MathDB

Problem 4

Part of 2012 VJIMC

Problems(2)

n^k starts/ends with same digit

Source: VJIMC 2012 1.4

5/31/2021
Find all positive integers nn for which there exists a positive integer kk such that the decimal representation of nkn^k starts and ends with the same digit.
number theoryDigits
inequality in 7 variables, [1,4]

Source: VJIMC 2012 2.4

6/1/2021
Let a,b,c,x,y,z,ta,b,c,x,y,z,t be positive real numbers with 1≤x,y,z≤41\le x,y,z\le4. Prove that x(2a)t+y(2b)t+z(2c)t≥y+z−x(b+c)t+z+x−y(c+a)t+x+y−z(a+b)t.\frac x{(2a)^t}+\frac y{(2b)^t}+\frac z{(2c)^t}\ge\frac{y+z-x}{(b+c)^t}+\frac{z+x-y}{(c+a)^t}+\frac{x+y-z}{(a+b)^t}.
inequalities