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?
S=13+29+327+481+…cap S equals one-third plus two-nineths plus 3 over 27 end-fraction plus 4 over 81 end-fraction plus …
Forget basic area formulas. You will need mass points, Ptolemy’s Theorem, power of a point, and advanced coordinate geometry to solve the final third of the test. Real Mathcounts National Sprint Problems & Solutions
While the Team Round permits calculators, the Sprint Round requires clean geometric logic. Common topics include cyclic quadrilaterals, similarities in right triangles, mass point geometry, and the application of Ptolemy’s or Stewart’s theorems. The Value of Step-by-Step Solutions Mathcounts National Sprint Round Problems And Solutions
The best solution manuals offer multiple pathways to a single answer—such as an algebraic proof paired alongside a visual or geometric interpretation. This dual-lens approach equips students with backup strategies if their primary method stalls during the live test. Exemplar Problem & Solution Analysis
be the probability that the sum of the rolls up to the current point is a multiple of 3 (congruent to P1cap P sub 1 be the probability that the sum leaves a remainder of 1 ( P2cap P sub 2 be the probability that the sum leaves a remainder of 2 (
Finding comprehensive text-based archives for MATHCOUNTS National Sprint Round problems can be tricky since the organization often protects this content behind its official store or registration. However, there are several official and reliable ways to access these problems and their solutions for practice. Problem 3: A circle with center O has a radius of 5 cm
Modular arithmetic is a fundamental tool at the national level. Problems heavily test prime factorization traits, the Chinese Remainder Theorem, Euler's Totient Function, and trailing zeros in base systems. 4. Geometry
22−2=r(2+1)2 the square root of 2 end-root minus 2 equals r open paren the square root of 2 end-root plus 1 close paren
Calculators are strictly prohibited.Points are awarded only for correct answers.There is no penalty for incorrect guesses.The problems generally increase in difficulty as the round progresses. such as advanced number theory
The number of total positive divisors is found by adding 1 to each exponent and multiplying the results:
The problems start relatively straightforward and become increasingly difficult, with the final 5–10 problems often featuring concepts rarely taught in middle school curriculum, such as advanced number theory, complex combinatorics, or intricate geometry.
The AoPS Wiki is the most extensive community-driven resource, featuring an archive of problems and solutions for past National Sprint Rounds.
Modular arithmetic, divisibility rules, the Chinese Remainder Theorem, and prime factorization properties populate the exam. Problems often ask for the units digit of a massive exponent or the number of trailing zeros of a factorial. 4. Euclidean Geometry
Unlike the Target Round, calculators are not allowed during the Sprint Round.