Difference between revisions of "2018 UNM-PNM Statewide High School Mathematics Contest II Problems"
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==Problem 2== | ==Problem 2== | ||
− | + | Determine all positive integers <math>a</math> such that <math>a < 100</math> and <math>a^3 + 23</math> is divisible by <math>24</math>. | |
[[2018 UNM-PNM Statewide High School Mathematics Contest II Problems/Problem 2|Solution]] | [[2018 UNM-PNM Statewide High School Mathematics Contest II Problems/Problem 2|Solution]] | ||
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==Problem 3== | ==Problem 3== | ||
− | + | Let <math>a_1 < a_2 < a_3</math> be three positive integers in the interval <math>[1,14]</math> satisfying <math>a_2-a_1>=3</math> and <math>a_3-a_2>=3</math>. How many different choices of <math>(a_1,a_2,a_3)</math> exist? | |
[[2018 UNM-PNM Statewide High School Mathematics Contest II Problems/Problem 3|Solution]] | [[2018 UNM-PNM Statewide High School Mathematics Contest II Problems/Problem 3|Solution]] | ||
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==Problem 4== | ==Problem 4== | ||
− | + | Suppose ABCD is a parallelogram with area <math>39\sqrt{95}</math> square units and <math>\angle{DAC}</math> is a right angle. If the lengths of all the sides of ABCD are integers, what is the perimeter of ABCD? | |
[[2018 UNM-PNM Statewide High School Mathematics Contest II Problems/Problem 4|Solution]] | [[2018 UNM-PNM Statewide High School Mathematics Contest II Problems/Problem 4|Solution]] | ||
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==Problem 5== | ==Problem 5== | ||
+ | Let <math>x</math> and <math>y</math> be two real numbers satisfying <math>x-4\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 <math>(a + b)^{2018}-a^{2018}-b^{2018}>= 2^{2\cdot 2018}-2^{2019}</math>. | |
[[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]] | ||
==Problem 8== | ==Problem 8== | ||
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+ | 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.) | ||
[[2018 UNM-PNM Statewide High School Mathematics Contest II Problems/Problem 8|Solution]] | [[2018 UNM-PNM Statewide High School Mathematics Contest II Problems/Problem 8|Solution]] | ||
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==Problem 9== | ==Problem 9== | ||
+ | Find the number of <math>4</math>-tuples <math>(a,b,c,d)</math> with <math>a, b, c</math> and <math>d</math> positive integers, such that <math>x^2-ax + b = 0, x^2-bx + c = 0, x^2 -cx + d = 0</math> and <math>x^2-dx + a = 0</math> have integer roots | ||
[[2018 UNM-PNM Statewide High School Mathematics Contest II Problems/Problem 9|Solution]] | [[2018 UNM-PNM Statewide High School Mathematics Contest II Problems/Problem 9|Solution]] | ||
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==Problem 10== | ==Problem 10== | ||
+ | Let <math>A,B,C</math> and <math>D</math> be points in the Cartesian plane each a distance <math>1</math> from the origin <math>(0,0)</math>. We define addition of points in the plane componentwise (If <math>P = (p_x,p_y)</math> and <math>Q = (q_x,q_y)</math>, then <math>P + Q = (p_x + q_x,p_y + q_y))</math>. Show that <math>A + B + C + D = (0,0)</math> if and only if <math>A,B,C</math> and <math>D</math> are the vertices of a rectangle | ||
[[2018 UNM-PNM Statewide High School Mathematics Contest II Problems/Problem 10|Solution]] | [[2018 UNM-PNM Statewide High School Mathematics Contest II Problems/Problem 10|Solution]] | ||
==See Also == | ==See Also == |
Latest revision as of 02:25, 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 .
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