messages listlengths 2 2 | answer stringlengths 0 1.22k | has_detailed_solution bool 2
classes |
|---|---|---|
[
{
"role": "user",
"content": "The operation $\\otimes$ is defined for all nonzero numbers by $a \\otimes b = \\frac{a^{2}}{b}$. Determine $[(1 \\otimes 2) \\otimes 3] - [1 \\otimes (2 \\otimes 3)]$."
},
{
"role": "assistant",
"content": "We are given the operation:\n\n$$\na \\otimes b = \\frac{a... | -\frac{2}{3} | true |
[
{
"role": "user",
"content": "Doug constructs a square window using $8$ equal-size panes of glass. The ratio of the height to width for each pane is $5 : 2$, and the borders around and between the panes are $2$ inches wide. In inches, what is the side length of the square window?"
},
{
"role": "assi... | 26 | true |
[
{
"role": "user",
"content": "Let $P(x)$ be a polynomial of degree $3n$ such that\n\\begin{align*} P(0) = P(3) = \\dots = P(3n) &= 2, \\\\ P(1) = P(4) = \\dots = P(3n+1-2) &= 1, \\\\ P(2) = P(5) = \\dots = P(3n+2-2) &= 0. \\end{align*}\nAlso, $P(3n+1) = 730$. Determine $n$."
},
{
"role": "assistant"... | 1 | true |
[
{
"role": "user",
"content": "Let $f$ be the function defined by $f(x)=ax^2-\\sqrt{2}$ for some positive $a$. If $f(f(\\sqrt{2}))=-\\sqrt{2}$ then $a=$"
},
{
"role": "assistant",
"content": "We are given a function:\n\n$$\nf(x) = ax^2 - \\sqrt{2}\n$$\n\nand a condition:\n\n$$\nf(f(\\sqrt{2})) = ... | \frac{\sqrt{2}}{2} | true |
[
{
"role": "user",
"content": "At Euclid Middle School the mathematics teachers are Mrs. Germain, Mr. Newton, and Mrs. Young. There are $11$ students in Mrs. Germain's class, $8$ students in Mr. Newton's class, and $9$ students in Mrs. Young's class taking the AMC $8$ this year. How many mathematics students... | 28 | true |
[
{
"role": "user",
"content": "If $991+993+995+997+999=5000-N$, then $N=$"
},
{
"role": "assistant",
"content": "We are given:\n\n$$\n991 + 993 + 995 + 997 + 999 = 5000 - N\n$$\n\nWe are to find the value of $ N $.\n\n---\n\n### Step 1: Add the numbers on the left side\n\nLet’s add the five conse... | 25 | true |
[
{
"role": "user",
"content": "The total in-store price for an appliance is $99.99$. A television commercial advertises the same product for three easy payments of $29.98$ and a one-time shipping and handling charge of $9.98$. How many cents are saved by buying the appliance from the television advertiser?"
... | 7 | true |
[
{
"role": "user",
"content": "Points $A,B,C,D,E$ and $F$ lie, in that order, on $\\overline{AF}$, dividing it into five segments, each of length 1. Point $G$ is not on line $AF$. Point $H$ lies on $\\overline{GD}$, and point $J$ lies on $\\overline{GF}$. The line segments $\\overline{HC}, \\overline{JE},$ a... | \frac{5}{3} | true |
[
{
"role": "user",
"content": "During the softball season, Judy had $35$ hits. Among her hits were $1$ home run, $1$ triple and $5$ doubles. The rest of her hits were single. What percent of her hits were single?"
},
{
"role": "assistant",
"content": "We are given:\n\n- Total hits: **35**\n- 1... | 80\% | true |
[
{
"role": "user",
"content": "The graph, $G$ of $y=\\log_{10}x$ is rotated $90^{\\circ}$ counter-clockwise about the origin to obtain a new graph $G'$. What is the equation for $G'$?"
},
{
"role": "assistant",
"content": "We are given the graph of the function:\n\n$$\ny = \\log_{10} x\n$$\n\nand... | 10^{-x} | true |
[
{
"role": "user",
"content": "A set of consecutive positive integers beginning with $1$ is written on a blackboard. One number is erased. The average (arithmetic mean) of the remaining numbers is $35\\frac{7}{17}$. What number was erased?"
},
{
"role": "assistant",
"content": "We are given:\n\n-... | 7 | true |
[
{
"role": "user",
"content": "A rectangular yard contains two flower beds in the shape of congruent isosceles right triangles. The remainder of the yard has a trapezoidal shape, as shown. The parallel sides of the trapezoid have lengths $15$ and $25$ meters. What fraction of the yard is occupied by the flow... | \frac{1}{5} | true |
[
{
"role": "user",
"content": "What is the sum of all possible values of $t$ between $0$ and $360$ such that the triangle in the coordinate plane whose vertices are $(\\cos 40^\\circ,\\sin 40^\\circ)$, $(\\cos 60^\\circ,\\sin 60^\\circ)$, and $(\\cos t^\\circ,\\sin t^\\circ)$ is isosceles?"
},
{
"rol... | 380 | true |
[
{
"role": "user",
"content": "In the adjoining figure, points $B$ and $C$ lie on line segment $AD$, and $AB, BC$, and $CD$ are diameters of circle $O, N$, and $P$, respectively. Circles $O, N$, and $P$ all have radius $15$ and the line $AG$ is tangent to circle $P$ at $G$. If $AG$ intersects circle $N$ at p... | 20 | true |
[
{
"role": "user",
"content": "The first three terms of an arithmetic progression are $x - 1, x + 1, 2x + 3$, in the order shown. The value of $x$ is:"
},
{
"role": "assistant",
"content": "We are given the first three terms of an **arithmetic progression (AP)**:\n\n$$\nx - 1, \\quad x + 1, \\qua... | 0 | true |
[
{
"role": "user",
"content": "Alicia had two containers. The first was $\\frac{5}{6}$ full of water and the second was empty. She poured all the water from the first container into the second container, at which point the second container was $\\frac{3}{4}$ full of water. What is the ratio of the volume of ... | \frac{9}{10} | true |
[
{
"role": "user",
"content": "An architect is building a structure that will place vertical pillars at the vertices of regular hexagon $ABCDEF$, which is lying horizontally on the ground. The six pillars will hold up a flat solar panel that will not be parallel to the ground. The heights of pillars at $A$, ... | 17 | true |
[
{
"role": "user",
"content": "The points $(2,-3)$, $(4,3)$, and $(5, k/2)$ are on the same straight line. The value(s) of $k$ is (are):"
},
{
"role": "assistant",
"content": "We are given three points: \n$$\n(2, -3),\\quad (4, 3),\\quad \\left(5, \\frac{k}{2}\\right)\n$$ \nThese points lie on ... | 12 | true |
[
{
"role": "user",
"content": "In a certain year the price of gasoline rose by $20\\%$ during January, fell by $20\\%$ during February, rose by $25\\%$ during March, and fell by $x\\%$ during April. The price of gasoline at the end of April was the same as it had been at the beginning of January. To the near... | 17 | true |
[
{
"role": "user",
"content": "Quadrilateral $ABCD$ satisfies $\\angle ABC = \\angle ACD = 90^{\\circ}, AC=20,$ and $CD=30.$ Diagonals $\\overline{AC}$ and $\\overline{BD}$ intersect at point $E,$ and $AE=5.$ What is the area of quadrilateral $ABCD?$"
},
{
"role": "assistant",
"content": "We are ... | 360 | true |
[
{
"role": "user",
"content": "The angle bisector of the acute angle formed at the origin by the graphs of the lines $y = x$ and $y=3x$ has equation $y=kx.$ What is $k?$"
},
{
"role": "assistant",
"content": "We are given two lines:\n\n- $ y = x $\n- $ y = 3x $\n\nThese lines intersect at the ori... | \frac{1+\sqrt{5}}{2} | true |
[
{
"role": "user",
"content": "A set $S$ of points in the $xy$-plane is symmetric about the origin, both coordinate axes, and the line $y=x$. If $(2,3)$ is in $S$, what is the smallest number of points in $S$?"
},
{
"role": "assistant",
"content": "We are given that a set $ S $ of points in the $... | 8 | true |
[
{
"role": "user",
"content": "Five positive consecutive integers starting with $a$ have average $b$. What is the average of $5$ consecutive integers that start with $b$?"
},
{
"role": "assistant",
"content": "We are given that five **positive consecutive integers starting with $ a $** have an **... | $a+4$ | true |
[
{
"role": "user",
"content": "At noon on a certain day, Minneapolis is $N$ degrees warmer than St. Louis. At $4{:}00$ the temperature in Minneapolis has fallen by $5$ degrees while the temperature in St. Louis has risen by $3$ degrees, at which time the temperatures in the two cities differ by $2$ degrees. ... | 60 | true |
[
{
"role": "user",
"content": "Consider all 1000-element subsets of the set $\\{1, 2, 3, \\dots , 2015\\}$. From each such subset choose the least element. The arithmetic mean of all of these least elements is $\\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p + q$."
},
... | 2016 | true |
[
{
"role": "user",
"content": "A man on his way to dinner shortly after $6:00$ p.m. observes that the hands of his watch form an angle of $110^{\\circ}$. Returning before $7:00$ p.m. he notices that again the hands of his watch form an angle of $110^{\\circ}$. The number of minutes that he has been away is:"... | 40 | true |
[
{
"role": "user",
"content": "A 3x3x3 cube is made of $27$ normal dice. Each die's opposite sides sum to $7$. What is the smallest possible sum of all of the values visible on the $6$ faces of the large cube?"
},
{
"role": "assistant",
"content": "We are given a **3x3x3 cube** made of **27 dice*... | 90 | true |
[
{
"role": "user",
"content": "In the multiplication problem below $A$, $B$, $C$, $D$ are different digits. What is $A+B$?\n$\\begin{array}{cccc} & A & B & A\\\\ \\times & & C & D\\\\ \\hline C & D & C & D\\\\ \\end{array}$"
},
{
"role": "assistant",
"content": "We are given the following multipl... | 1 | true |
[
{
"role": "user",
"content": "Andrea and Lauren are $20$ kilometers apart. They bike toward one another with Andrea traveling three times as fast as Lauren, and the distance between them decreasing at a rate of $1$ kilometer per minute. After $5$ minutes, Andrea stops biking because of a flat tire and waits... | 65 | true |
[
{
"role": "user",
"content": "The sum of two natural numbers is $17402$. One of the two numbers is divisible by $10$. If the units digit of that number is erased, the other number is obtained. What is the difference of these two numbers?"
},
{
"role": "assistant",
"content": "We are given the fo... | 14238 | true |
[
{
"role": "user",
"content": "What is the value of $2^{0^{1^9}} + (2^0)^{1^9}$?"
},
{
"role": "assistant",
"content": "We are asked to evaluate the expression:\n\n$$\n2^{0^{1^9}} + (2^0)^{1^9}\n$$\n\n---\n\n### Step 1: Simplify the exponents\n\nLet’s work from the inside out.\n\n- $ 1^9 = 1 $\n-... | 2 | true |
[
{
"role": "user",
"content": "If $a$ and $b$ are digits for which\n$\\begin{array}{ccc}& 2 & a\\ \\times & b & 3\\ \\hline & 6 & 9\\ 9 & 2 & \\ \\hline 9 & 8 & 9\\end{array}$\nthen $a+b =$"
},
{
"role": "assistant",
"content": "We are given a multiplication of two two-digit numbers:\n\n$$\n\\beg... | 7 | true |
[
{
"role": "user",
"content": "If $x, 2x+2, 3x+3, \\dots$ are in geometric progression, the fourth term is:"
},
{
"role": "assistant",
"content": "We are given a sequence:\n\n$$\nx,\\ 2x+2,\\ 3x+3,\\ \\dots\n$$\n\nand it's stated that these terms are in **geometric progression (GP)**.\n\n---\n\n#... | -13\frac{1}{2} | true |
[
{
"role": "user",
"content": "At $2:15$ o'clock, the hour and minute hands of a clock form an angle of:"
},
{
"role": "assistant",
"content": "We are asked to find the angle between the hour and minute hands of a clock at **2:15**.\n\n---\n\n### Step 1: Understand the positions of the hands\n\n-... | 22\frac {1}{2}^{\circ} | true |
[
{
"role": "user",
"content": "A finite sequence of three-digit integers has the property that the tens and units digits of each term are, respectively, the hundreds and tens digits of the next term, and the tens and units digits of the last term are, respectively, the hundreds and tens digits of the first t... | 37 | true |
[
{
"role": "user",
"content": "Square $EFGH$ has one vertex on each side of square $ABCD$. Point $E$ is on $AB$ with $AE=7\\cdot EB$. What is the ratio of the area of $EFGH$ to the area of $ABCD$?"
},
{
"role": "assistant",
"content": "We are given a square $ABCD$ and another square $EFGH$ with o... | \frac{25}{32} | true |
[
{
"role": "user",
"content": "Chandler wants to buy a $500$ dollar mountain bike. For his birthday, his grandparents send him $50$ dollars, his aunt sends him $35$ dollars and his cousin gives him $15$ dollars. He earns $16$ dollars per week for his paper route. He will use all of his birthday money and all... | 25 | true |
[
{
"role": "user",
"content": "The harmonic mean of a set of non-zero numbers is the reciprocal of the average of the reciprocals of the numbers. What is the harmonic mean of 1, 2, and 4?"
},
{
"role": "assistant",
"content": "To find the **harmonic mean** of the numbers $ 1, 2, $ and $ 4 $, we u... | \frac{12}{7} | true |
[
{
"role": "user",
"content": "Julie is preparing a speech for her class. Her speech must last between one-half hour and three-quarters of an hour. The ideal rate of speech is 150 words per minute. If Julie speaks at the ideal rate, what number of words would be an appropriate length for her speech?"
},... | 5650 | true |
[
{
"role": "user",
"content": "Let $ABC$ be an equilateral triangle. Extend side $\\overline{AB}$ beyond $B$ to a point $B'$ so that $BB'=3 \\cdot AB$. Similarly, extend side $\\overline{BC}$ beyond $C$ to a point $C'$ so that $CC'=3 \\cdot BC$, and extend side $\\overline{CA}$ beyond $A$ to a point $A'$ so ... | 16 | true |
[
{
"role": "user",
"content": "If the following instructions are carried out by a computer, what value of \\(X\\) will be printed because of instruction \\(5\\)?\n\n1. START \\(X\\) AT \\(3\\) AND \\(S\\) AT \\(0\\). \n2. INCREASE THE VALUE OF \\(X\\) BY \\(2\\). \n3. INCREASE THE VALUE OF \\(S\\) BY THE ... | 23 | true |
[
{
"role": "user",
"content": "Letters $A, B, C,$ and $D$ represent four different digits selected from $0, 1, 2, \\ldots ,9.$ If $(A+B)/(C+D)$ is an integer that is as large as possible, what is the value of $A+B$?"
},
{
"role": "assistant",
"content": "We are given:\n\n- $ A, B, C, D $ are **di... | 17 | true |
[
{
"role": "user",
"content": "A shopper plans to purchase an item that has a listed price greater than $\\$100$ and can use any one of the three coupons. Coupon A gives $15\\%$ off the listed price, Coupon B gives $\\$30$ off the listed price, and Coupon C gives $25\\%$ off the amount by which the listed pr... | 50 | true |
[
{
"role": "user",
"content": "In $\\triangle ABC$, $\\angle ABC=45^\\circ$. Point $D$ is on $\\overline{BC}$ so that $2 \\cdot BD=CD$ and $\\angle DAB=15^\\circ$. Find $\\angle ACB.$"
},
{
"role": "assistant",
"content": "We are given the following in triangle $ \\triangle ABC $:\n\n- $ \\angle ... | 75^\circ | true |
[
{
"role": "user",
"content": "Mary's top book shelf holds five books with the following widths, in centimeters: $6$, $\\dfrac{1}{2}$, $1$, $2.5$, and $10$. What is the average book width, in centimeters?"
},
{
"role": "assistant",
"content": "We are given the widths of five books in centimeters:... | 4 | true |
[
{
"role": "user",
"content": "The sum of the greatest integer less than or equal to $x$ and the least integer greater than or equal to $x$ is $5$. The solution set for $x$ is"
},
{
"role": "assistant",
"content": "We are given the following:\n\n> The **sum** of the **greatest integer less than o... | \{x \mid 2 < x < 3\} | true |
[
{
"role": "user",
"content": "A powderman set a fuse for a blast to take place in $30$ seconds. He ran away at a rate of $8$ yards per second. Sound travels at the rate of $1080$ feet per second. When the powderman heard the blast, he had run approximately:"
},
{
"role": "assistant",
"content": ... | 245 yd. | true |
[
{
"role": "user",
"content": "A student council must select a two-person welcoming committee and a three-person planning committee from among its members. There are exactly $10$ ways to select a two-person team for the welcoming committee. It is possible for students to serve on both committees. In how many... | 10 | true |
[
{
"role": "user",
"content": "Two years ago Pete was three times as old as his cousin Claire. Two years before that, Pete was four times as old as Claire. In how many years will the ratio of their ages be $2$ : $1$ ?"
},
{
"role": "assistant",
"content": "We are given:\n\n- Two years ago, Pete w... | 4 | true |
[
{
"role": "user",
"content": "A square piece of paper, 4 inches on a side, is folded in half vertically. Both layers are then cut in half parallel to the fold. Three new rectangles are formed, a large one and two small ones. What is the ratio of the perimeter of one of the small rectangles to the perimeter ... | \frac{5}{6} | true |
[
{
"role": "user",
"content": "Positive integers $a$ and $b$ are such that the graphs of $y=ax+5$ and $y=3x+b$ intersect the $x$-axis at the same point. What is the sum of all possible $x$-coordinates of these points of intersection?"
},
{
"role": "assistant",
"content": "We are given that the gr... | -8 | true |
[
{
"role": "user",
"content": "Four circles, no two of which are congruent, have centers at $A$, $B$, $C$, and $D$, and points $P$ and $Q$ lie on all four circles. The radius of circle $A$ is $\\frac{5}{8}$ times the radius of circle $B$, and the radius of circle $C$ is $\\frac{5}{8}$ times the radius of cir... | 192 | true |
[
{
"role": "user",
"content": "Medians $AD$ and $CE$ of $\\triangle ABC$ intersect in $M$. The midpoint of $AE$ is $N$. \nLet the area of $\\triangle MNE$ be $k$ times the area of $\\triangle ABC$. Then $k$ equals:"
},
{
"role": "assistant",
"content": "We are given a triangle $ \\triangle ABC $ ... | \frac{1}{6} | true |
[
{
"role": "user",
"content": "Find the minimum value of $\\sqrt{x^2+y^2}$ if $5x+12y=60$."
},
{
"role": "assistant",
"content": "We are asked to **minimize** the expression:\n\n$$\n\\sqrt{x^2 + y^2}\n$$\n\nsubject to the constraint:\n\n$$\n5x + 12y = 60\n$$\n\n---\n\n### Step 1: Understand the P... | \frac{60}{13} | true |
[
{
"role": "user",
"content": "On average, for every 4 sports cars sold at the local dealership, 7 sedans are sold. The dealership predicts that it will sell 28 sports cars next month. How many sedans does it expect to sell?"
},
{
"role": "assistant",
"content": "We are given that:\n\n- For **eve... | 49 | true |
[
{
"role": "user",
"content": "Two fair dice, each with at least $6$ faces are rolled. On each face of each die is printed a distinct integer from $1$ to the number of faces on that die, inclusive. The probability of rolling a sum of $7$ is $\\frac34$ of the probability of rolling a sum of $10,$ and the prob... | 17 | true |
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