2006 SMT/Team Problems
Contents
[hide]Problem 1
Given , where is at , is at , and is on the positive axis. Cone is formed when is rotated about the axis, and cone is formed when is rotated about the axis. If the volume of cone minus the volume fo cone is , find the length of .
Problem 2
In a given sequence , for terms , . For example, if the first two elements are and , respectively, the third entry would be , and the fourth would be , and so on. Given that a sequence of integers having this form starts with , and the element is , what is the second element?
Problem 3
A triangle has altitudes of lengths and . What is the maximum length of the third altitude?
Problem 4
Let and . The expression can be written as a polynomial in terms of and . What is this polynomial?
Problem 5
There exist two positive numbers such that . Find the product of the two possible .
Problem 6
The expression is equivalent to the expression for all positive integers where and are functions and is constant. Find .
Problem 7
Let be the set of all -tuples that satisfy and . If one of these -tuples is chosen at random, what's the probability that or is greater than or equal to ?
Problem 8
Evaluate:
Problem 9
has . Points and are midpoints of and , respectively. The medians and intersect at a right angle. Find .
Problem 10
Find the smallest positive integer for which there are at least even and odd positive integers so that is an integer.
Problem 11
Polynomial has roots . The coefficients satisfy for all pairs of integers . Given that , determine .
(Note: The original problem asked for , but the official solution makes it clear that the actual desired sum is .)
Problem 12
Find the total number of -tuples of positive integers so that for each , and regular polygons with numbers of sides respectively will fit into a tesselation at a point. That is, the sum of one interior angle from each of the polygons is .
Problem 13
A ray is drawn from the origin tangent to the graph of the upper part of the hyperbola in the first quadrant. This ray makes an angle of with the positive axis. Compute .
Problem 14
Find the smallest nonnegative integer for which is divisible by .
Problem 15
Let denote the composite integer so that . Compute
(Hint: )
(Note: The original hint stated that .)