Home > Olympiad > Vojtech Jarnik Competition 2015 – Category 1 Problems

Vojtech Jarnik Competition 2015 – Category 1 Problems

Problem 1. Let {f:\Bbb{R} \rightarrow \Bbb{R}} be differentiable on {\Bbb{R}}. Prove that there exists {x \in [0,1]} such that

\displaystyle \frac{4}{\pi}(f(1)-f(0))=(1+x^2)f'(x).

Problem 2. Consider the infinite chessboard whose rows and columns are indexed by positive integers. Is it possible to put a single positive rational number into each cell of the chessboard so that each positive rational number appears exactly once and the sum of every row and of every column is finite?

Problem 3. Let {P(x)=x^{2015}-2x^{2014}+1} and {Q(x) = x^{2015}-2x^{2014}-1}. Determine for each of the polynomials {P} and {Q} whether it is a divisor of some nonzero polynomial {c_0+c_1x+...+c_nx^n} whose coefficients {c_i} are all in the set {\{1,-1\}}.

Problem 4. Let {m} be a positive integer and let {p} be a prime divisor of {m}. Suppose that the complex polynomial {a_0+a_1x+...+a_nx^n} with {n< \displaystyle \frac{p}{p-1}\varphi(m)} and {a_n \neq 0} is divisible by the cyclotomic polynomial {\Phi_m(x)}. Prove that there at least {p} non-zero coefficients {a_i}.

The cyclotomic polynomial {\Phi_m(x)} is the monic polynomial whose roots are the {m}-th primitive complex roots of unity. Euler’s totient function {\varphi(m)} denotes the number of positive integers less than or eual to {m} which are coprime to {m}.

Categories: Olympiad Tags: , ,
  1. No comments yet.
  1. No trackbacks yet.

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s

%d bloggers like this: