input stringlengths 43 5.73k | output stringlengths 1 6.77k |
|---|---|
### Instruction:
Each block on the grid shown in the Figure is 1 unit by 1 unit. Suppose we wish to walk from $A$ to $B$ via a 7 unit path, but we have to stay on the grid -- no cutting across blocks. How many different paths can we take?[asy]size(3cm,3cm);int w=5;int h=4;int i;for (i=0; i<h; ++i){draw((0,i) -- (w-1,... | We know that we must take a 7 unit path. If we look at the grid a little more carefully, we can see that our path must consist of 4 steps to the right and 3 steps up, and we can take those steps in any order. So in order to specify a path, we must choose 3 of our 7 steps to be `up' (and the other 4 steps will thus be... |
### Instruction:
Let $S$ be the set of all ordered triple of integers $(a_1,a_2,a_3)$ with $1 \le a_1,a_2,a_3 \le 10$. Each ordered triple in $S$ generates a sequence according to the rule $a_n=a_{n-1}\cdot | a_{n-2}-a_{n-3} |$ for all $n\ge 4$. Find the number of such sequences for which $a_n=0$ for some $n$.
### Re... | Let $a_1=x, a_2=y, a_3=z$. First note that if any absolute value equals 0, then $a_n=0$. Also note that if at any position, $a_n=a_{n-1}$, then $a_{n+2}=0$. Then, if any absolute value equals 1, then $a_n=0$. Therefore, if either $|y-x|$ or $|z-y|$ is less than or equal to 1, then that ordered triple meets the criteria... |
### Instruction:
In a sequence of coin tosses, one can keep a record of instances in which a tail is immediately followed by a head, a head is immediately followed by a head, and etc. We denote these by TH, HH, and etc. For example, in the sequence TTTHHTHTTTHHTTH of 15 coin tosses we observe that there are two HH, thr... | Let's consider each of the sequences of two coin tosses as an operation instead; this operation takes a string and adds the next coin toss on (eg, THHTH + HT = THHTHT). We examine what happens to the last coin toss. Adding HH or TT is simply an identity for the last coin toss, so we will ignore them for now. However, a... |
### Instruction:
A sequence of ten $0$s and/or $1$s is randomly generated. If the probability that the sequence does not contain two consecutive $1$s can be written in the form $\dfrac{m}{n}$, where $m,n$ are relatively prime positive integers, find $m+n$.
### Response:
| Let $a_n$ denote the number of sequences of length $n$ that do not contain consecutive $1$s. A sequence of length $n$ must either end in a $0$ or a $1$. If the string of length $n$ ends in a $0$, this string could have been formed by appending a $0$ to any sequence of length $n-1$, of which there are $a_{n-1}$ such str... |
### Instruction:
A soccer team has $22$ available players. A fixed set of $11$ players starts the game, while the other $11$ are available as substitutes. During the game, the coach may make as many as $3$ substitutions, where any one of the $11$ players in the game is replaced by one of the substitutes. No player remo... | There are $0-3$ substitutions. The number of ways to sub any number of times must be multiplied by the previous number. This is defined recursively. The case for $0$ subs is $1$, and the ways to reorganize after $n$ subs is the product of the number of new subs ($12-n$) and the players that can be ejected ($11$). The f... |
### Instruction:
The diagram below shows a $4\times4$ rectangular array of points, each of which is $1$ unit away from its nearest neighbors.
[asy] unitsize(0.25inch); defaultpen(linewidth(0.7)); int i, j; for(i = 0; i < 4; ++i) for(j = 0; j < 4; ++j) dot(((real)i, (real)j)); [/asy]
Define a growing path to be a se... | We label our points using coordinates $0 \le x,y \le 3$, with the bottom-left point being $(0,0)$. By the Pythagorean Theorem, the distance between two points is $\sqrt{d_x^2 + d_y^2}$ where $0 \le d_x, d_y \le 3$; these yield the possible distances (in decreasing order)\[\sqrt{18},\ \sqrt{13},\ \sqrt{10},\ \sqrt{9},\ ... |
### Instruction:
Let $\mathcal{S}_{n}$ be the set of strings with only 0's or 1's with length $n$ such that any 3 adjacent place numbers sum to at least 1. For example, $00100$ works, but $10001$ does not. Find the number of elements in $\mathcal{S}_{11}$.
### Response:
| We will solve this problem by constructing a recursion satisfied by $\mathcal{S}_n$.
Let $A_1(n)$ be the number of such strings of length $n$ ending in 1, $A_2(n)$ be the number of such strings of length $n$ ending in a single 0 and $A_3(n)$ be the number of such strings of length $n$ ending in a double zero. Then $A_1... |
### Instruction:
The inhabitants of the island of Jumble use the standard Kobish alphabet ($20$ letters, A through T). Each word in their language is $4$ letters or less, and for some reason, they insist that all words contain the letter A at least once. How many words are possible?
### Response:
| Let's break this up into cases. Ok. First, let's look at the number of words that are exactly 4 letters long. Right. So, one of the letters must be an A. There are 4 choices for that. After we pick where the A is going to go, we have to pick what the other 3 letters are going to be. For each of those 3 letters, we have... |
### Instruction:
A scanning code consists of a $7 \times 7$ grid of squares, with some of its squares colored black and the rest colored white. There must be at least one square of each color in this grid of $49$ squares. A scanning code is called $\textit{symmetric}$ if its look does not change when the entire square ... | Draw a $7 \times 7$ square.
$\begin{tabular}{|c|c|c|c|c|c|c|} \hline K & J & H & G & H & J & K \\ \hline J & F & E & D & E & F & J \\ \hline H & E & C & B & C & E & H \\ \hline G & D & B & A & B & D & G \\ \hline H & E & C & B & C & E & H \\ \hline J & F & E & D & E & F & J \\ \hline K & J & H & G & H & J & K \\ \hline... |
### Instruction:
There is a set of 1000 switches, each of which has four positions, called $A, B, C$, and $D$. When the position of any switch changes, it is only from $A$ to $B$, from $B$ to $C$, from $C$ to $D$, or from $D$ to $A$. Initially each switch is in position $A$. The switches are labeled with the 1000 diffe... | For each $i$th switch (designated by $x_{i},y_{i},z_{i}$), it advances itself only one time at the $i$th step; thereafter, only a switch with larger $x_{j},y_{j},z_{j}$ values will advance the $i$th switch by one step provided $d_{i}= 2^{x_{i}}3^{y_{i}}5^{z_{i}}$ divides $d_{j}= 2^{x_{j}}3^{y_{j}}5^{z_{j}}$. Let $N = 2... |
### Instruction:
The Gnollish language consists of 3 words, ``splargh,'' ``glumph,'' and ``amr.'' In a sentence, ``splargh'' cannot come directly before ``glumph''; all other sentences are grammatically correct (including sentences with repeated words). How many valid 3-word sentences are there in Gnollish?
### Resp... | We proceed by counting the complement, or the number of invalid 3-word sentences. A sentence is invalid precisely when it is of the form ``(word) splargh glumph'' or ``splargh glumph (word).'' There are 3 choices for the missing word in each sentence, and since each case is exclusive, we have a total of 6 invalid sen... |
### Instruction:
There are 5 yellow pegs, 4 red pegs, 3 green pegs, 2 blue pegs, and 1 orange peg to be placed on a triangular peg board. In how many ways can the pegs be placed so that no (horizontal) row or (vertical) column contains two or more pegs of the same color? (Any two pegs of the same color are indistingui... | To avoid having two yellow pegs in the same row or column, there must be exactly one yellow peg in each row and in each column. Hence, starting at the top of the array, the peg in the first row must be yellow, the second peg of the second row must be yellow, the third peg of the third row must be yellow, etc. To avoid ... |
### Instruction:
Consider sequences that consist entirely of $A$'s and $B$'s and that have the property that every run of consecutive $A$'s has even length, and every run of consecutive $B$'s has odd length. Examples of such sequences are $AA$, $B$, and $AABAA$, while $BBAB$ is not such a sequence. How many such sequen... | Let $a_n$ and $b_n$ denote, respectively, the number of sequences of length $n$ ending in $A$ and $B$. If a sequence ends in an $A$, then it must have been formed by appending two $A$s to the end of a string of length $n-2$. If a sequence ends in a $B,$ it must have either been formed by appending one $B$ to a string o... |
### Instruction:
Matt will arrange four identical, dotless dominoes (shaded 1 by 2 rectangles) on the 5 by 4 grid below so that a path is formed from the upper left-hand corner $A$ to the lower righthand corner $B$. In a path, consecutive dominoes must touch at their sides and not just their corners. No domino may be p... | The shortest possible path from $A$ to $B$ requires $4$ dominoes, which is all we have, so we must use them to make only down and right movements - we have none to waste going up or left. We need to make $3$ movements to the right and $4$ down, and we can arrange them however we wish. So there are
$$\binom{7}{3}=35$... |
### Instruction:
A triangular array of squares has one square in the first row, two in the second, and in general, $k$ squares in the $k$th row for $1 \leq k \leq 11.$ With the exception of the bottom row, each square rests on two squares in the row immediately below (illustrated in the given diagram). In each square o... | Label each of the bottom squares as $x_0, x_1 \ldots x_9, x_{10}$.
Through induction, we can find that the top square is equal to ${10\choose0}x_0 + {10\choose1}x_1 + {10\choose2}x_2 + \ldots {10\choose10}x_{10}$. (This also makes sense based on a combinatorial argument: the number of ways a number can "travel" to the ... |
### Instruction:
There are two distinguishable flagpoles, and there are $19$ flags, of which $10$ are identical blue flags, and $9$ are identical green flags. Let $N$ be the number of distinguishable arrangements using all of the flags in which each flagpole has at least one flag and no two green flags on either pole a... | The well known problem of ordering $x$ elements of a string of $y$ elements such that none of the $x$ elements are next to each other has ${y-x+1\choose x}$ solutions. (1)
We generalize for $a$ blues and $b$ greens. Consider a string of $a+b$ elements such that we want to choose the greens such that none of them are ne... |
### Instruction:
A mail carrier delivers mail to the nineteen houses on the east side of Elm Street. The carrier notices that no two adjacent houses ever get mail on the same day, but that there are never more than two houses in a row that get no mail on the same day. How many different patterns of mail delivery are po... | Let $0$ represent a house that does not receive mail and $1$ represent a house that does receive mail. This problem is now asking for the number of $19$-digit strings of $0$'s and $1$'s such that there are no two consecutive $1$'s and no three consecutive $0$'s.
The last two digits of any $n$-digit string can't be $11$... |
### Instruction:
A given sequence $r_1, r_2, \dots, r_n$ of distinct real numbers can be put in ascending order by means of one or more "bubble passes". A bubble pass through a given sequence consists of comparing the second term with the first term, and exchanging them if and only if the second term is smaller, then c... | If any of $r_1, \ldots, r_{19}$ is larger than $r_{20}$, one of these numbers will be compared with $r_{20}$ on the 19th step of the first bubble pass and $r_{20}$ will be moved back to the 19th position. Thus, $r_{20}$ must be the largest of the first 20 terms. In addition, $r_{20}$ must be larger than $r_{21}, r_{22}... |
### Instruction:
A bored student walks down a hall that contains a row of closed lockers, numbered $1$ to $1024$. He opens the locker numbered 1, and then alternates between skipping and opening each locker thereafter. When he reaches the end of the hall, the student turns around and starts back. He opens the first clo... | On his first pass, he opens all of the odd lockers. So there are only even lockers closed. Then he opens the lockers that are multiples of $4$, leaving only lockers $2 \pmod{8}$ and $6 \pmod{8}$. Then he goes ahead and opens all lockers $2 \pmod {8}$, leaving lockers either $6 \pmod {16}$ or $14 \pmod {16}$. He then go... |
### Instruction:
On the refrigerator, MATHCOUNTS is spelled out with 10 magnets, one letter per magnet. Two vowels and three consonants fall off and are put away in a bag. If the Ts are indistinguishable, how many distinct possible collections of letters could be put in the bag?
### Response:
| Let's divide the problem into two cases: one where 0 or 1 T's fall off and one where both T's fall off:
0 or 1 T's: \[\dbinom{3}{2}\dbinom{6}{3}=3\times20=60\]
2 T's: \[\dbinom{3}{2}\dbinom{5}{1}=3\times5=15\]
Total: $60+15=75$ |
### Instruction:
A collection of 8 cubes consists of one cube with edge-length $k$ for each integer $k, 1 \le k \le 8.$ A tower is to be built using all 8 cubes according to the rules:
Any cube may be the bottom cube in the tower.
The cube immediately on top of a cube with edge-length $k$ must have edge-length at most ... | We proceed recursively. Suppose we can build $T_m$ towers using blocks of size $1, 2, \ldots, m$. How many towers can we build using blocks of size $1, 2, \ldots, m, m + 1$? If we remove the block of size $m + 1$ from such a tower (keeping all other blocks in order), we get a valid tower using blocks $1, 2, \ldots, m$.... |
### Instruction:
Define a $\it{good\ word}$ as a sequence of letters that consists only of the letters $A$, $B$, and $C$ --- some of these letters may not appear in the sequence --- and in which $A$ is never immediately followed by $B$, $B$ is never immediately followed by $C$, and $C$ is never immediately followed by ... | This problem involves counting the number of sequences that satisfy certain conditions, which suggests using the multiplication principle of counting: if we can break down the task of forming a sequence into a series of steps, and we know how many choices we have for each step, then we can multiply those choices to get... |
### Instruction:
A sequence consists of $2010$ terms. Each term after the first is 1 larger than the previous term. The sum of the $2010$ terms is $5307$. When every second term is added up, starting with the first term and ending with the second last term, what is the sum?
### Response:
| We label the terms $x_1, x_2, x_3, \ldots, x_{2009},x_{2010}$.
Suppose that $S$ is the sum of the odd-numbered terms in the sequence; that is, \[ S = x_1 + x_3 + x_5 + \cdots + x_{2007}+x_{2009} \]We know that the sum of all of the terms is 5307; that is, \[ x_1 + x_2 + x_3 + \cdots + x_{2009}+x_{2010} = 5307 \]Next, ... |
### Instruction:
Define a function $A(m, n)$ by \[ A(m,n) = \left\{ \begin{aligned} &n+1& \text{ if } m = 0 \\ &A(m-1, 1) & \text{ if } m > 0 \text{ and } n = 0 \\ &A(m-1, A(m, n-1))&\text{ if } m > 0 \text{ and } n > 0. \end{aligned} \right.\]Compute $A(2, 1).$
### Response:
| I notice that this function is recursive, meaning that it calls itself with smaller arguments until it reaches a base case. The base case is when $m = 0$, in which case the function just adds one to $n$. So I want to keep track of how the arguments change as I apply the function repeatedly. I start with $A(2, 1)$, whic... |
### Instruction:
Consider the triangular array of numbers with 0, 1, 2, 3, $\dots$ along the sides and interior numbers obtained by adding the two adjacent numbers in the previous row. Rows 1 through 6 are shown.
\[
\begin{array}{ccccccccccc}
& & & & & 0 & & & & & \\
& & & & 1 & & 1 & & & & \\
& & & 2 & & 2 & & 2 & &... | Let $f(n)$ denote the sum of the numbers in the $n$th row. We start by looking at an example.
Suppose we take the 5th row, make a copy of every number, and send each copy to the fifth row.
[asy]
unitsize (1 cm);
pair A, B;
int i;
for (i = 1; i <= 5; ++i) {
A = (2*i - 1,1);
B = (2*i - 2,0);
draw(interp(A,B,0.2)... |
### Instruction:
A sequence $a_1,$ $a_2,$ $a_3,$ $\dots,$ is defined recursively by $a_1 = 1,$ $a_2 = 1,$ and for $k \ge 3,$
\[a_k = \frac{1}{3} a_{k - 1} + \frac{1}{4} a_{k - 2}.\]Evaluate $a_1 + a_2 + a_3 + \dotsb.$
### Response:
| Let $S = a_ 1 + a_2 + a_3 + \dotsb.$ Then
\begin{align*}
S &= a_1 + a_2 + a_3 + a_4 + a_5 + \dotsb \\
&= 1 + 1 + \left( \frac{1}{3} a_2 + \frac{1}{4} a_1 \right) + \left( \frac{1}{3} a_3 + \frac{1}{4} a_2 \right) + \left( \frac{1}{3} a_4 + \frac{1}{4} a_3 \right) + \dotsb \\
&= 2 + \frac{1}{3} (a_2 + a_3 + a_4 + \dots... |
### Instruction:
A sequence of integers is defined as follows: $a_i = i$ for $1 \le i \le 5,$ and
\[a_i = a_1 a_2 \dotsm a_{i - 1} - 1\]for $i > 5.$ Evaluate $a_1 a_2 \dotsm a_{2011} - \sum_{i = 1}^{2011} a_i^2.$
### Response:
| Hey, what's the value of $a_6$? I see. So $f(5) = 1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 - (1^2 + 2^2 + 3^2 + 4^2 + 5^2) = 120 - 55 = 65.$ Right, and $f(6) = 1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 \cdot (1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 - 1) - (1^2 + 2^2 + 3^2 + 4^2 + 5^2 + (1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 - 1)^2)$. Right. So... |
### Instruction:
Let $n$ be a positive integer. The sequence $(x_k)$ is defined by $x_0 = 0,$ $x_1 = 1,$ and
\[x_{k + 2} = \frac{(n - 1) x_{k + 1} - (n - k) x_k}{k + 1}\]for $k \ge 0.$ Find $x_0 + x_1 + x_2 + \dotsb$ as a function of $n.$
### Response:
| The first few terms are
\begin{align*}
x_2 &= \frac{(n - 1) \cdot 1 - (n - k) \cdot 0}{1} = n - 1, \\
x_3 &= \frac{(n - 1)(n - 1) - (n - 1) \cdot 1}{2} = \frac{(n - 1)(n - 2)}{2}, \\
x_4 &= \frac{(n - 1) \cdot \frac{(n - 1)(n - 2)}{2} - (n - 2)(n - 1)}{3} = \frac{(n - 1)(n - 2)(n - 3)}{6}.
\end{align*}It looks like
\[x... |
### Instruction:
The function $f$ is defined on the set of integers and satisfies \[f(n)= \begin{cases} n-3 & \mbox{if }n\ge 1000 \\ f(f(n+5)) & \mbox{if }n<1000. \end{cases}\]Find $f(84)$.
### Response:
| Denote by (1) and (2) the two parts of the definition of $f$, respectively. If we begin to use the definition of $f$ to compute $f(84)$, we use (2) until the argument is at least $1000$: \[f(84) = f(f(89)) = f(f(f(94))) = \dots = f^N(1004)\](where $f^N$ denotes composing $f$ with itself $N$ times, for some $N$). The nu... |
### Instruction:
Shown below are rows 1, 2, and 3 of Pascal's triangle.
\[
\begin{array}{ccccccc}
& & 1 & & 1 & & \\
& 1 & & 2 & & 1 & \\
1 & & 3 & & 3 & & 1
\end{array}
\]Let $(a_i),$ $(b_i),$ $(c_i)$ be the sequence, from left to right, of elements in the 2005th, 2006th, and 2007th rows, respectively, with the leftm... | More generally, suppose $(a_i),$ $(b_i),$ $(c_i)$ represent the entries in rows $n - 1,$ $n,$ $n + 1$ of Pascal's triangle. Then
\[a_i = \binom{n - 1}{i}, \ b_i = \binom{n}{i}, \ c_i = \binom{n + 1}{i},\]so
\begin{align*}
\frac{a_i}{b_i} &= \frac{\binom{n - 1}{i}}{\binom{n}{i}} \\
&= \frac{\frac{(n - 1)!}{i! (n - i - ... |
### Instruction:
The squares of a chessboard are labelled with numbers, as shown below.
[asy]
unitsize(0.8 cm);
int i, j;
for (i = 0; i <= 8; ++i) {
draw((i,0)--(i,8));
draw((0,i)--(8,i));
}
for (i = 0; i <= 7; ++i) {
for (j = 0; j <= 7; ++j) {
label("$\frac{1}{" + string(i + 8 - j) + "}$", (i + 0.5, j + 0.5)... | Numbers the rows 1, 2, 3, $\dots,$ 8 from top to bottom. Let $r_1$ be the row number of the chosen square in the first column. (For example, if the 5th square is chosen in the first column, then $r_1 = 5.$) Then the label of that square is $\frac{1}{r_1}.$
Similarly, if $r_2$ is the row number of the chosen square ... |
### Instruction:
Let $x$ and $y$ be two distinct positive real numbers. We define three sequences $(A_n),$ $(G_n),$ and $(H_n)$ as follows. First, $A_1,$ $G_1,$ and $H_1$ are the arithmetic mean, geometric mean, and harmonic mean of $x$ and $y,$ respectively. Then for $n \ge 2,$ $A_n,$ $G_n,$ $H_n$ are the arithmeti... | By AM-GM-HM,
\[A_1 \ge G_ 1 \ge H_1.\]Since $x$ and $y$ are distinct, equality cannot occur, so $A_1 > G_1 > H_1.$ Note that $G_1 = \sqrt{xy},$ and
\[A_1 H_1 = \frac{x + y}{2} \cdot \frac{2}{\frac{1}{x} + \frac{1}{y}} = \frac{x + y}{2} \cdot \frac{4xy}{x + y} = xy,\]so $G_1^2 = A_1 H_1.$
Now, suppose $A_n > G_n > H_n... |
### Instruction:
Let $u_n$ be the $n^\text{th}$ term of the sequence
\[1,\,\,\,\,\,\,2,\,\,\,\,\,\,5,\,\,\,\,\,\,6,\,\,\,\,\,\,9,\,\,\,\,\,\,12,\,\,\,\,\,\,13,\,\,\,\,\,\,16,\,\,\,\,\,\,19,\,\,\,\,\,\,22,\,\,\,\,\,\,23,\ldots,\]
where the first term is the smallest positive integer that is $1$ more than a multiple of $... | First, observe that the difference between consecutive terms within a grouping will always equal $3.$ Second, since all terms in a group with $n$ terms are congruent to $n$ modulo $3$ and all terms in a group with $n+1$ terms are congruent to $n+1$ modulo $3,$ the difference between the first term of the group with $n+... |
### Instruction:
In Zuminglish, all words consist only of the letters $M, O,$ and $P$. As in English, $O$ is said to be a vowel and $M$ and $P$ are consonants. A string of $M's, O's,$ and $P's$ is a word in Zuminglish if and only if between any two $O's$ there appear at least two consonants. Let $N$ denote the number o... | Let $a_n$ denote the number of $n$-letter words ending in two constants (CC), $b_n$ denote the number of $n$-letter words ending in a constant followed by a vowel (CV), and let $c_n$ denote the number of $n$-letter words ending in a vowel followed by a constant (VC - the only other combination, two vowels, is impossibl... |
### Instruction:
In a certain sequence the first term is $a_1=2007$ and the second term is $a_2=2008$. Furthermore, the values of the remaining terms are chosen so that $a_n+a_{n+1}+a_{n+2}=n$ for all $n\ge 1$. Determine $a_{1000}$.
### Response:
| To get started, we compute the first ten terms as: \[ 2007, 2008, -4014, 2008, 2009, -4013, 2009, 2010, -4012, 2010, \ldots \]It appears that each term is 1 greater than the number three terms previous. We can demonstrate that this will always occur using the given recurrence relation. We know that $a_n+a_{n+1}+a_{n+... |
### Instruction:
Mady has an infinite number of balls and empty boxes available to her. The empty boxes, each capable of holding four balls, are arranged in a row from left to right. At the first step, she places a ball in the first box (the leftmost box) of the row. At each subsequent step, she places a ball in the fi... | After trying the first few steps, we notice that the boxes resemble the set of positive integers in quinary (base $5$). In particular, the first box corresponds to the units digit, the second corresponds to the fives digit, and so forth. An empty box corresponds to the digit $0$ and a box with $k$ balls, $1 \le k \le 4... |
### Instruction:
Zan has created this iterative rule for generating sequences of whole numbers:
1) If a number is 25 or less, double the number.
2) If a number is greater than 25, subtract 12 from it.
Let $F$ be the first number in a sequence generated by the rule above. $F$ is a "sweet number" if 16 is not a term i... | Consider the remainders of numbers in one of these sequences modulo 12. The first step doubles the remainder, but second step does not change it. So, if repeatedly doubling a number modulo 12 does not give $16 \equiv 4$, the number 16 cannot be a term in the sequence. On the other hand, if there is a term congruent to ... |
### Instruction:
Sherlock Holmes and Dr. Watson recover a suitcase with a three-digit combination lock from a mathematician turned criminal. Embedded in the suitcase above the lock is the cryptic message "AT SEA BASE. SEAS EBB SEA: BASS. "
Dr. Watson comments, "This probably isn't about ocean fish. Perhaps it is an en... | We must solve the addition problem $$ \begin{array}{@{}c@{\;}c@{}c@{}c@{}c@{}c} & & S & E & A & S_d \\ & & & E & B & B_d \\ + & & & S & E & A_d\\ \cline{1-6} & & B & A& S& S_d\end{array},$$ where $d$ is an unknown base. It follows that $S + B + A$ leaves a residue of $S$ upon division by $d$. Thus, $B+A$ must be divisi... |
### Instruction:
In a rectangular array of points, with 5 rows and $N$ columns, the points are numbered consecutively from left to right beginning with the top row. Thus the top row is numbered 1 through $N,$ the second row is numbered $N + 1$ through $2N,$ and so forth. Five points, $P_1, P_2, P_3, P_4,$ and $P_5,$ ar... | Let each point $P_i$ be in column $c_i$. The numberings for $P_i$ can now be defined as follows.\begin{align*}x_i &= (i - 1)N + c_i\\ y_i &= (c_i - 1)5 + i \end{align*}
We can now convert the five given equalities.\begin{align}x_1&=y_2 & \Longrightarrow & & c_1 &= 5 c_2-3\\ x_2&=y_1 & \Longrightarrow & & N+c_2 &= 5 c_1... |
### Instruction:
In SHORT BINGO, a $5\times5$ card is filled by marking the middle square as WILD and placing 24 other numbers in the remaining 24 squares.
Specifically a card is made by placing 5 distinct numbers from the set $1-10$ in the first column, 5 distinct numbers from $11-20$ in the second column, 4 distinc... | There are 10 choices for the top number. That leaves 9 for the second number. Once those are chosen, there are 8 possibilities for the third number, then 7 for the fourth and 6 for the fifth. That gives a total of \[10\times9\times 8 \times 7\times 6 = 30240\] possible first columns. |
### Instruction:
Carolyn and Paul are playing a game starting with a list of the integers $1$ to $n.$ The rules of the game are:
$\bullet$ Carolyn always has the first turn.
$\bullet$ Carolyn and Paul alternate turns.
$\bullet$ On each of her turns, Carolyn must remove one number from the list such that this numb... | The list starts as $1,$ $2,$ $3,$ $4,$ $5,$ $6.$
If Carolyn removes $2,$ then Paul removes the remaining positive divisor of $2$ (that is, $1$) to leave the list $3,$ $4,$ $5,$ $6.$ Carolyn must remove a number from this list that has at least one positive divisor other than itself remaining. The only such number is $... |
### Instruction:
The points on this graph show the end-of-year sales amounts for each year. During what year after 1994 did sales increase the most number of dollars? Express your answer as a 4-digit year.
[asy]unitsize(101/10);
real textsize = 10, largetext = 12;
draw((10,0)--(0,0)--(0,10),Arrows(4,20));
path vertic... | The chart shows the amounts of sales at the end of each year. Therefore we're looking for a year on the chart that has the largest difference from the previous year. Determining the sales that increased the most number of dollars is equivalent to finding the right endpoint of the segment with the steepest slope. F... |
### Instruction:
The scores on a $110$-point test were organized in the stem-and-leaf plot shown. $9 | 6$ represents $96$ points. What is the mode of the scores? \begin{tabular}{c|lllllll}
\multicolumn{8}{c}{\underline{Points on the Test}}\\
5 &0 & 0 & & & & &\\
6 &3 & & & & & &\\
7 &7 & 8 & & & & &\\
8 &2 & 6 & 7 & 9 ... | In the fourth row, the digit $9$ appears $4$ times. This is the greatest number of appearances of a digit in any row, so the mode is $89.$ |
### Instruction:
In this Number Wall, you add the numbers next to each other and write the sum in the block directly above the two numbers. Which number will be the block labeled '$n$'? [asy]
draw((0,0)--(8,0)--(8,2)--(0,2)--cycle);
draw((2,0)--(2,2));
draw((4,0)--(4,2));
draw((6,0)--(6,2));
draw((1,2)--(7,2)--(7,4)--(... | Filling in the squares from the bottom of the pyramid to the top, we have the following: [asy]
draw((0,0)--(8,0)--(8,2)--(0,2)--cycle);
draw((2,0)--(2,2));
draw((4,0)--(4,2));
draw((6,0)--(6,2));
draw((1,2)--(7,2)--(7,4)--(1,4)--cycle);
draw((3,2)--(3,4));
draw((5,2)--(5,4));
draw((2,4)--(2,6)--(6,6)--(6,4)--cycle);
d... |
### Instruction:
A $\textit{palindrome}$ is a number which reads the same forward as backward, for example 313 or 1001. Ignoring the colon, how many different palindromes are possible on a 12-hour digital clock displaying only the hours and minutes? (Notice a zero may not be inserted before a time with a single-digit h... | First, consider the three-digit palindromes. There are $9$ choices for the first digit (the hour): $1$, $2$,..., $9$. There are $6$ choices for the second digit (the tens digit of the minutes): $0$, $1$, ..., $5$. The last digit (the units digit of the minutes) has to be the same as the first digit. So there are $9... |
### Instruction:
A square carpet of side length 9 feet is designed with one large shaded square and eight smaller, congruent shaded squares, as shown. [asy]
draw((0,0)--(9,0)--(9,9)--(0,9)--(0,0));
fill((1,1)--(2,1)--(2,2)--(1,2)--cycle,gray(.8));
fill((4,1)--(5,1)--(5,2)--(4,2)--cycle,gray(.8));
fill((7,1)--(8,1)-... | We are given that $\frac{9}{\text{S}}=\frac{\text{S}}{\text{T}}=3.$ \[\frac{9}{\text{S}}=3\] gives us $S=3,$ so \[\frac{\text{S}}{\text{T}}=3\] gives us $T=1$. There are 8 shaded squares with side length $\text{T}$ and there is 1 shaded square with side length $\text{S},$ so the total shaded area is $8\cdot(1\cdot1)+1\... |
### Instruction:
A circular cylindrical post with a circumference of 4 feet has a string wrapped around it, spiraling from the bottom of the post to the top of the post. The string evenly loops around the post exactly four full times, starting at the bottom edge and finishing at the top edge. The height of the post is ... | Each time the string spirals around the post, it travels 3 feet up and 4 feet around the post. If we were to unroll this path, it would look like: [asy]
size(150);
draw((0,0)--(0,3)--(4,3)--(4,0)--cycle, linewidth(.7));
draw((0,0)--(4,3),linewidth(.7));
label("3",(0,1.5),W);
label("4",(2,3),N);
[/asy] Clearly, a 3-4-5... |
### Instruction:
Below is a portion of the graph of a function, $y=f(x)$:
[asy]
import graph; size(8cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-3.25,xmax=5.25,ymin=-3.25,ymax=4.25;
pen cqcqcq=rgb(0.75,0.75,0.75);
/*grid*/ pen gs=linewidth(0.7)+cqcqcq+linetype("2 ... | Note that the graph of $g(x)$ is identical to the graph of $f(x)$ shifted $a$ units to the left. (This is true because if $(x,f(x))$ is a point on the graph of $f$, then $(x-a,f(x))$ is the corresponding point on the graph of $g$.)
The graph of a function and its inverse are reflections of each other across the line $... |
### Instruction:
Below is a portion of the graph of a function, $y=p(x)$:
[asy]
import graph; size(6cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-3.5,xmax=3.5,ymin=-4.25,ymax=6.25;
pen cqcqcq=rgb(0.75,0.75,0.75);
/*grid*/ pen gs=linewidth(0.7)+cqcqcq+linetype("2 2")... | The point $(1.5,4)$ is on the graph. This means that $p(1.5)=4$. |
### Instruction:
The first three stages of a pattern are shown below, in which each line segment represents a toothpick. If the pattern continues such that at each successive stage, three toothpicks are added to the previous arrangement, how many toothpicks are necessary to create the arrangement for the 250th stage? [... | The number of toothpicks in each stage form an arithmetic sequence. The first term in this arithmetic sequence is 4, and the common difference is 3 (the number of toothpicks added to get to the next stage), so the number of toothpicks used in the 250th stage is $4 + 3 \cdot 249 = 751$. |
### Instruction:
In the function machine shown, the input is 10. What is the output?
[asy]
size(200); currentpen = fontsize(10pt); picture a,b,c,d,e,f;
real height = 3, width1 = 10, width2 = 11, width3 = 10, width4 = 10;
real widthC = 20,heightC = 6;
real widthE = 10, lengthE = 4.5,angleE = 60;
draw(a,(0,0)--(width1,0... | We just follow the flowchart. First, we double 10 to get 20. Since 20 is greater than 18, we follow the chart to the right and subtract 5, giving a final output of $15$. |
### Instruction:
The function $f$ is graphed below. Each small box has width and height 1.
[asy]
size(150);
real ticklen=3;
real tickspace=2;
real ticklength=0.1cm;
real axisarrowsize=0.14cm;
pen axispen=black+1.3bp;
real vectorarrowsize=0.2cm;
real tickdown=-0.5;
real tickdownlength=-0.15inch;
real tickdownbase=0.3... | Reading from the graph we see that $f(3)=6$. Therefore Larry writes 6 on his second finger. Since $f(6)=5$ we see that Larry writes 5 on this third finger. If we apply $f$ again, we see that Larry writes \[f(5)=4\] on the fourth finger. After this Larry writes $f(4)=3$ on his fifth finger. Now the process repeats!... |
### Instruction:
Below is a portion of the graph of an invertible function, $y=f(x)$:
[asy]
import graph; size(8cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-3.25,xmax=3.25,ymin=-6.25,ymax=7.25;
pen cqcqcq=rgb(0.75,0.75,0.75);
/*grid*/ pen gs=linewidth(0.7)+cqcqcq+... | Since $f(b)=4$, the point $(b,4)$ is on the graph of $y=f(x)$. By inspection, $(2,4)$ is on the graph, so $b=2$ (there is no other candidate since $f$ is stated to be an invertible function).
Similarly, since $f(a)=2$, the point $(a,2)$ is on the graph of $y=f(x)$. By inspection, $(0,2)$ is on the graph, so $a=0$.
Th... |
### Instruction:
A street has 20 houses on each side, for a total of 40 houses. The addresses on the south side of the street form an arithmetic sequence, as do the addresses on the north side of the street. On the south side, the addresses are 4, 10, 16, etc., and on the north side they are 3, 9, 15, etc. A sign paint... | Let's take the south side of the street first. The sequence begins with 4 and increases by 6 each time. So we can write it as 4, 10, 16, ... And we know there are 20 houses, so we need to find the 20th number in the sequence. We could write out the sequence and look for the 20th number, but there's a better way. We can... |
### Instruction:
Below is a portion of the graph of a function, $y=E(x)$:
[asy]
import graph; size(8cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-4.5,xmax=4.5,ymin=-0.99,ymax=6.5;
pen cqcqcq=rgb(0.75,0.75,0.75);
/*grid*/ pen gs=linewidth(0.7)+cqcqcq+linetype("2 2");... | The point $(3,5)$ is on the graph. This means that $E(3)=5$. |
### Instruction:
Below is a portion of the graph of a function, $y=h(x)$:
[asy]
import graph; size(8cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-4.25,xmax=4.25,ymin=-7.25,ymax=6.25;
pen cqcqcq=rgb(0.75,0.75,0.75);
/*grid*/ pen gs=linewidth(0.7)+cqcqcq+linetype("2 ... | If the graphs intersect at $(a,b)$, then we have $$h(a) = h(a-3) \qquad(= b).$$Thus, $(a,b)$ and $(a-3,b)$ are both on the original graph of $y=h(x)$. Looking for two points on the original graph which are separated by $3$ units horizontally, we find $(-2,3)$ and $(1,3)$. Thus $a-3=-2,$ $a=1,$ and $b=3;$ the graphs of ... |
### Instruction:
How many paths are there from $A$ to $B$ on the lines of the grid shown, if every step must be up or to the right?[asy]size(3cm,3cm);int w=6;int h=3;int i;for (i=0; i<h; ++i){ draw((0,i) -- (w-1,i));}for (i=0; i<w; ++i){ draw((i, 0)--(i,h-1));}label("$B$", (w-1,h-1), NE);label("$A$", (0,0), SW);[/asy]
... | There are 5 steps to the right, and 2 steps up. These 7 steps can be made in any order, so the answer is $\dbinom{7}{2} = \dfrac{7 \times 6}{2 \times 1} = 21$. |
### Instruction:
Three different one-digit positive integers are placed in the bottom row of cells. Numbers in adjacent cells are added and the sum is placed in the cell above them. In the second row, continue the same process to obtain a number in the top cell. What is the difference between the largest and smallest n... | If the lower cells contain $A$, $B$ and $C$, then the second row will contain $A + B$ and $B + C$, and the top cell will contain $A + 2B+C$. To obtain the smallest sum, place 1 in the center cell and 2 and 3 in the outer ones. The top number will be 7. For the largest sum, place 9 in the center cell and 7 and 8 in the... |
### Instruction:
If
\[f(n + 1) = (-1)^{n + 1} n - 2f(n)\]for $n \ge 1,$ and $f(1) = f(1986),$ compute
\[f(1) + f(2) + f(3) + \dots + f(1985).\]
### Response:
| We can list the equations
\begin{align*}
f(2) &= 1 - 2f(1), \\
f(3) &= -2 - 2f(2), \\
f(4) &= 3 - 2f(3), \\
f(5) &= -4 - 2f(4), \\
&\dots, \\
f(1985) &= -1984 - 2f(1984), \\
f(1986) &= 1985 - 2f(1985).
\end{align*}Adding these equations, we get
\[f(2) + f(3) + \dots + f(1986) = (1 - 2 + 3 - 4 + \dots + 1983 - 1984 + 19... |
### Instruction:
An infinite sequence of real numbers $a_1, a_2, \dots$ satisfies the recurrence
\[ a_{n+3} = a_{n+2} - 2a_{n+1} + a_n \]for every positive integer $n$. Given that $a_1 = a_3 = 1$ and $a_{98} = a_{99}$, compute $a_1 + a_2 + \dots + a_{100}$.
### Response:
| Writing down the recursion for $n = 1, 2, 3, \dots, 97,$ we have
\[\begin{aligned}
a_4 &= a_3 - 2a_2 + a_1 \\
a_5 &= a_4 - 2a_3 + a_2 \\
a_6 &= a_5 - 2a_4 + a_3 \\
&\;\,\vdots \\
a_{100} &= a_{99} - 2a_{98} + a_{97}.
\end{aligned}\]Summing all $97$ of these equations, we have \[a_4 + \dots + a_{100} = (a_3 + \dots + a... |
### Instruction:
The function defined by
\[f(x) = \left\{
\begin{array}{cl}
x + k & \text{if $x < 4$}, \\
2x - 3 & \text{if $x \ge 4$}
\end{array}
\right.\]has an inverse, and the inverse is defined for all real numbers. Enter all possible values of $k,$ separated by commas.
### Response:
| Note that $f(4) = 5.$ If we set $k$ so that $x + 4 = 5,$ then $k = 1.$ This value of $k$ makes the function continuous, as shown below.
[asy]
unitsize(0.3 cm);
int i;
for (i = -8; i <= 8; ++i) {
draw((i,-8)--(i,8),gray(0.7));
draw((-8,i)--(8,i),gray(0.7));
}
draw((-8,0)--(8,0),Arrows(6));
draw((0,-8)--(0,8),A... |
### Instruction:
The sequence $a_0$, $a_1$, $a_2$, $\ldots\,$ satisfies the recurrence equation
\[
a_n = 2 a_{n-1} - 2 a_{n - 2} + a_{n - 3}
\]for every integer $n \ge 3$. If $a_{20} = 1$, $a_{25} = 10$, and $a_{30} = 100$, what is the value of $a_{1331}$?
### Response:
| We can calculate the first few terms of the sequence and look for a pattern. For $n=3$,
$$a_3 = 2a_2 - 2a_1 + a_0.$$For $n=4$ we get
$$a_4 = 2a_3 - 2a_2 + a_1 = 2(2a_2 - 2a_1 + a_0) - 2a_2+a_1 = 2a_2 - 3a_1+2a_0.$$With $n=5$ we have
$$a_5 = 2a_4 - 2a_3 + a_2 = 2(2a_2 - 3a_1+2a_0) - 2(2a_2 - 2a_1 + a_0) +a_2 = a_2 - 2a_... |
### Instruction:
Given that \begin{align*}x_{1}&=211,\\ x_{2}&=375,\\ x_{3}&=420,\\ x_{4}&=523,\ \text{and}\\ x_{n}&=x_{n-1}-x_{n-2}+x_{n-3}-x_{n-4}\ \text{when}\ n\geq5, \end{align*}find the value of $x_{531}+x_{753}+x_{975}$.
### Response:
| This problem involves a recursive sequence, which means that each term is defined in terms of previous terms. A common technique for solving problems with recursive sequences is to look for a pattern or a formula that describes the general term of the sequence. To do this, I will write out a few more terms of the seque... |
### Instruction:
Let \[A = \lceil \log_2 2 \rceil + \lceil \log_2 3 \rceil + \dots + \lceil \log_2 1000 \rceil\]and \[B = \lfloor \log_2 2 \rfloor + \lfloor \log_2 3 \rfloor + \dots + \lfloor \log_2 1000 \rfloor.\]Compute $A-B.$
### Response:
| Grouping the corresponding terms in $A-B,$ we can write \[A-B = \left(\lceil \log_2 2 \rceil - \lfloor \log_2 2 \rfloor\right) + \left(\lceil \log_2 3 \rceil - \lfloor \log_2 3 \rfloor\right) + \dots + \left(\lceil \log_2 1000 \rceil - \lfloor \log_2 1000 \rfloor\right). \]For a real number $x,$ we have $\lceil x \rcei... |
### Instruction:
The Fibonacci sequence is defined by $F_1 = F_2 = 1$ and $F_n = F_{n - 1} + F_{n - 2}$ for $n \ge 3.$
Compute
\[\sum_{j = 1}^{2004} i^{2004 - F_j}.\]
### Response:
| Since $i^4 = 1,$ $i^{2004 - F_j} = \frac{1}{i^{F_j}}$ depends only on the value of $F_j$ modulo 4.
We compute the first few Fibonacci numbers modulo 4:
\[
\begin{array}{c|c}
n & F_n \pmod{4} \\ \hline
1 & 1 \\
2 & 1 \\
3 & 2 \\
4 & 3 \\
5 & 1 \\
6 & 0 \\
7 & 1 \\
8 & 1
\end{array}
\]Since $F_7 \equiv F_1 \equiv 1 \pmo... |
### Instruction:
The sequence 1, 2, 4, 5, 10, 11, 22, 23, 46, 47, $\dots$ is formed as follows:
$\bullet$ Start with the number 1.
$\bullet$ Add one to get 2.
$\bullet$ Double that to get 4.
$\bullet$ Add one to get 5.
$\bullet$ Double that to get 10, and so on.
We repeat the steps of "add one" and "double that",... | If we take every other term, starting with the second term 2, we get
\[2, 5, 11, 23, 47, \dots.\]If we add one to each of these terms, we get
\[3, 6, 12, 24, 48, \dots.\]Each term appear to be double the previous term.
To confirm this, let one term in the original sequence be $x - 1,$ after we have added 1. Then the ... |
### Instruction:
A "slackrope walker" is much like a tightrope walker except that the rope on which he performs is not pulled tight. Paul, a slackrope walker, has a rope tied to two $15\text{ m}$ high poles which are $14\text{ m}$ apart. When he is standing on the rope $5\text{ m}$ away from one of the poles, he is $3\... | Label the points $A,$ $B,$ $C,$ and $D,$ as shown. Through $P,$ draw a line parallel to $DC$ as shown. The points $X$ and $Y$ are where this line meets $AD$ and $BC.$ From this, we see that $$AX=BY=15-3=12.$$ Also, $PY=14-5=9.$
[asy]
draw((0,0)--(14,0)--(14,15)--(5,3)--(0,15)--cycle,black+linewidth(1));
draw((0,3)--(5... |
### Instruction:
A solid $5\times 5\times 5$ cube is composed of unit cubes. Each face of the large, solid cube is partially painted with gray paint, as shown. [asy]
fill((0,0)--(0,1)--(1,1)--(1,0)--cycle,gray);
fill((0,4)--(0,5)--(1,5)--(1,4)--cycle,gray);
fill((4,1)--(5,1)--(5,0)--(4,0)--cycle,gray);
fill((1,2)--... | We know that each of the unit cubes in the $3\times3\times3$ cube in the center of the $5\times5\times5$ cube has no paint on it. On the surface of the cube, three of the unit cubes on each edge of the big cube have no paint on them, and the center unit cube of each face of the big cube has no paint on it. Since a cube... |
### Instruction:
A standard deck of playing cards with $26$ red cards and $26$ black cards is split into two piles, each having at least one card. In pile $A$ there are six times as many black cards as red cards. In pile $B,$ the number of red cards is a multiple of the number of black cards. How many red cards are in ... | Let pile $A$ have $r_A$ red cards and $b_A$ black cards, and let pile $B$ have $r_B$ red cards and $b_B$ black cards. From the given information, we have $$\left\{ \begin{array}{ll}
r_A+r_B & = 26 \\
b_A+b_B & = 26 \\
b_A &= 6\cdot r_A \\
r_B &= m\cdot b_B \\
\end{array} \right.$$ for some positive integer $m.$ Substi... |
### Instruction:
Below is a drawing of all $26$ letters in the English alphabet. As drawn below, some of these letters could be portions of the graph of a function, and some cannot. For example, the $\textsf{O}$ resembles an ellipse, which cannot be part of the graph of a function.
As drawn below, which of these lette... | To be part of the graph of a function, a figure must have at most one point of intersection with any vertical line. Only two letters (as drawn in the problem) have this property: $\textsf{V}$ and $\textsf{W}.$ (Following the instructions, your answer should be formatted as $\text{VW}.$) |
### Instruction:
Below are the graphs of two functions, $f(x)$ and $g(x)$, defined on the domain $0\le x\le 18$:
[asy]
import graph; size(8cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-1.5,xmax=18.5,ymin=-1.5,ymax=12.5;
pen cqcqcq=rgb(0.75,0.75,0.75);
/*grid*/ pen ... | We can measure $|f(x)-g(x)|$ as the vertical distance between the two graphs at $x$. The sign of $f(x)-g(x)$ is positive if $f(x)>g(x)$, so we focus on the part of the graph where the dashed orange line lies above the solid dark green line. Within this part of the graph, the greatest vertical distance between the orang... |
### Instruction:
Below is a portion of the graph of a function, $y=h(x)$:
[asy]
import graph; size(8cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-0.75,xmax=8.25,ymin=-1.25,ymax=10.25;
pen cqcqcq=rgb(0.75,0.75,0.75);
/*grid*/ pen gs=linewidth(0.7)+cqcqcq+linetype("2... | We could check $h(x)$ separately for each integer $x$ from $0$ to $8$: for example, $h(0)\approx 3.8$, so $h(0)>0$, but $h(1)\approx -0.7$, so $h(1)\not>1$, and so on.
However, it is easier to see at a glance which $x$ satisfy $h(x)>x$ by superimposing a graph of $y=x$ on the graph of $y=h(x)$:
[asy]
draw((-0.75,-0.7... |
### Instruction:
Let $M$ be a subset of $\{1,2,3, \ldots, 15\}$ such that the product of any three distinct elements of $M$ is not a square. Determine the maximum number of elements in $M$.
### Response:
| Note that the product of the three elements in each of the sets $\{1,4,9\},\{2,6,12\}$, $\{3,5,15\}$, and $\{7,8,14\}$ is a square. Hence none of these sets is a subset of $M$. Because they are disjoint, it follows that $M$ has at most $15-4=11$ elements.Since 10 is not an element of the aforementioned sets, if $10 \no... |
### Instruction:
How many labeled graphs with a score of (6, 2, 2, 2, 2, 2, 2) are there?
Relevant Theorem: In graph theory, the Score Theorem, also known as the Havel-Hakimi Theorem, is a method used to determine if a given degree sequence can be realized by a simple, undirected graph. A degree sequence is a list of n... | 15 |
### Instruction:
Consider the following graph, with links costs listed, and assume we are using shortest-path (or lowest-cost) routing, and that routing has equilibrated to a constant set of routing tables. The routing algorithm uses poisoned reverse, advertising an infinite weight for the poisoned paths. is the distan... | True |
### Instruction:
How many triangles are there whose sides are all integers and whose maximum side length equals 11?
Relevant Theorem: Integer Programming (IP) is a mathematical optimization technique that deals with linear programming problems where some or all of the variables are restricted to take integer values. It... | 36 |
### Instruction:
Suppose the codeword that we use to describe a random variable X always starts with a symbol chosen from the set {7, 8, 9} , followed by binary digits {0, 1}. Thus we have a ternary code for the first symbol and binary thereafter. Give the optimal uniquely decodeable code (minimum expected number of sy... | [7, 8, 9, 70, 80, 90] |
### Instruction:
Let a undirected graph G with edges E = {<0,2>,<2,1>,<2,3>,<3,4>,<4,1>}, which <A,B> represent Node A is connected to Node B. What is the shortest path from node 4 to node 0? Represent the path as a list.
Relevant Theorem: In graph theory, the Shortest Path refers to the problem of finding the path wit... | [4, 1, 2, 0] |
### Instruction:
How many distinct directed trees can be constructed from a undirected tree with 100 nodes?
Relevant Theorem: An Acyclic Graph in graph theory is a type of graph that does not contain any cycles. In other words, it is a graph where you cannot traverse through the vertices and edges and return to the sta... | 100 |
### Instruction:
Figure Q8 shows the contour of an object. Represent it with an 8-directional chain code. Represent the answer as a list with each digit as a element.
Relevant Theorem: Chain Code is a technique used in computer vision and image processing, rather than machine learning. It is a method for representing t... | [6, 7, 0, 6, 6, 4, 3, 4, 3, 1, 1] |
### Instruction:
Let a undirected graph G with edges E = {<0,3>, <1,3>, <2,3>}, which <A,B> represent Node A is connected to Node B. What is the minimum vertex cover of G? Represent the vertex cover in a list of ascending order.
Relevant Theorem: Vertex Cover in graph theory is a set of vertices in a graph such that ea... | [3] |
### Instruction:
Calculate the Hamming pairwise distances and determine the minimum Hamming distance among the following codewords: 00000,10101,01010
Relevant Theorem: Hamming distance is a concept in computer networking and information theory that measures the difference between two strings of equal length by counting... | 2 |
### Instruction:
Let $X_0, X_1, X_2, \ldots$ be drawn i.i.d. from $p(x)$, and $x\in\{1,2,3,\ldots,100\}. Let $N$ be the waiting time to the next occurrence of $X_0$. Compute $E(N)$.
Relevant Theorem: Expected waiting time in information theory refers to the average time one has to wait before a specific event or messag... | 100.0 |
### Instruction:
Assuming we are underground, and the only thing we can observe is whether a person brings an umbrella or not. The weather could be either rain or sunny. Assuming the P(rain)=0.6 and P(sunny)=0.4. Assuming the weather on day $k$ is dependent on the weather on day $k-1$. We can write the transition proba... | 0.5167 |
### Instruction:
If the sum-product algorithm is run on a factor graph with a tree structure (no loops), then after a finite number of messages have been sent, there will be no pending messages. True or false?
Relevant Theorem: Message Passing algorithm, also known as Belief Propagation or Sum-Product algorithm, is a t... | True |
### Instruction:
Let M be the set of bounded functions (i.e. \sup_{x\in[a,b]}|f(x)|<\infty) in C[0,1]. Is the set ${F(x)=\int_0^x f(t) dt | f \in M }$ a sequentially compact set? Answer 1 for yes and 0 for no. Furthermore, it can be proved using 1. Arzelà-Ascoli theorem, 2. Riesz representation theorem, 3. Banach fixed... | [1, 1] |
### Instruction:
A distribution represented by a directed tree can be written as an equivalent distribution over the corresponding undirected tree. True or false?
Relevant Theorem: An Acyclic Graph in graph theory is a type of graph that does not contain any cycles. In other words, it is a graph where you cannot traver... | True |
### Instruction:
Suppose $E \subset(0,2 \pi) is a measurable set. \left\{\xi_n
ight\}$ is an arbitrary sequence of real numbers. If the Lebesgue measure of E is 2, what is $\lim _{n
ightarrow \infty} \int_E \cos ^2 (n x+\xi_n ) dx$? Return the numeric.
Relevant Theorem: Lebesgue measure is a fundamental concept in rea... | 1.0 |
### Instruction:
./mingyin/mdp.png shows a rectangular gridworld representation of a simple finite MDP. The cells of the grid correspond to the states of the environment. At each cell, four actions are possible: north, south, east, and west, which deterministically cause the agent to move one cell in the respective dir... | [8.8, 24.4] |
### Instruction:
Given the following equation: x^4 - x - 10 = 0. determine the initial approximations for finding the smallest positive root. Use these to find the root correct to three decimal places with Secant method.
Relevant Theorem: A scent algorithm in numerical analysis is a computational method used to find th... | 1.856 |
### Instruction:
If there exists an ordered numbering of the nodes such that for each node there are no links going to a lower-numbered node, then there are no directed cycles in a directed graph. True or false?
Relevant Theorem: An Acyclic Graph in graph theory is a type of graph that does not contain any cycles. In o... | True |
### Instruction:
Given an array of integers `nums` and an integer `target`, return _indices of the two numbers such that they add up to `target`_.
You may assume that each input would have **_exactly_ one solution**, and you may not use the _same_ element twice.
You can return the answer in any order.
**Example 1:**... | ```python
def twoSum(nums, target):
map = {}
for i, num in enumerate(nums):
complement = target - num
if complement in map:
return [map[complement], i]
map[num] = i
return []
```
explanation
The algorithm leverages a hash map (unordered_map in C++, HashMap in Java, dictio... |
### Instruction:
You are given two **non-empty** linked lists representing two non-negative integers. The digits are stored in **reverse order**, and each of their nodes contains a single digit. Add the two numbers and return the sum as a linked list.
You may assume the two numbers do not contain any leading zero, exc... | ```python
class ListNode:
def __init__(self, val=0, next=None):
self.val = val
self.next = next
def addTwoNumbers(l1, l2):
dummy = ListNode(0)
current = dummy
carry = 0
while l1 or l2 or carry:
sum_val = (l1.val if l1 else 0) + (l2.val if l2 else 0) + carry
carry = ... |
### Instruction:
Given a string `s`, find the length of the **longest** **substring** without repeating characters.
**Example 1:**
**Input:** s = "abcabcbb "
**Output:** 3
**Explanation:** The answer is "abc ", with the length of 3.
**Example 2:**
**Input:** s = "bbbbb "
**Output:** 1
**Explanation:** The answer... | ```python
def length_of_longest_substring(s: str) -> int:
left = 0
right = 0
max_length = 0
characters = set()
while right < len(s):
if s[right] not in characters:
characters.add(s[right])
max_length = max(max_length, right - left + 1)
right += 1
... |
### Instruction:
Given two sorted arrays `nums1` and `nums2` of size `m` and `n` respectively, return **the median** of the two sorted arrays.
The overall run time complexity should be `O(log (m+n))`.
**Example 1:**
**Input:** nums1 = \[1,3\], nums2 = \[2\]
**Output:** 2.00000
**Explanation:** merged array = \[1,2,3... | ```python
def findMedianSortedArrays(nums1, nums2):
if len(nums1) > len(nums2):
return findMedianSortedArrays(nums2, nums1)
x, y = len(nums1), len(nums2)
low, high = 0, x
while low <= high:
partition_x = (low + high) // 2
partition_y = (x + y + 1) // 2 - partition_x
... |
### Instruction:
Given a string `s`, return _the longest_ _palindromic_ _substring_ in `s`.
**Example 1:**
**Input:** s = "babad "
**Output:** "bab "
**Explanation:** "aba " is also a valid answer.
**Example 2:**
**Input:** s = "cbbd "
**Output:** "bb "
**Constraints:**
* `1 <= s.length <= 1000`
* `s` co... | ```python
def longest_palindromic_substring(s: str) -> str:
n = len(s)
if n == 0: return ""
start, max_length = 0, 1
for i in range(n):
l, r = i, i
while r < n - 1 and s[r] == s[r + 1]:
r += 1
i = r
while l > 0 and r < n - 1 and s[l - 1] == s[r + 1]:
... |
### Instruction:
The string `"PAYPALISHIRING "` is written in a zigzag pattern on a given number of rows like this: (you may want to display this pattern in a fixed font for better legibility)
P A H N
A P L S I I G
Y I R
And then read line by line: `"PAHNAPLSIIGYIR "`
Write the code that will take a string... | ```python
def convert(s: str, numRows: int) -> str:
if numRows == 1 or numRows >= len(s):
return s
rows = [""] * numRows
curRow = 0
goingDown = False
for c in s:
rows[curRow] += c
if curRow == 0 or curRow == numRows - 1:
goingDown = not goingDown
curRow ... |
### Instruction:
Given a signed 32-bit integer `x`, return `x` _with its digits reversed_. If reversing `x` causes the value to go outside the signed 32-bit integer range `[-231, 231 - 1]`, then return `0`.
**Assume the environment does not allow you to store 64-bit integers (signed or unsigned).**
**Example 1:**
**... | ```python
def reverse(x: int) -> int:
sign = -1 if x < 0 else 1
x = abs(x)
res = 0
while x:
res = res * 10 + x % 10
x //= 10
res *= sign
return res if -2**31 <= res <= 2**31 - 1 else 0
```
explanation
The algorithm initializes `res` as 0. Then, we determine the sign of the input ... |
### Instruction:
Implement the `myAtoi(string s)` function, which converts a string to a 32-bit signed integer (similar to C/C++'s `atoi` function).
The algorithm for `myAtoi(string s)` is as follows:
1. Read in and ignore any leading whitespace.
2. Check if the next character (if not already at the end of the stri... | ```python
def myAtoi(s):
result, i, sign = 0, 0, 1
while i < len(s) and s[i] == ' ': i += 1
if i < len(s) and (s[i] == '-' or s[i] == '+'):
sign = -1 if s[i] == '-' else 1
i += 1
while i < len(s) and s[i].isdigit():
result = result * 10 + int(s[i])
i += 1
if resul... |
### Instruction:
Given an integer `x`, return `true` _if_ `x` _is a_ _**palindrome**__, and_ `false` _otherwise_.
**Example 1:**
**Input:** x = 121
**Output:** true
**Explanation:** 121 reads as 121 from left to right and from right to left.
**Example 2:**
**Input:** x = -121
**Output:** false
**Explanation:** From... | ```python
def is_palindrome(x: int) -> bool:
if x < 0:
return False
original, reversed = x, 0
while x > 0:
reversed = reversed * 10 + x % 10
x //= 10
return original == reversed
```
explanation
Here is the algorithm:
1. If the integer is negative, immediately return false, as n... |
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