Monday, January 30, 2012

Successive reflection

Given 2012 points on the plane: $A_1, \dots, A_{2012}$, and a point $P$. Suppose $B_1, \dots, B_{2012}$ is a permutation of the $A_i$s, we determine the shadow of $P$ as follows:

Reflect $P$ with respect to $B_1$ to obtain $P_1$. Reflect $P_1$ with respect to $B_2$ to obtain $P_2$, and so on, to arrive with $P_{2012}$. We call this last point the shadow of $P$.

Obviously, depending on the permutation of $B_i$s, one may arrive at different shadows of $P$. Find the maximal numbers of shadows of $P$ over all possible permutations.

Saturday, January 28, 2012

Graph with degree 3

A graph where each vertex has degree 3 has all of its edges colored with red, green, or blue. How many colorings are there such that every 3 edges that meet in a vertex are either of the same color or have 3 different colors?

Replacing number on the board

In a blackboard, the number 2012 is initially written. On each turn, the student can choose on of the three numbers: 6, 1006, 1509, and multiply it with the existing number on the board. The existing number is then erased and replaced with the new product.

The teacher has a specific integer $k > 1$ in mind, and the goal for the student is to get the number of the board to have form $n^k$ where $n$ is an integer.

Determine all values of $k$ such that, after a finite number of turns, the number of the board will have the desired form.