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94 problems match “counting”. Clear

Unit complex numbers with a Re-Im condition ●●●○○ 50 ptsHow many complex numbers $z$ satisfy the following two conditions: $|z|=1$ and $\operatorname{Re}\left(z^{100}\right)=\operatorname{Im}\left(z^{200}\right)$ ?
Gaussian integer pairs with product 10 ●●●○○ 50 ptsCompute the number of ordered pairs of complex numbers $(u, v)$ such that $uv = 10$ and such that the real and imaginary parts of $u$ and $v$ are integers.
Rolling more 1s than 6s ●●●○○ 50 ptsIf Greg rolls four fair six-sided dice, what is the probability that he rolls more 1s than 6s? Round to two decimal places.
Each player gets one ace ●●●○○ 50 ptsA deck of 52 cards containing four aces is shuffled thoroughly and the cards are dealt among four players so that each player receives 13 cards. Find the probability that each player receives exactly one ace. Give the answer as a percentage rounded to one decimal place.
Partition of a set with size conditions ●●●○○ 50 ptsLet $N$ be the number of ordered pairs of nonempty sets $\mathcal{A}$ and $\mathcal{B}$ such that $\mathcal{A} \cup \mathcal{B} = \{1,2,3,4,5,6,7,8,9,10,11,12\}$ , $\mathcal{A} \cap \mathcal{B} = \emptyset$ , the number of elements of $\mathcal{A}$ is not an element of $\mathcal{A}$ , and the number of elements of $\mathcal{B}$ is not an element of $\mathcal{B}$ . Find $N$ .
Blue segments in a two-colored grid ●●●○○ 50 ptsA $16\times16$ square grid of dots is colored red and blue. Two dots that are adjacent in the grid (horizontally or vertically) are joined by a line segment. A segment joining two dots of the same color has that color, and a segment joining dots of different colors is yellow. There are 133 red dots in total. Exactly 2 of the 4 corner dots are red, and 32 other red dots lie on the boundary of the grid (not corners). There are 196 yellow segments in total. How many blue segments are there?
Maximum red beads on a string ●●●○○ 50 ptsA string is threaded with 150 beads, each red, blue or green. Among any six consecutive beads there is at least one green bead, and among any eleven consecutive beads there is at least one blue bead. What is the maximum number of red beads on the string?
Counting topologies on three points ●●●○○ 50 ptsLet $X$ be a set with $3$ elements. How many topologies can be defined on $X$ ? Equivalently, how many subsets of the power set $\mathcal{P}(X)$ are topologies on $X$ ?
Least n with many unit-fraction solutions ●●●○○ 50 ptsLet $x$ , $y$ and $n$ be positive integers, and count the solutions $(x, y)$ of $\frac{1}{x}+\frac{1}{y}=\frac{1}{n}$ with $x \le y$ . For $n=4$ there are exactly three such solutions. Find the least value of $n$ for which the number of solutions exceeds one thousand.
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 $X$ .