Datasets:

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aaabiao

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RL-dataset

like 0

[ { "content": "A red ball and a green ball are randomly and independently tossed into bins numbered with the positive integers so that for each ball, the probability that it is tossed into bin $k$ is $2^{-k}$ for $k = 1,2,3....$ What is the probability that the red ball is tossed into a higher-numbered bin than the green ball? Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]

[ { "content": "Granny Smith has $63. Elberta has $2 more than Anjou and Anjou has one-third as much as Granny Smith. How many dollars does Elberta have? Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]

[ { "content": "In the figure, the outer equilateral triangle has area $16$, the inner equilateral triangle has area $1$, and the three trapezoids are congruent. What is the area of one of the trapezoids? Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]

[ { "content": "A number $x$ is $2$ more than the product of its reciprocal and its additive inverse. In which interval does the number lie? Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]

[ { "content": "A $1 \\times 2$ rectangle is inscribed in a semicircle with the longer side on the diameter. What is the area of the semicircle? Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]

[ { "content": "Patty has $20$ coins consisting of nickels and dimes. If her nickels were dimes and her dimes were nickels, she would have $70$ cents more. How much are her coins worth? Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]

[ { "content": "A carton contains milk that is $2\\%$ fat, an amount that is $40\\%$ less fat than the amount contained in a carton of whole milk. What is the percentage of fat in whole milk? Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]

[ { "content": "Let $\\triangle A_0B_0C_0$ be a triangle whose angle measures are exactly $59.999^\\circ$, $60^\\circ$, and $60.001^\\circ$. For each positive integer $n$, define $A_n$ to be the foot of the altitude from $A_{n-1}$ to line $B_{n-1}C_{n-1}$. Likewise, define $B_n$ to be the foot of the altitude from $B_{n-1}$ to line $A_{n-1}C_{n-1}$, and $C_n$ to be the foot of the altitude from $C_{n-1}$ to line $A_{n-1}B_{n-1}$. What is the least positive integer $n$ for which $\\triangle A_nB_nC_n$ is obtuse? Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]

[ { "content": "$3^3 + 3^3 + 3^3 =$ Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]

[ { "content": "A long piece of paper $5$ cm wide is made into a roll for cash registers by wrapping it $600$ times around a cardboard tube of diameter $2$ cm, forming a roll $10$ cm in diameter. Approximate the length of the paper in meters. (Pretend the paper forms $600$ concentric circles with diameters evenly spaced from $2$ cm to $10$ cm.) Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]

[ { "content": "A number is called flippy if its digits alternate between two distinct digits. For example, $2020$ and $37373$ are flippy, but $3883$ and $123123$ are not. How many five-digit flippy numbers are divisible by $15?$ Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]

[ { "content": "A small bottle of shampoo can hold $35$ milliliters of shampoo, whereas a large bottle can hold $500$ milliliters of shampoo. Jasmine wants to buy the minimum number of small bottles necessary to completely fill a large bottle. How many bottles must she buy? Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]

[ { "content": "The polygon enclosed by the solid lines in the figure consists of 4 congruent squares joined edge-to-edge. One more congruent square is attached to an edge at one of the nine positions indicated. How many of the nine resulting polygons can be folded to form a cube with one face missing? Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]

[ { "content": "In a high school with $500$ students, $40\\%$ of the seniors play a musical instrument, while $30\\%$ of the non-seniors do not play a musical instrument. In all, $46.8\\%$ of the students do not play a musical instrument. How many non-seniors play a musical instrument? Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]

[ { "content": "If $f(x)=\\frac{x(x-1)}{2}$, then $f(x+2)$ equals: Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]

[ { "content": "In the table shown, the formula relating \\(x\\) and \\(y\\) is:\n\\[\\begin{array}{|c|c|c|c|c|c|}\\hline x & 1 & 2 & 3 & 4 & 5\\\\ \\hline y & 3 & 7 & 13 & 21 & 31\\\\ \\hline\\end{array}\\] Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]

[ { "content": "What is the median of the following list of $4040$ numbers?\n\\[1, 2, 3, \\ldots, 2020, 1^2, 2^2, 3^2, \\ldots, 2020^2\\] Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]

[ { "content": "When Cheenu was a boy, he could run $15$ miles in $3$ hours and $30$ minutes. As an old man, he can now walk $10$ miles in $4$ hours. How many minutes longer does it take for him to walk a mile now compared to when he was a boy? Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]

[ { "content": "The sum of all numbers of the form $2k + 1$, where $k$ takes on integral values from $1$ to $n$ is: Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]

[ { "content": "Equilateral triangle $ABP$ (see figure) with side $AB$ of length $2$ inches is placed inside square $AXYZ$ with side of length $4$ inches so that $B$ is on side $AX$. The triangle is rotated clockwise about $B$, then $P$, and so on along the sides of the square until $P$ returns to its original position. The length of the path in inches traversed by vertex $P$ is equal to Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]

[ { "content": "Jack and Jill run 10 km. They start at the same point, run 5 km up a hill, and return to the starting point by the same route. Jack has a 10 minute head start and runs at the rate of 15 km/hr uphill and 20 km/hr downhill. Jill runs 16 km/hr uphill and 22 km/hr downhill. How far from the top of the hill are they when they pass each other going in opposite directions (in km)? Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]

[ { "content": "A straight line joins the points $(-1,1)$ and $(3,9)$. Its $x$-intercept is: Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]

[ { "content": "The number of points common to the graphs of\n$(x-y+2)(3x+y-4)=0$ and $(x+y-2)(2x-5y+7)=0$ is: Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]

[ { "content": "At the grocery store last week, small boxes of facial tissue were priced at 4 boxes for $5. This week they are on sale at 5 boxes for $4. The percent decrease in the price per box during the sale was closest to Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]

[ { "content": "The sum of all integers between 50 and 350 which end in 1 is Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]

[ { "content": "If $\\log_6 x=2.5$, the value of $x$ is: Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]

[ { "content": "For each positive integer $n$, let \n\n$a_n = \\frac{(n+9)!}{(n-1)!}$.\nLet $k$ denote the smallest positive integer for which the rightmost nonzero digit of $a_k$ is odd. The rightmost nonzero digit of $a_k$ is Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]

[ { "content": "Point $F$ is taken in side $AD$ of square $ABCD$. At $C$ a perpendicular is drawn to $CF$, meeting $AB$ extended at $E$. The area of $ABCD$ is $256$ square inches and the area of $\\triangle CEF$ is $200$ square inches. Then the number of inches in $BE$ is: Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]

[ { "content": "At the end of 1994, Walter was half as old as his grandmother. The sum of the years in which they were born was 3838. How old will Walter be at the end of 1999? Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]

[ { "content": "A rise of $600$ feet is required to get a railroad line over a mountain. The grade can be kept down by lengthening the track and curving it around the mountain peak. The additional length of track required to reduce the grade from $3\\%$ to $2\\%$ is approximately: Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]

[ { "content": "Let $A$ be the area of the triangle with sides of length $25, 25$, and $30$. Let $B$ be the area of the triangle with sides of length $25, 25,$ and $40$. What is the relationship between $A$ and $B$? Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]

[ { "content": "Three one-inch squares are placed with their bases on a line. The center square is lifted out and rotated 45 degrees, as shown. Then it is centered and lowered into its original location until it touches both of the adjoining squares. How many inches is the point $B$ from the line on which the bases of the original squares were placed? Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]

[ { "content": "The product of $\\sqrt[3]{4}$ and $\\sqrt[4]{8}$ equals Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]

[ { "content": "Two fair coins are to be tossed once. For each head that results, one fair die is to be rolled. What is the probability that the sum of the die rolls is odd? Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]

[ { "content": "In the figure, it is given that angle $C = 90^{\\circ}$, $\\overline{AD} = \\overline{DB}$, $DE \\perp AB$, $\\overline{AB} = 20$, and $\\overline{AC} = 12$. The area of quadrilateral $ADEC$ is: Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]

[ { "content": "A ticket to a school play cost $x$ dollars, where $x$ is a whole number. A group of 9th graders buys tickets costing a total of $48, and a group of 10th graders buys tickets costing a total of $64. How many values for $x$ are possible? Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]

[ { "content": "On Halloween Casper ate $\\frac{1}{3}$ of his candies and then gave $2$ candies to his brother. The next day he ate $\\frac{1}{3}$ of his remaining candies and then gave $4$ candies to his sister. On the third day he ate his final $8$ candies. How many candies did Casper have at the beginning? Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]

[ { "content": "The polynomial $(x+y)^9$ is expanded in decreasing powers of $x$. The second and third terms have equal values \nwhen evaluated at $x=p$ and $y=q$, where $p$ and $q$ are positive numbers whose sum is one. What is the value of $p$? Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]

[ { "content": "Let $A$, $B$ and $C$ be three distinct points on the graph of $y=x^2$ such that line $AB$ is parallel to the $x$-axis and $\\triangle ABC$ is a right triangle with area $2008$. What is the sum of the digits of the $y$-coordinate of $C$? Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]

[ { "content": "The Incredible Hulk can double the distance he jumps with each succeeding jump. If his first jump is 1 meter, the second jump is 2 meters, the third jump is 4 meters, and so on, then on which jump will he first be able to jump more than 1 kilometer (1,000 meters)? Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]

[ { "content": "When three different numbers from the set $\\{ -3, -2, -1, 4, 5 \\}$ are multiplied, the largest possible product is Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]

[ { "content": "Three times Dick's age plus Tom's age equals twice Harry's age. \nDouble the cube of Harry's age is equal to three times the cube of Dick's age added to the cube of Tom's age. \nTheir respective ages are relatively prime to each other. The sum of the squares of their ages is Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]

[ { "content": "How many unordered pairs of edges of a given cube determine a plane? Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]

[ { "content": "The two wheels shown below are spun and the two resulting numbers are added. The probability that the sum is even is Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]

[ { "content": "At Euler Middle School, $198$ students voted on two issues in a school referendum with the following results: $149$ voted in favor of the first issue and $119$ voted in favor of the second issue. If there were exactly $29$ students who voted against both issues, how many students voted in favor of both issues? Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]

[ { "content": "It takes Clea 60 seconds to walk down an escalator when it is not operating, and only 24 seconds to walk down the escalator when it is operating. How many seconds does it take Clea to ride down the operating escalator when she just stands on it? Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]

[ { "content": "If $1998$ is written as a product of two positive integers whose difference is as small as possible, then the difference is Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]

[ { "content": "Let $S$ be the set of lattice points in the coordinate plane, both of whose coordinates are integers between $1$ and $30,$ inclusive. Exactly $300$ points in $S$ lie on or below a line with equation $y=mx.$ The possible values of $m$ lie in an interval of length $\\frac ab,$ where $a$ and $b$ are relatively prime positive integers. What is $a+b?$ Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]

[ { "content": "A box contains $11$ balls, numbered $1, 2, 3, \\dots 11$. If $6$ balls are drawn simultaneously at random, what is the probability that the sum of the numbers on the balls drawn is odd? Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]

[ { "content": "The graph shows the number of minutes studied by both Asha (black bar) and Sasha (grey bar) in one week. On the average, how many more minutes per day did Sasha study than Asha? Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]

[ { "content": "Supposed that $x$ and $y$ are nonzero real numbers such that $\\frac{3x+y}{x-3y}=-2$. What is the value of $\\frac{x+3y}{3x-y}$? Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]

[ { "content": "If $A*B$ means $\\frac{A+B}{2}$, then $(3*5)*8$ is Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]

[ { "content": "If $x=1+2^p$ and $y=1+2^{-p}$, then $y$ in terms of $x$ is Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]

[ { "content": "At Euclid High School, the number of students taking the AMC 10 was $60$ in 2002, $66$ in 2003, $70$ in 2004, $76$ in 2005, $78$ in 2006, and is $85$ in 2007. Between what two consecutive years was there the largest percentage increase? Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]

[ { "content": "What is the smallest sum of two $3$-digit numbers that can be obtained by placing each of the six digits $4,5,6,7,8,9$ in one of the six boxes in this addition problem?\n[asy]\nunitsize(12);\ndraw((0,0)--(10,0));\ndraw((-1.5,1.5)--(-1.5,2.5));\ndraw((-1,2)--(-2,2));\ndraw((1,1)--(3,1)--(3,3)--(1,3)--cycle);\ndraw((1,4)--(3,4)--(3,6)--(1,6)--cycle);\ndraw((4,1)--(6,1)--(6,3)--(4,3)--cycle);\ndraw((4,4)--(6,4)--(6,6)--(4,6)--cycle);\ndraw((7,1)--(9,1)--(9,3)--(7,3)--cycle);\ndraw((7,4)--(9,4)--(9,6)--(7,6)--cycle);\n[/asy] Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]

[ { "content": "Let \\[f(n)=\\dfrac{5+3\\sqrt{5}}{10}\\left(\\dfrac{1+\\sqrt{5}}{2}\\right)^n+\\dfrac{5-3\\sqrt{5}}{10}\\left(\\dfrac{1-\\sqrt{5}}{2}\\right)^n.\\]\nThen $f(n+1)-f(n-1)$, expressed in terms of $f(n)$, equals: Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]

[ { "content": "A magazine printed photos of three celebrities along with three photos of the celebrities as babies. The baby pictures did not identify the celebrities. Readers were asked to match each celebrity with the correct baby pictures. What is the probability that a reader guessing at random will match all three correctly? Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]

[ { "content": "If $3x^3-9x^2+kx-12$ is divisible by $x-3$, then it is also divisible by: Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]

[ { "content": "A class collects 50 dollars to buy flowers for a classmate who is in the hospital. Roses cost 3 dollars each, and carnations cost 2 dollars each. No other flowers are to be used. How many different bouquets could be purchased for exactly 50 dollars? Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]

[ { "content": "A ream of paper containing $500$ sheets is $5$ cm thick. Approximately how many sheets of this type of paper would there be in a stack $7.5$ cm high? Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]

[ { "content": "There is more than one integer greater than $1$ which, when divided by any integer $k$ such that $2 \\le k \\le 11$, has a remainder of $1$. \nWhat is the difference between the two smallest such integers? Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]

[ { "content": "The root(s) of $\\frac {15}{x^2 - 4} - \\frac {2}{x - 2} = 1$ is (are): Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]

[ { "content": "If four times the reciprocal of the circumference of a circle equals the diameter of the circle, then the area of the circle is Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]

[ { "content": "A number $N$ has three digits when expressed in base $7$. When $N$ is expressed in base $9$ the digits are reversed. Then the middle digit is: Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]

[ { "content": "Four identical squares and one rectangle are placed together to form one large square as shown. The length of the rectangle is how many times as large as its width?\n\n[asy]\nunitsize(8mm); defaultpen(linewidth(.8pt));\ndraw((0,0)--(4,0)--(4,4)--(0,4)--cycle);\ndraw((0,3)--(0,4)--(1,4)--(1,3)--cycle);\ndraw((1,3)--(1,4)--(2,4)--(2,3)--cycle);\ndraw((2,3)--(2,4)--(3,4)--(3,3)--cycle);\ndraw((3,3)--(3,4)--(4,4)--(4,3)--cycle);\n[/asy] Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]

[ { "content": "In this figure $\\angle RFS = \\angle FDR$, $FD = 4$ inches, $DR = 6$ inches, $FR = 5$ inches, $FS = 7\\dfrac{1}{2}$ inches. The length of $RS$, in inches, is: Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]

[ { "content": "The real numbers $c, b, a$ form an arithmetic sequence with $a \\geq b \\geq c \\geq 0$. The quadratic $ax^2+bx+c$ has exactly one root. What is this root? Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]

[ { "content": "Three generous friends, each with some money, redistribute the money as followed:\nAmy gives enough money to Jan and Toy to double each amount has.\nJan then gives enough to Amy and Toy to double their amounts. \nFinally, Toy gives enough to Amy and Jan to double their amounts.\nIf Toy had 36 dollars at the beginning and 36 dollars at the end, what is the total amount that all three friends have? Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]

[ { "content": "Arjun and Beth play a game in which they take turns removing one brick or two adjacent bricks from one \"wall\" among a set of several walls of bricks, with gaps possibly creating new walls. The walls are one brick tall. For example, a set of walls of sizes $4$ and $2$ can be changed into any of the following by one move: $(3,2),(2,1,2),(4),(4,1),(2,2),$ or $(1,1,2).$ \n\nArjun plays first, and the player who removes the last brick wins. For which starting configuration is there a strategy that guarantees a win for Beth? Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]

[ { "content": "A laser is placed at the point $(3,5)$. The laser beam travels in a straight line. Larry wants the beam to hit and bounce off the $y$-axis, then hit and bounce off the $x$-axis, then hit the point $(7,5)$. What is the total distance the beam will travel along this path? Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]

[ { "content": "Placing no more than one X in each small square, what is the greatest number of X's that can be put on the grid shown without getting three X's in a row vertically, horizontally, or diagonally?\n\n[asy] for(int a=0; a<4; ++a) { draw((a,0)--(a,3)); } for(int b=0; b<4; ++b) { draw((0,b)--(3,b)); } [/asy] Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]

[ { "content": "A sphere with center $O$ has radius $6$. A triangle with sides of length $15, 15,$ and $24$ is situated in space so that each of its sides is tangent to the sphere. What is the distance between $O$ and the plane determined by the triangle? Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]

[ { "content": "If $\\log_{10}{m}= b-\\log_{10}{n}$, then $m=$ Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]

[ { "content": "In a small pond there are eleven lily pads in a row labeled 0 through 10. A frog is sitting on pad 1. When the frog is on pad $N$, $0<N<10$, it will jump to pad $N-1$ with probability $\\frac{N}{10}$ and to pad $N+1$ with probability $1-\\frac{N}{10}$. Each jump is independent of the previous jumps. If the frog reaches pad 0 it will be eaten by a patiently waiting snake. If the frog reaches pad 10 it will exit the pond, never to return. What is the probability that the frog will escape without being eaten by the snake? Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]

[ { "content": "Alicia had two containers. The first was $\\frac{5}{6}$ full of water and the second was empty. She poured all the water from the first container into the second container, at which point the second container was $\\frac{3}{4}$ full of water. What is the ratio of the volume of the first container to the volume of the second container? Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]

[ { "content": "Eight points are chosen on a circle, and chords are drawn connecting every pair of points. No three chords intersect in a single point inside the circle. How many triangles with all three vertices in the interior of the circle are created? Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]

[ { "content": "The numbers $1,2,\\dots,9$ are randomly placed into the $9$ squares of a $3 \\times 3$ grid. Each square gets one number, and each of the numbers is used once. What is the probability that the sum of the numbers in each row and each column is odd? Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]

[ { "content": "Karl bought five folders from Pay-A-Lot at a cost of $\\$ 2.50$ each.\nPay-A-Lot had a 20%-off sale the following day. How much could\nKarl have saved on the purchase by waiting a day? Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]

[ { "content": "Let $n$ be a positive integer greater than 4 such that the decimal representation of $n!$ ends in $k$ zeros and the decimal representation of $(2n)!$ ends in $3k$ zeros. Let $s$ denote the sum of the four least possible values of $n$. What is the sum of the digits of $s$? Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]

[ { "content": "The sale ad read: \"Buy three tires at the regular price and get the fourth tire for 3 dollars.\" Sam paid 240 dollars for a set of four tires at the sale. What was the regular price of one tire? Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]

[ { "content": "Right triangle $ACD$ with right angle at $C$ is constructed outwards on the hypotenuse $\\overline{AC}$ of isosceles right triangle $ABC$ with leg length $1$, as shown, so that the two triangles have equal perimeters. What is $\\sin(2\\angle BAD)$? Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]

[ { "content": "How many positive integers $n$ satisfy the following condition:\n$(130n)^{50} > n^{100} > 2^{200}$? Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]

[ { "content": "Two circles lie outside regular hexagon $ABCDEF$. The first is tangent to $\\overline{AB}$, and the second is tangent to $\\overline{DE}$. Both are tangent to lines $BC$ and $FA$. What is the ratio of the area of the second circle to that of the first circle? Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]

[ { "content": "In a given plane, points $A$ and $B$ are $10$ units apart. How many points $C$ are there in the plane such that the perimeter of $\\triangle ABC$ is $50$ units and the area of $\\triangle ABC$ is $100$ square units? Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]

[ { "content": "What is equivalent to $\\sqrt{\\frac{x}{1-\\frac{x-1}{x}}}$ when $x < 0$? Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]

[ { "content": "In a far-off land three fish can be traded for two loaves of bread and a loaf of bread can be traded for four bags of rice. How many bags of rice is one fish worth? Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]

[ { "content": "Given a finite sequence $S=(a_1,a_2,\\ldots ,a_n)$ of $n$ real numbers, let $A(S)$ be the sequence \n$\\left(\\frac{a_1+a_2}{2},\\frac{a_2+a_3}{2},\\ldots ,\\frac{a_{n-1}+a_n}{2}\\right)$\nof $n-1$ real numbers. Define $A^1(S)=A(S)$ and, for each integer $m$, $2\\le m\\le n-1$, define $A^m(S)=A(A^{m-1}(S))$. Suppose $x>0$, and let $S=(1,x,x^2,\\ldots ,x^{100})$. If $A^{100}(S)=\\left(\\frac{1}{2^{50}}\\right)$, then what is $x$? Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]

[ { "content": "Inside a right circular cone with base radius $5$ and height $12$ are three congruent spheres with radius $r$. Each sphere is tangent to the other two spheres and also tangent to the base and side of the cone. What is $r$? Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]

[ { "content": "A circle is inscribed in a triangle with side lengths $8, 13$, and $17$. Let the segments of the side of length $8$, made by a point of tangency, be $r$ and $s$, with $r<s$. What is the ratio $r:s$? Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]

[ { "content": "An arithmetic sequence is a sequence in which each term after the first is obtained by adding a constant to the previous term. For example, $2,5,8,11,14$ is an arithmetic sequence with five terms, in which the first term is $2$ and the constant added is $3$. Each row and each column in this $5\\times5$ array is an arithmetic sequence with five terms. The square in the center is labelled $X$ as shown. What is the value of $X$? Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]

[ { "content": "Square $EFGH$ is inside the square $ABCD$ so that each side of $EFGH$ can be extended to pass through a vertex of $ABCD$. Square $ABCD$ has side length $\\sqrt {50}$ and $BE = 1$. What is the area of the inner square $EFGH$? Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]

[ { "content": "When Ringo places his marbles into bags with 6 marbles per bag, he has 4 marbles left over. When Paul does the same with his marbles, he has 3 marbles left over. Ringo and Paul pool their marbles and place them into as many bags as possible, with 6 marbles per bag. How many marbles will be leftover? Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]

[ { "content": "If $y+4 = (x-2)^2$ and $x+4 = (y-2)^2$, and $x \\neq y$, what is the value of $x^2+y^2$? Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]

[ { "content": "Ike and Mike go into a sandwich shop with a total of $30.00 to spend. Sandwiches cost $4.50 each and soft drinks cost $1.00 each. Ike and Mike plan to buy as many sandwiches as they can, and use any remaining money to buy soft drinks. Counting both sandwiches and soft drinks, how many items will they buy? Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]

[ { "content": "The smallest positive integer $x$ for which $1260x=N^3$, where $N$ is an integer, is: Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]

[ { "content": "Each of the $39$ students in the eighth grade at Lincoln Middle School has one dog or one cat or both a dog and a cat. Twenty students have a dog and $26$ students have a cat. How many students have both a dog and a cat? Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]

[ { "content": "How many primes less than $100$ have $7$ as the ones digit? Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]

[ { "content": "The real number $x$ satisfies the equation $x+\\frac{1}{x} = \\sqrt{5}$. What is the value of $x^{11}-7x^{7}+x^3?$ Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]

[ { "content": "Let $P$ equal the product of 3,659,893,456,789,325,678 and 342,973,489,379,256. The number of digits in $P$ is: Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]

[ { "content": "If $\\log_M{N}=\\log_N{M}$, $M \\ne N$, $MN>0$, $M \\ne 1$, $N \\ne 1$, then $MN$ equals: Let's think step by step and output the final answer within \\boxed{}.", "role": "user" } ]