Let be a real number

MATHEMATICS · Multi-part · AoPS · Mathematics · Special Functions · Special Functions

Problem

Let xx be a real number. Let [x][x] denote the greatest integer function, {x}\{x\} the fractional part, and ⌈x⌉\lceil x\rceil the least integer function. Solve the following. (i) ⌈x⌉2=[x]2+2x\lceil x\rceil^2=[x]^2+2x (ii) [2x]−2x=[x+1][2x]-2x=[x+1] (iii) [x2]+2[x]=3x[x^2]+2[x]=3x, where 0≤x≤20\leq x\leq2 (iv) y=4−[x]2y=4-[x]^2 and [y]+y=6[y]+y=6 (v) [x]+∣x−2∣≤0[x]+|x-2|\leq0 and −1≤x≤3-1\leq x\leq3

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