UCSC Team Contest: Round 2

August 16, 2013

Algebra

1. Let a, b be real numbers such that 1 a2 ab + b2 2. Prove that 2 9(a4 + b4) 72.

2. Prove that the value of the expressionn +

0 +n +

1 +n +

2 + +n +n2 1 +

n +n2

n0 +n1 +

n2 + +

nn2 1 +

nn2

is a constant for all positive integer values of n.

3. Find all polynomials P (x) for which

P (x)P (x 1) + P (x2) = 0.

Combinatorics

1. A frog lives on a triangle ABC. Every turn, she jumps to one of the adjacent vertices. In n hops, inhow many ways can she get from vertex A back to vertex A?

2. Alice and Bob play a variant of the game hot or cold. First, Bob places a box of candies at a locationwithin 2001 feet from Alice. Alice is myopic and can only see the box if it is a distance less than equalto 1 foot from her. Alices objective is to find the box by taking a sequence of steps of length lessthan or equal to 1 foot in any direction. After each step, Alice can ask Bob Hot or cold? Bob mustanswer hot if Alices step took her closer to the box and cold otherwise. Show that Alice can findthe box taking at most 2016 steps and asking at most 13 questions.

3. The Republic of Santa Cruz is a country with at least one village in which some pairs of villages areconnected by roads. Bob founded a bakery company and wants to build stores in several villages (atmost one store per village) in such a way that every citizen of The Republic of Santa Cruz who cantbuy Bobs croissants in their own village can buy them in one of the neighbouring ones. Prove thatthe number of ways to do so is odd.

Geometry

1. Let ABC be a triangle with points K and L on its sides AB, AC, respectively. Let P = BL CKand assume that BC2 = BK BA + CL CA. Prove that AKLP is cyclic.

2. Let ABC be a triangle inscribed in a circle and P a variable point on the arc BC of not containingvertex A. Denote by I, J the incenters of triangles PAB,PAC respectively. Prove that as P varies,

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(a) the circles with diameters IJ all have a common point,

(b) the midpoints of the segments IJ all lie on a single circle,

(c) the circumcircles of the triangles PIJ all have a common point.

3. Let ABC be a triangle and let M , N , P be the midpoints of sides BC, CA, AB, respectively. Denoteby X, Y , Z the midpoints of the altitudes emerging from the vertices A, B, C, respectively. Provethat the radical center of the circles AMX, BNY , CPZ is the nine-point center of triangle ABC.

Number Theory

1. Solve in integers the equation1! + 2! + 3! + + x! = y2.

2. Let n 2 and let P (x) = xn + an1xn1 + + a1x + 1 be a polynomial with positive integercoefficients such that ak = ank for k = 1, 2, , n 1. Prove that there exist infinitely many pairs(x, y) of positive integers such that both x|P (y) and y|P (x).

3. Find all functions f : {1, 2, ...} {1, 2, ...} such that m n divides f(m) f(n) for all m 6= n andsuch that gcd(f(a), f(b)) = 1 whenever gcd(a, b) = 1.

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