MathDB

6.

Part of 2020 Saint Petersburg Mathematical Olympiad

Problems(3)

a_{n+2}=a_n(a_{n+1}+1) then (a_n)^n |a_{a_n}

Source: Saint Petersburg MO 2020 Grade 11 Problem 6

5/7/2020
The sequence ana_n is given as a_1=1, a_2=2 \;\;\; \text{and} \;\;\;\; a_{n+2}=a_n(a_{n+1}+1)   \forall n\geq 1 Prove that aana_{a_n} is divisible by (an)n(a_n)^n for n≥100n\geq 100.
number theory
a connected social network

Source: Saint Petersburg MO 2020 Grade 10 Problem 6

5/7/2020
On a social network, no user has more than ten friends ( the state "friendship" is symmetrical). The network is connected: if, upon learning interesting news a user starts sending it to its friends, and these friends to their own friends and so on, then at the end, all users hear about the news. Prove that the network administration can divide users into groups so that the following conditions are met:
[*] each user is in exactly one group [*] each group is connected in the above sense [*] one of the groups contains from 11 to 100100 members and the remaining from 100100 to 900900.
combinatorics
(1,1),(2,3),(4,5) and (999,111) exists point on y=2x

Source: Saint Petersburg MO 2020 Grade 9 Problem 6

5/7/2020
The points (1,1),(2,3),(4,5)(1,1),(2,3),(4,5) and (999,111)(999,111) are marked in the coordinate system. We continue to mark points in the following way :
[*]If points (a,b)(a,b) are marked then (b,a)(b,a) and (a−b,a+b)(a-b,a+b) can be marked [*]If points (a,b)(a,b) and (c,d)(c,d) are marked then so can be (ad+bc,4ac−4bd)(ad+bc, 4ac-4bd).
Can we, after some finite number of these steps, mark a point belonging to the line y=2xy=2x.
algebracombinatoricsinvariantanalytic geometrynumber theory