The center of the incircle, the center of the smaller circle, and the vertex are collinear, lying on the angle bisector of .The distance from to the center of the large incircle is Set up the Ratio: Let the radius of the smaller circle be . The distance from to the center of the smaller circle is .The distance from to the large center can also be written as the sum of: The distance from to the small center ( The radius of the small circle ( The radius of the large circle ( Therefore:

: Every problem is worth the same point value. It's far better to get the first 20 problems 100% correct than to rush through and make careless mistakes. A common piece of advice from past participants is to always read the problem carefully to avoid "silly mistakes".

To understand the rigor of the competition, let us analyze three representative problems inspired by the upper-tier difficulty (Problems 20–30) of historical Mathcounts National Sprint Rounds. Problem 1: Number Theory (Divisibility & Factorization)

23S=1323=12two-thirds cap S equals one-third over two-thirds end-fraction equals one-half Now, isolate by multiplying both sides by 32three-halves

a3+b3+c3−15=18a cubed plus b cubed plus c cubed minus 15 equals 18 a3+b3+c3=33a cubed plus b cubed plus c cubed equals 33 33 National-Level Preparation Strategies

The "no calculator" rule is the great equalizer. The Mathcounts National Sprint Round problems and solutions rely heavily on number sense, algebraic manipulation, spatial reasoning, and clever shortcuts—not computational brute force.

If a problem requires a long case-by-case analysis, skip it. The points for #2 and #30 are worth the exact same.