MathDB

19

Part of 2017 AMC 10

Problems(2)

Picky Friends

Source: 2017 AMC 12A #14

2/8/2017
Alice refuses to sit next to either Bob or Carla. Derek refuses to sit next to Eric. How many ways are there for the five of them to sit in a row of 55 chairs under these conditions?
<spanclass=′latex−bold′>(A)</span> 12<spanclass=′latex−bold′>(B)</span> 16<spanclass=′latex−bold′>(C)</span> 28<spanclass=′latex−bold′>(D)</span> 32<spanclass=′latex−bold′>(E)</span> 40<span class='latex-bold'>(A)</span>\ 12\qquad<span class='latex-bold'>(B)</span>\ 16\qquad<span class='latex-bold'>(C)</span>\ 28\qquad<span class='latex-bold'>(D)</span>\ 32\qquad<span class='latex-bold'>(E)</span>\ 40
AMCAMC 12AMC 12 A2017 AMC 12AAMC 10AMC 10 Acombinatorics
Never seen this before

Source: 2017 AMC 10B #19, 12B #15

2/16/2017
Let ABCABC be an equilateral triangle. Extend side AB‾\overline{AB} beyond BB to a point B′B' so that BB′=3ABBB' = 3AB. Similarly, extend side BC‾\overline{BC} beyond CC to a point C′C' so that CC′=3BCCC' = 3BC, and extend side CA‾\overline{CA} beyond AA to a point A′A' so that AA′=3CAAA' = 3CA. What is the ratio of the area of △A′B′C′\triangle A'B'C' to the area of △ABC\triangle ABC?
<spanclass=′latex−bold′>(A)</span>9:1<spanclass=′latex−bold′>(B)</span>16:1<spanclass=′latex−bold′>(C)</span>25:1<spanclass=′latex−bold′>(D)</span>36:1<spanclass=′latex−bold′>(E)</span>37:1<span class='latex-bold'>(A) </span>9:1\qquad<span class='latex-bold'>(B) </span>16:1\qquad<span class='latex-bold'>(C) </span>25:1\qquad<span class='latex-bold'>(D) </span>36:1\qquad<span class='latex-bold'>(E) </span>37:1
ratiogeometryAMCAMC 10AMC 10 B2017 AMC 10B2017 AMC 12B