International Mathematics Competition
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IMC 2019
  Problems & Solutions

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.)


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 ext{for} quad n=0,1,2,ldots )

Prove that the following series is convergent:

(displaystyle sum_{n=0}^infty|a_{n+1}-a_n|. )(displaystyle (1) )

(Proposed by Orif Ibrogimov, National University of Uzbekistan)


Problem 8. Let (displaystyle Omega={(x,y,z)in mathbb{Z}^3: y+1ge xge yge zge 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)


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)


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

(displaystyle lim_{R ightarrow infty} sum_{(a,b) in mathcal{D}_R} rac{(-1)^{a+b}}{a^2+b^2}.)

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