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

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

## Problem 1

Let $x \ne y$ be two real numbers. Let $x,a_1,a_2,a_3,y$ and $b_1,x,b_2,b_3,y,b_4$ be two arithmetic sequences.

Calculate $\frac{b_4-b_3}{a_2-a_1}$.

## Problem 2

Determine all positive integers $a$ such that $a < 100$ and $a^3 + 23$ is divisible by $24$.

## Problem 3

Let $a_1 < a_2 < a_3$ be three positive integers in the interval $[1,14]$ satisfying $a_2-a_1>=3$ and $a_3-a_2>=3$. How many diﬀerent choices of $(a_1,a_2,a_3)$ exist?

## Problem 4

Suppose ABCD is a parallelogram with area $39\sqrt{95}$ square units and $\angle{DAC}$ is a right angle. If the lengths of all the sides of ABCD are integers, what is the perimeter of ABCD?

## Problem 5

Let $x$ and $y$ be two real numbers satisfying $x-4\sqrt{y} = 2\sqrt{x-y}$. What are all the possible values of $x$?

## Problem 6

A round robin chess tournament took place between $16$ players. In such a tournament, each player plays each of the other players exactly once. A win results in a score of $1$ for the player, a loss results in a score of $-1$ for the player and a tie results in a score of $0$. If at least $75$ 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 $a,b$ be positive real numbers such that $\frac{1}{a}+ \frac{1}{b} = 1$. Show that $(a + b)^{2018}-a^{2018}-b^{2018}>= 2^{2\cdot 2018}-2^{2019}$.

## 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 $4$-tuples $(a,b,c,d)$ with $a, b, c$ and $d$ positive integers, such that $x^2-ax + b = 0, x^2-bx + c = 0, x^2 -cx + d = 0$ and $x^2-dx + a = 0$ have integer roots

## Problem 10

Let $A,B,C$ and $D$ be points in the Cartesian plane each a distance $1$ from the origin $(0,0)$. We deﬁne addition of points in the plane componentwise (If $P = (p_x,p_y)$ and $Q = (q_x,q_y)$, then $P + Q = (p_x + q_x,p_y + q_y))$. Show that $A + B + C + D = (0,0)$ if and only if $A,B,C$ and $D$ are the vertices of a rectangle