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Part of 1994 IMC

Problems(2)

IMC 1994 D1 P3

Source:

3/6/2017
Given a set SS of 2n−12n-1, n∈Nn\in \mathbb N, different irrational numbers. Prove that there are nn different elements x1,x2,…,xn∈Sx_1, x_2, \ldots, x_n\in S such that for all non-negative rational numbers a1,a2,…,ana_1, a_2, \ldots, a_n with a1+a2+…+an>0a_1+a_2+\ldots + a_n>0 we have that a1x1+a2x2+⋯+anxna_1x_1+a_2x_2+\cdots +a_nx_n is an irrational number.
IMCirrational numbercollege contestsreal analysis
IMC 1994 D2 P3

Source:

3/6/2017
Let ff be a real-valued function with n+1n+1 derivatives at each point of R\mathbb R. Show that for each pair of real numbers aa, bb, a<ba<b, such that ln⁡(f(b)+f′(b)+⋯+f(n)(b)f(a)+f′(a)+⋯+f(n)(a))=b−a\ln\left( \frac{f(b)+f'(b)+\cdots + f^{(n)} (b)}{f(a)+f'(a)+\cdots + f^{(n)}(a)}\right)=b-a there is a number cc in the open interval (a,b)(a,b) for which f(n+1)(c)=f(c)f^{(n+1)}(c)=f(c)
IMCreal analysisfunctioncalculusderivative