MathDB

24

Part of 2014 AMC 12/AHSME

Problems(2)

Iterated Absolute Value Function

Source: 2014 AMC 12A #24

2/5/2014
Let f0(x)=x+∣x−100∣−∣x+100∣f_0(x)=x+|x-100|-|x+100|, and for n≥1n\geq 1, let fn(x)=∣fn−1(x)∣−1f_n(x)=|f_{n-1}(x)|-1. For how many values of xx is f100(x)=0f_{100}(x)=0?
<spanclass=′latex−bold′>(A)</span>299<spanclass=′latex−bold′>(B)</span>300<spanclass=′latex−bold′>(C)</span>301<spanclass=′latex−bold′>(D)</span>302<spanclass=′latex−bold′>(E)</span>303<span class='latex-bold'>(A) </span>299\qquad <span class='latex-bold'>(B) </span>300\qquad <span class='latex-bold'>(C) </span>301\qquad <span class='latex-bold'>(D) </span>302\qquad <span class='latex-bold'>(E) </span>303\qquad
functionAMC
Diagonal Sum in Cyclic Pentagon

Source: 2014 AMC 12B #24

2/20/2014
Let ABCDEABCDE be a pentagon inscribed in a circle such that AB=CD=3AB=CD=3, BC=DE=10BC=DE=10, and AE=14AE=14. The sum of the lengths of all diagonals of ABCDEABCDE is equal to mn\frac{m}{n}, where mm and nn are relatively prime positive integers. What is m+nm+n? <spanclass=′latex−bold′>(A)</span>129<spanclass=′latex−bold′>(B)</span>247<spanclass=′latex−bold′>(C)</span>353<spanclass=′latex−bold′>(D)</span>391<spanclass=′latex−bold′>(E)</span>421<span class='latex-bold'>(A) </span>129\qquad <span class='latex-bold'>(B) </span>247\qquad <span class='latex-bold'>(C) </span>353\qquad <span class='latex-bold'>(D) </span>391\qquad <span class='latex-bold'>(E) </span>421\qquad
geometrytrapezoidAsymptotenumber theoryrelatively primeAMC