MathDB

25

Part of 2015 AMC 10

Problems(2)

Distance is Greater than One Half

Source: 2015 AMC 12A #23 / 10A #25

2/4/2015
Let SS be a square of side length 11. Two points are chosen independently at random on the sides of SS. The probability that the straight-line distance between the points is at least 12\tfrac12 is a−bπc\tfrac{a-b\pi}c, where aa, bb, and cc are positive integers and gcd⁡(a,b,c)=1\gcd(a,b,c)=1. What is a+b+ca+b+c?
<spanclass=′latex−bold′>(A)</span>59<spanclass=′latex−bold′>(B)</span>60<spanclass=′latex−bold′>(C)</span>61<spanclass=′latex−bold′>(D)</span>62<spanclass=′latex−bold′>(E)</span>63<span class='latex-bold'>(A) </span>59\qquad<span class='latex-bold'>(B) </span>60\qquad<span class='latex-bold'>(C) </span>61\qquad<span class='latex-bold'>(D) </span>62\qquad<span class='latex-bold'>(E) </span>63
probabilityintegrationgeometrytrigonometrycalculusrectanglegeometric probability
Volume and Surface area

Source: 2015 AMC10B #25, AMC12B #23

2/26/2015
A rectangular box measures a×b×ca \times b \times c, where a,a, b,b, and cc are integers and 1≤a≤b≤c1 \leq a \leq b \leq c. The volume and surface area of the box are numerically equal. How many ordered triples (a,b,c)(a,b,c) are possible?
<spanclass=′latex−bold′>(A)</span>4<spanclass=′latex−bold′>(B)</span>10<spanclass=′latex−bold′>(C)</span>12<spanclass=′latex−bold′>(D)</span>21<spanclass=′latex−bold′>(E)</span>26 <span class='latex-bold'>(A) </span>4\qquad<span class='latex-bold'>(B) </span>10\qquad<span class='latex-bold'>(C) </span>12\qquad<span class='latex-bold'>(D) </span>21\qquad<span class='latex-bold'>(E) </span>26
geometryAMCAMC 10 BAMC 12