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Part of 2017 CentroAmerican

Problems(2)

Centroamerican Math Olympiad - 2017 P2

Source: OMCC 2017

6/23/2017
We call a pair (a,b)(a,b) of positive integers, a<391a<391, pupusa if
lcm(a,b)>lcm(a,391)\textup{lcm}(a,b)>\textup{lcm}(a,391)
Find the minimum value of bb across all pupusa pairs.
Fun Fact: OMCC 2017 was held in El Salvador. Pupusa is their national dish. It is a corn tortilla filled with cheese, meat, etc.
OMCCOMCC 2017CENTROalgebranumber theory
Centroamerican Math Olympiad 2017 - P5

Source: OMCC 2017

6/23/2017
Susana and Brenda play a game writing polynomials on the board. Susana starts and they play taking turns.
1) On the preparatory turn (turn 0), Susana choose a positive integer n0n_0 and writes the polynomial P0(x)=n0P_0(x)=n_0.
2) On turn 1, Brenda choose a positive integer n1n_1, different from n0n_0, and either writes the polynomial
P1(x)=n1x+P0(x) or P1(x)=n1x−P0(x)P_1(x)=n_1x+P_0(x) \textup{ or } P_1(x)=n_1x-P_0(x)
3) In general, on turn kk, the respective player chooses an integer nkn_k, different from n0,n1,…,nk−1n_0, n_1, \ldots, n_{k-1}, and either writes the polynomial
Pk(x)=nkxk+Pk−1(x) or Pk(x)=nkxk−Pk−1(x)P_k(x)=n_kx^k+P_{k-1}(x) \textup{ or } P_k(x)=n_kx^k-P_{k-1}(x)
The first player to write a polynomial with at least one whole whole number root wins. Find and describe a winning strategy.
OMCCOMCC 2017CENTROGame Theoryalgebrapolynomialcombinatorics