MathDB

20

Part of 2018 AMC 12/AHSME

Problems(2)

Cyclic Quad AIME

Source: 2018 AMC 12A #20

2/8/2018
Triangle ABCABC is an isosceles right triangle with AB=AC=3AB=AC=3. Let MM be the midpoint of hypotenuse BC‾\overline{BC}. Points II and EE lie on sides AC‾\overline{AC} and AB‾\overline{AB}, respectively, so that AI>AEAI>AE and AIMEAIME is a cyclic quadrilateral. Given that triangle EMIEMI has area 22, the length CICI can be written as a−bc\frac{a-\sqrt{b}}{c}, where aa, bb, and cc are positive integers and bb is not divisible by the square of any prime. What is the value of a+b+ca+b+c?
<spanclass=′latex−bold′>(A)</span>9<spanclass=′latex−bold′>(B)</span>10<spanclass=′latex−bold′>(C)</span>11<spanclass=′latex−bold′>(D)</span>12<spanclass=′latex−bold′>(E)</span>13 <span class='latex-bold'>(A) </span>9 \qquad <span class='latex-bold'>(B) </span>10 \qquad <span class='latex-bold'>(C) </span>11 \qquad <span class='latex-bold'>(D) </span>12 \qquad <span class='latex-bold'>(E) </span>13 \qquad
geometrycyclic quadrilateralAMCAMC 12AMC 12 AAIME
Triangle Madness

Source: 2018 AMC 12B #20

2/16/2018
Let ABCDEFABCDEF be a regular hexagon with side length 11. Denote by XX, YY, and ZZ the midpoints of sides AB‾,CD‾,EF‾\overline{AB},\overline{CD},\overline{EF}, respectively. What is the area of the convex hexagon whose interior is the intersection of the interiors of △ACE\triangle{ACE} and △XYZ\triangle{XYZ}?
<spanclass=′latex−bold′>(A)</span>383<spanclass=′latex−bold′>(B)</span>7163<spanclass=′latex−bold′>(C)</span>15323<spanclass=′latex−bold′>(D)</span>123<spanclass=′latex−bold′>(E)</span>9163<span class='latex-bold'>(A) </span>\dfrac{3}{8}\sqrt{3}\qquad<span class='latex-bold'>(B) </span>\dfrac{7}{16}\sqrt{3}\qquad<span class='latex-bold'>(C) </span>\dfrac{15}{32}\sqrt{3}\qquad<span class='latex-bold'>(D) </span>\dfrac{1}{2}\sqrt{3}\qquad<span class='latex-bold'>(E) </span>\dfrac{9}{16}\sqrt{3}
AMC 12AMCAMC 12 B2018 AMC 12B2018 AMC