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MATHCOUNTS 2026 State Target Round (Problem 5)

For every positive integer nn, let Q(n)Q(n) be the sum of the greatest integers less than or equal to the square roots of the integers 1,2,3,…,n1,2,3,\ldots,n. For example, Q(3)=1+1+1=3Q(3)=1+1+1=3 and Q(10)=1+1+1+2+2+2+2+2+3+3=19Q(10)=1+1+1+2+2+2+2+2+3+3=19. What is the least positive integer nn for which Q(1)+Q(2)+Q(3)+⋯+Q(n)≥2026?Q(1)+Q(2)+Q(3)+\cdots+Q(n)\ge2026?