ITMC 2026 Grade 8 National Preliminary sample-r1 Question 13

For real $x$, $[x]$ denotes the greatest integer not exceeding $x$ (the integer part). The decimal or fractional part is $\{x\}=x-[x]$, so $x=[x]+\{x\}$. Examples: $[3]=3$, $[6.25]=6$, $[-6.25]=-7$, $[22/7]=3$. The source lists: (1) $0\lt \{x\}\lt 1$ for every real $x$; (2) $\{x\}=0$ exactly when $x$ is an integer; (3) $x-1\lt [x]\le x\lt [x]+1$; (4) $[n+x]=n+[x]$ for positive integers $n$; (5) $[-x]=-[x]-1$ if $x$ is not an integer, and $[-x]=-[x]$ if $x$ is an integer. Editorial correction: property (1) must read $0\le\{x\}\lt 1$; its printed strict lower bound contradicts (2). The definition of $[x]$ is used below.

Find $[x]$ if $x-\dfrac1{2020}\lt -2019\lt x$.