Mathcounts National Sprint Round Problems And Solutions May 2026

The Mysterious Sprint Round

While the MATHCOUNTS syllabus is broad, the National Sprint Round consistently focuses on four primary pillars of competitive middle school math:

Problem 3: A circle with center O has a radius of 5 cm. Two chords, AB and CD, intersect at point E. If AE = 8 and EB = 4, what is the length of CD? Mathcounts National Sprint Round Problems And Solutions

Cyclic Quadrilaterals: Utilizing Ptolemy’s Theorem or power of a point.

Look for symmetry or sequences in geometry and number theory problems to simplify calculations. No Rounding: The Mysterious Sprint Round While the MATHCOUNTS syllabus

Answer: (\boxed108)

How many positive integers less than 100 are divisible by 3 or 5 but not by both? Problem 3: The Perfect Square Sneak (Difficulty: Hard)

Problem 3: The Perfect Square Sneak (Difficulty: Hard)

Problem (based on 2018 Sprint #25):
How many three-digit integers ( \overlineabc ) (with ( a \neq 0 )) are such that ( \overlineab + \overlinebc ) is a perfect square?

Solution Approach:
Do not square 25 and 24 separately (that wastes time). Use the difference of squares:
[ a^2 - b^2 = (a-b)(a+b) ]
Here, ( a=25, b=24 ):
[ (25-24)(25+24) = (1)(49) = 49 ]
Answer: 49