Mathcounts National Sprint Round Problems And Solutions Review

Number theory in the Sprint Round rewards knowledge of divisor function and prime factorization. Category 2: Algebra – Systems and Symmetry Problem (Modeled after 2017 National Sprint #27): If (x + y = 8) and (x^2 + y^2 = 34), find the value of (x^3 + y^3).

Now 289 = 17^2. Positive integer factor pairs: (1,289), (17,17), (289,1). Case 1: 3a-17=1 → a=6, then 3b-17=289 → b=102 → sum=108. Case 2: 3a-17=17 → a=34/3 no. Case 3: 3a-17=289 → a=102, then b=6 → same sum 108. Also negative factors? a,b positive so 3a-17> -? Actually if a=1, 3-17=-14, product with negative to get 289, but then b negative. So only positive pairs. Mathcounts National Sprint Round Problems And Solutions

A number with exactly 5 divisors must be of the form (p^4) where (p) is prime (since divisor count = exponent+1, so exponent=4). (p^4 < 100) → (p^4 < 100). (2^4=16), (3^4=81), (5^4=625) (too big). So (n = 16) and (81). That’s 2 numbers. Number theory in the Sprint Round rewards knowledge