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19 problems match “modular arithmetic”. Clear

Smallest solution of a linear congruence ●●○○○ 20 ptsWhat is the smallest positive integer $x$ that satisfies the congruence $30x \equiv 42 \pmod{47}$ ?
Maximum size of a special set ●●●○○ 50 ptsA set $\mathcal{S}$ of distinct positive integers has the following property: for every integer $x$ in $\mathcal{S}$ , the arithmetic mean of the set of values obtained by deleting $x$ from $\mathcal{S}$ is an integer. Given that $1$ belongs to $\mathcal{S}$ and that $2002$ is the largest element of $\mathcal{S}$ , what is the greatest number of elements that $\mathcal{S}$ can have?
Color of the last ball ●●●○○ 50 ptsMilan has a bag of 2020 red balls and 2021 green balls. He repeatedly draws 2 balls out of the bag uniformly at random. If they are the same color, he changes them both to the opposite color and returns them to the bag. If they are different colors, he discards them. Eventually the bag has 1 ball left. Let $p$ be the probability that it is green. Compute $\lfloor 2021 p \rfloor$ .
101st digit of a period-200 reciprocal ●●●○○ 50 ptsLet $p$ be a prime number such that the period of the decimal expansion of $\frac{1}{p}$ is 200. That is, $\frac{1}{p}=0.XXXX\ldots$ for some block $X$ of 200 digits, but $\frac{1}{p}\neq 0.YYYY\ldots$ for every block $Y$ with fewer than 200 digits. Find the 101st digit, counting from the left, of $X$ .
Library visits of two friends ●●●○○ 50 ptsXiao Ming goes to the library on the first Wednesday of January, and then again every 4 days, meaning there are 4 days between visits (so the second visit is on a Monday). Xiao Qiang goes to the library on the first Thursday of January, and then again every 3 days, meaning there are 3 days between visits. If in January the two went to the library together only once, on which day of January was this?
Integers with distinct remainders ●●●○○ 50 ptsHow many positive integers at most 420 leave different remainders when divided by each of 5, 6, and 7?
Five Sundays in February ●●●○○ 50 ptsApril 16, 2011 was a Saturday. Find all years in the 21st century (2001–2100) in which February has five Sundays.
Solving a four-congruence CRT system ●●●○○ 50 ptsFind the least nonnegative integer $x$ satisfying $x \equiv 4 \pmod{5}$ , $x \equiv 2 \pmod{7}$ , $x \equiv 2 \pmod{11}$ and $x \equiv 12 \pmod{13}$ .
Coefficient of a quadratic from its values ●●●●○ 100 ptsThe function $f(x)$ is of the form $ax^{2}+bx+c$ for some integers $a$ , $b$ and $c$ . Given that $\{f(177883), f(348710), f(796921), f(858522)\} = \{1324754875645, 1782225466694, 1984194627862, 4388794883485\}$ , compute $a$ .