MathDB

7

Part of 2013 Saint Petersburg Mathematical Olympiad

Problems(3)

St.Peterburg, P7 Grade 11, 2013

Source:

4/17/2014
In the language of wolves has two letters FF and PP, any finite sequence which forms a word. А word YY is called 'subpart' of word XX if Y is obtained from X by deleting some letters (for example, the word FFPFFFPF has 8 'subpart's: F, P, FF, FP, PF, FFP, FPF, FFF). Determine nn such that the nn is the greatest number of 'subpart's can have n-letter word language of wolves.
F. Petrov, V. Volkov
combinatorics proposedcombinatorics
Divisors of naturals

Source: St Petersburg Olympiad 2013, Grade 10, P7

10/13/2017
Let a1,a2a_1,a_2 - two naturals, and 1<b1<a1,1<b2<a21<b_1<a_1,1<b_2<a_2 and b1∣a1,b2∣a2b_1|a_1,b_2|a_2. Prove that a1b1+a2b2−1a_1b_1+a_2b_2-1 is not divided by a1a2a_1a_2
number theory
St.Peterburg, P7 Grade 9, 2013

Source:

4/27/2014
Given is a natural number aa with 5454 digits, each digit equal to 00 or 11. Prove the remainder of aa when divide by 33⋅34⋯39 33\cdot 34\cdots 39 is larger than 100000100000. (It's mean: a≡r(mod33⋅34⋯39)a \equiv r \pmod{33\cdot 34\cdots 39 } with 0<r<33⋅34⋯39 0<r<33\cdot 34\cdots 39 then prove that r>100000r>100000 ) M. Antipov
modular arithmeticnumber theory proposednumber theory