Problem 1 (Reid, David S.)
The Killer

Show that, for all nonnegative integers m and n,

lies between 0 and 1, inclusive. Hint: This is equal to

Problem 2 (Reid)
4BOding

Let ABCD be a cyclic quadrilateral, and let O be the intersection of its diagonals. Suppose

AB * OC = AC * OB = BC * OA

Prove BD >= 4BO.

Problem 3 (Sasha)
But there are no even primes!

Let S be the set of positive integers n such that every prime dividing n is 1 or 2 mod 4. Solve the system

2a⁶ = c² + d²
2b³ = c² - d²

for a, b, c, d in S.

Problem 4 (Sasha)
Let's play bughouse QBOB!

The game of QBOB is played as follows. A "bob" is a piece that is placed in the center of an n x n board, or as close to it is as possible. On alternating turns, one player moves the bob one square up, down, left, or right into an unoccupied square. The other player places a "q-stone" into an unoccupied square. The former player wins if the bob reaches the edge of the board. The latter wins if the bob is entirely surrounded by q-stones. Find all n such that both players have a forced win if they go first.

Problem 5 (Sasha)
GGAA!

Given a pentagon GREAT, with points M, O, and P inside such that

• GE and GA are trisectors of angle RGT
  
• M lies inside the quadrilateral REAT
  
• MO bisects angle GMT
  
• MP bisects angle GMR
  
• R, P, O, and T are collinear
  
• MPRE and MOTA are cyclic
  
• angle GRT = 3 * angle GAT
  
• angle GTR = 3 * angle GER

prove that AM * GE = GA * ME.

Problem 6 (Sasha)
An f-ing hard problem

Let S be the set of nonnegative integers. Find all functions f: S -> S such that

f (x² + y²) = x f(x) + y f(y)

for all x, y in S.

Problem 7 (Yogesh)
This problem speaks for itself

Let

Prove that S_n = 1 mod 3 if and only if there is a 2 in the base 3 representation of n + 1.

Problem 8 (Vekin)
A problem with practical applications

At the Math Olympiad Winter Program, there are mn students and mn team problems. On the day of the team contest, the evil instructor Narik will divide the students into m teams and give each team a disjoint set of n problems. Each student must present a different one of their team's problems. The students, on the night before the contest, each learn problems so that no matter what Nirak says, everyone will be able to present a problem. It so happens that no student solves more problems than Melody. What is the least number of problems that Melody could possibly know?

Problem 9 (David S.)
I always knew Lincoln was special!

We have n points plus 1 more special point called Lincoln. Let b_n be the number of trees on these points such that only Lincoln has degree >= 3 (the n other points are distinguishable). Show

b_n < n!e^{2n1/2 + 1}(1.00000001)

for sufficiently large n.

Problem 10 (Kevin)
What does abstruse mean?

Define the abstruse mean of two rationals (in lowest terms) to be the weighted arithmetic mean with each number weighted proportionally to its denominator. For any a₀ and a₁, define a_i for i >= 2 as the abstruse mean of a_i-1 and a_i-2. Let f(x, y) be the limit of the sequence with a₀ = x and a₁ = y. Find the range of f.

Problem 11 (Gabriel)
S_n-tials

Given an initial nonnegative integer s₁, define the sequence {s_n} by s_n+1 = |s_n - n|. Determine, as an explicit function of the positive integer k, the least s₁ for which the sequence will eventually assume the form a, a + k, a + 1, a + k + 1, a + 2, a + k + 2, a + 3, ..., for some constant a, and prove that infinitely many such s₁ exist for each k.

Problem 12 (Lawrence)
Try to draw the diagram

1. Prove that it is not possible to fill d-dimensional space with regular polyhedra that are not all hypercubes if d >= 9.

2. Prove that it is possible if d = 3 or d = 4.

Problem 13 (Reid, David S., Sasha)
Back to 3D

Say that a point P is above a face F of a polyhedron iff the perpendicular from P to the plane containing F meets the plane inside or on the boundary of F. For a polyhedron Z, define f (Z) to be the maximum over all points P inside Z of the number of faces which P is above. For each n >= 4, find the minimum value of f over all convex polyhedra with n faces.

Problem 14 (Paul)
Don't you wish that said "number of factors"?

Are there any positive integers n such that n is one fifth the sum of its factors?

Problem 15 (Po)
Po's Mess #3

Solve the following inequality:

a²b + a² + ac² + 115a + b²c + b² + c² + 27c + 176 < 6ab + 22ac + 14bc + 5b

for all ordered triples (a, b, c) of positive integral primes.

Problem 16 (Gabriel)
Going off on a tangent

For a triangle, ABC, let A₁, B₁, C₁ be the points where the incircle is tangent to BC, AC, AB respectively, and let A₂, B₂, C₂ be the points where the respective excircles are tangent to BC, AC, AB. Prove that the lines through A₂, B₂, C₂ parallel to AA₁, BB₁, CC₁ respectively are concurrent.

Problem 17 (Gabriel)
Try it

Given a row of n squares (n > 1), we can place stones in some squares so that no square contains more than one stone. We define the following operation:

First, for each square containing a stone (other than the rightmost one), we add a stone to the square to its right. Then, if any square contains more than one stone, remove two stones from it and add a stone to the square on its right (if there is such a square). If another square contains more than one, do the same thing; continue this process until no square contains more than one stone.

Now, we start with an initial configuration C of stones and apply the operation repeatedly. After an odd number of iterations, C reappears. Prove that C contains at most one stone.

Problem 18 (Po)
Po's Football

Define a football to be a geometrical figure that is the intersection of two circular arcs each less than 20 degrees in measure, of equal radii.

Let us define the endpoints of a football to be the intersection of the circles. Let the center of the football be the midpoint of the line between the endpoints of the football.

Now, let us say that we have a football F. Let its center be X, and its endpoints be B and C. Select a point A on F. Let A₁ be the point on the arc of the football not containing A, such that angle BAX and angle BA₁X are supplementary. Define A₂ similarly: let A₂ be the point on the arc of the football not containing A such that angle CAX and angle CA₂X are supplementary. Extend A₁X and A₂X to meet F. Let the points of intersection be P₁ and P₂, respectively.

After you've got all that, prove that

AP₁ * XP₂ = AP₂ * XP₁

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