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13 problems match “arithmetic sequences”. Clear

Integer trapped between two sequences ●●●○○ 50 ptsLet $\{a_n\}$ and $\{b_n\}$ be sequences with $a_1=0$ and $b_1=2013$ , such that for every positive integer $n$ the triples $(a_n, a_{n+1}, b_n)$ and $(a_{n+1}, b_{n+1}, b_n)$ are arithmetic sequences. Find the unique positive integer $c$ such that $a_n < c < b_n$ for all positive integers $n$ .
Argument of a sum of cis terms ●●●○○ 50 ptsThe number $\operatorname{cis} 75^\circ + \operatorname{cis} 83^\circ + \operatorname{cis} 91^\circ + \dots + \operatorname{cis} 147^\circ$ is expressed in the form $r \operatorname{cis} \theta$ , where $r > 0$ and $0^\circ \le \theta < 360^\circ$ . Find $\theta$ in degrees.
Possible values of the largest removed integer ●●●●○ 100 ptsYann writes down the first $n$ consecutive positive integers, $1,2,3,4,\ldots,n-1,n$ . He removes four different integers $p,q,r,s$ from the list. At least three of $p,q,r,s$ are consecutive and $100<p<q<r<s$ . The average of the integers remaining in the list is $89.5625$ . What is the number of possible values of $s$ ?