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Summary: Estimating Sums
June 27, 2006
1. Given positive real numbers x1, x2, . . . , xn whose sum is an integer, prove that one can
choose a nonempty proper sublist of the xi such that the fractional part of the sum of
this sublist is at most 1/n.
2. (IMO, 1987) Let x1, x2, . . . , xn be real numbers satisfying x2
1 + x2
2 + · · · + x2
n = 1.
Prove that for every integer k 2 there are integers a1, a2, . . . , an, not all zero, such
that |ai| k - 1 for all i, and |a1x1 + a2x2 + · · · + anxn| (k - 1)
n/(kn
- 1).
3. Given a set of n nonnegative numbers whose sum is 1, prove that there exist two
disjoint subsets, not both empty, whose sums differ by at most 1/(2n
- 1). Is this
bound optimal for every n?
4. Let n be an odd positive integer and let x1, . . . , xn, y1, . . . , yn be nonnegative real
numbers satisfying x1 + · · · + xn = y1 + · · · + yn. Show that there exists a proper,
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