Practice (68)
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Chad has $100$ cookies that he wants to distribute among four friends. Two of them, Jeff and Qiao, are rivals; neither wants the other to receive more cookies than they do. The other two, Jim and Townley, don't care about how many cookies they receive. In how many ways can Chad distribute all $100$ cookies to his four friends so that everyone is satisfied? (Some of his four friends may receive zero cookies.)
How many length ten strings consisting of only $A$s and $B$s contain neither "$BAB$" nor "$BBB$" as a substring?
A $3 \times 3$ grid is filled with positive integers and has the property that each integer divides both the integer directly above it and directly to the right of it. Given that the number in the top-right corner is 30, how many distinct grids are possible?
Consider a non-empty set of segments of length 1 in the plane which do not intersect except at their endpoints. (In other words, if point $P$ lies on distinct segments $a$ and $b$, then $P$ is an endpoint of both $a$ and $b$.) This set is called $3-amazing$ if each endpoint of a segment is the endpoint of exactly
three segments in the set. Find the smallest possible size of a 3-amazing set of segments.
Let $a_n=\binom{2020}{3n-1}$. Find the vale of $\displaystyle\sum_{n=1}^{673}a_n$.
A code consists of four different digits from $1$ to $9$, inclusive. What is the probability of selection a code that consists of four consecutive digits but not necessarily in order? Express your answer as a common fraction.
How many different ways to express $13$ as the sum of several positive odd numbers? Order
matters. For example, $1 + 1 + 3 + 3 + 5$ is treated as a different expression as 1 + 3 + 1 + 3 + 5
A bug stands at point $A$ of $\triangle{ABC}$. In each move, it craws randomly to one of the other two vertices. Let the probability of it returns on point $A$ in 2015 steps be $\frac{m}{n}$, where $m$ and $n$ are co-prime. Find the last three digits of $m+n$.
A spider has one sock and one shoe for each of its eight legs. In how many different orders can the spider put on its socks and shoes, assuming that, on each leg, the sock must be put on before the shoe.
A binary palindrome is a positive integer whose standard base 2 (binary) representation is a palindrome (reads the same backward or forward). (Leading zeros are not permitted in the standard representation.) For example, 2015 is a binary palindrome, because in base 2 it is 11111011111. How many positive integers less than 2015 are binary palindromes?
In the diagram below, how many different routes are there from point $M$ to point $P$ using only the ling segments shown? A route is not allowed to intersect itself, not even at a single point.
Say that a rational number is special if its decimal expression is of the form $0.\overline{abcdef}$, where $a, b, c, d, e$ and $f$ are digits (possibly equal) that include each of the digits $2, 0, 1$, and $5$ at least once (in some order). How many special rational numbers are there?
Among all pairs of real numbers $(x, y)$ such that $\sin\sin x=\sin\sin y$ with $-10\pi \le x, y \le 10\pi$. Oleg randomly selected a pair $(X, Y)$. Compute the probability that $X = Y$.
A $\textit{permutation}$ of a finite set is a one-to-one function from the set onto itself. A $\textit{cycle}$ in a permutation $P$ is a nonempty sequence of distinct items $x_1, \cdots, x_n$ such that $P(x_1)=x_2$, $P(x_2)=x_3$, $\cdots$, $P(x_n)=x_1$. Note that we allow the 1-cycle $x_1$ where $P(x_1)=x_1$ and the 2-cycle $x_1, x_2$ where $P(x_1)=x_2$ and $P(x_2)=x_1$. Every permutation of a finite site splits the set into a finite number of disjoint cycles. If this number equal to 2, then the permutation is called $\textit{bi-cyclic}$. Computer the number of bi-cyclic permutation of the 7-element set formed by letters "PROBLEMS".
An ant begins at a vertex of a convex regular icosahedron (a figure with 20 triangular faces and 12 vertices). The ant moves along one edge at a time. Each time, the ant reaches a vertex, it randomly choose to next walk along any of the edges extending from that vertex (including the edge it just arrived from). Find the probability that after walking along exactly six (not necessarily distinct) edges, the ant finds itself at its starting vertex.
Sabrina has a fair tetrahedral die whose faces are numbered 1, 2, 3, and 4, respectively. She creates a sequence by rolling the die and recording the number on its bottom face. However, she discards (without recording) any role such that appending its number to the sequence would result in two consecutive terms that sum to 5. Sabrina stops the moment that all four numbers appear in the sequence. Find the expected (average) number of terms in Sabrina's sequence.
Ten unfair coins with probability of $1, \frac{1}{2}, \frac{1}{3}, \dots, \frac{1}{10}$ of showing heads are flipped. What is the probability that odd number of heads are shown?
Ten unfair coins with probability of $1, \frac{1}{3}, \frac{1}{4}, \dots, \frac{1}{11}$ of showing heads are flipped. What is the probability that odd number of heads are shown?
Tina randomly selects two distinct numbers from the set {1, 2, 3, 4, 5}, and Sergio randomly selects a number from the set {1, 2, ..., 10}. What is the probability that Sergio's number is larger than the sum of the two numbers chosen by Tina?
Find all $n\in\mathbb{N}$ such that $$\binom{n}{k-1} = 2 \binom{n}{k} + \binom{n}{k+1}$$
for some natural number $k < n$.
If $n$ is an integer such that the values of $(3n+1)$ and $(4n+1)$ are both squares, prove that $n$ is a multiple of $56$.
Let $a$, $b$, $c$, $d$, and $e$ be five positive integers. If $ab+c=3115$, $c^2+d^2=e^2$, both $a$ and $c$ are prime numbers, $b$ is even and has $11$ divisors. Find these five numbers
25 boys and 8 girls sit in a circle. If there are at least two boys between any two girls, how many different sitting plans are there? (Two sitting plans will be considered as the same if they differ just by rotating.)
Joe is playing with a set of $6$ masses: $1$g, $2$g, $4$g, $8$g, $16$g, and $32$g. He found that some weights can be measured in more than one way. For example, $7$g can be measure by putting $1$g, $2$g, and $4$g on one side of a balance. It can also be achieved by putting $1$g and $8$g on different sides of a balance. He therefore wonder which weight can be measured using these masses in the most number of different ways? Can you help him to find it out? Describe how will you approach this problem. The final answer is optional.
Mary typed a six-digit number, but the two 1s she typed didn't show. What appeared was 2002. How many different six-digit numbers could she have typed?