Grossman Math Olympiad, 2006

• Question 1 - The Confrontation

  Ehud and Benjamin are participating in a public debate. Each one presents a question to his opponent in turn. Ehud is chosen to be the first to present a question. A "tricky question" is a question that the opponent has no answer to. A contestant who manages to ask a tricky question immediately wins the debate. The probability of each of the two contestants finding (in turn) a tricky question is exactly 1/2. Also, it is known that there is no dependence between the questions. What is the probability of Ehud winning the debate?

  Topics:

  Algebra -> Sequences Probability Theory

• Question 2 - Divisible by 13

  All natural numbers from 1 to 2006 are written on a sheet of paper, and a series of operations is performed as described below. At each step, any number of numbers are deleted from the list and their sum is denoted by S. Instead of the deleted numbers, a single number is added, which is the remainder obtained from the division of S by 13. After some number of such steps, only two numbers remain on the paper. One of them is 100. Find the other number.

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  Topics:

  Number Theory -> Modular Arithmetic / Remainder Arithmetic Number Theory -> Division -> Parity (Even/Odd) Algebra -> Sequences -> Arithmetic Progression / Arithmetic Sequence

• Question 3 - Sets in the Plane

  A. Does there exist a set A in the plane such that its intersection with every circle contains exactly two points?

  B. Does there exist a set B in the plane such that its intersection with every circle of radius 1 contains exactly two points?

  Topics:

  Geometry -> Plane Geometry -> Circles Proof and Example -> Constructing an Example / Counterexample Set Theory Proof and Example -> Proof by Contradiction Minimum and Maximum Problems / Optimization Problems

• Question 4 - Hexagonal Tiling

  Given two types of tiles. The shape of each tile of the first type is a regular hexagon with a side of length 1. The shape of each tile of the second type is a regular hexagon with a side of length 2. An unlimited supply of tiles of each type is given. Is it possible to tile the entire plane using these tiles, using both types of tiles?

  Topics:

  Logic -> Reasoning / Logic Proof and Example -> Proof by Contradiction Geometry -> Plane Geometry -> Angle Calculation

• Question 5 - Triangle Side Lengths

  Let `n > 2` be an integer, and let ` t_1,t_2,...,t_n` be positive real numbers such that

  `(t_1+t_2+...+t_n)(1/t_1 + 1/t_2 + ... + 1/t_n) < n^2+1`

  Prove that for all i,j,k such that `1<=i<j<k<=n`, the triple of numbers `t_i,t_j,t_k` are the side lengths of a triangle.

  Topics:

  Geometry -> Plane Geometry -> Triangles Algebra -> Inequalities Proof and Example -> Proof by Contradiction Geometry -> Plane Geometry -> Triangle Inequality

• Question 6 - Polynomial with Integer Coefficients

  Let `p(x)` be a polynomial with integer coefficients such that `p(-2006) < p(2006)=2005`. Prove that `p(-2006)<=-2007`.
   

  Topics:

  Number Theory Algebra -> Algebraic Techniques Algebra -> Inequalities

• Question 7 - A Walk in the Plane

  Given a Cartesian coordinate system x-y in the plane. You need to get from the point (1,0) to the point (2006,2005), where in each step you are allowed to move one unit up (in the positive direction of y) or one unit to the right (in the positive direction of the x-axis).

  a. In how many different paths can the task be performed?

  b. In how many different paths can the task be performed if it is forbidden at any stage to pass through a point on the line x=y?

  Topics:

  Combinatorics -> Combinatorial Geometry Combinatorics -> Binomial Coefficients and Pascal's Triangle Geometry -> Plane Geometry -> Plane Transformations -> Congruence Transformations (Isometries) -> Reflection