MathDB

2

Part of 2010 Mathcenter Contest

Problems(2)

exists n such that the k-th digit from the right of 2^n is d

Source: Mathcenter Contest / Oly - Thai Forum 2010 R1 p2 https://artofproblemsolving.com/community/c3196914_mathcenter_contest

11/15/2022
Let kk and dd be integers such that k>1k>1 and 0≤d<90\leq d<9. Prove that there exists some integer nn such that the kkth digit from the right of 2n2^n is dd.
(tatari/nightmare)
number theoryDigitspower of 2
|{x^n}-a|&lt;=1/ 2(p+q)

Source: Mathcenter Contest / Oly - Thai Forum 2010 R2 p2 https://artofproblemsolving.com/community/c3196914_mathcenter_contest

11/15/2022
A positive rational number xx is called banzai if the following conditions are met: ∙\bullet x=\frac{p}{q}>1 where p,qp,q are comprime natural numbers ∙\bullet exist constants α,N\alpha,N such that for all integers n≥Nn\geq N,∣{ xn}−α∣≤12(p+q).\mid \left\{\,x^n\right\} -\alpha\mid \leq \dfrac{1}{2(p+q)}. Find the total number of banzai numbers.
Note:{ x}\left\{\,x\right\} means fractional part of xx
(tatari/nightmare)
fractional partnumber theoryalgebra