(a1 + ... + am)/m ≥ (n + 1)/2.
|
A1. Let m and n be positive integers. Let a1, a2, ... , am be distinct elements of {1, 2, ... , n} such that whenever ai + aj ≤ n for some i, j (possibly the same) we have ai + aj = ak for some k. Prove that:
(a1 + ... + am)/m ≥ (n + 1)/2. |
|
| A2. ABC is an isosceles triangle with AB = AC. M is the midpoint of BC and O is the point on the line AM such that OB is perpendicular to AB. Q is an arbitrary point on BC different from B and C. E lies on the line AB and F lies on the line AC such that E, Q, F are distinct and collinear. Prove that OQ is perpendicular to EF if and only if QE = QF. |
|
| A3. For any positive integer k, let f(k) be the number of elements in the set {k+1, k+2, ... , 2k} which have exactly three 1s when written in base 2. Prove that for each positive integer m, there is at least one k with f(k) = m, and determine all m for which there is exactly one k. |
|
| B1. Determine all ordered pairs (m, n) of positive integers for which (n3 + 1)/(mn - 1) is an integer. |
|
| B2. Let S be the set of all real numbers greater than -1. Find all functions f from S into S such that f(x + f(y) + xf(y)) = y + f(x) + yf(x) for all x and y, and f(x)/x is strictly increasing on each of the intervals -1 < x < 0 and 0 < x. |
|
| B3. Show that there exists a set A of positive integers with the following property: for any infinite set S of primes, there exist two positive integers m in A and n not in A, each of which is a product of k distinct elements of S for some k ≥ 2. |
|