## Archive for the ‘**Round 5 (In-Person)**’ Category

## Round 5: Problem 6

Let’s play a game.

A fair die is rolled repeatedly until two consecutive rolls are equal, at which point the game ends.

If we let S be the sum of the results of all of the rolls, is S more likely to be even or odd, or is it equally likely to be both?

## Round 5: Problem 5

Each of nine lines cut a unit square into two quadrilaterals whose areas are in a ratio of 2:1. Prove that at least three lines must be concurrent (i.e. they intersect at a common point).

## Round 5: Problem 4

Let be an infinite arithmetic progression of positive integers with the special property that the sum of the first n terms of is a perfect square for all n. If 2009 is the k-th term of the sequence, how small can k be?

## Round 5: Problem 3

What is the most number of non-overlapping tiles of the given form that can be used to cover a 6×6 board?

## Round 5: Problem 2

A pile of identical spheres (each of radius 1 meter) are placed in a pile so that they form a triangular pyramid that culminates with one lone sphere at the top of the pile. If the pile of spheres is higher than 2009 centimeters, what is the minimum number of spheres that can be assembled to form the pyramid?

## Round 5: Problem 1

A set of distinct positive integers is called “sour” if it has the property that, for every 3 distinct elements—a,b and c—of the set, a+b+c is always a power of 2. What is the largest size a “sour” set can have?