Difference between revisions of "1980 USAMO Problems/Problem 3"
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Newton Sums gives the following relationship between sums of the form <math>P_k=a^k+b^k+c^k</math> | Newton Sums gives the following relationship between sums of the form <math>P_k=a^k+b^k+c^k</math> | ||
− | <cmath>P_k | + | <cmath>P_k-S_1P_{k-1}+S_2P_{k-2}-S_3P_{k-3}=0</cmath> |
Where <math>S_1=a+b+c</math>, <math>S_2=ab+bc+ac</math>, and <math>S_3=abc</math>. It is given that <math>P_0, P_1, P_2</math> are real. Note that if <math>P_{k-1}, P_{k-2}, P_{k-3}</math> are real, then clearly <math>P_k</math> is real because all other parts of the above equation are real, completing the induction. | Where <math>S_1=a+b+c</math>, <math>S_2=ab+bc+ac</math>, and <math>S_3=abc</math>. It is given that <math>P_0, P_1, P_2</math> are real. Note that if <math>P_{k-1}, P_{k-2}, P_{k-3}</math> are real, then clearly <math>P_k</math> is real because all other parts of the above equation are real, completing the induction. | ||
Latest revision as of 11:20, 6 May 2017
Problem
is an integral multiple of
.
and
are real numbers. If
, show that
for any positive integer
.
Solution
Let ,
,
be numbers in the complex plane.
Note that implies
which is real. Also note that
are the imaginary parts of
and that
are the imaginary parts of
by de Moivre's Theorem. Therefore,
and
are real because their imaginary parts sum to zero.
Finally, note that is real as well.
It suffices to show that is real for all positive integer
, which can be shown by induction.
Newton Sums gives the following relationship between sums of the form
Where
,
, and
. It is given that
are real. Note that if
are real, then clearly
is real because all other parts of the above equation are real, completing the induction.
See Also
1980 USAMO (Problems • Resources) | ||
Preceded by Problem 2 |
Followed by Problem 4 | |
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.