MathDB

6

Part of 2015 AIME Problems

Problems(2)

Angles in Minor Arcs

Source: 2015 AIME I Problem 6

3/20/2015
Point A,B,C,D,A,B,C,D, and EE are equally spaced on a minor arc of a circle. Points E,F,G,H,IE,F,G,H,I and AA are equally spaced on a minor arc of a second circle with center CC as shown in the figure below. The angle ∠ABD\angle ABD exceeds ∠AHG\angle AHG by 12∘12^\circ. Find the degree measure of ∠BAG\angle BAG.[asy] pair A,B,C,D,E,F,G,H,I,O; O=(0,0); C=dir(90); B=dir(70); A=dir(50); D=dir(110); E=dir(130); draw(arc(O,1,50,130)); real x=2*sin(20*pi/180); F=x*dir(228)+C; G=x*dir(256)+C; H=x*dir(284)+C; I=x*dir(312)+C; draw(arc(C,x,200,340)); label("AA",A,dir(0)); label("BB",B,dir(75)); label("CC",C,dir(90)); label("DD",D,dir(105)); label("EE",E,dir(180)); label("FF",F,dir(225)); label("GG",G,dir(260)); label("HH",H,dir(280)); label("II",I,dir(315)); [/asy]
AMCAIMEAIME I
Polynomial

Source: 2015 AIME 2 Problem 6

3/26/2015
Steve says to Jon, "I am thinking of a polynomial whose roots are all positive integers. The polynomial has the form P(x)=2x3−2ax2+(a2−81)x−cP(x)=2x^3-2ax^2+(a^2-81)x-c for some positive integers aa and cc. Can you tell me the values of aa and cc?"
After some calculations, Jon says, "There is more than one such polynomial."
Steve says, "You’re right. Here is the value of aa." He writes down a positive integer and asks, "Can you tell me the value of cc?"
Jon says, "There are still two possible values of cc."
Find the sum of the two possible values of cc.
algebrapolynomial