conversations listlengths 2 2 |
|---|
[
{
"from": "user",
"value": "The operation $\\otimes$ is defined for all nonzero numbers by $a\\otimes b =\\frac{a^{2}}{b}$. Determine $[(1\\otimes 2)\\otimes 3]-[1\\otimes (2\\otimes 3)]$.\n$\\text{(A)}\\ -\\frac{2}{3}\\qquad\\text{(B)}\\ -\\frac{1}{4}\\qquad\\text{(C)}\\ 0\\qquad\\text{(D)}\\ \\frac{1}{4}\... |
[
{
"from": "user",
"value": "Doug constructs a square window using $8$ equal-size panes of glass, as shown. 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?\n[asy] fill((0,0)--... |
[
{
"from": "user",
"value": "$P(x)$ is a polynomial of degree $3n$ such that\n\\begin{eqnarray*} P(0) = P(3) = \\cdots &=& P(3n) = 2, \\\\ P(1) = P(4) = \\cdots &=& P(3n-2) = 1, \\\\ P(2) = P(5) = \\cdots &=& P(3n-1) = 0, \\quad\\text{ and }\\\\ && P(3n+1) = 730.\\end{eqnarray*}\nDetermine $n$.\nPlease reaso... |
[
{
"from": "user",
"value": "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 $a=$\n$\\text{(A) } \\frac{2-\\sqrt{2}}{2}\\quad \\text{(B) } \\frac{1}{2}\\quad \\text{(C) } 2-\\sqrt{2}\\quad \\text{(D) } \\frac{\\sqrt{2}}{2}\\quad \\text{(E) ... |
[
{
"from": "user",
"value": "The total in-store price for an appliance is $\\textdollar 99.99$. A television commercial advertises the same product for three easy payments of $\\textdollar 29.98$ and a one-time shipping and handling charge of $\\textdollar 9.98$. How many cents are saved by buying the applia... |
[
{
"from": "user",
"value": "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 $\\overline{HC}, \\overline{JE},$ and... |
[
{
"from": "user",
"value": "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?\n$\\text{(A)}\\ 28\\% \\qquad \\text{(B)}\\ 35\\% \\qquad \\text{(C)}\\ 70\\% \\qquad \\text{(D... |
[
{
"from": "user",
"value": "The graph, $G$ of $y=\\log_{10}x$ is rotated $90^{\\circ}$ counter-clockwise about the origin to obtain a new graph $G'$. Which of the following is an equation for $G'$?\n(A) $y=\\log_{10}\\left(\\frac{x+90}{9}\\right)$ (B) $y=\\log_{x}10$ (C) $y=\\frac{1}{x+1}$ (D) $y=10^{-x}$ (... |
[
{
"from": "user",
"value": "Jose, Thuy, and Kareem each start with the number 10. Jose subtracts 1 from the number 10, doubles his answer, and then adds 2. Thuy doubles the number 10, subtracts 1 from her answer, and then adds 2. Kareem subtracts 1 from the number 10, adds 2 to his number, and then doubl... |
[
{
"from": "user",
"value": "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? \n$\\textbf{(A)}\\ 6\\qquad \\textbf{(B)}\\ 7 \\qquad \\textbf{(C)}\\... |
[
{
"from": "user",
"value": "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 flower... |
[
{
"from": "user",
"value": "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), \\text{ and } (\\cos t^\\circ,\\sin t^\\circ)\\]\nis isosceles? \n$\\text... |
[
{
"from": "user",
"value": "Given: $x > 0, y > 0, x > y$ and $z\\ne 0$. The inequality which is not always correct is:\n$\\textbf{(A)}\\ x + z > y + z \\qquad\\textbf{(B)}\\ x - z > y - z \\qquad\\textbf{(C)}\\ xz > yz$\n$\\textbf{(D)}\\ \\frac {x}{z^2} > \\frac {y}{z^2} \\qquad\\textbf{(E)}\\ xz^2 > yz^2$\... |
[
{
"from": "user",
"value": "The harmonic mean of two numbers can be calculated as twice their product divided by their sum. The harmonic mean of $1$ and $2016$ is closest to which integer?\n$\\textbf{(A)}\\ 2 \\qquad \\textbf{(B)}\\ 45 \\qquad \\textbf{(C)}\\ 504 \\qquad \\textbf{(D)}\\ 1008 \\qquad \\textb... |
[
{
"from": "user",
"value": "The first three terms of an arithmetic progression are $x - 1, x + 1, 2x + 3$, in the order shown. The value of $x$ is: \n$\\textbf{(A)}\\ -2\\qquad\\textbf{(B)}\\ 0\\qquad\\textbf{(C)}\\ 2\\qquad\\textbf{(D)}\\ 4\\qquad\\textbf{(E)}\\ \\text{undetermined}$\nPlease reason step by... |
[
{
"from": "user",
"value": "Let a geometric progression with n terms have first term one, common ratio $r$ and sum $s$, where $r$ and $s$ are not zero. The sum of the geometric progression formed by replacing each term of the original progression by its reciprocal is\n$\\textbf{(A) }\\frac{1}{s}\\qquad \\te... |
[
{
"from": "user",
"value": "Alicia had two containers. The first was $\\tfrac{5}{6}$ 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 $\\tfrac{3}{4}$ full of water. What is the ratio of the volume of ... |
[
{
"from": "user",
"value": "Misha rolls a standard, fair six-sided die until she rolls $1-2-3$ in that order on three consecutive rolls. The probability that she will roll the die an odd number of times is $\\dfrac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$.\nPlease reason s... |
[
{
"from": "user",
"value": "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 $A$, $B... |
[
{
"from": "user",
"value": "Diameter $AB$ of a circle has length a $2$-digit integer (base ten). Reversing the digits gives the length of the perpendicular chord $CD$. The distance from their intersection point $H$ to the center $O$ is a positive rational number. Determine the length of $AB$.\nPlease reason... |
[
{
"from": "user",
"value": "Find the smallest positive number from the numbers below.\n$\\textbf{(A)} \\ 10-3\\sqrt{11} \\qquad \\textbf{(B)} \\ 3\\sqrt{11}-10 \\qquad \\textbf{(C)}\\ 18-5\\sqrt{13}\\qquad \\textbf{(D)}\\ 51-10\\sqrt{26}\\qquad \\textbf{(E)}\\ 10\\sqrt{26}-51$\nPlease reason step by step,... |
[
{
"from": "user",
"value": "The points $(2,-3)$, $(4,3)$, and $(5, k/2)$ are on the same straight line. The value(s) of $k$ is (are):\n$\\textbf{(A)}\\ 12\\qquad \\textbf{(B)}\\ -12\\qquad \\textbf{(C)}\\ \\pm 12\\qquad \\textbf{(D)}\\ {12}\\text{ or }{6}\\qquad \\textbf{(E)}\\ {6}\\text{ or }{6\\frac{2... |
[
{
"from": "user",
"value": "Suppose $q_1,q_2,...$ is an infinite sequence of integers satisfying the following two conditions:\n(a) $m - n$ divides $q_m - q_n$ for $m>n \\geq 0$\n(b) There is a polynomial $P$ such that $|q_n|<P(n)$ for all $n$.\nProve that there is a polynomial $Q$ such that $q_n = Q(n)$ fo... |
[
{
"from": "user",
"value": "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 neares... |
[
{
"from": "user",
"value": "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 $ABCD?$\n$\\textbf{(A) } 330 \\qquad \\textbf{(B) } 340 \\qquad \\te... |
[
{
"from": "user",
"value": "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 $k?$\n$\\textbf{(A)} \\ \\frac{1+\\sqrt{5}}{2} \\qquad \\textbf{(B)} \\ \\frac{1+\\sqrt{7}}{2} \\qquad \\textbf{(C)} \\ \\frac{2+\\sqrt{3}}{2} \\q... |
[
{
"from": "user",
"value": "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 $S$?\n$\\mathrm{(A) \\ } 1\\qquad \\mathrm{(B) \\ } 2\\qquad \\mathrm{(C) \\ } 4\\qquad \\mathrm{(D) \\ } ... |
[
{
"from": "user",
"value": "Five positive consecutive integers starting with $a$ have average $b$. What is the average of $5$ consecutive integers that start with $b$?\n$\\textbf{(A)}\\ a+3\\qquad\\textbf{(B)}\\ a+4\\qquad\\textbf{(C)}\\ a+5\\qquad\\textbf{(D)}\\ a+6\\qquad\\textbf{(E)}\\ a+7$\nPlease reaso... |
[
{
"from": "user",
"value": "Which of the following is the negation of the statement: For all $x$ of a certain set, $x^2>0$?\n$\\textbf{(A)}\\ \\text{For all x}, x^2 < 0\\qquad \\textbf{(B)}\\ \\text{For all x}, x^2 \\le 0\\qquad \\textbf{(C)}\\ \\text{For no x}, x^2>0\\qquad \\\\ \\textbf{(D)}\\ \\text{For ... |
[
{
"from": "user",
"value": "What is the greatest power of $2$ that is a factor of $10^{1002} - 4^{501}$?\n$\\textbf{(A) } 2^{1002} \\qquad\\textbf{(B) } 2^{1003} \\qquad\\textbf{(C) } 2^{1004} \\qquad\\textbf{(D) } 2^{1005} \\qquad\\textbf{(E) }2^{1006}$\nPlease reason step by step, and put your final answe... |
[
{
"from": "user",
"value": "Consider all 1000-element subsets of the set $\\{1, 2, 3, ... , 2015\\}$. 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 $p + q$.\nPlease reaso... |
[
{
"from": "user",
"value": "A man on his way to dinner short 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: \n... |
[
{
"from": "user",
"value": "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?\n$\\text{(A)}\\ 60 \\qquad \\text{(B)}\\ 72 \\qquad \\text{(C)}\\ 84 \\qquad \\text{(D)}\\ 90 \\qquad... |
[
{
"from": "user",
"value": "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}\\]\n$\\textbf{(A)}\\ 1\\qquad\\textbf{(B)}\\ 2\\qquad\\textbf{(C)}\\ 3\\qquad\\textbf{... |
[
{
"from": "user",
"value": "Let $\\, a =(m^{m+1} + n^{n+1})/(m^m + n^n), \\,$ where $\\,m\\,$ and $\\,n\\,$ are positive integers. Prove that $\\,a^m + a^n \\geq m^m + n^n$.\n[You may wish to analyze the ratio $\\,(a^N - N^N)/(a-N),$ for real $\\, a \\geq 0 \\,$ and integer $\\, N \\geq 1$.]\nPlease reason... |
[
{
"from": "user",
"value": "The sum of two natural numbers is $17{,}402$. 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?\n$\\textbf{(A)} ~10{,}272\\qquad\\textbf{(B)} ~11{,}700\\qquad\\textb... |
[
{
"from": "user",
"value": "What is the value of \\[2^{\\left(0^{\\left(1^9\\right)}\\right)}+\\left(\\left(2^0\\right)^1\\right)^9?\\]\n$\\textbf{(A) } 0 \\qquad\\textbf{(B) } 1 \\qquad\\textbf{(C) } 2 \\qquad\\textbf{(D) } 3 \\qquad\\textbf{(E) } 4$\nPlease reason step by step, and put your final answer w... |
[
{
"from": "user",
"value": "Which of the following describes the graph of the equation $(x+y)^2=x^2+y^2$?\n$\\textbf{(A) } \\text{the\\,empty\\,set}\\qquad \\textbf{(B) } \\textrm{one\\,point}\\qquad \\textbf{(C) } \\textrm{two\\,lines} \\qquad \\textbf{(D) } \\textrm{a\\,circle} \\qquad \\textbf{(E) } \\te... |
[
{
"from": "user",
"value": "If $\\texttt{a}$ and $\\texttt{b}$ 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 $\\texttt{a+b =}$\n$\\mathrm{(A)\\ } 3 \\qquad \\mathrm{(B) \\ }4 \\qquad \\mathrm{(C) \\ } 7 \\qquad \... |
[
{
"from": "user",
"value": "At $2: 15$ o'clock, the hour and minute hands of a clock form an angle of:\n$\\textbf{(A)}\\ 30^{\\circ} \\qquad\\textbf{(B)}\\ 5^{\\circ} \\qquad\\textbf{(C)}\\ 22\\frac {1}{2}^{\\circ} \\qquad\\textbf{(D)}\\ 7\\frac {1}{2} ^{\\circ} \\qquad\\textbf{(E)}\\ 28^{\\circ}$\nPlease r... |
[
{
"from": "user",
"value": "In [right triangle](https://artofproblemsolving.com/wiki/index.php/Right_triangle) $ABC$ with [right angle](https://artofproblemsolving.com/wiki/index.php/Right_angle) $C$, $CA = 30$ and $CB = 16$. Its legs $CA$ and $CB$ are extended beyond $A$ and $B$. [Points](https://artofpr... |
[
{
"from": "user",
"value": "A finite [sequence](https://artofproblemsolving.com/wiki/index.php/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, ... |
[
{
"from": "user",
"value": "What time was it $2011$ minutes after midnight on January 1, 2011? \n$\\textbf{(A)}\\ \\text{January 1 at 9:31PM}$\n$\\textbf{(B)}\\ \\text{January 1 at 11:51PM}$\n$\\textbf{(C)}\\ \\text{January 2 at 3:11AM}$\n$\\textbf{(D)}\\ \\text{January 2 at 9:31AM}$\n$\\textbf{(E)}\\ \\tex... |
[
{
"from": "user",
"value": "A basketball player made $5$ baskets during a game. Each basket was worth either $2$ or $3$ points. How many different numbers could represent the total points scored by the player?\n$\\textbf{(A)}\\ 2 \\qquad \\textbf{(B)}\\ 3 \\qquad \\textbf{(C)}\\ 4 \\qquad \\textbf{(D)}\\ 5 ... |
[
{
"from": "user",
"value": "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 $ABCD$?\n$\\text{(A)}\\,\\frac{49}{64} \\qquad\\text{(B)}\\,\\frac{25}{32} \\qquad\\text{(C)}\\,\\frac78 \\qquad... |
[
{
"from": "user",
"value": "A house worth $ $9000$ is sold by Mr. A to Mr. B at a $10$ % loss. Mr. B sells the house back to Mr. A at a $10$ % gain. \nThe result of the two transactions is: \n$\\textbf{(A)}\\ \\text{Mr. A breaks even} \\qquad \\textbf{(B)}\\ \\text{Mr. B gains }$900 \\qquad \\textbf{(C)}\\ ... |
[
{
"from": "user",
"value": "Chandler wants to buy a $500$ dollar mountain bike. For his birthday, his grandparents\nsend him $50$ dollars, his aunt sends him $35$ dollars and his cousin gives him $15$ dollars. He earns\n$16$ dollars per week for his paper route. He will use all of his birthday money and all... |
[
{
"from": "user",
"value": "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?\n$\\textbf{(A) }\\frac{3}{7}\\qquad\\textbf{(B) }\\frac{7}{12}\\qquad\\textbf{(C) }\\frac{12}{7}\\qquad\\textbf{(D) }\\frac... |
[
{
"from": "user",
"value": "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, which of the following number of words would be an appropriate length for h... |
[
{
"from": "user",
"value": "If the following instructions are carried out by a computer, which 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 VALUE OF $X$. \n4. IF $S$ IS AT LEAST... |
[
{
"from": "user",
"value": "Which of the following is equivalent to \"If P is true, then Q is false.\"?\nPlease reason step by step, and put your final answer within \\boxed{{}}."
},
{
"from": "assistant",
"value": "<think>\nOkay, let's tackle this logic problem. The question is asking which of ... |
[
{
"from": "user",
"value": "Let $m$ be the least positive integer divisible by $17$ whose digits sum to $17$. Find $m$.\nPlease reason step by step, and put your final answer within \\boxed{{}}."
},
{
"from": "assistant",
"value": "<think>\nOkay, let's see. I need to find the smallest positive i... |
[
{
"from": "user",
"value": "A shopper plans to purchase an item that has a listed price greater than $\\textdollar 100$ and can use any one of the three coupons. Coupon A gives $15\\%$ off the listed price, Coupon B gives $\\textdollar 30$ off the listed price, and Coupon C gives $25\\%$ off the amount by w... |
[
{
"from": "user",
"value": "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 $\\angle ACB.$\n$\\text{(A) }54^\\circ \\qquad \\text{(B) }60^\\circ \\qquad \\text{(C) }72^\\circ \\qquad \\text{(D) }75^\\circ \\qquad \\text... |
[
{
"from": "user",
"value": "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?\n$\\mathrm{(A)}\\ 1 \\qquad \\mathrm{(B)}\\ 2 \\qquad \\mathrm{(C)}\\ 3 \\qquad \\mathrm{(D)}\\ 4 \\qquad ... |
[
{
"from": "user",
"value": "The equation $2^{2x}-8\\cdot 2^x+12=0$ is satisfied by:\n$\\text{(A) } log(3)\\quad \\text{(B) } \\tfrac{1}{2}log(6)\\quad \\text{(C) } 1+log(\\tfrac{3}{2})\\quad \\text{(D) } 1+\\frac{log(3)}{log(2)}\\quad \\text{(E) none of these}$\nPlease reason step by step, and put your fina... |
[
{
"from": "user",
"value": "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\n$\\textbf{(A)}\\ \\Big\\{\\frac{5}{2}\\Big\\}\\qquad \\textbf{(B)}\\ \\big\\{x\\ |\\ 2 \\le x \\le 3\\big\\}\\qquad \\textbf{(C)}\\ \\... |
[
{
"from": "user",
"value": "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:\n$\\textbf{(A)}\\ \\text{200 yd.}\\qquad\\textbf{(B)... |
[
{
"from": "user",
"value": "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 d... |
[
{
"from": "user",
"value": "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 $2$ : $1$ ?\n$\\textbf{(A)}\\ 2 \\qquad\\textbf{(B)} \\ 4 \\qquad\\textbf{(C)} \\ 5 \\qquad\\textbf{(D)} ... |
[
{
"from": "user",
"value": "If $\\theta$ is a constant such that $0 < \\theta < \\pi$ and $x + \\dfrac{1}{x} = 2\\cos{\\theta}$, then for each positive integer $n$, $x^n + \\dfrac{1}{x^n}$ equals\n$\\textbf{(A)}\\ 2\\cos\\theta\\qquad \\textbf{(B)}\\ 2^n\\cos\\theta\\qquad \\textbf{(C)}\\ 2\\cos^n\\theta\\q... |
[
{
"from": "user",
"value": "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?\n$\\textbf{(A)}\\ {-20}\\qquad\\textbf{(B)}\\ {-18}\\qquad\\textbf{(C)}\\ {-1... |
[
{
"from": "user",
"value": "(Titu Andreescu)\nProve that for every positive integer $n$ there exists an $n$-digit number divisible by $5^n$ all of whose digits are odd.\nPlease reason step by step, and put your final answer within \\boxed{{}}."
},
{
"from": "assistant",
"value": "<think>\nOkay, ... |
[
{
"from": "user",
"value": "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 $\\tfrac{5}{8}$ times the radius of circle $B$, and the radius of circle $C$ is $\\tfrac{5}{8}$ times the radius of cir... |
[
{
"from": "user",
"value": "A figure is an equiangular parallelogram if and only if it is a\n$\\textbf{(A)}\\ \\text{rectangle}\\qquad \\textbf{(B)}\\ \\text{regular polygon}\\qquad \\textbf{(C)}\\ \\text{rhombus}\\qquad \\textbf{(D)}\\ \\text{square}\\qquad \\textbf{(E)}\\ \\text{trapezoid}$\nPlease reason... |
[
{
"from": "user",
"value": "Find the minimum value of $\\sqrt{x^2+y^2}$ if $5x+12y=60$. \n$\\textbf{(A)}\\ \\frac{60}{13}\\qquad \\textbf{(B)}\\ \\frac{13}{5}\\qquad \\textbf{(C)}\\ \\frac{13}{12}\\qquad \\textbf{(D)}\\ 1\\qquad \\textbf{(E)}\\ 0$\nPlease reason step by step, and put your final answer withi... |
[
{
"from": "user",
"value": "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? \n$\\textbf{(A)}\\ 7\\qquad\\textbf{(B)}\\ 32\\qquad\\textbf{(C)}\\ 35\\qquad\\textbf{... |
[
{
"from": "user",
"value": "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 probab... |
[
{
"from": "user",
"value": "Five towns are connected by a system of roads. There is exactly one road connecting each pair of towns. Find the number of ways there are to make all the roads one-way in such a way that it is still possible to get from any town to any other town using the roads (possibly passing... |
[
{
"from": "user",
"value": "The number $2.5252525\\ldots$ can be written as a fraction. \nWhen reduced to lowest terms the sum of the numerator and denominator of this fraction is:\n$\\textbf{(A) }7\\qquad \\textbf{(B) }29\\qquad \\textbf{(C) }141\\qquad \\textbf{(D) }349\\qquad \\textbf{(E) }\\text{none of... |
[
{
"from": "user",
"value": "For all non-zero numbers $x$ and $y$ such that $x = 1/y$, $\\left(x-\\frac{1}{x}\\right)\\left(y+\\frac{1}{y}\\right)$ equals\n$\\textbf{(A) }2x^2\\qquad \\textbf{(B) }2y^2\\qquad \\textbf{(C) }x^2+y^2\\qquad \\textbf{(D) }x^2-y^2\\qquad \\textbf{(E) }y^2-x^2$\nPlease reason ste... |
[
{
"from": "user",
"value": "The values of $k$ for which the equation $2x^2-kx+x+8=0$ will have real and equal roots are: \n$\\textbf{(A)}\\ 9\\text{ and }-7\\qquad\\textbf{(B)}\\ \\text{only }-7\\qquad\\textbf{(C)}\\ \\text{9 and 7}\\\\ \\textbf{(D)}\\ -9\\text{ and }-7\\qquad\\textbf{(E)}\\ \\text{only 9}$... |
[
{
"from": "user",
"value": "How many perfect cubes lie between $2^8+1$ and $2^{18}+1$, inclusive?\n$\\textbf{(A) }4\\qquad\\textbf{(B) }9\\qquad\\textbf{(C) }10\\qquad\\textbf{(D) }57\\qquad \\textbf{(E) }58$\nPlease reason step by step, and put your final answer within \\boxed{{}}."
},
{
"from": "a... |
[
{
"from": "user",
"value": "A line that passes through the origin intersects both the line $x = 1$ and the line $y=1+ \\frac{\\sqrt{3}}{3} x$. The three lines create an equilateral triangle. What is the perimeter of the triangle?\n$\\textbf{(A)}\\ 2\\sqrt{6} \\qquad\\textbf{(B)} \\ 2+2\\sqrt{3} \\qquad\\tex... |
[
{
"from": "user",
"value": "Alicia earns 20 dollars per hour, of which $1.45\\%$ is deducted to pay local taxes. How many cents per hour of Alicia's wages are used to pay local taxes?\n$\\mathrm{(A) \\ } 0.0029 \\qquad \\mathrm{(B) \\ } 0.029 \\qquad \\mathrm{(C) \\ } 0.29 \\qquad \\mathrm{(D) \\ } 2.9 \\q... |
[
{
"from": "user",
"value": "The numbers $-2, 4, 6, 9$ and $12$ are rearranged according to these rules:\n\n 1. The largest isn't first, but it is in one of the first three places. \n 2. The smallest isn't last, but it is in one of the last three places. \n 3. The median isn't first or l... |
[
{
"from": "user",
"value": "Mary thought of a positive two-digit number. She multiplied it by $3$ and added $11$. Then she switched the digits of the result, obtaining a number between $71$ and $75$, inclusive. What was Mary's number? \n$\\textbf{(A)}\\ 11\\qquad\\textbf{(B)}\\ 12\\qquad\\textbf{(C)}\\ 13\\... |
[
{
"from": "user",
"value": "Trapezoid $ABCD$ has $\\overline{AB}\\parallel\\overline{CD},BC=CD=43$, and $\\overline{AD}\\perp\\overline{BD}$. Let $O$ be the intersection of the diagonals $\\overline{AC}$ and $\\overline{BD}$, and let $P$ be the midpoint of $\\overline{BD}$. Given that $OP=11$, the length of... |
[
{
"from": "user",
"value": "The expression $\\frac{1^{4y-1}}{5^{-1}+3^{-1}}$ is equal to: \n$\\textbf{(A)}\\ \\frac{4y-1}{8}\\qquad\\textbf{(B)}\\ 8\\qquad\\textbf{(C)}\\ \\frac{15}{2}\\qquad\\textbf{(D)}\\ \\frac{15}{8}\\qquad\\textbf{(E)}\\ \\frac{1}{8}$\nPlease reason step by step, and put your final ans... |
[
{
"from": "user",
"value": "Carrie has a rectangular garden that measures $6$ feet by $8$ feet. She plants the entire garden with strawberry plants. Carrie is able to plant $4$ strawberry plants per square foot, and she harvests an average of $10$ strawberries per plant. How many strawberries can she expect... |
[
{
"from": "user",
"value": "When the natural numbers $P$ and $P'$, with $P>P'$, are divided by the natural number $D$, the remainders are $R$ and $R'$, respectively. When $PP'$ and $RR'$ are divided by $D$, the remainders are $r$ and $r'$, respectively. Then:\n$\\text{(A) } r>r' \\text{ always}\\quad \\tex... |
[
{
"from": "user",
"value": "The sides of a triangle have lengths $6.5$, $10$, and $s$, where $s$ is a whole number. What is the smallest possible value of $s$?\n\n$\\text{(A)}\\ 3 \\qquad \\text{(B)}\\ 4 \\qquad \\text{(C)}\\ 5 \\qquad \\text{(D)}\\ 6 \\qquad \\text{(E)}\\ 7$\nPlease reason step by step, a... |
[
{
"from": "user",
"value": "One proposal for new postage rates for a letter was $30$ cents for the first ounce and $22$ cents for each additional ounce (or fraction of an ounce). The postage for a letter weighing $4.5$ ounces was\n$\\text{(A)}\\ \\text{96 cents} \\qquad \\text{(B)}\\ \\text{1.07 dollars} \... |
[
{
"from": "user",
"value": "Let $(a_n)$ and $(b_n)$ be the sequences of real numbers such that\n\\[ (2 + i)^n = a_n + b_ni \\]for all integers $n\\geq 0$, where $i = \\sqrt{-1}$. What is\\[\\sum_{n=0}^\\infty\\frac{a_nb_n}{7^n}\\,?\\]\n$\\textbf{(A) }\\frac 38\\qquad\\textbf{(B) }\\frac7{16}\\qquad\\textbf{... |
[
{
"from": "user",
"value": "An $11\\times 11\\times 11$ wooden cube is formed by gluing together $11^3$ unit cubes. What is the greatest number of unit cubes that can be seen from a single point?\n$\\text{(A) 328} \\quad \\text{(B) 329} \\quad \\text{(C) 330} \\quad \\text{(D) 331} \\quad \\text{(E) 332}$\n... |
[
{
"from": "user",
"value": "If each of two intersecting lines intersects a hyperbola and neither line is tangent to the hyperbola, \nthen the possible number of points of intersection with the hyperbola is: \n$\\textbf{(A)}\\ 2\\qquad \\textbf{(B)}\\ 2\\text{ or }3\\qquad \\textbf{(C)}\\ 2\\text{ or }4\\qqu... |
[
{
"from": "user",
"value": "Are there integers $a$ and $b$ such that $a^5b+3$ and $ab^5+3$ are both perfect cubes of integers?\nPlease reason step by step, and put your final answer within \\boxed{{}}."
},
{
"from": "assistant",
"value": "<think>\nOkay, so I need to figure out if there are integ... |
[
{
"from": "user",
"value": "If $\\frac {1}{x} - \\frac {1}{y} = \\frac {1}{z}$, then $z$ equals:\n$\\textbf{(A)}\\ y - x\\qquad \\textbf{(B)}\\ x - y\\qquad \\textbf{(C)}\\ \\frac {y - x}{xy}\\qquad \\textbf{(D)}\\ \\frac {xy}{y - x}\\qquad \\textbf{(E)}\\ \\frac {xy}{x - y}$\nPlease reason step by step, an... |
[
{
"from": "user",
"value": "Ashley, Betty, Carlos, Dick, and Elgin went shopping. Each had a whole number of dollars to spend, and together they had $56$ dollars. The absolute difference between the amounts Ashley and Betty had to spend was $19$ dollars. The absolute difference between the amounts Betty and... |
[
{
"from": "user",
"value": "A palindrome between $1000$ and $10,000$ is chosen at random. What is the probability that it is divisible by $7$?\n$\\textbf{(A)}\\ \\dfrac{1}{10} \\qquad \\textbf{(B)}\\ \\dfrac{1}{9} \\qquad \\textbf{(C)}\\ \\dfrac{1}{7} \\qquad \\textbf{(D)}\\ \\dfrac{1}{6} \\qquad \\textbf{(... |
[
{
"from": "user",
"value": "Let $P$ be an interior point of circle $K$ other than the center of $K$. Form all chords of $K$ which pass through $P$, and determine their midpoints. The locus of these midpoints is \n$\\textbf{(A)} \\text{ a circle with one point deleted} \\qquad \\\\ \\textbf{(B)} \\text{ a ci... |
[
{
"from": "user",
"value": "What is the greatest three-digit positive integer $n$ for which the sum of the first $n$ positive integers is $\\underline{not}$ a divisor of the product of the first $n$ positive integers?\n$\\textbf{(A) } 995 \\qquad\\textbf{(B) } 996 \\qquad\\textbf{(C) } 997 \\qquad\\textbf{(... |
[
{
"from": "user",
"value": "In $\\triangle ABC$ with right angle at $C$, altitude $CH$ and median $CM$ trisect the right angle. If the area of $\\triangle CHM$ is $K$, then the area of $\\triangle ABC$ is \n $\\textbf{(A)}\\ 6K\\qquad\\textbf{(B)}\\ 4\\sqrt3\\ K\\qquad\\textbf{(C)}\\ 3\\sqrt3\\ K\\qquad\\t... |
[
{
"from": "user",
"value": "Claire adds the degree measures of the interior angles of a convex polygon and arrives at a sum of $2017$. She then discovers that she forgot to include one angle. What is the degree measure of the forgotten angle?\n$\\textbf{(A)}\\ 37\\qquad\\textbf{(B)}\\ 63\\qquad\\textbf{(C)}... |
[
{
"from": "user",
"value": "Zara has a collection of $4$ marbles: an Aggie, a Bumblebee, a Steelie, and a Tiger. She wants to display them in a row on a shelf, but does not want to put the Steelie and the Tiger next to one another. In how many ways can she do this?\n$\\textbf{(A) }6 \\qquad \\textbf{(B) }8 ... |
[
{
"from": "user",
"value": "Ray's car averages $40$ miles per gallon of gasoline, and Tom's car averages $10$ miles per gallon of gasoline. Ray and Tom each drive the same number of miles. What is the cars' combined rate of miles per gallon of gasoline?\n$\\textbf{(A)}\\ 10 \\qquad \\textbf{(B)}\\ 16 \\qqua... |
[
{
"from": "user",
"value": "Triangle $ABC$ has vertices $A = (3,0)$, $B = (0,3)$, and $C$, where $C$ is on the line $x + y = 7$. What is the area of $\\triangle ABC$?\n\n$\\mathrm{(A)}\\ 6\\qquad \\mathrm{(B)}\\ 8\\qquad \\mathrm{(C)}\\ 10\\qquad \\mathrm{(D)}\\ 12\\qquad \\mathrm{(E)}\\ 14$\nPlease reason... |
[
{
"from": "user",
"value": "A team won $40$ of its first $50$ games. How many of the remaining $40$ games must this team win so it will have won exactly $70 \\%$ of its games for the season?\n$\\text{(A)}\\ 20 \\qquad \\text{(B)}\\ 23 \\qquad \\text{(C)}\\ 28 \\qquad \\text{(D)}\\ 30 \\qquad \\text{(E)}\\ ... |
[
{
"from": "user",
"value": "For each real number $x$, let $\\textbf{[}x\\textbf{]}$ be the largest integer not exceeding $x$ \n(i.e., the integer $n$ such that $n\\le x<n+1$). Which of the following statements is (are) true?\n$\\textbf{I. [}x+1\\textbf{]}=\\textbf{[}x\\textbf{]}+1\\text{ for all }x \\\\ \\t... |
[
{
"from": "user",
"value": "Positive integers $a$ and $b$ are each less than $6$. What is the smallest possible value for $2 \\cdot a - a \\cdot b$?\n\n$\\textbf{(A)}\\ -20\\qquad\\textbf{{(B)}}\\ -15\\qquad\\textbf{{(C)}}\\ -10\\qquad\\textbf{{(D)}}\\ 0\\qquad\\textbf{{(E)}}\\ 2$\nPlease reason step by ste... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.