Mathcounts National Sprint Round Problems And Solutions !free! Jun 2026

How many three-digit integers are divisible by 3 or 5?

: For comprehensive step-by-step solutions to past National Sprint and Target rounds, several authoritative books are available: Mathcounts National Competition Solutions (2011–2016) Mathcounts National Sprint Round Problems And Solutions

Draw rectangle ABCD with AB = 12 (horizontal), AD = 5 (vertical). Let A = (0,0), B = (12,0), C = (12,5), D = (0,5). "From one corner to the midpoint of the opposite side": Choose corner A(0,0). Opposite side is BC? No – the side opposite A is side CD. The midpoint of CD: C(12,5) to D(0,5) → midpoint M = (6,5). Distance AM = (\sqrt{(6-0)^2 + (5-0)^2} = \sqrt{36 + 25} = \sqrt{61}). So answer is (\sqrt{61}) cm. How many three-digit integers are divisible by 3 or 5

The Sprint Round is a test of raw speed, accuracy, and mental arithmetic. Unlike the Target Round (which allows calculators and has fewer, multi-step problems), the Sprint Round demands rapid recognition of mathematical patterns. The problems increase in difficulty from #1 (warm-up) to #30 (notoriously tricky). A typical National competition sees top finishers solving 25–28 problems correctly; a perfect score of 30 is exceptionally rare. "From one corner to the midpoint of the

Similar triangles, coordinate geometry, and 3D volume ratios.