2006 iTest Problems/Problem U6
The following problem is from the Ultimate Question of the 2006 iTest, where solving this problem required the answer of a previous problem. When the problem is rewritten, the T-value is substituted.
Problem
and are nonzero real numbers such that
The smallest possible value of is equal to where and are relatively prime positive integers. Find .
Solutions
Solution 1
Notice that and . This suggests that we can factor the entire expression.
By rearranging terms, we have
By the Zero Product Property, either or . In the first case, . In the second case, the equation can be rearranged into , a circle with center and radius .
The values of and must be on the circle for the second case. In addition, the line connecting and the origin has a slope of . Therefore, we need to find the smallest slope of a line connecting a point on the circle to the origin. That line must be tangent to the circle.
By drawing tangent lines and letting be half the smaller angle formed between the two tangent lines (as seen in diagram), we find that . Using the Double Angle Identity yields , so .
Since the slope of the line tangent to the circle is , the smallest value of is , so .
Solution 2 (credit to NikoIsLife)
Let , and we want to be minimized. Substituting for results in Since , we must have or . In the latter case, The minimum value of is , so .
See Also
2006 iTest (Problems) | ||
Preceded by: Problem U5 |
Followed by: Problem U7 | |
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