MathDB

5

Part of 2023 Czech-Polish-Slovak Junior Match

Problems(2)

S<- S+1/S

Source: 2023 Czech-Polish-Slovak Match Junior, individual p5 CPSJ

5/5/2024
Bartek patiently performs operations on fractions. In each move, he adds its inverse to the current result, obtaining a new result. Bartek starts with the number 11: after the first move, he receives the result 2, after the second move, the result is 52\frac{5}{2}, after the third move 2910\frac{29}{10}, etc. After 300300 moves, Bartek receives the result xx. Determine the largest integer not greater than xx.
algebra
x,y,z -> ..., xy + yz + zx = 1000, digits related

Source: 2023 Czech-Polish-Slovak Match Junior, Team p5 CPSJ

5/5/2024
Mazo performs the following operation on triplets of non-negative integers: If at least one of them is positive, it chooses one positive number, decreases it by one, and replaces the digits in the units place with the other two numbers. It starts with the triple xx, yy, zz. Find a triple of positive integers xx, yy, zz such that xy+yz+zx=1000xy + yz + zx = 1000 (*) and the number of operations that Mazo can subsequently perform with the triple x,y,zx, y, z is
(a) maximal (i.e. there is no triple of positive integers satisfying (*) that would allow him to do more operations);
(b) minimal (i.e. every triple of positive integers satisfying (*) allows him to perform at least so many operations).
number theoryDigits