Difference between revisions of "1991 AIME Problems/Problem 15"
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− | == Solution == | + | == Solution 1 (Geometric Interpretation)== |
− | + | Consider <math>n</math> right triangles joined at their vertices, with bases <math>a_1,a_2,\ldots,a_n</math> and heights <math>1,3,\ldots, 2n - 1</math>. The sum of their hypotenuses is the value of <math>S_n</math>. The minimum value of <math>S_n</math>, then, is the length of the straight line connecting the bottom vertex of the first right triangle and the top vertex of the last right triangle, so | |
<cmath> | <cmath> | ||
S_n \ge \sqrt {\left(\sum_{k = 1}^n (2k - 1)\right)^2 + \left(\sum_{k = 1}^n a_k\right)^2}. | S_n \ge \sqrt {\left(\sum_{k = 1}^n (2k - 1)\right)^2 + \left(\sum_{k = 1}^n a_k\right)^2}. |
Revision as of 01:43, 25 September 2020
Problem
For positive integer , define
to be the minimum value of the sum
where
are positive real numbers whose sum is 17. There is a unique positive integer
for which
is also an integer. Find this
.
Solution 1 (Geometric Interpretation)
Consider right triangles joined at their vertices, with bases
and heights
. The sum of their hypotenuses is the value of
. The minimum value of
, then, is the length of the straight line connecting the bottom vertex of the first right triangle and the top vertex of the last right triangle, so
Since the sum of the first
odd integers is
and the sum of
is 17, we get
If this is an integer, we can write
, for an integer
. Thus,
The only possible value, then, for
is
, in which case
, and
.
Solution 2
The inequality
is a direct result of the Minkowski Inequality. Continue as above.
Solution 3
Let for
and
. We then have that
Note that that
.
Note that for any angle
, it is true that
and
are reciprocals. We thus have that
. By the AM-HM inequality on these
values, we have that:
This is thus the minimum value, with equality when all the tangents are equal. The only value for which
is an integer is
(see above solutions for details).
See also
1991 AIME (Problems • Answer Key • Resources) | ||
Preceded by Problem 14 |
Followed by Last question | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
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