MathDB

6

Part of 2022 LMT Spring

Problems(2)

2022 Spring LMT Team 6

Source:

9/1/2022
For all yy, define cubic fy(x)f_y (x) such that fy(0)=yf_y (0) = y, fy(1)=y+12f_y (1) = y +12, fy(2)=3y2f_y (2) = 3y^2, fy(3)=2y+4f_y (3) = 2y +4. For all yy, fy(4)f_y(4) can be expressed in the form ay2+by+cay^2 +by +c where a,b,ca,b,c are integers. Find a+b+ca +b +c.
algebra
2022 LMT Spring Accuracy Problem 6

Source:

9/29/2023
Jacob likes to watchMickeyMouse Clubhouse! One day, he decides to create his own MickeyMouse head shown below, with two circles ω1\omega_1 and ω2\omega_2 and a circle ω\omega, and centers O1O_1, O2O_2, and OO, respectively. Let ω1\omega_1 and ω\omega meet at points P1P_1 and Q1Q_1, and let ω2\omega_2 and ω\omega meet at points P2P_2 and Q2Q_2. Point P1P_1 is closer to O2O_2 than Q1Q_1, and point P2P_2 is closer to O1O_1 than Q2Q_2. Given that P1P_1 and P2P_2 lie on O1O2O_1O_2 such that O1P1=P1P2=P2O2=2O_1P_1 = P_1P_2 = P_2O_2 = 2, and Q1O1∥Q2O2Q_1O_1 \parallel Q_2O_2, the area of ω\omega can be written as nπn \pi. Find nn. https://cdn.artofproblemsolving.com/attachments/6/d/d98a05ee2218e80fd84d299d47201669736d99.png
geometryLMT