MathDB

4

Part of 1987 Bundeswettbewerb Mathematik

Problems(2)

Bundeswettbewerb Mathematik 1987 Problem 1.4

Source: Bundeswettbewerb Mathematik 1987 Round 1

10/9/2022
Place the integers 1,2,…,n31,2 , \ldots, n^{3} in the cells of a n×n×nn\times n \times n cube such that every number appears once. For any possible enumeration, write down the maximal difference between any two adjacent cells (adjacent means having a common vertex). What is the minimal number noted down?
geometry3D geometrycombinatoricsDifferencecubeGeneralization
Bundeswettbewerb Mathematik 1987 Problem 2.4

Source: Bundeswettbewerb Mathematik 1987 Round 2

10/9/2022
Let 1<k≤n1<k\leq n be positive integers and x1,x2,…,xkx_1 , x_2 , \ldots , x_k be positive real numbers such that x1⋅x2⋅…⋅xk=x1+x2+…+xk.x_1 \cdot x_2 \cdot \ldots \cdot x_k = x_1 + x_2 + \ldots +x_k.
a) Show that x1n−1+x2n−1+…+xkn−1≥kn.x_{1}^{n-1} +x_{2}^{n-1} + \ldots +x_{k}^{n-1} \geq kn.
b) Find all numbers k,nk,n and x1,x2,…,xkx_1, x_2 ,\ldots , x_k for which equality holds.
Inequalityequalityalgebrainequalities