Difference between revisions of "2018 UNM-PNM Statewide High School Mathematics Contest II Problems"
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==Problem 5== | ==Problem 5== | ||
+ | Let <math>x</math> and <math>y</math> be two real numbers satisfying <math>x-\sqrt{y} = 2\sqrt{x-y}</math>. What are all the possible values of <math>x</math>? | ||
[[2018 UNM-PNM Statewide High School Mathematics Contest II Problems/Problem 5|Solution]] | [[2018 UNM-PNM Statewide High School Mathematics Contest II Problems/Problem 5|Solution]] | ||
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==Problem 6== | ==Problem 6== | ||
+ | A round robin chess tournament took place between <math>16</math> players. In such a tournament, each player plays each of the other players exactly once. A win results in a score of <math>1</math> for the player, a loss results in a score of <math>-1</math> for the player and a tie results in a score of <math>0</math>. If at least <math>75</math> 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. | ||
[[2018 UNM-PNM Statewide High School Mathematics Contest II Problems/Problem 6|Solution]] | [[2018 UNM-PNM Statewide High School Mathematics Contest II Problems/Problem 6|Solution]] | ||
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==Problem 7== | ==Problem 7== | ||
− | + | Let <math>a,b</math> be positive real numbers such that <math>\frac{1}{a}+ \frac{1}{b} = 1</math>. Show that $(a + b)^{2018}-a^{2018}-b^{2018}>= 2^{2\cdot 2018}-2^{2019}. | |
[[2018 UNM-PNM Statewide High School Mathematics Contest II Problems/Problem 7|Solution]] | [[2018 UNM-PNM Statewide High School Mathematics Contest II Problems/Problem 7|Solution]] |
Revision as of 02:21, 20 January 2019
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}.