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?

In-Person Exam

The scores seem like they have been tallied, so if you’ve made it to the in-person exam you should have gotten an e-mail by now it seems.

The in-person exam is scheduled for Sunday, May 10 from 10:00 a.m. to 1:00 p.m. at Baruch College. It will be in the Vertical Campus, Room 3165.

Update: Note the room change.

Round 4: Problem 5 Solution

Tricoloring

This is a nice tiling problem in disguise. Let’s ignore the letters and symbols for now and just work with a blank 5×8 board. The 3×1 rectangular pieces are like straight triominoes that we will use to try to tile the board.

Essentially, our problem now is the following: with 13 triomino(e)s, how many different places can we place a monomino on a 5×8 board.

Tricolor the board as shown. Note that we now have 13 red squares, 13 yellow squares, but 14 black squares. Therefore, the monomino must be placed on one of the black squares. However, the monomino cannot be placed on just any black square. A monomino can only be placed on a black square if there is no way to flip our coloring so that the square in our original coloring is no longer black. Flipping the coloring disqualifies every black square besides those 2 on the third row.

We can verify that in fact we may place a monomino on either of these two squares by showing a tiling (try it, it’s trivial). Therefore we conclude that a monomino can only appear on one of these 2 squares. Plugging in these 2 black squares into the original board / cake, we see that on these two squares are the ‘2′ and the ‘9′ of ‘2009.’ These are the only two possible remaining pieces.

Round 4: Problem 2 Solution

Here’s a geometric construction proof:

• Extend and so that they intersect at an exterior point .
• Extend and in a similar fashion so they intersect at another exterior point, , forming quadrilateral .
• since each external angle is supplementary with the 60 degree internal angle.
• , so are equilateral triangles.
• in equilateral triangles.
• is a parallelogram (both pairs of opposite angles are congruent), so .
• , so by equilateral triangles as shown above.

Round 4: Problem 5

At the prize ceremony for the 2009 CUNY Math Challenge there will be a cake that is decorated so the message to the right appears on top. Each letter, digit, and symbol of the message forms one square of a 5 x 8 rectangle. Each of the 13 second prize winners cuts and eats a 3 x 1 rectangular piece of cake leaving just one square for the grand prize winner. What letter, digit, or symbol could possibly be on the grand prize winner’s piece of cake? (You must identify all possibilities and prove that no others could occur.)