MathDB

6

Part of 2011 AIME Problems

Problems(2)

Parabola

Source:

3/18/2011
Suppose that a parabola has vertex (14,−98)\left(\tfrac{1}{4},-\tfrac{9}{8}\right), and equation y=ax2+bx+cy=ax^2+bx+c, where a>0a>0 and a+b+ca+b+c is an integer. The minimum possible value of aa can be written as pq,\tfrac{p}{q}, where pp and qq are relatively prime positive integers. Find p+qp+q.
conicsparabolaanalytic geometrygeometrygeometric transformationnumber theoryrelatively prime
Interesting ordered quadruples

Source: 2011 AIME II #6

3/31/2011
Define an ordered quadruple of integers (a,b,c,d)(a, b, c, d) as interesting if 1≤a<b<c<d≤101 \le a<b<c<d \le 10, and a+d>b+ca+d>b+c. How many ordered quadruples are there?
probabilityAMCAIME