MathDB

4

Part of 2008 Saint Petersburg Mathematical Olympiad

Problems(3)

St Petersburgh 2008 #4

Source:

7/23/2011
The numbers x1,...x100x_1,...x_{100} are written on a board so that x1=12 x_1=\frac{1}{2} and for every nn from 11 to 9999, xn+1=1−x1x2x3∗...∗x100x_{n+1}=1-x_1x_2x_3*...*x_{100}. Prove that x100>0.99x_{100}>0.99.
inductioninequalities proposedinequalities
Guessing a number

Source: St Petersburg 2008 10th grade #4

7/28/2011
A wizard thinks of a number from 11 to nn. You can ask the wizard any number of yes/no questions about the number. The wizard must answer all those questions, but not necessarily in the respective order. What is the least number of questions that must be asked in order to know what the number is for sure. (In terms of nn.)
Fresh translation.
number theory proposednumber theory
Numbers on circle

Source: St Petersburg Olympiad 2008, Grade 11, P4

8/30/2017
There are 100100 numbers on circle, and no one number is divided by other. In same time for all numbers we make next operation: If (a,b)(a,b) are two neighbors (aa is left neighbor) , then we write between a,ba,b number a(a,b)\frac{a}{(a,b)} and erase a,ba,b This operation was repeated some times. What maximum number of 11 we can receive ?
Example: If we have circle with 33 numbers 4,5,64,5,6 then after operation we receive circle with numbers 4(4,5)=4,5(5,6)=5,6(6,4)=3\frac{4}{(4,5)}=4,\frac{5}{(5,6)}=5, \frac{6}{(6,4)}=3.
number theory