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

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==Problem 3==
 
==Problem 3==
  
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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==
  
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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]]

Revision as of 05:31, 19 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

Solution

Problem 6

Solution

Problem 7

Solution

Problem 8

Solution

Problem 9

Solution

Problem 10

Solution

See Also

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