1

Part of 1990 Bundeswettbewerb Mathematik

Problems(2)

f(x) =x^2 + 2bx + c >= for all integers n, then f(x) >= 0

Source: Germany Federal - Bundeswettbewerb Mathematik 1990, round 1, p1

Consider the trinomial

f(x) = x^2 + 2bx + c

with integer coefficients

b

c

. Prove that if

f(n) \ge 0

for all integers

n

f(x) \ge 0

even for all rational numbers

x

trinomialquadratic polynomialInteger Polynomialalgebra

Product of two integers is congruent 1 mod the third

Source: 1990 Bundeswettbewerb Mathematik, Round 2 - #1

Find all triples

(a,b,c)

of positive integers such that the product of any two of them when divided by the third leaves the remainder

1

remaindernumber theory