messages listlengths 3 3 |
|---|
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Let $A$, $B$, $C$ and $D$ be the vertices of a regular tetrahedron each of whose edges measures 1 meter. A bug, starting from vertex $A$, observes the following rule: at each vert... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "How many even integers between 4000 and 7000 have four different digits?\n",
"role": "user"
},
{
"content": "To determine the number of even integers between 4000 and ... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "For each permutation $a_1,a_2,a_3,\\cdots,a_{10}$ of the integers $1,2,3,\\cdots,10$, form the sum\n\\[|a_1-a_2|+|a_3-a_4|+|a_5-a_6|+|a_7-a_8|+|a_9-a_{10}|.\\]\nThe average value ... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "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 probabi... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "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 ar... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "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... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "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 tr... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "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 g... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "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 eac... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "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"... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "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 $\\fra... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "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, ... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "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 ... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "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 in... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "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 $\\t... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "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 (circl... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "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 l... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "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{... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "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 ... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "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... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "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 con... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "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... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "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) ... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "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 licen... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "How many rows of Pascal's Triangle contain the number $43$?",
"role": "user"
},
{
"content": "To determine how many rows of Pascal's Triangle contain the number \\(43\... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Call a set of integers \"spacy\" if it contains no more than one out of any three consecutive integers. How many subsets of $\\{1, 2,\n3, \\dots, 12\\}$, including the empty set,... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "A box contains 4 white balls and 4 black balls. I draw them out of the box, one at a time. What is the probability that all of my draws alternate colors?",
"role": "user"
... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Each outcome on the spinner below has equal probability. If you spin the spinner three times and form a three-digit number from the three outcomes, such that the first outcome is ... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "A rectangle has a perimeter of 64 inches and each side has an integer length. How many non-congruent rectangles meet these criteria?",
"role": "user"
},
{
"content": "... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "A license plate in a certain state consists of 4 digits, not necessarily distinct, and 2 letters, also not necessarily distinct. These six characters may appear in any order, exce... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "A cube is painted so that one side is blue, two sides are red, and three sides are green. How many different such cubes can be painted? Two cubes are considered the same if one cu... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "On the Saturday of a weekend softball tournament, Team A plays Team B and Team C plays Team D. Then on Sunday, the two Saturday winners play for first and second places while the ... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Nine people sit down at random seats around a round table. Four of them are math majors, three others are physics majors, and the two remaining people are biology majors. What is ... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "How many 3-digit numbers have the property that the units digit is at least twice the tens digit?",
"role": "user"
},
{
"content": "To solve the problem of finding how... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Allison, Brian and Noah each have a 6-sided cube. All of the faces on Allison's cube have a 5. The faces on Brian's cube are numbered 1, 2, 3, 4, 5 and 6. Three of the faces on No... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Monica tosses a fair 6-sided die. If the roll is a prime number, then she wins that amount of dollars (so that, for example, if she rolls 3, then she wins 3 dollars). If the rol... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "A Senate committee has 5 Democrats and 5 Republicans. Assuming all politicians are distinguishable, in how many ways can they sit around a circular table without restrictions? (T... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "On the game show $\\text{\\emph{Wheel of Fraction}}$, you see the following spinner. Given that each region is the same area, what is the probability that you will earn exactly $\... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "We are allowed to remove exactly one integer from the list $$-1,0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,11,$$and then we choose two distinct integers at random from the remaining list. ... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "The Rotokas of Papua New Guinea have twelve letters in their alphabet. The letters are: A, E, G, I, K, O, P, R, S, T, U, and V. Suppose license plates of five letters utilize only... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "A box contains six cards. Three of the cards are black on both sides, one card is black on one side and red on the other, and two of the cards are red on both sides. You pick a c... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "I have 6 shirts, 4 pairs of pants, and 6 hats. The pants come in tan, black, blue, and gray. The shirts and hats come in those colors, and also white and yellow. I refuse to wear ... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "How many different four-digit numbers can be formed by arranging the four digits in 2004?",
"role": "user"
},
{
"content": "To determine how many different four-digit ... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "If I choose four cards from a standard $52$-card deck, with replacement, what is the probability that I will end up with one card from each suit?",
"role": "user"
},
{
... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "My three-digit code is 023. Reckha can't choose a code that is the same as mine in two or more of the three digit-positions, nor that is the same as mine except for switching the ... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Two cards are chosen at random from a standard 52-card deck. What is the probability that the first card is a heart and the second card is a 10?",
"role": "user"
},
{
... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Jane and her brother each spin this spinner once. The spinner has five congruent sectors. If the non-negative difference of their numbers is less than 3, Jane wins. Otherwise, her... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "At the grocery store, I bought $4$ different items. I brought $3$ identical bags, and handed them to the cashier. How many ways are there for the cashier to put the items I bought... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Sue owns 11 pairs of shoes: six identical black pairs, three identical brown pairs and two identical gray pairs. If she picks two shoes at random, what is the probability that the... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "How many paths are there from $A$ to $B$, if every step must be up or to the right?[asy]size(4cm,4cm);int w=6;int h=5;int i;pen p=fontsize(9);for (i=0; i<h; ++i){draw((0,i) -- (w-... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "In the game Deal or No Deal, participants choose a box at random from a set of $26,$ one containing each of the following values: \\begin{tabular}{|c|c|}\\hline\\$.01&\\$1,000\\\\... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "How many odd numbers between $100$ and $999$ have distinct digits?",
"role": "user"
},
{
"content": "To determine how many odd numbers between 100 and 999 have distinc... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Alice and Bob play a game with a baseball. On each turn, if Alice has the ball, there is a 1/2 chance that she will toss it to Bob and a 1/2 chance that she will keep the ball. If... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "How many different routes are there from point $A$ to point $B$ if you can only move to the right or down along the drawn segments?\n\n[asy]\nunitsize(0.09inch);\ndraw((0,0)--(10,... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "In this square array of 16 dots, four dots are to be chosen at random. What is the probability that the four dots will be collinear? Express your answer as a common fraction.\n\n[... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "How many non-congruent triangles with perimeter 7 have integer side lengths?",
"role": "user"
},
{
"content": "To find the number of non-congruent triangles with integ... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "In the United States, coins have the following thicknesses: penny, 1.55 mm; nickel, 1.95 mm; dime, 1.35 mm; quarter, 1.75 mm. If a stack of these coins is exactly 14 mm high, how ... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "How many ways are there to put 4 indistinguishable balls into 2 indistinguishable boxes?",
"role": "user"
},
{
"content": "To solve the problem of placing 4 indistingu... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "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 indistingu... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "A bag contains 3 tan, 2 pink and 4 violet chips. If the 9 chips are randomly drawn from the bag, one at a time and without replacement, what is the probability that the chips are ... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "The deli has four kinds of bread, six kinds of meat, and five kinds of cheese. A sandwich consists of one type of bread, one type of meat, and one type of cheese. Ham, chicken, ch... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "In the geoboard shown, the points are evenly spaced vertically and horizontally. Segment $AB$ is drawn using two points, as shown. Point $C$ is to be chosen from the remaining $23... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Human beings have discovered a habitable planet and soon after, they find 10 more habitable planets. Of these 11, only 5 are deemed ``Earth-like'' in their resources and the rest ... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "How many distinct sequences of four letters can be made from the letters in PROBLEM if each letter can be used only once and each sequence must begin with L and not end with P?",
... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Four distinct points, $A$, $B$, $C$, and $D$, are to be selected from $1996$ points evenly spaced around a circle. All quadruples are equally likely to be chosen. What is the prob... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Three positive integers $a$, $b$, and $c$ satisfy $a\\cdot b\\cdot c=8!$ and $a<b<c$. What is the smallest possible value of $c-a$?",
"role": "user"
},
{
"content": "... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "How many integers from 1 through 9999, inclusive, do not contain any of the digits 2, 3, 4 or 5?",
"role": "user"
},
{
"content": "To solve this problem, we need to co... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "If I roll 5 standard 6-sided dice and multiply the number on the face of each die, what is the probability that the result is a composite number?",
"role": "user"
},
{
... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "How many numbers are in the first $20$ rows of Pascal's Triangle (from the $0$th row to the $19$th row)?",
"role": "user"
},
{
"content": "To determine how many number... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "A point is chosen at random on the number line between 0 and 1, and the point is colored green. Then, another point is chosen at random on the number line between 0 and 1, and thi... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "My school's Physics Club has 22 members. It needs to select 3 officers: chairman, vice-chairman, and sergeant-at-arms. Each person can hold at most one office. Two of the membe... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Find the smallest positive integer that is both an integer power of 11 and is not a palindrome.",
"role": "user"
},
{
"content": "To find the smallest positive integer... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "A zealous geologist is sponsoring a contest in which entrants have to guess the age of a shiny rock. He offers these clues: the age of the rock is formed from the six digits 2, 2,... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "In the rectangle below, line segment $MN$ separates the rectangle into $2$ sections. What is the largest number of sections into which the rectangle can be separated when $4$ line... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "How many even three-digit integers have the property that their digits, read left to right, are in strictly increasing order (each digit is greater than the previous digit)?",
... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "In the local frisbee league, teams have 7 members and each of the 4 teams takes turns hosting tournaments. At each tournament, each team selects two members of that team to be on ... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Two mathematicians were both born in the last 500 years. Each lives (or will live) to be 100 years old, then dies. Each mathematician is equally likely to be born at any point dur... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "The first six rows of Pascal's triangle are shown below, beginning with row zero. Except for the $1$ at each end, row $4$ consists of only even numbers, as does row $2.$ How many ... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "The integers $r$ and $k$ are randomly selected, where $-3 < r < 6$ and $1 < k < 8$. What is the probability that the division $r \\div k$ is an integer value? Express your answer ... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "In the diagram below, we have $\\sin \\angle RPQ = \\frac{7}{25}$. What is $\\sin \\angle RPS$?\n\n[asy]\npair R,P,Q,SS;\nSS = (-2,0);\nP = (0,0);\nQ = (2,0);\nR = rotate(aSin(7/... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "A triangle with sides of 5, 12, and 13 has both an inscribed and a circumscribed circle. What is the distance between the centers of those circles?",
"role": "user"
},
{
... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Regions I, II and III are bounded by squares. The perimeter of region I is 12 units and the perimeter of region II is 24 units. What is the ratio of the area of region I to the ar... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "The center of a circle has coordinates $(6,-5)$. The circle is reflected about the line $y=x$. What are the $x,y$ coordinates of the center of the image circle? State the $x$ coor... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "A right circular cylinder with its diameter equal to its height is inscribed in a right circular cone. The cone has diameter 10 and altitude 12, and the axes of the cylinder and c... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Piravena must make a trip from $A$ to $B$, then from $B$ to $C$, then from $C$ to $A$. Each of these three parts of the trip is made entirely by bus or entirely by airplane. The... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "In the diagram, $O$ is the center of a circle with radii $OP=OQ=5$. What is the perimeter of the shaded region?\n\n[asy]\nsize(100);\nimport graph;\nlabel(\"$P$\",(-1,0),W); labe... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "In the diagram, $\\triangle ABE$, $\\triangle BCE$ and $\\triangle CDE$ are right-angled, with $\\angle AEB=\\angle BEC = \\angle CED = 60^\\circ$, and $AE=24$. [asy]\npair A, B, ... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "In square $ABCD$, $AD$ is 4 centimeters, and $M$ is the midpoint of $\\overline{CD}$. Let $O$ be the intersection of $\\overline{AC}$ and $\\overline{BM}$. What is the ratio of $O... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "A cone has a volume of $12288\\pi$ cubic inches and the vertex angle of the vertical cross section is 60 degrees. What is the height of the cone? Express your answer as a decimal ... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Square $ABCD$ has side length $1$ unit. Points $E$ and $F$ are on sides $AB$ and $CB$, respectively, with $AE = CF$. When the square is folded along the lines $DE$ and $DF$, sid... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Let $\\triangle ABC$ be a right triangle such that $B$ is a right angle. A circle with diameter of $BC$ meets side $AC$ at $D.$ If $AD = 1$ and $BD = 4,$ then what is $CD$?",
... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Three faces of a right rectangular prism have areas of 48, 49 and 50 square units. What is the volume of the prism, in cubic units? Express your answer to the nearest whole number... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "The line $y = b-x$ with $0 < b < 4$ intersects the $y$-axis at $P$ and the line $x=4$ at $S$. If the ratio of the area of triangle $QRS$ to the area of triangle $QOP$ is 9:25, wha... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Two circles of radius 2 are centered at $(2,0)$ and at $(0,2)$. What is the area of the intersection of the interiors of the two circles? Express your answer in fully expanded for... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "What is the area of the shaded region in the figure below? Round your answer to the nearest square centimeter.\n\n[asy]\ndraw((0,0) -- (3,0) -- (3,3) -- (0,3)--cycle) ; draw((3,0)... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Which of the cones below can be formed from a $252^{\\circ}$ sector of a circle of radius 10 by aligning the two straight sides?\n\n[asy]\ndraw((5.8,8.1)..(-10,0)--(0,0)--(3.1,-9.... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "In the diagram, $\\triangle XYZ$ is right-angled at $X,$ with $YX=60$ and $XZ=80.$ The point $W$ is on $YZ$ so that $WX$ is perpendicular to $YZ.$ Determine the length of $WZ.$ [a... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "A square and a circle intersect so that each side of the square contains a chord of the circle equal in length to the radius of the circle. What is the ratio of the area of the sq... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "The radius of the inscribed circle is 6 cm. What is the number of centimeters in the length of $\\overline{AB}$? Express your answer in simplest radical form. [asy]\nimport olympi... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "In right triangle $ABC$, shown below, $\\cos{C}=\\frac{9\\sqrt{130}}{130}$. Find $AC$.\n\n[asy]\ndraw((0,0)--(7,0)--(0,9)--cycle,black+linewidth(1));\ndraw(rightanglemark((7,0),(... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.