Difference between revisions of "1980 AHSME Problems/Problem 29"
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== Solution == | == Solution == | ||
− | + | Sum of three equations, | |
+ | |||
+ | x^2-2xy+2y^2+6yz+9z^2 | ||
+ | = (x-y)^2+(y+3z)^2 = 175 | ||
+ | |||
+ | (x,y,z) are integers, ie. 175 = a^2 + b^2, | ||
+ | |||
+ | a^2: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169 | ||
+ | b^2: 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 == |
Revision as of 22:09, 28 March 2020
Problem
How many ordered triples (x,y,z) of integers satisfy the system of equations below?
Solution
Sum of three equations,
x^2-2xy+2y^2+6yz+9z^2 = (x-y)^2+(y+3z)^2 = 175
(x,y,z) are integers, ie. 175 = a^2 + b^2,
a^2: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169 b^2: 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 |
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