# 2018 UNM-PNM Statewide High School Mathematics Contest II Problems

UNM - PNM STATEWIDE MATHEMATICS CONTEST L. February 3, 2018. Second Round. Three Hours

## Contents

## Problem 1

Let be two real numbers. Let and be two arithmetic sequences.

Calculate .

## Problem 2

Determine all positive integers such that and is divisible by .

## Problem 3

Let be three positive integers in the interval satisfying and . How many diﬀerent choices of exist?

## Problem 4

Suppose ABCD is a parallelogram with area square units and is a right angle. If the lengths of all the sides of ABCD are integers, what is the perimeter of ABCD?

## Problem 5

Let and be two real numbers satisfying . What are all the possible values of ?

## Problem 6

A round robin chess tournament took place between players. In such a tournament, each player plays each of the other players exactly once. A win results in a score of for the player, a loss results in a score of for the player and a tie results in a score of . If at least percent of the games result in a tie, show that at least two of the players have the same score at the end of the tournament.

## Problem 7

Let be positive real numbers such that . Show that .

## Problem 8

Using red, blue and yellow colored toothpicks and marshmallows, how many ways are there to construct distinctly colored regular hexagons? (Note that two colored hexagons are the same if we can either rotate one of the hexagons and obtain the other or ﬂip one of the hexagons about some line and obtain the other.)

## Problem 9

Find the number of -tuples with and positive integers, such that and have integer roots

## Problem 10

Let and be points in the Cartesian plane each a distance from the origin . We deﬁne addition of points in the plane componentwise (If and , then . Show that if and only if and are the vertices of a rectangle