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