MathDB

14

Part of 2015 AIME Problems

Problems(2)

Integer Area in bounded by inequalities

Source: 2015 AIME I Problem 14

3/20/2015
For each integer n≥2n \ge 2, let A(n)A(n) be the area of the region in the coordinate plane defined by the inequalities 1≤x≤n1\le x \le n and 0≤y≤x⌊x⌋0\le y \le x \left\lfloor \sqrt x \right\rfloor, where ⌊x⌋\left\lfloor \sqrt x \right\rfloor is the greatest integer not exceeding x\sqrt x. Find the number of values of nn with 2≤n≤10002\le n \le 1000 for which A(n)A(n) is an integer.
geometryanalytic geometryinequalities
Polynomial Given Values Find Value of Another Polynomial

Source: 2015 AIME 2 Problem 14

3/26/2015
Let xx and yy be real numbers satisfying x4y5+y4x5=810x^4y^5+y^4x^5=810 and x3y6+y3x6=945x^3y^6+y^3x^6=945. Evaluate 2x3+(xy)3+2y32x^3+(xy)^3+2y^3.
algebrapolynomial