Mathcounts National Sprint Round Problems And Solutions Site

Spend roughly 1.5 to 2 minutes per problem. If you get stuck on a calculation for more than 60 seconds, circle it and move on.

s=5+7+82=10s equals the fraction with numerator 5 plus 7 plus 8 and denominator 2 end-fraction equals 10

N≡33(mod35)cap N triple bar 33 space open paren mod space 35 close paren

National-level combinatorics often requires tracking complex constraints. Expect to encounter problems involving the Principle of Inclusion-Exclusion (PIE), geometric probability, expected value, and advanced permutations where items are indistinguishable. 2. Number Theory and Modular Arithmetic

144=122=(22⋅3)2=24⋅32144 equals 12 squared equals open paren 2 squared center dot 3 close paren squared equals 2 to the fourth power center dot 3 squared Using the divisor formula, we add to each exponent and multiply the results: Mathcounts National Sprint Round Problems And Solutions

r=5+12−132r equals the fraction with numerator 5 plus 12 minus 13 and denominator 2 end-fraction r=42=2r equals four-halves equals 2 Key Strategies for Sprint Round Success

Never tackle a Sprint Round sequentially from problem 1 to 30 without a plan. Divide your 40 minutes using a three-pass system:

A circle is inscribed inside a right triangle with side lengths of 5, 12, and 13. What is the radius of the inscribed circle (inradius)? Solution: There are multiple ways to find the inradius (

You can't "study" for Nationals; you have to "train." Use these resources to find past National Sprint Rounds: 2025 Chapter Competition - Sprint Round Problems 1−30 Spend roughly 1

Ep(x!)=∑k=1∞⌊xpk⌋cap E sub p open paren x exclamation mark close paren equals sum from k equals 1 to infinity of the floor of the fraction with numerator x and denominator p to the k-th power end-fraction end-floor

Success at the National level demands specialized training systems. Individual brilliance must be paired with mechanical efficiency.

Unlike the Chapter or State levels, the National Sprint Round features problems that often blend multiple disciplines—geometry, number theory, and combinatorics—into a single question. You have exactly 80 seconds per problem.

The National Sprint Round separates the strong from the elite. Consistent practice with old MATHCOUNTS and AMC 8 problems is the best preparation. Focus on speed without sacrificing accuracy—every correct answer moves you up the leaderboard. Expect to encounter problems involving the Principle of

The Mathcounts National Competition represents the absolute pinnacle of middle school mathematics in the United States. For elite young mathematicians, reaching this level is the culmination of hundreds of hours of rigorous preparation. Among the various stages of the tournament, the Sprint Round is arguably the purest test of speed, accuracy, and mathematical intuition.

(4−(−r))2+(3−6)2=r2⟹(4+r)2+9=r2⟹16+8r+r2+9=r2⟹8r=-25open paren 4 minus open paren negative r close paren close paren squared plus open paren 3 minus 6 close paren squared equals r squared ⟹ open paren 4 plus r close paren squared plus 9 equals r squared ⟹ 16 plus 8 r plus r squared plus 9 equals r squared ⟹ 8 r equals negative 25 , which gives a negative radius. Let's re-verify the coordinates. . The perpendicular bisector of AMcap A cap M must intersect the line .Midpoint of AMcap A cap M . Slope of AMcap A cap M .Slope of perpendicular bisector is 43four-thirds .Equation of perpendicular bisector: .Find intersection with

Permutations and combinations at the national level go far beyond simple grid-walking or coin-tossing. You will encounter advanced casework, geometric probability, the Principle of Inclusion-Exclusion (PIE), and stars-and-bars techniques for distributing items. 3. High-Level Algebra

y1+y2+y3+y4+y5+5=10y sub 1 plus y sub 2 plus y sub 3 plus y sub 4 plus y sub 5 plus 5 equals 10