Difference between revisions of "1993 UNCO Math Contest II Problems/Problem 6"
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== Solution == | == Solution == | ||
+ | (a) We can rewrite the given equation as <math>y^2-x^2 = 25+36 = 61</math>. Use difference of squares to obtain <math>(y + x)(y - x) = 61</math>. Since <math>61</math> is a prime we conclude that <math>(y + x) = 61 \text{ and } (y - x) = 1</math>, giving us <math>x = 30 \text{ and } y = 31</math>. | ||
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+ | |||
+ | (b) It is not too hard to notice that the LHS above is <math>n^2 + (n+1)^2 + (n(n+1))^2</math> and the RHS above is <math>(n(n+1)+1)^2</math> for <math>n = 2, 3, 4 \text{ and } 5</math>. We will prove that the LHS <math>=</math> RHS for all integers (although the proof extends to real numbers) in (c). | ||
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+ | (c) We expand the LHS to obtain | ||
+ | <cmath>\begin{align*} | ||
+ | n^2 + n^2 + 2n + 1 + n^2(n^2 + 2n + 1) &= n^4 + 2n^3 + 3n^2 + 2n + 1 \\ | ||
+ | &= (n^4 + n^3 + n^2) + (n^3 + n^2 + n) + (n^2 + n + 1) \\ | ||
+ | &= (n^2 + n + 1)^2 \\ | ||
+ | \end{align*}</cmath> | ||
+ | Thus LHS = RHS and we are done. | ||
+ | ~AK2006 | ||
== See also == | == See also == | ||
{{UNCO Math Contest box|year=1993|n=II|num-b=5|num-a=7}} | {{UNCO Math Contest box|year=1993|n=II|num-b=5|num-a=7}} | ||
− | [[Category: | + | [[Category:Introductory Algebra Problems]] |
Latest revision as of 22:13, 19 August 2021
Problem
Observe that
(a) Find integers and
so that
(b) Conjecture a general rule that is being illustrated here.
(c) Prove your conjecture.
Solution
(a) We can rewrite the given equation as . Use difference of squares to obtain
. Since
is a prime we conclude that
, giving us
.
(b) It is not too hard to notice that the LHS above is and the RHS above is
for
. We will prove that the LHS
RHS for all integers (although the proof extends to real numbers) in (c).
(c) We expand the LHS to obtain
Thus LHS = RHS and we are done.
~AK2006
See also
1993 UNCO Math Contest II (Problems • Answer Key • Resources) | ||
Preceded by Problem 5 |
Followed by Problem 7 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 | ||
All UNCO Math Contest Problems and Solutions |