Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "2017 UNM-PNM Statewide High School Mathematics Contest II Problems/Problem 4"

(Created page with " == Problem == Find all real triples <math>(x,y,z)</math> which are solutions to the system: <math>x^3 + x^2y + x^2z = 40</math> <math>y^3 + y^2x + y^2z = 90</math> <math...")
 
(→‎Solution)
 
(One intermediate revision by the same user not shown)
Line 3: Line 3:
 
== Problem ==
 
== Problem ==
  
−
Find all real triples <math>(x,y,z)</math> which are solutions to the system:
+
Find a second-degree polynomial with integer coefficients, <math>p(x) = ax^2 + bx + c</math>, such that <math>p(1),p(3),p(5)</math>, and <math>p(7)</math> are perfect squares, but <math>p(2)</math> is not.
−
 
 
−
<math>x^3 + x^2y + x^2z = 40</math>
 
−
 
 
−
<math>y^3 + y^2x + y^2z = 90</math>
 
−
 
 
−
<math>z^3 + z^2x + z^2y = 250</math>
 
  
 
== Solution==
 
== Solution==
  
  
 +
Answer: A polynomial that satisfies the criteria is easily constructed by first centering it at <math>x = 4</math>, that is <math>p(x) = \overline{a}(x-4)^2 + \overline{c}</math>. Now we have two conditions: <math>n^2 = p(1) = 9\overline{a}+\overline{c}</math> and <math>m^2 = p(3) = \overline{a}+\overline{c}</math> that determines possible candidates that can then be checked against the condition that <math>p(2)</math> is not a perfect square and the condition that the coefficients are integers. A few possible answers are <math>(n,m,\overline{a},\overline{c}) = (0,4,-2,18),(1,3,-1,10),(2,6,-4,40),(3,5,-2,27),\ldots </math>
  
 
== See also ==
 
== See also ==

Latest revision as of 04:05, 19 January 2019


Problem

Find a second-degree polynomial with integer coefficients, $p(x) = ax^2 + bx + c$, such that $p(1),p(3),p(5)$, and $p(7)$ are perfect squares, but $p(2)$ is not.

Solution

Answer: A polynomial that satisfies the criteria is easily constructed by first centering it at $x = 4$, that is $p(x) = \overline{a}(x-4)^2 + \overline{c}$. Now we have two conditions: $n^2 = p(1) = 9\overline{a}+\overline{c}$ and $m^2 = p(3) = \overline{a}+\overline{c}$ that determines possible candidates that can then be checked against the condition that $p(2)$ is not a perfect square and the condition that the coefficients are integers. A few possible answers are $(n,m,\overline{a},\overline{c}) = (0,4,-2,18),(1,3,-1,10),(2,6,-4,40),(3,5,-2,27),\ldots$

See also

2017 UNM-PNM Contest II (Problems • Answer Key • Resources)
Preceded by
Problem 3 		Followed by
Problem 5
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10
All UNM-PNM Problems and Solutions
Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2017_UNM-PNM_Statewide_High_School_Mathematics_Contest_II_Problems/Problem_4&oldid=100608"