Home > Uncategorized > IMC 2017 – Day 1

IMC 2017 – Day 1

Problem 1. Determine all complex numbers {\lambda} for which there exists a positive integer {n} and a real {n\times n} matrix {A} such that {A^2 = A^T} and {\lambda} is an eigenvalue of {A}.

Problem 2. Let {f: \Bbb{R} \rightarrow (0,\infty)} be a differentiable function and suppose that there exists a constant {L>0} such that

\displaystyle |f'(x)-f'(y)|\leq L|x-y|

for all {x,y}. Prove that

\displaystyle (f'(x))^2 < 2Lf(x)

holds for all {x}.

Problem 3. For any positive integer {m} denote by {P(m)} the product of positive divisors of {m}. For every positive integer {n} define the sequence

\displaystyle a_1(n)=n, \ \ \ a_{k+1}(n) = P(a_k(n)) \ \ (k=1,2,...,2016).

Determine whether for every set {S \subseteq \{1,2,...,2017\}} there exists a positive integer {n} such that the following condition is satisfied:

For every {k} with {1\leq k \leq 2017} the number {a_k(n)} is a perfect square if and only if {k \in S}.

Problem 4. There are {n} people in a city and each of them has exactly {1000} friends (friendship is mutual). Prove that it is possible to select a group {S} of people such that at least {n/2017} persons in {S} have exactly two friends in {S}.

Problem 5. Let {k} and {n} be positive integers with {n \geq k^2-3k+4} and let

\displaystyle f(z) = z^{n-1}+c_{n-2}z^{n-2}+...+c_0

be a polynomial with complex coefficients such that

\displaystyle c_0c_{n-2} = c_1c_{n-3} = ... = c_{n-2}c_0 = 0.

Prove that {f(z)} and {z^n-1} have at most {n-k} common roots.

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