{a3}{a_4}, ... ,\frac{a_{99}}{a_{100}}$$ are all distinct. How many distinct numbers must there be, at least, in the sequence $a_1, a_2, a_3, . . . , a_{100}$ ?

In triangle $ABC$ the angular bisector from $A$ intersects the side $BC$ at the point $D$ , and the angular bisector from $B$ intersects the side $AC$ at the point $E$ . Furthermore $|AE| + |BD| = |AB|$ . Prove that $\angle C = 60^o$