MathDB

3

Part of 1993 Iran MO (2nd round)

Problems(2)

Find the smallest positive integer m[Iran Second Round 1993]

Source:

11/25/2010
Let n,rn, r be positive integers. Find the smallest positive integer mm satisfying the following condition. For each partition of the set {1,2,…,m}\{1, 2, \ldots ,m \} into rr subsets A1,A2,…,ArA_1,A_2, \ldots ,A_r, there exist two numbers aa and bb in some Ai,1≤i≤rA_i, 1 \leq i \leq r, such that 1<ab<1+1n. 1 < \frac ab < 1 +\frac 1n.
pigeonhole principlenumber theory proposednumber theory
The fraction f(x)/g(x) [Iran Second Round 1993]

Source: 1993 Iran Second Round

11/25/2010
Let f(x)f(x) and g(x)g(x) be two polynomials with real coefficients such that for infinitely many rational values of xx, the fraction f(x)g(x)\frac{f(x)}{g(x)} is rational. Prove that f(x)g(x)\frac{f(x)}{g(x)} can be written as the ratio of two polynomials with rational coefficients.
algebrapolynomialratioalgebra proposed