conversations listlengths 2 2 | source stringclasses 1
value | score int64 5 5 |
|---|---|---|
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: How many different $4\\times 4$ arrays whose entries are all 1's and -1's have the property that the sum of the entries in each row is 0 and the sum of the entries in each column is 0?\n [/INST] The problem is asking us for al... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Nine tiles are numbered $1, 2, 3, \\cdots, 9,$ respectively. Each of three players randomly selects and keeps three of the tiles, and sums those three values. The probability that all three players obtain an odd sum is $m/n,$ ... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Let $n$ be the number of ordered quadruples $(x_1,x_2,x_3,x_4)$ of positive odd integers that satisfy $\\sum_{i = 1}^4 x_i = 98.$ Find $\\frac n{100}.$\n [/INST] Define $x_i = 2y_i - 1$. Then $2\\left(\\sum_{i = 1}^4 y_i\\righ... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Two mathematicians take a morning coffee break each day. They arrive at the cafeteria independently, at random times between 9 a.m. and 10 a.m., and stay for exactly $m$ minutes. The probability that either one arrives while t... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Forty teams play a tournament in which every team plays every other team exactly once. No ties occur, and each team has a $50 \\%$ chance of winning any game it plays. The probability that no two teams win the same number of g... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: 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... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: A deck of forty cards consists of four $1$'s, four $2$'s,..., and four $10$'s. A matching pair (two cards with the same number) is removed from the deck. Given that these cards are not returned to the deck, let $m/n$ be the pr... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Given eight distinguishable rings, let $n$ be the number of possible five-ring arrangements on the four fingers (not the thumb) of one hand. The order of rings on each finger is significant, but it is not required that each fi... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Given that\n$\\frac 1{2!17!}+\\frac 1{3!16!}+\\frac 1{4!15!}+\\frac 1{5!14!}+\\frac 1{6!13!}+\\frac 1{7!12!}+\\frac 1{8!11!}+\\frac 1{9!10!}=\\frac N{1!18!}$\nfind the greatest integer that is less than $\\frac N{100}$.\n [/IN... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Let $S$ be the set of points whose coordinates $x,$ $y,$ and $z$ are integers that satisfy $0\\le x\\le2,$ $0\\le y\\le3,$ and $0\\le z\\le4.$ Two distinct points are randomly chosen from $S.$ The probability that the midpoint... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: 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... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: The numbers $1, 2, 3, 4, 5, 6, 7,$ and $8$ are randomly written on the faces of a regular octahedron so that each face contains a different number. The probability that no two consecutive numbers, where $8$ and $1$ are conside... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Club Truncator is in a soccer league with six other teams, each of which it plays once. In any of its 6 matches, the probabilities that Club Truncator will win, lose, or tie are each $\\frac {1}{3}$. The probability that Club ... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Each of the $2001$ students at a high school studies either Spanish or French, and some study both. The number who study Spanish is between $80$ percent and $85$ percent of the school population, and the number who study Frenc... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: A set of positive numbers has the triangle property if it has three distinct elements that are the lengths of the sides of a triangle whose area is positive. Consider sets $\\{4, 5, 6, \\ldots, n\\}$ of consecutive positive in... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Each unit square of a 3-by-3 unit-square grid is to be colored either blue or red. For each square, either color is equally likely to be used. The probability of obtaining a grid that does not have a 2-by-2 red square is $\\fr... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Many states use a sequence of three letters followed by a sequence of three digits as their standard license-plate pattern. Given that each three-letter three-digit arrangement is equally likely, the probability that such a li... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Jane is 25 years old. Dick is older than Jane. In $n$ years, where $n$ is a positive integer, Dick's age and Jane's age will both be two-digit number and will have the property that Jane's age is obtained by interchanging the ... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Compute $\\dbinom{8}{4}$. [/INST] \\begin{align*}\n\\dbinom{8}{4} &= \\dfrac{8!}{4!4!} \\\\\n&= \\dfrac{8\\times 7\\times 6\\times 5}{4\\times 3\\times 2\\times 1} \\\\\n&= \\dfrac{8}{4\\times 2\\times 1}\\times 7 \\times \\df... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Let $\\mathcal{S}$ be the set $\\lbrace1,2,3,\\ldots,10\\rbrace$ Let $n$ be the number of sets of two non-empty disjoint subsets of $\\mathcal{S}$. (Disjoint sets are defined as sets that have no common elements.) Find the rem... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Let $N$ be the number of positive integers that are less than or equal to $2003$ and whose base-$2$ representation has more $1$'s than $0$'s. Find the remainder when $N$ is divided by $1000$.\n [/INST] In base-$2$ representati... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: The sum of the areas of all triangles whose vertices are also vertices of a $1$ by $1$ by $1$ cube is $m + \\sqrt{n} + \\sqrt{p},$ where $m, n,$ and $p$ are integers. Find $m + n + p.$\n [/INST] [asy] size(120); defaultpen(lin... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: An integer between $1000$ and $9999$, inclusive, is called balanced if the sum of its two leftmost digits equals the sum of its two rightmost digits. How many balanced integers are there?\n [/INST] If the common sum of the fir... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: A bug starts at a vertex of an equilateral triangle. On each move, it randomly selects one of the two vertices where it is not currently located, and crawls along a side of the triangle to that vertex. Given that the probabili... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Define a regular $n$-pointed star to be the union of $n$ line segments $P_1P_2, P_2P_3,\\ldots, P_nP_1$ such that\nthe points $P_1, P_2,\\ldots, P_n$ are coplanar and no three of them are collinear,\neach of the $n$ line segme... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: A jar has $10$ red candies and $10$ blue candies. Terry picks two candies at random, then Mary picks two of the remaining candies at random. Given that the probability that they get the same color combination, irrespective of ... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: How many positive integers less than 10,000 have at most two different digits?\n [/INST] First, let's count numbers with only a single digit. We have nine of these for each length, and four lengths, so 36 total numbers.\nNow, ... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: A particle moves in the Cartesian plane according to the following rules:\nFrom any lattice point $(a,b),$ the particle may only move to $(a+1,b), (a,b+1),$ or $(a+1,b+1).$\nThere are no right angle turns in the particle's pat... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Robert has 4 indistinguishable gold coins and 4 indistinguishable silver coins. Each coin has an engraving of one face on one side, but not on the other. He wants to stack the eight coins on a table into a single stack so that... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: A hotel packed breakfast for each of three guests. Each breakfast should have consisted of three types of rolls, one each of nut, cheese, and fruit rolls. The preparer wrapped each of the nine rolls and once wrapped, the rolls... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: The cards in a stack of $2n$ cards are numbered consecutively from 1 through $2n$ from top to bottom. The top $n$ cards are removed, kept in order, and form pile $A.$ The remaining cards form pile $B.$ The cards are then resta... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: 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:\nAny cube may be the bottom cube in the tower.\nThe cube... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Let set $\\mathcal{A}$ be a 90-element subset of $\\{1,2,3,\\ldots,100\\},$ and let $S$ be the sum of the elements of $\\mathcal{A}.$ Find the number of possible values of $S.$\n [/INST] The smallest $S$ is $1+2+ \\ldots +90 =... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Seven teams play a soccer tournament in which each team plays every other team exactly once. No ties occur, each team has a $50\\%$ chance of winning each game it plays, and the outcomes of the games are independent. In each g... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Let $(a_1,a_2,a_3,\\ldots,a_{12})$ be a permutation of $(1,2,3,\\ldots,12)$ for which\n$a_1>a_2>a_3>a_4>a_5>a_6 \\mathrm{\\ and \\ } a_6<a_7<a_8<a_9<a_{10}<a_{11}<a_{12}.$\nAn example of such a permutation is $(6,5,4,3,2,1,7,... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Two cards are dealt from a standard deck of 52 cards. What is the probability that the first card dealt is a $\\diamondsuit$ and the second card dealt is a $\\spadesuit$? [/INST] Once a $\\diamondsuit$ is dealt, there are onl... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Find the number of ordered pairs of positive integers $(a,b)$ such that $a+b=1000$ and neither $a$ nor $b$ has a zero digit.\n [/INST] There are $\\left\\lfloor\\frac{999}{10}\\right\\rfloor = 99$ numbers up to 1000 that have ... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: 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... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: 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 $AA$... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: The diagram below shows a $4\\times4$ rectangular array of points, each of which is $1$ unit away from its nearest neighbors.\n[asy] unitsize(0.25inch); defaultpen(linewidth(0.7)); int i, j; for(i = 0; i < 4; ++i) \tfor(j = 0... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: 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 whic... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: The Annual Interplanetary Mathematics Examination (AIME) is written by a committee of five Martians, five Venusians, and five Earthlings. At meetings, committee members sit at a round table with chairs numbered from $1$ to $15... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: A game show offers a contestant three prizes A, B and C, each of which is worth a whole number of dollars from $$ 1$ to $$ 9999$ inclusive. The contestant wins the prizes by correctly guessing the price of each prize in the or... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Jackie and Phil have two fair coins and a third coin that comes up heads with probability $\\frac47$. Jackie flips the three coins, and then Phil flips the three coins. Let $\\frac {m}{n}$ be the probability that Jackie gets t... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Define an ordered triple $(A, B, C)$ of sets to be minimally intersecting if $|A \\cap B| = |B \\cap C| = |C \\cap A| = 1$ and $A \\cap B \\cap C = \\emptyset$. For example, $(\\{1,2\\},\\{2,3\\},\\{1,3,4\\})$ is a minimally i... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: The $52$ cards in a deck are numbered $1, 2, \\cdots, 52$. Alex, Blair, Corey, and Dylan each picks a card from the deck without replacement and with each card being equally likely to be picked, The two persons with lower numb... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Dave arrives at an airport which has twelve gates arranged in a straight line with exactly $100$ feet between adjacent gates. His departure gate is assigned at random. After waiting at that gate, Dave is told the departure gat... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Let $N$ be the number of ordered pairs of nonempty sets $\\mathcal{A}$ and $\\mathcal{B}$ that have the following properties:\n$\\mathcal{A} \\cup \\mathcal{B} = \\{1,2,3,4,5,6,7,8,9,10,11,12\\}$,\n$\\mathcal{A} \\cap \\mathca... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Nine delegates, three each from three different countries, randomly select chairs at a round table that seats nine people. Let the probability that each delegate sits next to at least one delegate from another country be $\\fr... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Let $P(x) = x^2 - 3x - 9$. A real number $x$ is chosen at random from the interval $5 \\le x \\le 15$. The probability that $\\lfloor\\sqrt{P(x)}\\rfloor = \\sqrt{P(\\lfloor x \\rfloor)}$ is equal to $\\frac{\\sqrt{a} + \\sqrt... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Define an ordered quadruple of integers $(a, b, c, d)$ as interesting if $1 \\le ab+c$. How many interesting ordered quadruples are there?\n [/INST] Rearranging the inequality we get $d-c > b-a$. Let $e = 11$, then $(a, b-a, c... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: A board game spinner is divided into three regions labeled $A$, $B$ and $C$. The probability of the arrow stopping on region $A$ is $\\frac{1}{2}$ and on region $B$ is $\\frac{1}{5}$. What is the probability of the arrow sto... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Ed has five identical green marbles, and a large supply of identical red marbles. He arranges the green marbles and some of the red ones in a row and finds that the number of marbles whose right hand neighbor is the same color... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Nine people sit down for dinner where there are three choices of meals. Three people order the beef meal, three order the chicken meal, and three order the fish meal. The waiter serves the nine meals in random order. Find the ... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: In a group of nine people each person shakes hands with exactly two of the other people from the group. Let $N$ be the number of ways this handshaking can occur. Consider two handshaking arrangements different if and only if a... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: At a certain university, the division of mathematical sciences consists of the departments of mathematics, statistics, and computer science. There are two male and two female professors in each department. A committee of six p... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Let $S$ be the increasing sequence of positive integers whose binary representation has exactly $8$ ones. Let $N$ be the 1000th number in $S$. Find the remainder when $N$ is divided by $1000$.\n [/INST] Okay, an exercise in co... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Melinda has three empty boxes and $12$ textbooks, three of which are mathematics textbooks. One box will hold any three of her textbooks, one will hold any four of her textbooks, and one will hold any five of her textbooks. If... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Let $A = \\{1, 2, 3, 4, 5, 6, 7\\}$, and let $N$ be the number of functions $f$ from set $A$ to set $A$ such that $f(f(x))$ is a constant function. Find the remainder when $N$ is divided by $1000$.\n [/INST] Any such function ... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: A $7\\times 1$ board is completely covered by $m\\times 1$ tiles without overlap; each tile may cover any number of consecutive squares, and each tile lies completely on the board. Each tile is either red, blue, or green. Let ... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Ten adults enter a room, remove their shoes, and toss their shoes into a pile. Later, a child randomly pairs each left shoe with a right shoe without regard to which shoes belong together. The probability that for every positi... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Arnold is studying the prevalence of three health risk factors, denoted by A, B, and C, within a population of men. For each of the three factors, the probability that a randomly selected man in the population has only this ri... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: A point $(x,y)$ is randomly picked from inside the rectangle with vertices $(0,0)$, $(4,0)$, $(4,1)$, and $(0,1)$. What is the probability that $x < y$? [/INST] The point $(x,y)$ satisfies $x < y$ if and only if it belongs ... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Charles has two six-sided die. One of the die is fair, and the other die is biased so that it comes up six with probability $\\frac{2}{3}$ and each of the other five sides has probability $\\frac{1}{15}$. Charles chooses one o... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Ten chairs are arranged in a circle. Find the number of subsets of this set of chairs that contain at least three adjacent chairs.\n [/INST] We know that a subset with less than $3$ chairs cannot contain $3$ adjacent chairs. T... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: 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\\... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: An urn contains $4$ green balls and $6$ blue balls. A second urn contains $16$ green balls and $N$ blue balls. A single ball is drawn at random from each urn. The probability that both balls are of the same color is $0.58$. Fi... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: A moving particle starts at the point $(4,4)$ and moves until it hits one of the coordinate axes for the first time. When the particle is at the point $(a,b)$, it moves at random to one of the points $(a-1,b)$, $(a,b-1)$, or $... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: A set of $n$ people participate in an online video basketball tournament. Each person may be a member of any number of $5$-player teams, but no two teams may have exactly the same $5$ members. The site statistics show a curiou... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: How many integers between $100$ and $999$, inclusive, have the property that some permutation of its digits is a multiple of $11$ between $100$ and $999?$ For example, both $121$ and $211$ have this property.\n$\\mathrm{\\text... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: How many triangles with positive area have all their vertices at points $(i,j)$ in the coordinate plane, where $i$ and $j$ are integers between $1$ and $5$, inclusive?\n$\\textbf{(A)}\\ 2128 \\qquad\\textbf{(B)}\\ 2148 \\qquad... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: 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 c... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Two different integers from 1 through 20 inclusive are chosen at random. What is the probability that both numbers are prime? Express your answer as a common fraction. [/INST] There are $\\binom{20}{2}$ pairs of distinct integ... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Consider polynomials $P(x)$ of degree at most $3$, each of whose coefficients is an element of $\\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\\}$. How many such polynomials satisfy $P(-1) = -9$?\n$\\textbf{(A) } 110 \\qquad \\textbf{(B) } 1... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Two quadrilaterals are considered the same if one can be obtained from the other by a rotation and a translation. How many different convex cyclic quadrilaterals are there with integer sides and perimeter equal to 32?\n$\\text... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: How many non- empty subsets $S$ of $\\{1,2,3,\\ldots ,15\\}$ have the following two properties?\n$(1)$ No two consecutive integers belong to $S$.\n$(2)$ If $S$ contains $k$ elements, then $S$ contains no number less than $k$.\... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Eight congruent equilateral triangles, each of a different color, are used to construct a regular octahedron. How many distinguishable ways are there to construct the octahedron? (Two colored octahedrons are distinguishable if... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: How many triangles with positive area are there whose vertices are points in the $xy$-plane whose coordinates are integers $(x,y)$ satisfying $1\\le x\\le 4$ and $1\\le y\\le 4$?\n$\\text{(A) } 496\\quad \\text{(B) } 500\\quad... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Suppose that 7 boys and 13 girls line up in a row. Let $S$ be the number of places in the row where a boy and a girl are standing next to each other. For example, for the row $\\text{GBBGGGBGBGGGBGBGGBGG}$ we have that $S=12$.... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: What is the value of the sum $S=\\sum_{k=0}^{49}(-1)^k\\binom{99}{2k}=\\binom{99}{0}-\\binom{99}{2}+\\binom{99}{4}-\\cdots -\\binom{99}{98}?$\n(A) $-2^{50}$ (B) $-2^{49}$ (C) 0 (D) $2^{49}$ (E) $2^{50}$\n [/INST] By the Binomi... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: In a certain cross country meet between 2 teams of 5 runners each, a runner who finishes in the $n$th position contributes $n$ to his teams score. The team with the lower score wins. If there are no ties among the runners, how... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Five people are sitting at a round table. Let $f\\geq 0$ be the number of people sitting next to at least 1 female and $m\\geq0$ be the number of people sitting next to at least one male. The number of possible ordered pairs $... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: A child has a set of 96 distinct blocks. Each block is one of 2 materials (plastic, wood), 3 sizes (small, medium, large), 4 colors (blue, green, red, yellow), and 4 shapes (circle, hexagon, square, triangle). How many blocks ... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: At the end of a professional bowling tournament, the top 5 bowlers have a playoff. First #5 bowls #4. The loser receives $5$th prize and the winner bowls #3 in another game. The loser of this game receives $4$th prize and the ... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Six distinct integers are picked at random from $\\{1,2,3,\\ldots,10\\}$. What is the probability that, among those selected, the second smallest is $3$?\n$\\textbf{(A)}\\ \\frac{1}{60}\\qquad \\textbf{(B)}\\ \\frac{1}{6}\\qqu... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: A drawer in a darkened room contains $100$ red socks, $80$ green socks, $60$ blue socks and $40$ black socks. A youngster selects socks one at a time from the drawer but is unable to see the color of the socks drawn. What is t... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: 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 $\\ma... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Let $\\star (x)$ be the sum of the digits of a positive integer $x$. $\\mathcal{S}$ is the set of positive integers such that for all elements $n$ in $\\mathcal{S}$, we have that $\\star (n)=12$ and $0\\le n< 10^{7}$. If $m$ i... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: An oreo shop sells $5$ different flavors of oreos and $3$ different flavors of milk. Alpha and Beta decide to purhcase some oreos. Since Alpha is picky, he will not order more than 1 of the same flavor. To be just as weird, Be... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: A mother purchases 5 blue plates, 2 red plates, 2 green plates, and 1 orange plate. How many ways are there for her to arrange these plates for dinner around her circular table if she doesn't want the 2 green plates to be adja... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: 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 integer... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: A paperboy delivers newspapers to 10 houses along Main Street. Wishing to save effort, he doesn't always deliver to every house, but to avoid being fired he never misses three consecutive houses. Compute the number of ways the... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Let $N$ denote the number of permutations of the $15$-character string $AAAABBBBBCCCCCC$ such that\nNone of the first four letters is an $A$.\nNone of the next five letters is a $B$.\nNone of the last six letters is a $C$.\nFi... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Let $N$ denote the number of $7$ digit positive integers have the property that their digits are in increasing order. Determine the remainder obtained when $N$ is divided by $1000$. (Repeated digits are allowed.)\n [/INST] Not... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: Albert starts to make a list, in increasing order, of the positive integers that have a first digit of 1. He writes $1, 10, 11, 12, \\ldots$ but by the 1,000th digit he (finally) realizes that the list would contain an infinit... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: How many 10-digit positive integers have all digits either 1 or 2, and have two consecutive 1's?\n [/INST] We take as our universe the set of 10-digit integers whose digits are all either 1 or 2, of which there are $2^{10}$, a... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: The number of increasing sequences of positive integers $a_1 \\le a_2 \\le a_3 \\le \\cdots \\le a_{10} \\le 2007$ such that $a_i-i$ is even for $1\\le i \\le 10$ can be expressed as ${m \\choose n}$ for some positive integers... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: License plates from different states follow different alpha-numeric formats, which dictate which characters of a plate must be letters and which must be numbers. Florida has license plates with an alpha-numeric format like the... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: In how many ways can I arrange 3 different math books and 5 different history books on my bookshelf, if I require there to be a math book on both ends? [/INST] Let's deal with the restriction first.\n\nThe restriction is that ... | custom_source | 5 |
[
{
"from": "human",
"value": "[INST] Compose a solution for the following problem: If $5! \\cdot 3! = n!$, what is the value of $n$? [/INST] This problem can be solved by the quick observation that $3! = 6$, and so $5! \\cdot 6 = 6!$. So $n=\\boxed{6}$."
},
{
"from": "gpt",
"value": "Solution pla... | custom_source | 5 |
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