MathDB

13

Part of 2019 AMC 10

Problems(2)

Triangles in Circles

Source: 2019 AMC 10A #13

2/8/2019
Let ΔABC\Delta ABC be an isosceles triangle with BC=ACBC = AC and ∠ACB=40∘\angle ACB = 40^{\circ}. Contruct the circle with diameter BC‾\overline{BC}, and let DD and EE be the other intersection points of the circle with the sides AC‾\overline{AC} and AB‾\overline{AB}, respectively. Let FF be the intersection of the diagonals of the quadrilateral BCDEBCDE. What is the degree measure of ∠BFC?\angle BFC ?
<spanclass=′latex−bold′>(A)</span>90<spanclass=′latex−bold′>(B)</span>100<spanclass=′latex−bold′>(C)</span>105<spanclass=′latex−bold′>(D)</span>110<spanclass=′latex−bold′>(E)</span>120<span class='latex-bold'>(A) </span> 90 \qquad<span class='latex-bold'>(B) </span> 100 \qquad<span class='latex-bold'>(C) </span> 105 \qquad<span class='latex-bold'>(D) </span> 110 \qquad<span class='latex-bold'>(E) </span> 120
AMCAMC 10AMC 10 A2019 AMC 10A2019 AMCgeometryTriangle
Median #7 Strikes Back

Source: 2019 AMC 12B #7

2/14/2019
What is the sum of all real numbers xx for which the median of the numbers 4,6,8,17,4,6,8,17, and xx is equal to the mean of those five numbers?
<spanclass=′latex−bold′>(A)</span>−5<spanclass=′latex−bold′>(B)</span>0<spanclass=′latex−bold′>(C)</span>5<spanclass=′latex−bold′>(D)</span>154<spanclass=′latex−bold′>(E)</span>354<span class='latex-bold'>(A) </span> -5 \qquad<span class='latex-bold'>(B) </span> 0 \qquad<span class='latex-bold'>(C) </span> 5 \qquad<span class='latex-bold'>(D) </span> \frac{15}{4} \qquad<span class='latex-bold'>(E) </span> \frac{35}{4}
2019 AMC 12BAMCAMC 12AMC 12 B2019 AMC 10BAMC 10AMC 10 B