IL-GILLIS 2016

Nina and Meir are walking on a $3$ km path towards grandma's house. They start walking at the same time from the same point. Meir's speed is $4$ km/h and Nina's speed is $3$ km/h.

Along the path there are several benches. Whenever Nina or Meir reaches a bench, they sit on it for some time and eat a cookie. Nina always takes $t$ minutes to eat a cookie, and Meir always takes $2t$ minutes to eat a cookie, where $t$ is a positive integer.

It turns out that Nina and Meir reached grandma's house at the same time. How many benches were there? Find all of the options.

We are given a cone with height 6, whose base is a circle with radius $\sqrt{2}$ . Inside the cone, there is an inscribed cube: Its bottom face on the base of the cone, and all of its top vertices lie on the cone. What is the length of the cube's edge?

Denote by $S(n)$ the sum of digits of $n$ . Given a positive integer $N$ , we consider the following process: We take the sum of digits $S(N)$ , then take its sum of digits $S(S(N))$ , then its sum of digits $S(S(S(N)))$ ... We continue this until we are left with a one-digit number.

We call the number of times we had to activate $S(\cdot)$ the depth of $N$ .

For example, the depth of 49 is 2, since $S(49)=13\rightarrow S(13)=4$ , and the depth of 45 is 1, since $S(45)=9$ .

[list=a] [] Prove that every positive integer $N$ has a finite depth, that is, at some point of the process we get a one-digit number. [] Define $x(n)$ to be the minimal positive integer with depth $n$ . Find the residue of $x(5776)\mod 6$ . [*] Find the residue of $x(5776)-x(5708)\mod 2016$ . [/list]

In the beginning, there is a circle with three points on it. The points are colored (clockwise): Green, blue, red. Jonathan may perform the following actions, as many times as he wants, in any order:

• Choose two adjacent points with different colors, and add a point between them with one of the two colors only.
• Choose two adjacent points with the same color, and add a point between them with any of the three colors.
• Choose three adjacent points, at least two of them having the same color, and delete the middle point.

Can Jonathan reach a state where only three points remain on the circle, colored (clockwise): Blue, green, red?

The Fibonacci sequence $F_n$ is defined by $F_1=F_2=1$ and the recurrence relation $F_n=F_{n-1}+F_{n-2}$ for all integers $n\geq3$ .

Let $m,n\geq1$ be integers. Find the minimal degree $d$ for which there exists a polynomial $f(x)=a_dx^d+a_{d-1}x^{d-1}+\dots+a_1x+a_0$ , which satisfies $f(k)=F_{m+k}$ for all $k=0,1,...,n$ .

Points $A_1,A_2,A_3,...,A_{12}$ are the vertices of a regular polygon in that order. The 12 diagonals $A_1A_6,A_2A_7,A_3A_8,...,A_{11}A_4,A_{12}A_5$ are marked, as in the picture below. Let $X$ be some point in the plane. From $X$ , we draw perpendicular lines to all 12 marked diagonals. Let $B_1,B_2,B_3,...,B_{12}$ be the feet of the perpendiculars, so that $B_1$ lies on $A_1A_6$ , $B_2$ lies on $A_2A_7$ and so on.

Evaluate the ratio $\frac{XA_1+XA_2+\dots+XA_{12}}{B_1B_6+B_2B_7+\dots+B_{12}B_5}$ .

Find all functions $f:\mathbb{Z}\rightarrow\mathbb{C}$ such that $f(x(2y+1))=f(x(y+1))+f(x)f(y)$ holds for any two integers $x,y$ .