MathDB

Problem 1

Part of 2019 LIMIT Category B

Problems(2)

product of sequence and limit

Source: LIMIT 2019 CBS1 P1

4/28/2021
Let a1=1a_1=1 and an=n(an−1+1)a_n=n(a_{n-1}+1) for n≥2n\ge2. Define pn=∏i=1n(1+1ai)p_n=\prod_{i=1}^n\left(1+\frac1{a_i}\right)Then lim⁡n→∞pn\lim_{n\to\infty}p_n is <spanclass=′latex−bold′>(A)</span> 1+e<span class='latex-bold'>(A)</span>~1+e <spanclass=′latex−bold′>(B)</span> e<span class='latex-bold'>(B)</span>~e <spanclass=′latex−bold′>(C)</span> 1<span class='latex-bold'>(C)</span>~1 <spanclass=′latex−bold′>(D)</span> ∞<span class='latex-bold'>(D)</span>~\infty
real analysis
inscribing n circles inside big circle

Source: LIMIT 2019 CBS2 P1

4/28/2021
Let n≥3n\ge3 be integer. Assume that inside a big circle, exactly nn small circles of radius rr can be drawn so that each small circle touches the big circle and also touches both its adjacent small circles. Then, the radius of big circle is <spanclass=′latex−bold′>(A)</span> rcsc⁡πn<span class='latex-bold'>(A)</span>~r\csc\frac{\pi}n <spanclass=′latex−bold′>(B)</span> rcsc⁡(1+2πn)<span class='latex-bold'>(B)</span>~r\csc\left(1+\frac{2\pi}n\right) <spanclass=′latex−bold′>(C)</span> rcsc⁡(1+π2n)<span class='latex-bold'>(C)</span>~r\csc\left(1+\frac{\pi}{2n}\right) <spanclass=′latex−bold′>(D)</span> rcsc⁡(1+πn)<span class='latex-bold'>(D)</span>~r\csc\left(1+\frac{\pi}n\right)
geometry