MathDB

20

Part of 2014 AMC 12/AHSME

Problems(2)

Minimum Length of Broken Line

Source: 2014 AMC 12A #20

2/5/2014
In △BAC\triangle BAC, ∠BAC=40∘\angle BAC=40^\circ, AB=10AB=10, and AC=6AC=6. Points DD and EE lie on AB‾\overline{AB} and AC‾\overline{AC} respectively. What is the minimum possible value of BE+DE+CDBE+DE+CD?
<spanclass=′latex−bold′>(A)</span>63+3<spanclass=′latex−bold′>(B)</span>272<spanclass=′latex−bold′>(C)</span>83<spanclass=′latex−bold′>(D)</span>14<spanclass=′latex−bold′>(E)</span>33+9<span class='latex-bold'>(A) </span>6\sqrt 3+3\qquad <span class='latex-bold'>(B) </span>\dfrac{27}2\qquad <span class='latex-bold'>(C) </span>8\sqrt 3\qquad <span class='latex-bold'>(D) </span>14\qquad <span class='latex-bold'>(E) </span>3\sqrt 3+9\qquad
geometrygeometric transformationreflectiontrigonometryinequalitiesAsymptotetrig identities
Logarithm Inequality

Source: 2014 AMC 12B #20

2/20/2014
For how many positive integers xx is log⁡10(x−40)+log⁡10(60−x)<2\log_{10}{(x-40)} + \log_{10}{(60-x)} < 2?
<spanclass=′latex−bold′>(A)</span> 10<spanclass=′latex−bold′>(B)</span> 18<spanclass=′latex−bold′>(C)</span> 19<spanclass=′latex−bold′>(D)</span> 20<spanclass=′latex−bold′>(E)</span> infinitely many{ <span class='latex-bold'>(A)</span>\ 10\qquad<span class='latex-bold'>(B)</span>\ 18\qquad<span class='latex-bold'>(C)</span>\ 19\qquad<span class='latex-bold'>(D)</span>}\ 20\qquad<span class='latex-bold'>(E)</span>\ \text{infinitely many}
logarithmsinequalitiesalgebrafunctiondomaincalculusintegration