Difference between revisions of "1985 USAMO Problems/Problem 1"
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==Solution== | ==Solution== | ||
− | {{ | + | Lemma: For a positive integer <math>n</math>, <math>1^3+2^3+\cdots +n^3 = (1+2+\cdots +n)^2</math> (Also known as Nicomachus's theorem) |
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+ | Proof by induction: The identity holds for <math>1</math>. Suppose the identity holds for a number <math>n</math>. It is well known that the sum of first <math>n</math> positive integers is <math>\frac{n(n+1)}{2} = \frac{n^2+n}{2}</math>. Thus its square is <math>\frac{n^4+2n^3+n^2}{4}</math>. Adding <math>(n+1)^3=n^3+3n^2+3n+1</math> to this we get <math>\frac{n^4+6n^3+13n^2+12n+4}{4}</math>, which can be rewritten as <math>\frac{(n^4+4n^3+6n^2+4n+1)+2(n^3+3n^2+3n+1)+(n^2+2n+1)}{4}</math> This simplifies to <math>\frac{(n+1)^4+2(n+1)^3+(n+1)^2}{4} = ({\frac{(n+1)^2+(n+1)}{2}})^2 = (1+2+\cdots +n+(n+1))^2</math>. The induction is complete. | ||
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+ | Let <math>j</math> be the sum <math>1+2+\cdots 1985</math>, and let <math>k</math> be the sum <math>1^2 + 2^2 + \cdots + 1985^2</math>. Then assign <math>x_i</math> the value <math>ik^4</math> for each <math>i = 1, 2,\cdots 1985</math>. Then: | ||
+ | <cmath> | ||
+ | \begin{align*} | ||
+ | x_1^2 +x_2^2 +\cdots +x_{1985}^2 & = 1^2k^8 +2^2k^8+\cdots +1985^2k^8 = k^8(1^2+2^2+\cdots +1985^2) = k^9 = {(k^3)}^3\\ | ||
+ | x_1^3 +x_2^3 +\cdots +x_{1985}^3 & = 1^3k^{12}+2^3k^{12}+\cdots 1985^3k^{12}=k^{12}(1^3+2^3+\cdots 1985^3) = k^{12}j^2 = ({k^6j})^2 | ||
+ | \end{align*} | ||
+ | </cmath> | ||
+ | |||
+ | Thus, a positive integral solution exists. | ||
+ | |||
+ | -Circling | ||
==See Also== | ==See Also== |
Latest revision as of 03:18, 23 October 2022
Problem
Determine whether or not there are any positive integral solutions of the simultaneous equations with distinct integers .
Solution
Lemma: For a positive integer , (Also known as Nicomachus's theorem)
Proof by induction: The identity holds for . Suppose the identity holds for a number . It is well known that the sum of first positive integers is . Thus its square is . Adding to this we get , which can be rewritten as This simplifies to . The induction is complete.
Let be the sum , and let be the sum . Then assign the value for each . Then:
Thus, a positive integral solution exists.
-Circling
See Also
1985 USAMO (Problems • Resources) | ||
Preceded by First Problem |
Followed by Problem 2 | |
1 • 2 • 3 • 4 • 5 | ||
All USAMO Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.