MathDB

24

Part of 2015 AMC 10

Problems(2)

Perimeter is an Integer

Source: 2015 AMC 12A #19/10A #24

2/4/2015
For some positive integers pp, there is a quadrilateral ABCDABCD with positive integer side lengths, perimeter pp, right angles at BB and CC, AB=2AB=2, and CD=ADCD=AD. How many different values of p<2015p<2015 are possible?
<spanclass=′latex−bold′>(A)</span>30<spanclass=′latex−bold′>(B)</span>31<spanclass=′latex−bold′>(C)</span>61<spanclass=′latex−bold′>(D)</span>62<spanclass=′latex−bold′>(E)</span>63<span class='latex-bold'>(A) </span>30\qquad<span class='latex-bold'>(B) </span>31\qquad<span class='latex-bold'>(C) </span>61\qquad<span class='latex-bold'>(D) </span>62\qquad<span class='latex-bold'>(E) </span>63
geometryperimetercalculusrhombusconicsparabola
Aaron the ant

Source: 2015 AMC 10 B Problem 24

2/26/2015
Aaron the ant walks on the coordinate plane according to the following rules. He starts at the origin p0=(0,0)p_0=(0,0) facing to the east and walks one unit, arriving at p1=(1,0)p_1=(1,0). For n=1,2,3,…n=1,2,3,\dots, right after arriving at the point pnp_n, if Aaron can turn 90∘90^\circ left and walk one unit to an unvisited point pn+1p_{n+1}, he does that. Otherwise, he walks one unit straight ahead to reach pn+1p_{n+1}. Thus the sequence of points continues p2=(1,1),p3=(0,1),p4=(−1,1),p5=(−1,0)p_2=(1,1), p_3=(0,1), p_4=(-1,1), p_5=(-1,0), and so on in a counterclockwise spiral pattern. What is p2015p_{2015}?
<spanclass=′latex−bold′>(A)</span>(−22,−13)<spanclass=′latex−bold′>(B)</span>(−13,−22)<spanclass=′latex−bold′>(C)</span>(−13,22)<spanclass=′latex−bold′>(D)</span>(13,−22)<spanclass=′latex−bold′>(E)</span>(22,−13) <span class='latex-bold'>(A) </span> (-22,-13)\qquad<span class='latex-bold'>(B) </span> (-13,-22)\qquad<span class='latex-bold'>(C) </span> (-13,22)\qquad<span class='latex-bold'>(D) </span> (13,-22)\qquad<span class='latex-bold'>(E) </span> (22,-13)
analytic geometryAMC