Week 1 Challenging Problems

Problems for the week

Deadline 19.04.2020

Problem 1.

Given any set S of 9 points within a unit square, show that there always exists three distinct points in S such that the area of the triangle formed by the three points is less than or equal to \frac{1}{8}. (Combinatorics)

Problem 2.

For every positive integer n, prove that

    \[\frac{\sigma(1)}{1} + \frac{\sigma(2)}{2}+\dots+\frac{\sigma(n)}{n} \leq 2n\]

where \sigma(n) is the sum of all the divisors of n. (Number Theory)

Problem 3.

Does there exist non-linear polynomials P and Q such that P(Q(x))=(x-1)(x-2)\dots (x-15)? (Polynomials)

Problem 4.

If f is a twice differentiable function and f'' is continuous on an open interval in \mathbb{R}, then prove that

    \[\lim_{h\to{0}} \frac{f(x+h)-2f(x)+f(x-h)}{h^2} = f''(x).\]


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