MathDB

1

Part of 2019 South East Mathematical Olympiad

Problems(2)

Non-homogeneous and asymmetric

Source: 2019 CSMO Grade 10 P1

7/30/2019
Find the largest real number kk, such that for any positive real numbers a,ba,b, (a+b)(ab+1)(b+1)≥kab2(a+b)(ab+1)(b+1)\geq kab^2
algebraInequalityChina
Does there exists sequence satisfying (mod 10) relation?

Source: 2019 CSMO Grade 11 Problem 1

7/30/2019
Let [a][a] represent the largest integer less than or equal to aa, for any real number aa. Let {a}=a−[a]\{a\} = a - [a].
Are there positive integers m,nm,n and n+1n+1 real numbers x_0,x_1,\hdots,x_n such that x0=428x_0=428, xn=1928x_n=1928, xk+110=[xk10]+m+{xk5}\frac{x_{k+1}}{10} = \left[\frac{x_k}{10}\right] + m + \left\{\frac{x_k}{5}\right\} holds?
Justify your answer.
algebra