6

Part of 2013 Tuymaada Olympiad

Problems(2)

A Diophantine equation in prime numbers

Source: Tuymaada 2013, Day 2, Problem 7 Juniors and 6 Seniors

Solve the equation

p^2-pq-q^3=1

in prime numbers.
A. Golovanov

LaTeXmodular arithmeticnumber theory proposednumber theory

Quadratic trinomials in an array

Source: Tuymaada 2013, Day 2, Problem 6 Juniors

Quadratic trinomials with positive leading coefficients are arranged in the squares of a

6 \times 6

108

coefficients are all integers from

-60

47

(each number is used once). Prove that at least in one column the sum of all trinomials has a real root.
K. Kokhas & F. Petrov

quadraticsabsolute valuealgebra proposedalgebra