Part A
1. Find all values of a for which the curves y = ax^2 + ax +1/24 and x = ay^2 + ay +1/24 are tangent to each other.
2. Find the least possible area of a convex set in the plane that intersects both branches of the hyperbola xy = 1 and both branches of the hyperbola xy = -1.
3. Let k be a positive integer. Suppose that the integers 1, 2, 3, ..., 3k+1 are written down in random order. What is the probability that at no time during this process, the sum of the integers that have been written up to that time is a positive integer divisible by 3? Your answer should be in closed form, but may include factorials.
4 A repunit is a positive integer whose digits in base 10 are all ones. Find all polynomials f with real coefficients such that if n is a repunit, then so is f(n)
5 Suppose that a finite group has exactly n elements of order p, where p is a prime. Prove that either n = 0 or p divides n+1.
6 A triangulation T of a polygon P is a finite collection of triangles whose union is P, and such that the intersection of any two triangles is either empty, or a shared vertex, or a shared side. Moreover, each side of P is a side of exactly one triangle in T. Say that T is admissible if every internal vertex is shared by or more triangles. Prove that there is an integer M(n), depending only on n, such that any admissible triangulation of a polygon P with n sides has at most M(n) triangles.
Part B
1. Let f be a polynomial with positive integer coefficients. Prove that if n is a positive integer, then f(n) divides f(f(n)+1) if and only if n = 1
2 Suppose that f:[0,1] -> R has a continuous derivative and that (Integration from 0 to 1) f(x) dx = 0.
Prove that for every a in (0, 1), |(Integration from 0 to a) f(x) dx | <= (1/8) (max from 0 to 1) |f'(x)|.
3 Let x0=1 and for n >= 0 let x(n+1) = 3xn + floor(xn*sqrt(5)). Find a closed-form expression for x2007.
4 Let n be a positive integer. Find the number of pairs P, Q of polynomials with real coefficients such that P^2 + Q^2 = x^(2n) + 1, and deg(P) < deg(Q)
5 Let k be a positive integer. Prove that there exist polynomials P_0, P_1, ..., P_(k-1) such that for any integer n,
(floor(n/k))^k = P_0(n) + P_1(n)*(floor(n/k)) + ..., P_(k-1)(n)*(floor(n/k))^(k-1)
6 For each positive integer n let f(n) be the number of ways to make n! cents using an unordered collection of coins, each worth k! cents for some k, 1 <= k <= n. Prove that for some constant C independent of n,
n^(n^2/2 - Cn)* e^(-n^2/4) <= f(n) <= n^(n^2/2 + Cn)* e^(-n^2/4)
1. Find all values of a for which the curves y = ax^2 + ax +1/24 and x = ay^2 + ay +1/24 are tangent to each other.
2. Find the least possible area of a convex set in the plane that intersects both branches of the hyperbola xy = 1 and both branches of the hyperbola xy = -1.
3. Let k be a positive integer. Suppose that the integers 1, 2, 3, ..., 3k+1 are written down in random order. What is the probability that at no time during this process, the sum of the integers that have been written up to that time is a positive integer divisible by 3? Your answer should be in closed form, but may include factorials.
4 A repunit is a positive integer whose digits in base 10 are all ones. Find all polynomials f with real coefficients such that if n is a repunit, then so is f(n)
5 Suppose that a finite group has exactly n elements of order p, where p is a prime. Prove that either n = 0 or p divides n+1.
6 A triangulation T of a polygon P is a finite collection of triangles whose union is P, and such that the intersection of any two triangles is either empty, or a shared vertex, or a shared side. Moreover, each side of P is a side of exactly one triangle in T. Say that T is admissible if every internal vertex is shared by or more triangles. Prove that there is an integer M(n), depending only on n, such that any admissible triangulation of a polygon P with n sides has at most M(n) triangles.
Part B
1. Let f be a polynomial with positive integer coefficients. Prove that if n is a positive integer, then f(n) divides f(f(n)+1) if and only if n = 1
2 Suppose that f:[0,1] -> R has a continuous derivative and that (Integration from 0 to 1) f(x) dx = 0.
Prove that for every a in (0, 1), |(Integration from 0 to a) f(x) dx | <= (1/8) (max from 0 to 1) |f'(x)|.
3 Let x0=1 and for n >= 0 let x(n+1) = 3xn + floor(xn*sqrt(5)). Find a closed-form expression for x2007.
4 Let n be a positive integer. Find the number of pairs P, Q of polynomials with real coefficients such that P^2 + Q^2 = x^(2n) + 1, and deg(P) < deg(Q)
5 Let k be a positive integer. Prove that there exist polynomials P_0, P_1, ..., P_(k-1) such that for any integer n,
(floor(n/k))^k = P_0(n) + P_1(n)*(floor(n/k)) + ..., P_(k-1)(n)*(floor(n/k))^(k-1)
6 For each positive integer n let f(n) be the number of ways to make n! cents using an unordered collection of coins, each worth k! cents for some k, 1 <= k <= n. Prove that for some constant C independent of n,
n^(n^2/2 - Cn)* e^(-n^2/4) <= f(n) <= n^(n^2/2 + Cn)* e^(-n^2/4)
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