10

Part of 2013 AIME Problems

Problems(2)

Sum of All Possible Sums of Roots of Cubic

Source: 2013 AIME I Problem 10

There are nonzero integers

a

b

r

s

such that the complex number

r+si

is a zero of the polynomial

P(x) = x^3 - ax^2 + bx - 65

. For each possible combination of

a

b

p_{a,b}

be the sum of the zeroes of

P(x)

. Find the sum of the

p_{a,b}

's for all possible combinations of

a

b

algebrapolynomialAMCAIMEAIME I

Maximize [BKL] where KL passes through a fixed point A

Source: AIME II 2013, Problem 10

Given a circle of radius

\sqrt{13}

A

be a point at a distance

4 + \sqrt{13}

from the center

O

of the circle. Let

B

be the point on the circle nearest to point

A

. A line passing through the point

A

intersects the circle at points

K

L

. The maximum possible area for

\triangle BKL

can be written in the form

\tfrac{a-b\sqrt{c}}{d}

a

b

c

d

are positive integers,

a

d

are relatively prime, and

c

is not divisible by the square of any prime. Find

a+b+c+d

geometryLaTeXnumber theoryrelatively primeAMCAIME