Difference between revisions of "1980 AHSME Problems/Problem 29"
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How many ordered triples (x,y,z) of integers satisfy the system of equations below? | How many ordered triples (x,y,z) of integers satisfy the system of equations below? | ||
| − | <cmath>\begin{array}{l} x^2-3xy+ | + | <cmath>\begin{array}{l} x^2-3xy+2y^2-z^2=31 \\ -x^2+6yz+2z^2=44 \\ x^2+xy+8z^2=100\\ \end{array} </cmath> |
<math>\text{(A)} \ 0 \qquad | <math>\text{(A)} \ 0 \qquad | ||
\text{(B)} \ 1 \qquad | \text{(B)} \ 1 \qquad | ||
| − | \text{(C)} \ 2 \qquad | + | \text{(C)} \ 2 \qquad \\ |
| − | \text{(D)}\ \text{a finite number greater than 2}\qquad | + | \text{(D)}\ \text{a finite number greater than 2}\qquad\\ |
| − | \text{(E)}\ \text{infinitely many} </math> | + | \text{(E)}\ \text{infinitely many} </math> |
| − | |||
== Solution == | == Solution == | ||
| − | <math> | + | Sum of three equations, |
| + | |||
| + | <math>x^2-2xy+2y^2+6yz+9z^2 | ||
| + | = (x-y)^2+(y+3z)^2 = 175</math> | ||
| + | |||
| + | (x,y,z) are integers, ie. <math>175 = a^2 + b^2</math>, | ||
| + | |||
| + | <math>a^2</math>: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169 | ||
| + | <math>b^2</math>: 174, 171, 166, 159, 150, 139, 126, 111, 94, 75, 54, 31, 6 | ||
| + | |||
| + | so there is NO solution | ||
| + | |||
| + | [[User:Wwei.yu|Wwei.yu]] ([[User talk:Wwei.yu|talk]]) 22:09, 28 March 2020 (EDT)Wei | ||
== See also == | == See also == | ||
Latest revision as of 09:14, 15 June 2021
Problem
How many ordered triples (x,y,z) of integers satisfy the system of equations below?
![\[\begin{array}{l} x^2-3xy+2y^2-z^2=31 \\ -x^2+6yz+2z^2=44 \\ x^2+xy+8z^2=100\\ \end{array}\]](http://latex.artofproblemsolving.com/8/7/5/875243b329d45c8a54367576d9f82e8999df15c7.png)
Solution
Sum of three equations,
(x,y,z) are integers, ie. 
,
: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169
: 174, 171, 166, 159, 150, 139, 126, 111, 94, 75, 54, 31, 6
so there is NO solution
Wwei.yu (talk) 22:09, 28 March 2020 (EDT)Wei
See also
| 1980 AHSME (Problems • Answer Key • Resources) | ||
| Preceded by Problem 28 |
Followed by Problem 30 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 | ||
| All AHSME Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
