Here are examples of the type of problems found on the National Sprint Round, demonstrating the logic needed to solve them. Problem 1: Number Theory (Advanced)
: Maintain a dedicated notebook specifically for mistakes. Copy down every problem missed during practice, rewrite the step-by-step solution by hand, and revisit that exact problem one week later to ensure the concept has stuck.
Line AE: from A(0,0) to E(3,15): slope = 15/3=5, equation y=5x. Line BD: from B(8,0) to D(0,15): slope = (15-0)/(0-8) = -15/8, equation: y = (-15/8)(x-8) = (-15/8)x + 15. Mathcounts National Sprint Round Problems And Solutions
23S=13+19+127+181+…two-thirds cap S equals one-third plus one-nineth plus 1 over 27 end-fraction plus 1 over 81 end-fraction plus …
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A cube with side length 4 is painted red on all sides. It is then cut into 64 small 1×1×1 cubes. How many of the small cubes have exactly two red faces? Solution: The small cubes with exactly two red faces are located along the edges of the original 4×4×4 cube, excluding the corner cubes. A cube has 12 edges. Along each edge, there are 4 small cubes.
A bag contains 5 red marbles, 3 blue marbles, and 2 green marbles. If two marbles are drawn at random without replacement, what is the probability that both are the same color? Line AE: from A(0,0) to E(3,15): slope =
Mastering the MATHCOUNTS National Sprint Round: Problems, Strategies, and Solutions
┌────────────────────────────────────────────────────────┐ │ MATHCOUNTS SPRINT ROUND │ ├───────────────────┬────────────────────────────────────┤ │ Number of Items │ 30 Problems │ ├───────────────────┼────────────────────────────────────┤ │ Time Allowed │ 40 Minutes │ ├───────────────────┼────────────────────────────────────┤ │ Average Time │ 80 Seconds Per Problem │ ├───────────────────┼────────────────────────────────────┤ │ Calculator Use │ Strictly Prohibited │ └───────────────────┴────────────────────────────────────┘
N≡33(mod35)cap N triple bar 33 space open paren mod space 35 close paren