What is your when practicing under timed conditions? Share public link
How many positive integers less than 100 are divisible by 3 or 5 but not by both?
Let ( a_1 = 3 ). ( a_2 = 2(3) + 4 = 10 ) ( a_3 = 2(10) + 4 = 24 ) ( a_4 = 2(24) + 4 = 52 ) ( a_5 = 2(52) + 4 = 108 )
The right side of the equation is now a standard, infinite geometric series with a first term ( 13one-third and a common ratio ( 13one-third . We apply the infinite geometric sum formula Mathcounts National Sprint Round Problems And Solutions
Here are a few more challenging problems from the Mathcounts National Sprint Round:
A=s(s−a)(s−b)(s−c)=21(21−14)(21−13)(21−15)cap A equals the square root of s open paren s minus a close paren open paren s minus b close paren open paren s minus c close paren end-root equals the square root of 21 open paren 21 minus 14 close paren open paren 21 minus 13 close paren open paren 21 minus 15 close paren end-root
National-level problems are distinct from school or chapter problems because they frequently require: What is your when practicing under timed conditions
This is an arithmetico-geometric series. A standard and highly efficient algebraic trick to solve this involves setting the sum equal to a variable, shifting it, and subtracting. be the sum of the series:
Mastering the Mathcounts National Sprint Round: Strategies, Problems, and Solutions
The AoPS Wiki is the most extensive community-driven resource, featuring an archive of problems and solutions for past National Sprint Rounds. ( a_2 = 2(3) + 4 = 10
On any given roll, there are 6 possible outcomes, each with a 16one-sixth
First, look at the first two congruences. From (1), we can write for some integer . Substitute this expression for
Hard — Algebra / clever substitution Problem: Solve for real x: x + sqrt(1 + x^2) = 3. Key insight: Let y = sqrt(1 + x^2). Then y - x = 1/ (x + y) *? (Better: isolate: sqrt(1 + x^2) = 3 - x. Square both sides carefully.) Square: 1 + x^2 = 9 - 6x + x^2 → 1 = 9 - 6x → 6x = 8 → x = 4/3. Check: RHS sqrt = sqrt(1 + 16/9) = sqrt(25/9)=5/3; LHS sum = 4/3 + 5/3 = 3 ✓. Answer: 4/3