MathDB

3

Part of 1998 Hungary-Israel Binational

Problems(2)

trinomial and consecutive natural numbers

Source: 9th-th Hungary-Israel Binational Mathematical Competition 1998

7/13/2007
Let a,b,c,m,n a, b, c, m, n be positive integers. Consider the trinomial f(x)=ax2+bx+c f (x) = ax^{2}+bx+c. Show that there exist n n consecutive natural numbers a1,a2,...,an a_{1}, a_{2}, . . . , a_{n} such that each of the numbers f(a1),f(a2),...,f(an) f (a_{1}), f (a_{2}), . . . , f (a_{n}) has at least m m different prime factors.
calculusintegrationnumber theory proposednumber theory
partitions of n into a sum of positive integers

Source: 9th-th Hungary-Israel Binational Mathematical Competition 1998

7/13/2007
Let n n be a positive integer. We consider the set P P of all partitions of n n into a sum of positive integers (the order is irrelevant). For every partition α \alpha, let ak(α) a_{k}(\alpha) be the number of summands in α \alpha that are equal to k,k=1,2,...,n. k, k = 1,2,...,n. Prove that ∑α∈P11a1(α)a1(α)!⋅2a2(α)a2(α)!...nan(α)an(α)!=1. \sum_{\alpha\in P}\frac{1}{1^{a_{1}(\alpha)}a_{1}(\alpha)!\cdot 2^{a_{2}(\alpha)}a_{2}(\alpha)!...n^{a_{n}(\alpha)}a_{n}(\alpha)!}=1.
combinatorics proposedcombinatorics