International Mathematics Competition
for University Students
2018

IMC2018: Problems on Day 2

Problem 6. Let $\displaystyle k$ be a positive integer. Find the smallest positive integer $\displaystyle n$ for which there exist $\displaystyle k$ nonzero vectors $\displaystyle v_1,\ldots,v_k$ in $\displaystyle \mathbb R^n$ such that for every pair $\displaystyle i,j$ of indices with $\displaystyle |i-j|> 1$ the vectors $\displaystyle v_i$ and $\displaystyle v_j$ are orthogonal.

(Proposed by Alexey Balitskiy, Moscow Institute of Physics and Technology and M.I.T.)

  Hint    Solution  

Problem 7. Let $\displaystyle (a_n)_{n=0}^\infty$ be a sequence of real numbers such that $\displaystyle a_0=0$ and

$\displaystyle a_{n+1}^3=a_n^2-8 \quad \text{for} \quad n=0,1,2,\ldots$

Prove that the following series is convergent:

(Proposed by Orif Ibrogimov, National University of Uzbekistan)

  Hint    Solution  

Problem 8. Let $\displaystyle \Omega=\{(x,y,z)\in \mathbb{Z}^3: y+1\ge x\ge y\ge z\ge 0\}$. A frog moves along the points of $\displaystyle \Omega$ by jumps of length $\displaystyle 1$. For every positive integer $\displaystyle n$, determine the number of paths the frog can take to reach $\displaystyle (n,n,n)$ starting from $\displaystyle (0,0,0)$ in exactly $\displaystyle 3n$ jumps.

(Proposed by Fedor Petrov and Anatoly Vershik, St. Petersburg State University)

  Hint    Solution  

Problem 9. Determine all pairs $\displaystyle P(x)$, $\displaystyle Q(x)$ of complex polynomials with leading coefficient $\displaystyle 1$ such that $\displaystyle P(x)$ divides $\displaystyle Q(x)^2+1$ and $\displaystyle Q(x)$ divides $\displaystyle P(x)^2+1$.

(Proposed by Rodrigo Angelo, Princeton University and Matheus Secco, PUC, Rio de Janeiro)

  Hint    Solution  

Problem 10. For $\displaystyle R>1$ let $\displaystyle \mathcal{D}_R = \{(a,b)\in\mathbb{Z}^2 \colon 0<a^2+b^2<R\}$. Compute

$\displaystyle \lim_{R\rightarrow \infty} \sum_{(a,b) \in \mathcal{D}_R} \frac{(-1)^{a+b}}{a^2+b^2}.$

(Proposed by Rodrigo Angelo, Princeton University and Matheus Secco, PUC, Rio de Janeiro)

  Hint    Solution