# Mathematical Competition

a_{1} Let f : R^2 -> R be a function such that f(x, y) +

f(y, z) + f(z, x) = 0 for all real numbers x, y, and z.

Prove that there exists a function g : R -> R such that

f(x, y) = g(x) − g(y) for all real numbers x and y .

a_{2} Alan and Barbara play a game in which they ta_{k}e turns

filling entries of an initially empty 2008 × 2008 array.

Alan plays first. At each turn, a player chooses a real

number and places it in a vacant entry. The game ends

when all the entries are filled. Alan wins if the determinant

of the resulting matrix is nonzero; Barbara wins if

it is zero . Which player has a winning strategy?

a_{3} Start with a finite sequence a_{1}, a_{2}, . . . , an of
positive

integers. If possible, choose two indices j < k such

that a_{j} does not divide a_{k}, and replace a_{j} and a_{k} by

gcd(a_{j} , a_{k}) and lcm(a_{j} , a_{k}), respectively. Prove that if

this process is repeated, it must eventually stop and the

final sequence does not depend on the choices made.

(Note: gcd means greatest common divisor and lcm

means least common multiple.)

A4 Define f : R -> R by

Does converge

A5 Let n≥3 be an integer. Let f(x) and g(x) be

polynomials with real coefficients such that the points

(f(1), g(1)), (f(2), g(2)), . . . , (f(n), g(n)) in R^2 are

the vertices of a regular n-gon in counterclockwise order .

Prove that at least one of f(x) and g(x) has degree

greater than or equal to n − 1.

A6 Prove that there exists a constant c > 0 such that in
every

nontrivial finite group G there exists a sequence of

length at most c ln |G| with the property that each element

of G equals the product of some subsequence.

(The elements of G in the sequence are not required to

be distinct. A subsequence of a sequence is obtained

by selecting some of the terms , not necessarily consecutive,

without reordering them; for example, 4, 4, 2 is a

subsequence of 2, 4, 6, 4, 2, but 2, 2, 4 is not.)

B1 What is the maximum number of rational points that can

lie on a circle in R ^2 whose center is not a rational point?

(A rational point is a point both of whose coordinates

are rational numbers.)

B2 Let Evaluate

B3 What is the largest possible radius of a circle
contained

in a 4-dimensional hypercube of side length 1?

B4 Let p be a prime number. Let h(x) be a

polynomial with integer coefficients such that

h(0), h(1), . . . , h(p^2 − 1) are distinct modulo p^2.

Show that h(0), h(1), . . . , h(p^3 − 1) are distinct

modulo p^3.

B5 Find all continuously differentiable functions f : R ->

R such that for every rational number q, the number

f(q) is rational and has the same denominator as q .

(The denominator of a rational number q is the unique

positive integer b such that q = a/b for some integer a

with gcd(a, b) = 1.) (Note: gcd means greatest common

divisor .)

B6 Let n and k be positive integers. Say that a
permutation

of {1, 2, . . . , n} is k-limited if
for all

i. Prove that the number of k-limited permutations of

{1, 2, . . . , n} is odd if and only if n 0 or 1 (mod

2k + 1).

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