Difference between revisions of "1989 AHSME Problems/Problem 27"
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<math>=k^5(1+k)(\tbinom{2}{2}+\tbinom{3}{2}k^2+\tbinom{4}{2}k^4+\tbinom{5}{2}k^6+...)</math>, the last part having general term <math>\tbinom{j}{2}k^{2j-4}</math> | <math>=k^5(1+k)(\tbinom{2}{2}+\tbinom{3}{2}k^2+\tbinom{4}{2}k^4+\tbinom{5}{2}k^6+...)</math>, the last part having general term <math>\tbinom{j}{2}k^{2j-4}</math> | ||
from which it is easy to see that, since <math>\tbinom{8}{2}=28</math>, the last part contains the term <math>28k^{12}</math> and the whole result <math>28k^{17}+28k^{18}</math>. So the answer is <math>\mathrm{(D)}</math>. | from which it is easy to see that, since <math>\tbinom{8}{2}=28</math>, the last part contains the term <math>28k^{12}</math> and the whole result <math>28k^{17}+28k^{18}</math>. So the answer is <math>\mathrm{(D)}</math>. | ||
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+ | ==Solution 2== | ||
+ | Suppose <math>n</math> is even. Thus <math>z</math> is even, so let <math>z=2t</math>. Hence <math>x+y+t=\frac{n}{2}</math>. By stars and bars, this has | ||
+ | <cmath>\binom{\frac{n}{2}-1}{3-1}</cmath> | ||
+ | solutions. This implies that <math>\frac{n}{2}-1=8\to n=18</math>. If <math>z</math> is odd, let <math>z=2t-1</math>. Then <math>x+y+t=\frac{n+1}{2}</math>, in which <math>n=17</math> is a solution. The answer is then <math>\mathrm{(D)}</math>. | ||
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+ | ~yofro | ||
== See also == | == See also == |
Latest revision as of 11:39, 5 July 2021
Contents
Problem
Let be a positive integer. If the equation has 28 solutions in positive integers , , and , then must be either
Solution
This is equivalent to finding the powers of with coefficient in the expansion of .
But this is
, the last part having general term from which it is easy to see that, since , the last part contains the term and the whole result . So the answer is .
Solution 2
Suppose is even. Thus is even, so let . Hence . By stars and bars, this has solutions. This implies that . If is odd, let . Then , in which is a solution. The answer is then .
~yofro
See also
1989 AHSME (Problems • Answer Key • Resources) | ||
Preceded by Problem 26 |
Followed by Problem 28 | |
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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