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 $(a + b)^{2018}-a^{2018}-b^{2018}>= 2^{2\cdot 2018}-2^{2019}.