MathDB

Problem 1

Part of 2002 Federal Math Competition of S&M

Problems(3)

x that satisfy floor inequality

Source: S&M 2002 1st Grade P1

5/14/2021
Determine all real numbers xx such that 2002⌊x⌋⌊−x⌋+x>⌊2x⌋x−⌊1+x⌋.\frac{2002\lfloor x\rfloor}{\lfloor-x\rfloor+x}>\frac{\lfloor2x\rfloor}{x-\lfloor1+x\rfloor}.
algebrafloor functioninequalities
inequality with natural parameters

Source: S&M 2002 3&4th Grade P1

5/15/2021
For any positive numbers a,b,ca,b,c and natural numbers n,kn,k prove the inequality an+kbn+bn+kcn+cn+kan≥ak+bk+ck.\frac{a^{n+k}}{b^n}+\frac{b^{n+k}}{c^n}+\frac{c^{n+k}}{a^n}\ge a^k+b^k+c^k.
inequalities
max{z} if x^2≥y+z & cyclic

Source: S&M 2002 2nd Grade P1

5/14/2021
Real numbers x,y,zx,y,z satisfy the inequalities x2≤y+z,y2≤z+xz2≤x+y.x^2\le y+z,\qquad y^2\le z+x\qquad z^2\le x+y.Find the minimum and maximum possible values of zz.
inequalities