MathDB

11

Part of 2021 AIME Problems

Problems(2)

2021 AIME I/11

Source: 2021 AIME I/11

3/11/2021
Let ABCDABCD be a cyclic quadrilateral with AB=4,BC=5,CD=6,AB=4,BC=5,CD=6, and DA=7DA=7. Let A1A_1 and C1C_1 be the feet of the perpendiculars from AA and CC, respectively, to line BD,BD, and let B1B_1 and D1D_1 be the feet of the perpendiculars from BB and D,D, respectively, to line ACAC. The perimeter of A1B1C1D1A_1B_1C_1D_1 is mn\frac mn, where mm and nn are relatively prime positive integers. Find m+nm+n.
AMCAIME
Class full of MOPpers

Source: 2021 AIME II Problem 11

3/19/2021
A teacher was leading a class of four perfectly logical students. The teacher chose a set SS of four integers and gave a different number in SS to each student. Then the teacher announced to the class that the numbers in SS were four consecutive two-digit positive integers, that some number in SS was divisible by 66, and a different number in SS was divisible by 77. The teacher then asked if any of the students could deduce what SS is, but in unison, all of the students replied no.
However, upon hearing that all four students replied no, each student was able to determine the elements of SS. Find the sum of all possible values of the greatest element of SS.
AIMEAIME II