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94 problems match “counting”. Clear

Counting z with integer parts of f(z) ●●●●○ 100 ptsLet $f : \mathbb{C} \to \mathbb{C}$ be defined by $f(z) = z^2 + iz + 1$ . How many complex numbers $z$ are there such that $\text{Im}(z) > 0$ and both the real and the imaginary parts of $f(z)$ are integers with absolute value at most $10$ ?
Minimum red balloons in a string ●●●●○ 100 ptsDavid is making a string of red, white, and blue balloons by these rules: - no red balloon is adjacent to another red balloon; - white balloons appear in groups of exactly two, and groups of white balloons are separated by at least two non-white balloons; - blue balloons appear in groups of exactly three, and groups of blue balloons are separated by at least three non-blue balloons. If David uses more than 600 balloons, find the smallest number of red balloons he can use.
Counting case-switching letter sequences ●●●●○ 100 ptsBob writes a sequence of letters of the alphabet, each of which is either uppercase or lowercase, under these rules. If he has just written an uppercase letter, he either writes the same letter in lowercase next, or the next letter of the alphabet in uppercase. If he has just written a lowercase letter, he either writes the same letter in uppercase next, or the preceding letter of the alphabet in lowercase. For example, $a\,A\,a\,A\,B\,C\,D\,d\,c\,b\,B\,C$ is such a sequence. How many sequences of 32 letters can he write that start at lowercase $a$ and end at lowercase $z$ ?
Possible second terms of a sequence ●●●●○ 100 ptsA sequence $a_1, a_2, \ldots$ of non-negative integers is defined by the rule $a_{n+2}=|a_{n+1}-a_n|$ for $n\geq1$ . If $a_1=999$ , $a_2<999$ , and $a_{2006}=1$ , how many different values of $a_2$ are possible?
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 $s$ ?
Count complex solutions of an exponential equation ●●●●○ 100 ptsHow many complex numbers $z$ such that $\left| z \right| < 30$ satisfy the equation $e^z = \frac{z - 1}{z + 1}$ ?
Integers reaching one in twenty steps ●●●●○ 100 ptsFor every positive integer $x$ , let $f(x)=\begin{cases}1 & \text{if }x = 1\\ \frac x{10} & \text{if }x\text{ is divisible by }10\\ x+1 & \text{otherwise}\end{cases}$ and define a sequence by $x_1=x$ and $x_{n+1}=f(x_n)$ for all positive integers $n$ . Let $d(x)$ be the smallest $n$ such that $x_n=1$ . (For example, $d(100)=3$ and $d(87)=7$ .) Let $m$ be the number of positive integers $x$ such that $d(x)=20$ . Find the sum of the distinct prime factors of $m$ .
Smallest n with many solutions ●●●●○ 100 ptsLet $f(x)=|2\{x\}-1|$ , where $\{x\}$ denotes the fractional part of $x$ . Find the smallest positive integer $n$ such that the equation $n f(x f(x))=x$ has at least $2012$ real solutions.
Minimum red beads in a necklace ●●●●○ 100 ptsA necklace consists of 50 blue beads and a certain number of red beads. In any segment of the necklace containing 8 blue beads, there are at least 4 red beads. What is the minimum number of red beads that can be in this necklace? The beads are arranged cyclically, so the last bead is adjacent to the first.
Roots in [0,1] after sine substitution ●●●●○ 100 ptsUsing a suitable substitution, determine the number of roots of the equation $8x(1-2x^2)(8x^4-8x^2+1)=1$ that lie in the interval $[0,1]$ .