MathDB

9

Part of 2016 AIME Problems

Problems(2)

Maximize Circumscribing Rectangle

Source: 2016 AIME I #9

3/4/2016
Triangle ABCABC has AB=40AB = 40, AC=31AC = 31, and sin⁡A=15\sin A = \tfrac15. This triangle is inscribed in rectangle AQRSAQRS with BB on QR‾\overline{QR} and CC on RS‾\overline{RS}. Find the maximum possible area of AQRSAQRS.
2016 AIME IgeometryAIME
Sequence of Events

Source: 2016 AIME II #9

3/17/2016
The sequences of positive integers 1,a2,a3,…1,a_2,a_3,\ldots and 1,b2,b3,…1,b_2,b_3,\ldots are an increasing arithmetic sequence and an increasing geometric sequence, respectively. Let cn=an+bnc_n=a_n+b_n. There is an integer kk such that ck−1=100c_{k-1}=100 and ck+1=1000c_{k+1}=1000. Find ckc_k.
AIMEAIME II2016 AIME II2016 AIME II Problem 9Sequences