MathDB

17

Part of 2012 AMC 12/AHSME

Problems(2)

Subsets with No Pairwise Sums Divisible by Five

Source: 2012 AMC 12A Problem #17

2/8/2012
Let SS be a subset of {1,2,3,…,30}\{1,2,3,\dots,30\} with the property that no pair of distinct elements in SS has a sum divisible by 55. What is the largest possible size of SS?
<spanclass=′latex−bold′>(A)</span> 10<spanclass=′latex−bold′>(B)</span> 13<spanclass=′latex−bold′>(C)</span> 15<spanclass=′latex−bold′>(D)</span> 16<spanclass=′latex−bold′>(E)</span> 18 <span class='latex-bold'>(A)</span>\ 10\qquad<span class='latex-bold'>(B)</span>\ 13\qquad<span class='latex-bold'>(C)</span>\ 15\qquad<span class='latex-bold'>(D)</span>\ 16\qquad<span class='latex-bold'>(E)</span>\ 18
modular arithmeticprobabilityAMC
Square in the first quadrant

Source: 2012 AMC12B #17

2/23/2012
Square PQRSPQRS lies in the first quadrant. Points (3,0),(5,0),(7,0),(3,0), (5,0), (7,0), and (13,0)(13,0) lie on lines SP,RQ,PQSP, RQ, PQ, and SRSR, respectively. What is the sum of the coordinates of the center of the square PQRSPQRS?
<spanclass=′latex−bold′>(A)</span> 6<spanclass=′latex−bold′>(B)</span> 6.2<spanclass=′latex−bold′>(C)</span> 6.4<spanclass=′latex−bold′>(D)</span> 6.6<spanclass=′latex−bold′>(E)</span> 6.8 <span class='latex-bold'>(A)</span>\ 6\qquad<span class='latex-bold'>(B)</span>\ 6.2\qquad<span class='latex-bold'>(C)</span>\ 6.4\qquad<span class='latex-bold'>(D)</span>\ 6.6\qquad<span class='latex-bold'>(E)</span>\ 6.8
analytic geometrygraphing linesslopetrigonometrygeometrygeometric transformationdilation