MathDB

4

Part of 2023 Bundeswettbewerb Mathematik

Problems(2)

Bundeswettbewerb Mathematik 2023

Source: Bundeswettbewerb Mathematik 2023 Round 1 P4

3/9/2023
Given a real number α\alpha in whose decimal representation α=0,a1a2a3…\alpha=0,a_1a_2a_3\dots each decimal digit aia_i (i=1,2,3,… )(i=1,2,3,\dots) is a prime number. The decimal digits are arranged along the path indicated by arrows in the accompanying figure, which can be thought of as continuing infinitely to the right and downward. For each m≥1m\geq 1, the decimal representation of a real number zmz_m is formed by writing before the decimal point the digit 0 and after the decimal point the sequence of digits of the mm-th row from the top read from left to right from the adjacent arrangement. In an analogous way, for all n≥1n\geq 1, the real numbers sns_n are formed with the digits of the nn-th column from the left to be read from top to bottom. For example, z3=0,a5a6a7a12a23a28…z_3=0,a_5a_6a_7a_{12}a_{23}a_{28}\dots and s2=0,a2a3a6a15a18a35…s_2=0,a_2a_3a_6a_{15}a_{18}a_{35}\dots.
Show:
(a) If α\alpha is rational, then all zmz_m and all sns_n are rational. (b) The converse of the statement formulated in (a) is false.
bundeswettbewercombinatoricsnumber theory
Partitioning vertices of 2n gons in n pairs with distinct distances

Source: Bundeswettbewerb Mathematik 2023, Round 2 - Problem 4

9/8/2023
Exactly nn chords (i.e. diagonals and edges) of a regular 2n2n-gon are coloured red, satisfying the following two conditions:
(1) Each of the 2n2n vertices occurs exactly once as the endpoint of a red chord. (2) No two red chords have the same length.
For which positive integers n≥2n \ge 2 is this possible?
combinatoricscombinatorics proposedconstruction