Mathcounts National Sprint Round Problems And Solutions -

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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). Mathcounts National Sprint Round Problems And Solutions

Since (b>0), (3a-17 >0 \Rightarrow a \ge 6). Also integer: (3a-17) divides (17a). Use division: (17a = 17/3*(3a-17) + 289/3) – messy. Instead, rewrite: (b = \frac17a3a-17 = 5 + \frac853a-17) after polynomial division? Let’s check: Divide 17a by (3a-17): quotient 5 (since 5*(3a-17)=15a-85, remainder 2a+85? No, do carefully: (17a) / (3a-17) = 5 + (2a+85)/(3a-17)? That doesn’t help. Better: Set (k = 3a-17), then (a = (k+17)/3), substitute into b: (b = \frac17(k+17)/3k = \frac17k+2893k = \frac173 + \frac2893k). For b integer, (3k) must divide 289 = 17^2. Thus (3k) is a divisor of 289: 1, 17, 289. But 3k positive, k = (3a-17) >0. 3k=1→k=1/3 no. 3k=17→k=17/3 no. 3k=289→k=289/3 no. So no integer k? That means I made an algebraic slip. (\boxed4464) Number theory in the Sprint Round rewards

Memorize symmetric polynomial identities. They save precious seconds. Category 3: Geometry – The Diagram is a Trap Problem (Modeled after 2016 National Sprint #28): In rectangle ABCD, AB = 8, BC = 15. Point E lies on side CD such that CE = 5. Lines AE and BD intersect at F. Find the area of triangle BEF. Also integer: (3a-17) divides (17a)

Then (x^3 + y^3 = (x+y)(x^2 - xy + y^2) = 8 \cdot (34 - 15) = 8 \cdot 19 = 152).