MathDB

13

Part of 2019 Durer Math Competition Finals

Problems(2)

x_1 +k/x_2=x_2 +k/x_3=...=x_n +k/x_1

Source: 2019 Dürer Math Competition Finals Day2 E13 https://artofproblemsolving.com/community/c1621835_2019_

1/5/2022
Let k>1k > 1 be a positive integer and n≥2019n \ge 2019 be an odd positive integer. The non-zero rational numbers x1,x2,...,xnx_1, x_2,..., x_n are not all equal, and satisfy the following chain of equalities: x1+kx2=x2+kx3=x3+kx4=...=xn−1+kxn=xn+kx1.x_1 +\frac{k}{x_2}= x_2 +\frac{k}{x_3}= x_3 +\frac{k}{x_4}= ... = x_{n-1} +\frac{k}{x_n}= x_n +\frac{k}{x_1}. What is the smallest possible value of kk?
algebrasystem of equationsnumber theorySystem
12 numbered chairs arranged in a circle

Source: 2019 Dürer Math Competition Finals Day2 E+13 https://artofproblemsolving.com/community/c1621835_2019_

1/5/2022
There are 1212 chairs arranged in a circle, numbered from 1 1 to 12 12. How many ways are there to select some of the chairs in such a way that our selection includes 33 consecutive chairs somewhere?
combinatorics