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Difference between revisions of "2018 UNM-PNM Statewide High School Mathematics Contest II Problems"

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==Problem 5==
 
==Problem 5==
  
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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==
  
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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==
  
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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

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


Solution

Problem 2

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

Solution

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?

Solution

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?

Solution

Problem 5

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

Solution

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.

Solution

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}.

Solution

Problem 8

Solution

Problem 9

Solution

Problem 10

Solution

See Also

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