1. Show that infinitely many positive integers cannot be written as a sum of three fourth powers of integers.
2. 6 open disks in the plane are such that the center of no disk lies inside another. Show that no point lies inside all 6 disks.
3. A polynomial with integer coefficients takes the value 5 at five distinct integers. Show that it does not take the value 9 at any integer.
4. Let p(x) = (x - x1)(x - x2)(x - x3), where x1, x2 and x3 are real. Show that p(x) p''(x) ≤ p'(x)2 for all x.
5. A 3 x 1 paper rectangle is folded twice to give a square side 1. The square is folded along a diagonal to give a right-angled triangle. A needle is driven through an interior point of the triangle, making 6 holes in the paper. The paper is then unfolded. Where should the point be in order to maximise the smallest distance between any two holes?
6. Show that (n - m)!/m! ≤ (n/2 + 1/2)n-2m for positive integers m, n with 2m ≤ n.
Solutions
Problem 1
Show that infinitely many positive integers cannot be written as a sum of three fourth powers of integers.
Solution
Squares must be 0 or 1 mod 4, so any number = 3 mod 4 is not even the sum of three squares.
Problem 3
A polynomial with integer coefficients takes the value 5 at five distinct integers. Show that it does not take the value 9 at any integer.
Solution
The polynomial can be written as (x-a)(x-b)(x-c)(x-d)(x-e)q(x) + 5 for some distinct integers a, b, c, d, e. Now let x be any other integer (not a, b, c, d, e). If q(x) = 0, then the value is 5, not 9. If q(x) ≠ 0, then |q(x)| ≥ 1. At most two of |x-a|, |x-b|, |x-c|, |x-d|, |x-e| can be 1 and at most two can be 2. So their product is certainly > 4. But to get 9 we must have (x-a)(x-b)(x-c)(x-d)(x-e)q(x) = 4.
Problem 6
Show that (n - m)!/m! ≤ (n/2 + 1/2)n-2m for positive integers m, n with 2m ≤ n.
Solution
Put n = 2m+r, so we have to show that (m+r)(m+r-1)...(m+1) ≤ (m + (r+1)/2)r. But this is just AM/GM applied to m+r, ... m+1.
Thanks to Suat Namli
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