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2023 Bundeswettbewerb Mathematik

Part of Bundeswettbewerb Mathematik

Subcontests

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algebra problem

Determine all triples (x,y,z)(x, y, z) of integers that satisfy the equation x2+y2+z2−xy−yz−zx=3x^2+ y^2+ z^2 - xy - yz - zx = 3

Maximal diameter of graph with minimum degree 42

A hilly island has 20232023 lookouts. It is known that each of them is in line of sight with at least 4242 of the other lookouts. For any two distinct lookouts XX and YY there is a positive integer nn and lookouts A1,A2,…,An+1A_1,A_2,\dots,A_{n+1} such that A1=XA_1=X and An+1=YA_{n+1}=Y and A1A_1 is in line of sight with A2A_2, A2A_2 with A3A_3, …\dots and AnA_n with An+1A_{n+1}. The smallest such number nn is called the viewing distance of XX and YY.
Determine the largest possible viewing distance that can exist between two lookouts under these conditions.
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Bundeswettbewerb Mathematik 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.

Partitioning vertices of 2n gons in n pairs with distinct distances

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?
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2023 Bundeswettbewerb Mathematik

Tick, Trick and Track have 20, 23 and 25 tickets respectively for the carousel at the fair in Duckburg. They agree that they will only ride all three together, for which they must each give up one of their tickets. Also, before a ride, if they want, they can redistribute tickets among themselves as many times as they want according to the following rule: If one has an even number of tickets, he can give half of his tickets to any of the other two. Can it happen that after any trip: (a) exactly one has no ticket left, (b) exactly two have no ticket left, (c) all tickets are given away?

GCD of p^6-7p^2+6 where p runs through prime numbers

Determine the greatest common divisor of the numbers p6−7p2+6p^6-7p^2+6 where pp runs through the prime numbers p≥11p \ge 11.