MathDB

2

Part of 1992 Vietnam Team Selection Test

Problems(2)

|f(P(x))| <= c

Source: Vietnam TST 1992 for the 33nd IMO, problem 2

6/25/2005
Let a polynomial f(x)f(x) be given with real coefficients and has degree greater or equal than 1. Show that for every real number c>0c > 0, there exists a positive integer n0n_0 satisfying the following condition: if polynomial P(x)P(x) of degree greater or equal than n0n_0 with real coefficients and has leading coefficient equal to 1 then the number of integers xx for which ∣f(P(x))∣≤c|f(P(x))| \leq c is not greater than degree of P(x)P(x).
algebrapolynomialalgebra unsolved
x^2 + y^2 - 5xy + 5 = 0

Source: Vietnam TST 1992 for the 33nd IMO, problem 5

6/25/2005
Find all pair of positive integers (x,y)(x, y) satisfying the equation x2+y2−5⋅x⋅y+5=0.x^2 + y^2 - 5 \cdot x \cdot y + 5 = 0.
quadraticsinvariantsymmetrynumber theory unsolvednumber theory