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94 problems match “counting”. Clear

Real roots of a sine-power equation ●●●●○ 100 ptsFind the number of real roots of the equation $\sin(2013\pi x)=x^{2013}$ .
Maximum smallest ball in bowl Jia ●●●●○ 100 ptsPlace the 27 balls numbered $1$ to $27$ into three bowls, Jia, Yi and Bing, so that the average ball number in bowl Jia is $15$ , in bowl Yi is $3$ , and in bowl Bing is $18$ , and each bowl contains at least $4$ balls. Find the maximum possible value of the smallest ball number in bowl Jia.
Birthdays that are equal or adjacent ●●●●○ 100 ptsAssume 23 people have independent birthdays, each equally likely to be any of 365 days (February 29 is not considered), and January 1 and December 31 are considered adjacent. Find the probability that at least two people share the same birthday or have adjacent birthdays. Round to seven decimal places.
Subsets with smallest element m ●●●●○ 100 ptsDetermine all values of $m$ with $1 \leq m \leq 47$ for which there are exactly $2^{15}$ subsets $X$ of the set $\{1,2,3,\ldots,47\}$ such that $m$ is the smallest element of $X$ , and for every $x \in X$ , either $x+m \in X$ or $x+m>47$ .