MathDB

2

Part of 1996 Irish Math Olympiad

Problems(2)

sum of digits

Source: Ireland 1996

7/1/2009
Let S(n) S(n) denote the sum of the digits of a natural number n n (in base 10 10). Prove that for every n n, S(2n)≤2S(n)≤10S(2n) S(2n) \le 2S(n) \le 10S(2n). Prove also that there is a positive integer n n with S(n)\equal{}1996S(3n).
number theory proposednumber theory
inequality with integers

Source: Ireland 1996

7/1/2009
Show that for every positive integer n n, 212⋅414⋅818⋅...⋅(2n)12n<4 2^{\frac{1}{2}} \cdot 4^{\frac{1}{4}} \cdot 8^{\frac{1}{8}} \cdot ... \cdot (2^n)^{\frac{1}{2^n}}<4.
inequalitiesinequalities proposed