MathDB

15

Part of 2019 AMC 10

Problems(2)

Recursive sequence

Source: 2019 AMC 10A #15 / AMC 12A #9

2/8/2019
A sequence of numbers is defined recursively by a1=1a_1 = 1, a2=37a_2 = \frac{3}{7}, and an=an−2⋅an−12an−2−an−1a_n=\frac{a_{n-2} \cdot a_{n-1}}{2a_{n-2} - a_{n-1}}for all n≥3n \geq 3 Then a2019a_{2019} can be written as pq\frac{p}{q}, where pp and qq are relatively prime positive inegers. What is p+q?p+q ?
<spanclass=′latex−bold′>(A)</span>2020<spanclass=′latex−bold′>(B)</span>4039<spanclass=′latex−bold′>(C)</span>6057<spanclass=′latex−bold′>(D)</span>6061<spanclass=′latex−bold′>(E)</span>8078<span class='latex-bold'>(A) </span> 2020 \qquad<span class='latex-bold'>(B) </span> 4039 \qquad<span class='latex-bold'>(C) </span> 6057 \qquad<span class='latex-bold'>(D) </span> 6061 \qquad<span class='latex-bold'>(E) </span> 8078
AMCAMC 10AMC 122019 AMC 12A2019 AMC 10A2019 AMCrecursion
Annoying Areas

Source: 2019 AMC 10B #15

2/14/2019
Two right triangles, T1T_1 and T2T_2, have areas of 1 and 2, respectively. One side length of one triangle is congruent to a different side length in the other, and another side length of the first triangle is congruent to yet another side length in the other. What is the square of the product of the third side lengths of T1T_1 and T2T_2?
<spanclass=′latex−bold′>(A)</span>283<spanclass=′latex−bold′>(B)</span>10<spanclass=′latex−bold′>(C)</span>323<spanclass=′latex−bold′>(D)</span>343<spanclass=′latex−bold′>(E)</span>12<span class='latex-bold'>(A) </span>\frac{28}3\qquad<span class='latex-bold'>(B) </span>10\qquad<span class='latex-bold'>(C) </span>\frac{32}3\qquad<span class='latex-bold'>(D) </span>\frac{34}3\qquad<span class='latex-bold'>(E) </span>12
AMCAMC 10AMC 10 B