MathDB

4

Part of 2022 Cyprus JBMO TST

Problems(3)

Find number in 2022-th position out of several four-digit numbers

Source: Cyprus 2022 Junior TST-1 Problem 4

2/21/2022
Consider the digits 1,2,3,4,5,6,71, 2, 3, 4, 5, 6, 7. (a) Determine the number of seven-digit numbers with distinct digits that can be constructed using the digits above. (b) If we place all of these seven-digit numbers in increasing order, find the seven-digit number which appears in the 2022th2022^{\text{th}} position.
combinatoricscounting
Maximal subset with |1/x-1/y| >= 1/1000

Source: Cyprus 2022 Junior TST-2 Problem 4

2/21/2022
Let AA be a subset of {1,2,3,…,50}\{1, 2, 3, \ldots, 50\} with the property: for every x,y∈Ax,y\in A with x≠yx\neq y, it holds that ∣1x−1y∣>11000.\left| \frac{1}{x}- \frac{1}{y}\right|>\frac{1}{1000}. Determine the largest possible number of elements that the set AA can have.
number theorypigeonhole principle
Replace a and b with a-2b

Source: Cyprus 2022 Junior TST-3 Problem 4

3/26/2022
The numbers 1,2,3,…,101, 2, 3, \ldots , 10 are written on the blackboard. In each step, Andrew chooses two numbers a,ba, b which are written on the blackboard such that a⩾2ba\geqslant 2b, he erases them, and in their place writes the number a−2ba-2b.
Find all numbers nn, such that after a sequence of steps as above, at the end only the number nn will remain on the blackboard.
combinatoricsinvariant