MathDB

2

Part of 1993 Irish Math Olympiad

Problems(2)

good numbers

Source: Ireland 1993

6/29/2009
A positive integer n n is called good good if it can be uniquely written simultaneously as a_1\plus{}a_2\plus{}...\plus{}a_k and as a1a2...ak a_1 a_2...a_k, where ai a_i are positive integers and k≥2 k \ge 2. (For example, 10 10 is good because 10\equal{}5\plus{}2\plus{}1\plus{}1\plus{}1\equal{}5 \cdot 2 \cdot 1 \cdot 1 \cdot 1 is a unique expression of this form). Find, in terms of prime numbers, all good natural numbers.
number theoryprime numbersnumber theory unsolved
real numbers

Source: Ireland 1993

6/29/2009
Let ai,bi a_i,b_i (i\equal{}1,2,...,n) be real numbers such that the ai a_i are distinct, and suppose that there is a real number α \alpha such that the product (a_i\plus{}b_1)(a_i\plus{}b_2)...(a_i\plus{}b_n) is equal to α \alpha for each i i. Prove that there is a real number β \beta such that (a_1\plus{}b_j)(a_2\plus{}b_j)...(a_n\plus{}b_j) is equal to β \beta for each j j.
algebrapolynomialalgebra unsolved