Mathcounts National Sprint Round Problems And Solutions [cracked] Jun 2026

When practicing, never use $x$. Use numbers. If a problem asks for the probability of rolling a sum of 7 on two dice, don't derive a formula. List the pairs: $(1,6), (2,5), (3,4), (4,3), (5,2), (6,1)$. There are 6 ways. $6/36 = 1/6$. Speed comes from concrete examples, not abstract variables.

⌊20/3⌋+⌊20/9⌋=6+2=8the floor of 20 / 3 end-floor plus the floor of 20 / 9 end-floor equals 6 plus 2 equals 8 So, 3⁸ divides 20!. ⌊20/5⌋=4the floor of 20 / 5 end-floor equals 4 So, 5⁴ divides 20!. For p = 7: ⌊20/7⌋=2the floor of 20 / 7 end-floor equals 2 So, 7² divides 20!. Mathcounts National Sprint Round Problems And Solutions

National problems frequently feature advanced counting techniques. You must master permutations, combinations, casework analysis, complementary counting, and the Principle of Inclusion-Exclusion (PIE). Probability questions often involve geometric probability, conditional probability, or expected value. 3. Number Theory When practicing, never use $x$

Square the original equation: $(x + \frac1x)^2 = 5^2$ $x^2 + 2(x)(\frac1x) + \frac1x^2 = 25$ $x^2 + 2 + \frac1x^2 = 25$ $x^2 + \frac1x^2 = 23$. This takes roughly 15 seconds if a student recognizes the "perfect square" structure. List the pairs: $(1,6), (2,5), (3,4), (4,3), (5,2), (6,1)$