| {"messages": [{"role": "user", "content": "How many integer solutions does the equation \\(\\sqrt{x} + \\sqrt{y} = \\sqrt{1960}\\) have?"}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "Initially, there were $n$ numbers written on the board: $1, 2, \\ldots, n$. It is allowed to erase any two numbers on the board and write the absolute value of their difference instead. What is the smallest number that can be on the board after $(n-1)$ such operations:\n\na) for $n=111$ \nb) for $n=110$"}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "Given that \\(x = 2a^5 = 3b^2\\) where \\(a\\) and \\(b\\) are positive integers, find the least possible value of \\(x\\)."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "The glass gauge on a cylindrical coffee maker shows that there are $45$ cups left when the coffee maker is $36\\%$ full. How many cups of coffee does it hold when it is full? [asy] draw((5,0)..(0,-1.3)..(-5,0)); draw((5,0)--(5,10)); draw((-5,0)--(-5,10)); draw(ellipse((0,10),5,1.3)); draw(circle((.3,1.3),.4)); draw((-.1,1.7)--(-.1,7.9)--(.7,7.9)--(.7,1.7)--cycle); fill((-.1,1.7)--(-.1,4)--(.7,4)--(.7,1.7)--cycle,black); draw((-2,11.3)--(2,11.3)..(2.6,11.9)..(2,12.2)--(-2,12.2)..(-2.6,11.9)..cycle); [/asy]"}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "A half-sector of a circle with a radius of $6$ inches, together with its interior, can be rolled up to form the lateral surface area of a right circular cone by taping along the two radii. Calculate the volume of the cone in cubic inches."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "Triangle $QRS$ is an isosceles triangle with an area of 180 square inches. A line segment $MN$ cuts the triangle, forming a smaller isosceles triangle and an isosceles trapezoid, where $MN$ is parallel to the base $RS$. The area of the isosceles trapezoid is 120 square inches. If the altitude of triangle $QRS$ from vertex $Q$ is 30 inches, determine the length of $\\overline{MN}$."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "If \\( a \\) is the sum of all the positive factors of 2002, find the value of \\( a \\).\n\nIt is given that \\( x>0, y>0 \\) and \\( \\sqrt{x}(\\sqrt{x}+\\sqrt{y})=3 \\sqrt{y}(\\sqrt{x}+5 \\sqrt{y}) \\). If \\( b=\\frac{2 x+\\sqrt{x y}+3 y}{x+\\sqrt{x y}-y} \\), find the value of \\( b \\).\n\nGiven that the equation \\(||x-2|-1|=c\\) has only 3 integral solutions, find the value of \\( c \\).\n\nIf \\( d \\) is the positive real root of the equation \\(\\frac{1}{2}\\left\\{\\frac{1}{2}\\left[\\frac{1}{2}\\left(\\frac{1}{2} x^{2}+2\\right)+2\\right]+2\\right\\}=2\\), find the value of \\( d \\)."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "Let $\\alpha$ and $\\beta$ be conjugate complex numbers such that $\\frac{\\alpha}{\\beta^3}$ is a real number and $|\\alpha - \\beta| = 6$. Find $|\\alpha|$."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "Math City plans to add more streets and now has 10 streets, but two pairs of these streets are parallel to each other. No other streets are parallel, and no street is parallel to more than one other street. What is the greatest number of police officers needed at intersections, assuming that each intersection has exactly one police officer stationed?"}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "It is required to construct a right triangle in the coordinate plane with its legs parallel to the axes, and the medians to the midpoints of these legs should lie on the lines $y = 2x + 1$ and $y = nx + 3$. The task is to find the number of different constants $n$ for which this configuration is possible."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "Suppose you have a 8-gon, with vertices numbered 1 through 8 in counterclockwise order. Draw the diagonal from vertex 5 to vertex 3, from vertex 4 to vertex 6, from vertex 3 to vertex 6, from vertex 5 to vertex 8, from vertex 2 to vertex 8, from vertex 3 to vertex 8, from vertex 4 to vertex 1, and from vertex 2 to vertex 6. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?"}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "Find the number of times the graphs of f(x) = -1(-2cos(1π(2|x - 3| - 2) + 3) + 2)^2 - 5(-2cos(1π(2|x - 3| - 2) + 3) + 2) - 1 and g(x) = -1e^(1(2|x + 1|)) + 3 intersect in the interval [-10, 10]."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "For each real number \\(x\\), let \\(f(x)\\) be the minimum of the numbers \\(4x+1\\), \\(x+2\\), and \\(-2x+4\\). Find the maximum value of \\(f(x)\\). The original answer is in the form \\(\\frac{k}{m}\\). Please provide the value of \\(k + m\\)."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "Given $x \\gt 0$, $y \\gt 0$, $x+2y=1$, calculate the minimum value of $\\frac{{(x+1)(y+1)}}{{xy}}$."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "The polynomial\n\\[ax^4 + bx^3 + cx^2 + dx + e = 0\\] \nhas integer coefficients and roots $-3$, $6$, $10$, and $-\\frac{1}{4}$. If $e$ is a positive integer, find its smallest possible value."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "3. It is known that the numbers $x, y, z$ form an arithmetic progression in the given order with a common difference $\\alpha=\\arccos \\frac{5}{9}$, and the numbers $1+\\cos x, 1+\\cos y, 1+\\cos z$ form a non-constant geometric progression in the given order. Find $\\cos y$."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "A quadrilateral is inscribed in a circle with radius 1. Two opposite sides are parallel. The difference between their lengths is \\( d > 0 \\). The distance from the intersection of the diagonals to the center of the circle is \\( h \\). Find \\(\\sup \\frac{d}{h}\\) and describe the cases in which it is attained."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "The symbol $\\lfloor x \\rfloor$ denotes the largest integer not exceeding $x$. For example, $\\lfloor 3 \\rfloor = 3,$ and $\\lfloor 9/2 \\rfloor = 4.$ Compute \\[\\lfloor \\sqrt{1} \\rfloor + \\lfloor \\sqrt{2} \\rfloor + \\lfloor \\sqrt{3} \\rfloor + \\cdots + \\lfloor \\sqrt{16} \\rfloor.\\]"}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "A line passes through $A(1,1)$ and $B(100,1000)$. Find the number of other points with integer coordinates that are on the line and strictly between $A$ and $B$."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "12. (20 points) As shown in Figure 2, given $k>0$ and $k \\neq 1$, the line $l: y=k x+1$ is symmetric to the line $l_{1}: y=k_{1} x+1$ with respect to the line $y=x+1$. The lines $l$ and $l_{1}$ intersect the ellipse $E$: $\\frac{x^{2}}{4}+y^{2}=1$ at points $A, M$ and $A, N$ respectively.\n(1) Find the value of $k k_{1}$;\n(2) Prove that for any $k$, the line $M N$ always passes through a fixed point."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "A theorem states that for any five points in the plane, with no three collinear, there exists a unique conic section that passes through all of them. Consider the five points \\[(0, 0),\\; (0, 4),\\; (6, 0),\\; (6, 4),\\; (-3, 2)\\]. Determine the type of conic section and find the length of its minor axis assuming it is an ellipse with axes parallel to the coordinate axes."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "You have {'j': 3, 'z': 2} and need to distribute them into 2 different boxes with capacities [2, 3]. In how many ways can this be done?"}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "Find the determinant of the following matrix. Round to the nearest integer, then give your answer modulo 1000:\n[[0, -4, -7],\n [-4, -3, -3],\n [-7, -3, 5]]"}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "An infinite sheet of paper is divided into equal squares, some of which are colored red. In each $2\\times3$ rectangle, there are exactly two red squares. Now consider an arbitrary $9\\times11$ rectangle. How many red squares does it contain? (The sides of all considered rectangles go along the grid lines.)"}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "At Lincoln High School, there are 30 players on the soccer team. All players are participating in at least one of the three science courses: biology, chemistry, or physics. There are 15 players taking biology, 10 players taking chemistry, 5 players taking physics, and 2 players are taking all three sciences. How many unique players are taking chemistry, considering some might be taking more than one science subject?"}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "Given the sequence $\\left\\{a_{n}\\right\\}$ defined by $a_{n}=(\\sqrt{2}+1)^{n}-(\\sqrt{2}-1)^{n}(n \\in \\mathbb{N})$, find the units digit of $\\left[a_{2017}\\right]$. Express your answer as a single digit between 0 and 9."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "6 Given complex numbers $z_{1}$ and $z_{2}$ satisfy $\\left|z_{2}\\right|=4,4 z_{1}^{2}-2 z_{1} z_{2}+z_{2}^{2}=0$, then the maximum value of $\\left|\\left(z_{1}+1\\right)^{2}\\left(z_{1}-2\\right)\\right|$ is $\\qquad$ ."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "Given that $x$ is a multiple of $23478$, what is the greatest common divisor of $f(x)=(2x+3)(7x+2)(13x+7)(x+13)$ and $x$?"}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "Let $P(x)$ be a polynomial of degree 10 with non-negative integer coefficients. The remainder when $P(x)$ is divided by $(x - 1)$ is 3. How many such polynomials are there?"}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "Let $f(x)=x^{2}-2$, and let $f^{n}$ denote the function $f$ applied $n$ times. Compute the remainder when $f^{24}(18)$ is divided by 89."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "Calculate the greatest common factor of 19 and 2167."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "Consider two lines: line $r$ parametrized as\n\\begin{align*} \nx &= 2 + 5t,\\\\\ny &= 3 - 2t,\n\\end{align*}\nand the line $s$ parametrized as\n\\begin{align*} \nx &= 1 + 5u,\\\\\ny &= -2 - 2u.\n\\end{align*}\nLet $C$ be a point on line $r$, $D$ be a point on line $s$, and let $Q$ be the foot of the perpendicular from $C$ to line $s$.\n\nThen $\\overrightarrow{QC}$ is the projection of $\\overrightarrow{DC}$ onto some vector $\\begin{pmatrix} w_1\\\\w_2\\end{pmatrix}$ such that $w_1 - w_2 = 3$. Find $\\begin{pmatrix}w_1 \\\\ w_2 \\end{pmatrix}$."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "The *Olimpia* country is formed by $n$ islands. The most populated one is called *Panacenter*, and every island has a different number of inhabitants. We want to build bridges between these islands, which we'll be able to travel in both directions, under the following conditions:\r\n\r\na) No pair of islands is joined by more than one bridge.\r\n\r\nb) Using the bridges we can reach every island from Panacenter.\r\n\r\nc) If we want to travel from Panacenter to every other island, in such a way that we use each bridge at most once, the number of inhabitants of the islands we visit is strictly decreasing.\r\n\r\nDetermine the number of ways we can build the bridges."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "What is the probability of such event happening: Form a word by randomly choosing 3 letters from the multiset {e: 2, j: 3, t: 1, c: 2}, shuffle the letters in the word, what is the probability of exact 2 letter 'c' remains in the same position?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "Define M to be the number of ordered triples of positive integers (a, b, c) where a, b, c <= 3^2 and 2a^3 + 0b^3 + -2c^3 = 0 (mod 3^3). Find the remainder of M when divided by 1000."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "Calculate the determinant of the matrix, round to the nearest integer, then find the result modulo 1000:\n[[-7, 0, -3],\n [-4, -2, 5],\n [-8, 6, -1]]"}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "At a round table, 30 people are seated - knights and liars (knights always tell the truth, and liars always lie). It is known that each of them at this table has exactly one friend, and for knights, this friend is a liar, and for liars, this friend is a knight (friendship is always mutual). When asked, \"Is your friend sitting next to you?\", those sitting every other seat answered \"Yes\". How many of the rest could also have answered \"Yes\"?"}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "In the vertices of a regular 300-gon, the numbers from 1 to 300 are arranged in some order, each number appearing exactly once. It turns out that for each number \\(a\\), there are as many numbers smaller than \\(a\\) among the 15 closest numbers to it clockwise as there are among the 15 closest numbers to it counterclockwise. A number that is larger than all 30 numbers closest to it is called \"huge.\" What is the smallest possible number of huge numbers?"}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "A perpendicular dropped from the vertex $C$ of parallelogram $A B C D$ to the line $C D$ intersects at point $F$ a perpendicular dropped from vertex $A$ to the diagonal $B D$. A perpendicular dropped from point $B$ to the line $A B$ intersects at point $E$ the perpendicular bisector of segment $A C$. In what ratio does segment $E F$ divide side $B C$?"}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "Find the integers \\( n \\geq 1 \\) for which \\( n \\) divides \\( 2^{n} - 1 \\)."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "Let $ABF$ be a right-angled triangle with $\\angle AFB = 90$ , a square $ABCD$ is externally to the triangle. If $FA = 6$ , $FB = 8$ and $E$ is the circumcenter of the square $ABCD$ , determine the value of $EF$ "}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "Find the number of integers \\( n \\) that satisfy\n\\[ 50 < n^2 < 200. \\]"}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "There is a $40\\%$ chance of rain on Saturday and a $30\\%$ chance of rain on Sunday. However, it is twice as likely to rain on Sunday if it rains on Saturday than if it does not rain on Saturday. The probability that it rains at least one day this weekend is $\\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. Find $a+b$."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "What is the sum of the greatest common divisor of $60$ and $45045$ and the least common multiple of $60$ and $45045$?"}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "On a $12$-hour clock, an elapsed time of five hours looks the same as an elapsed time of $25$ hours. Because of this, we can say that five hours is \"clock equivalent'' to its square number of hours. What is the least whole number of hours that is greater than $5$ hours and is \"clock equivalent'' to its square number of hours?"}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "Evaluate $\\sqrt[3]{1+27} + \\sqrt[3]{1+\\sqrt[3]{27}}$."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "函数 $f(x)=\\lg 2 \\cdot \\lg 5-\\lg 2 x \\cdot \\lg 5 x$ 的最大值为 $\\qquad$.\n原始的答案是\\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值"}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "Let N denote the number of ordered pairs of positive integers (a, b) such that a, b <= 2^4 and a^2 + b^2 is a multiple of 2^4. Find N mod 1000."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "In a 4x6 grid, how many different paths are there from the bottom left (0, 0) to the top right (3, 5), if you can only move right or up at each step, subject to the constraint that you cannot move through the following cells: (3, 1), (3, 2)?"}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "In convex quadrilateral $ABCD$, $\\angle A \\cong \\angle C$, $AB = CD = 200$, and $AD \\neq BC$. The perimeter of $ABCD$ is $780$. Find $\\lfloor 1000 \\cos A \\rfloor.$"}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "Given parallelogram $ABCD$ with $E$ the midpoint of diagonal $AC$. Point $E$ is connected to a point $F$ in $DA$ so that $DF=\\frac{2}{5}DA$. Determine the ratio of the area of $\\triangle DFE$ to the area of $\\triangle ABE$."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "Find all prime numbers \\( p \\) such that \\( 2p^4 - p^2 + 16 \\) is a perfect square."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "In a $5 \\times 5$ grid, place an \"L\" shape composed of 4 small squares, which can be rotated and flipped. No two \"L\" shapes can overlap. What is the maximum number of \"L\" shapes that can be placed?"}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "Given the equation $\\sin^2 x + \\sin^2 3x + \\sin^2 5x + \\sin^2 7x = 2$, reduce it to an equivalent form \\[\\cos ax \\cos bx \\cos cx = 0,\\] for some integers $a, b, c$. Find the sum $a+b+c$."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "There are 3 math teams in the area, with 5, 7, and 8 students respectively. Each team has two co-captains. If I randomly select a team, and then randomly select two members of that team to give a copy of $\\emph{Introduction to Geometry}$, what is the probability that both of the people who receive books are co-captains?"}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "\nA truck left the settlement of Mirny at a speed of 40 km/h. At the same time, a car left the town of Tikhiy in the same direction as the truck. In the first hour of the journey, the car covered 50 km, and in each subsequent hour, it covered 5 km more than in the previous hour. After how many hours will the car catch up with the truck if the distance between the settlement and the town is 175 km?"}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "How many ordered triples $(a,b,c)$ of integers with $1\\le a\\le b\\le c\\le 60$ satisfy $a\\cdot b=c$ ?"}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "What is the highest common factor of 532 and 2546?"}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "In triangle $ABC$, let $D$ be the foot of the altitude from $A$. Suppose that $AD = 4$, $BD = 3$, $CD = 2$, and $AB$ is extended past $B$ to a point $E$ such that $BE = 5$. Determine the value of $CE^2$."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "In triangle \\(PQR\\), the angle \\(QRP\\) is \\(60^\\circ\\). Find the distance between the points of tangency on side \\(QR\\) of the inscribed circle with radius 2 and the circle with radius 3 that is tangent to the extensions of sides \\(PQ\\) and \\(PR\\)."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "Define M to be the number of ordered triples of positive integers (a, b, c) where a, b, c <= 3^3 and 0a^3 + 0b^3 + -1c^3 = 0 (mod 3^2). Find the remainder of M when divided by 1000."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "In a 4x3 grid, how many different paths are there from the bottom left (0, 0) to the top right (3, 2), if you can only move right or up at each step, subject to the constraint that you cannot move through the following cells: (3, 1), (3, 0), (2, 0), (1, 0)?"}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "An ellipse has major and minor axes of lengths 12 and 6 respectively. A circle is drawn with one focus as a center, and it is just tangent to the ellipse inside it. Find the radius of this circle."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "What is the probability of such event happening: Form a word by randomly choosing 3 letters from the multiset {t: 5, j: 3}, shuffle the letters in the word, what is the probability of at least 2 letter 'j' remains in the same position?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "Let $A_1, A_2, A_3, \\ldots , A_{12}$ be the vertices of a regular $12-$gon (dodecagon). Find the number of points in the plane that are equidistant to at least $3$ distinct vertices of this $12-$gon."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "Let $ABC$ be a triangle with $AB = 3, AC = 4,$ and $BC = 5$ , let $P$ be a point on $BC$ , and let $Q$ be the point (other than $A$ ) where the line through $A$ and $P$ intersects the circumcircle of $ABC$ . Prove that\n\\[PQ\\le \\frac{25}{4\\sqrt{6}}.\\]"}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "A triangle was cut into two triangles. Find the maximum value of $N$ such that among the 6 angles of these two triangles, exactly $N$ are the same."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "There are several (more than one) consecutive natural numbers written on a board, the sum of which equals 2016. What can be the smallest of these numbers?"}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "Compute $\\log_2 6 \\cdot \\log_3 72 - \\log_2 9 - \\log_3 8$."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "Given that $|\\cos\\theta|= \\frac {1}{5}$ and $\\frac {5\\pi}{2}<\\theta<3\\pi$, find the value of $\\sin \\frac {\\theta}{2}$."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "A square with an area of $40$ is inscribed in a semicircle. What is the area of a square that could be inscribed in the entire circle with the same radius? Provide your answer as a number."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "A box contains eight cards. Four of the cards are black on both sides, two cards are black on one side and red on the other, and two of the cards are red on both sides. You pick a card uniformly at random from the box and look at a random side. Given that the side you see is red, what is the probability that the other side is also red?"}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "A $\\pm 1$ -*sequence* is a sequence of $2022$ numbers $a_1, \\ldots, a_{2022},$ each equal to either $+1$ or $-1$ . Determine the largest $C$ so that, for any $\\pm 1$ -sequence, there exists an integer $k$ and indices $1 \\le t_1 < \\ldots < t_k \\le 2022$ so that $t_{i+1} - t_i \\le 2$ for all $i$ , and $$ \\left| \\sum_{i = 1}^{k} a_{t_i} \\right| \\ge C. $$ "}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "The members of a band are arranged in a rectangular formation. When they are arranged in 8 rows, there are 2 positions unoccupied in the formation. When they are arranged in 9 rows, there are 3 positions unoccupied. How many members are in the band if the membership is between 100 and 200?"}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "The 12 numbers from 1 to 12 on a clock face divide the circumference into 12 equal parts. Using any 4 of these division points as vertices to form a quadrilateral, find the total number of rectangles that can be formed."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "Suppose you have a 12-gon, with vertices numbered 1 through 12 in counterclockwise order. Draw the diagonal from vertex 12 to vertex 5, from vertex 8 to vertex 3, from vertex 1 to vertex 9, from vertex 7 to vertex 3, from vertex 7 to vertex 12, and from vertex 10 to vertex 6. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?"}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2|(-1(2((-1x + 1)/(1x))^2 + 1((-1x + 1)/(1x)) + 2) + 3)| - 2 is strictly increasing."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "Two cards are dealt at random from a standard deck of 52 cards. What is the probability that the first card is a $\\clubsuit$ and the second card is a Queen?"}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "Find the largest positive integer $n$ such that\n\\[\\sin^n x + \\cos^n x \\ge \\frac{1}{n}\\]for all real numbers $x.$"}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "Solve -3*z + 2*u + 12 = 0, -4*z - 23*u = -z + 33 + 30 for z."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "There are two prime numbers $p$ so that $5p$ can be expressed in the form $\\left\\lfloor \\dfrac{n^2}{5}\\right\\rfloor$ for some positive integer $n.$ What is the sum of these two prime numbers?"}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "There are three boxes labeled A, B, and C, each containing six cards labeled with the numbers 1, 2, 3, 4, 5, and 6, respectively. Calculate the total number of ways to draw one card from each of the boxes A, B, and C in sequence such that the numbers on the cards form an arithmetic sequence."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "What is the probability of such event happening: Form a word by randomly choosing 3 letters from the multiset {q: 1, i: 7}, shuffle the letters in the word, what is the probability of at least 1 letter 'q' remains in the same position?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "Find the maximum value of\n\\[\\cos \\theta_1 \\sin \\theta_2 + \\cos \\theta_2 \\sin \\theta_3 + \\cos \\theta_3 \\sin \\theta_4 + \\cos \\theta_4 \\sin \\theta_5 + \\cos \\theta_5 \\sin \\theta_1,\\]over all real numbers $\\theta_1,$ $\\theta_2,$ $\\theta_3,$ $\\theta_4,$ and $\\theta_5.$"}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "Chips, colored either black or white, are placed in the cells of a 4x4 grid such that: a) each cell contains at most one chip, b) all chips in the same row and all chips in the same column have the same colour, c) any additional chip placed on the grid would violate one or more of the previous two conditions. Furthermore, we have the following constraints (with the cells 0-indexed): cell (0, 3) is black, cell (3, 0) is black, cell (3, 1) is black, cell (2, 2) is white, cell (1, 2) is white. How many chips are placed on the grid?"}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "3. Fill the numbers $1,2, \\cdots, 36$ into a $6 \\times 6$ grid, with each cell containing one number, such that the numbers in each row are in increasing order from left to right. The minimum value of the sum of the six numbers in the third column is $\\qquad$\n(2015, National Junior High School Mathematics League Competition)"}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "Eleven distinct points, $P$, $Q$, $R$, and $S$, are to be selected from $2023$ points evenly spaced around a circle. All quadruples are equally likely to be chosen. What is the probability that the chord $\\overline{PQ}$ intersects the chord $\\overline{RS}$?"}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "Compute the rank of the given 5x5 matrix:\n[[-19, -10, 9, 19, 23],\n [16, -20, -16, -10, -16],\n [49, -27, -3, -15, -19],\n [-39, 18, 5, 15, 19],\n [46, -17, -10, -22, -28]]"}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "In the mobile below, the two cross beams and the seven supporting strings are all massless. The hanging objects are $M_1 = 400 \\, \\text{g}$, $M_2 = 200 \\, \\text{g}$, and $M_4 = 500 \\, \\text{g}$. What is the value of $M_3$ for the system to be in static equilibrium? If the answer is $x \\, \\text{g}$, what is the value of $x$?"}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "What is the highest common divisor of 1074100 and 1100?"}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "GD = 11. Let R be the midpoint of segment DG. Let W be the midpoint of segment RG. Point H is obtained by rotating point R by angle 7pi/12 about point W. Point L is the midpoint of segment WH. Circle O has center H and passes through point L. Point K is the inversion of point R with respect to circle O. Point M is the midpoint of segment KW. What is the length of segment RM? Express your answer as a decimal number rounded to 2 decimal places."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "In a competition consisting of $n$ true/false questions, 8 participants are involved. It is known that for any ordered pair of true/false questions $(A, B)$, there are exactly two participants whose answers are (true, true); exactly two participants whose answers are (true, false); exactly two participants whose answers are (false, true); and exactly two participants whose answers are (false, false). Find the maximum value of $n$ and explain the reasoning."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "Jamie owns a plumbing business and is passionate about training new apprentices. Each apprentice requires 3 hours of training per day. Jamie currently has 4 apprentices and wants to take on 2 more. If Jamie spends a total of 8 hours a day running the business and can work up to 14 hours a day, how many more hours can Jamie dedicate to training once the new apprentices join?"}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "Let $ABC$ be a right triangle with $\\angle{ACB}=90^{\\circ}$ . $D$ is a point on $AB$ such that $CD\\perp AB$ . If the area of triangle $ABC$ is $84$ , what is the smallest possible value of $$ AC^2+\\left(3\\cdot CD\\right)^2+BC^2? $$ *2016 CCA Math Bonanza Lightning #2.3*"}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "A four-digit square number has the sum of its two middle digits equal to its first digit from the left, and the sum of its first and third digits is ten times its last digit. Which is this square number?"}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "Evaluate $\\frac{2^{5}-2^{3}}{2^{2}}$.\n(A) 6\n(B) 1\n(C) $\\frac{1}{4}$\n(D) 0\n(E) 30"}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "Find the exact value of $1+\\frac{1}{1+\\frac{2}{1+\\frac{1}{1+\\frac{2}{1+\\ldots}}}}$."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "On Qingming Festival, students take a bus to the Martyrs' Cemetery. If the bus increases its speed by one-fifth after traveling for 1 hour at its original speed, it can arrive 20 minutes earlier than the scheduled time. If the bus first travels 72 km at its original speed and then increases its speed by one-third, it can arrive 30 minutes earlier than the scheduled time. How many kilometers is it from the school to the Martyrs' Cemetery?"}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "Given that the total distance from $0$ to $40$ consists of ten steps, and Mark moves forward eight steps but then takes one step back, determine Mark's final position."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "Let $a$ and $b$ be positive integers such that $a$ has $4$ factors and $b$ has $2a$ factors. If $b$ is divisible by $a$, what is the least possible value of $b$?"}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "Two equal cones with a common vertex touch each other and some plane $\\alpha$. Let $l$ be the line along which the planes of the bases of the cones intersect. Find the angle between the line $l$ and the plane $\\alpha$ if the height of each cone is 2 and the radius of the base is 1."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:\n - Each person has a unique name: `Alice`, `Peter`, `Eric`, `Bob`, `Arnold`, `Carol`\n - Everyone has a unique favorite cigar: `pall mall`, `yellow monster`, `dunhill`, `blue master`, `prince`, `blends`\n - People have unique favorite music genres: `hip hop`, `jazz`, `country`, `pop`, `classical`, `rock`\n - Each person has a unique favorite drink: `water`, `milk`, `boba tea`, `tea`, `root beer`, `coffee`\n - The mothers' names in different houses are unique: `Kailyn`, `Penny`, `Janelle`, `Holly`, `Sarah`, `Aniya`\n - Everyone has something unique for lunch: `soup`, `pizza`, `spaghetti`, `stir fry`, `stew`, `grilled cheese`\n\n## Clues:\n1. Carol is directly left of the person who loves eating grilled cheese.\n2. Eric is not in the second house.\n3. The person whose mother's name is Holly is somewhere to the right of Carol.\n4. The person who loves eating grilled cheese is somewhere to the right of the person who loves rock music.\n5. Eric is directly left of Carol.\n6. The person who loves pop music is not in the third house.\n7. Eric is the person who loves country music.\n8. The person who loves classical music is in the sixth house.\n9. The coffee drinker is Bob.\n10. The person who smokes many unique blends is Peter.\n11. The person who loves the stew is not in the fifth house.\n12. The root beer lover is directly left of The person whose mother's name is Janelle.\n13. There are two houses between The person whose mother's name is Sarah and the person who smokes Yellow Monster.\n14. Eric is the tea drinker.\n15. The person partial to Pall Mall is somewhere to the right of the person who loves stir fry.\n16. The person who loves the soup is Bob.\n17. The person who loves hip-hop music is directly left of The person whose mother's name is Kailyn.\n18. Arnold is somewhere to the right of The person whose mother's name is Kailyn.\n19. The one who only drinks water is directly left of the person who smokes Blue Master.\n20. The person who loves the spaghetti eater is somewhere to the left of the person who smokes many unique blends.\n21. The person whose mother's name is Sarah is directly left of the person who loves jazz music.\n22. The person who loves hip-hop music is directly left of the root beer lover.\n23. The one who only drinks water is the person who loves the stew.\n24. The Dunhill smoker is not in the second house.\n25. The person who likes milk is The person whose mother's name is Janelle.\n26. Eric is The person whose mother's name is Aniya.\n What is the value of attribute House for the person whose attribute Name is Eric? Please reason step by step, and put your final answer within \\boxed{}"}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "KX = 1. Point D is the midpoint of segment XK. Let G be the perpendicular bisector of segment XK. Circle U has center X and passes through point K. Let A be the vector from point D to point X. Let I be point X translated by vector A. Let Y be the rotation of point D by angle 3pi/4 around point K. Point L is the inversion of point D with respect to circle U. Let M be the line through point X perpendicular to line G. Circle S passes through points I, Y, and L. Point W is the reflection of point Y across line M. Let F be the inversion of point D in circle S. Point N is the reflection of point W across line G. What is the distance between points N and F? Express your answer as a decimal number rounded to 2 decimal places."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "The factorial of an integer $m$ is denoted as $m!$ and is the product of all positive integers up to $m$. For example, $2!=1 \\times 2=2$, and $(3!)!=1 \\times 2 \\times 3 \\times 4 \\times 5 \\times 6=720$. Given that $((n!)!)!$ is a factor of $(2021!)!$, what is the maximum value of $n$?"}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "If 25,197,631 hot dogs are packaged in sets of 17, how many will be left over?"}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "Compute the rank of the given 5x5 matrix:\n[[-13, -11, 16, -7, 55],\n [-1, -5, -28, 35, 30],\n [-47, 27, 8, -9, 1],\n [25, -21, -17, 36, 38],\n [-30, 15, -34, 15, 2]]"}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "There are 5 houses, numbered 1 to 5 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:\n - Each person has a unique name: `Eric`, `Peter`, `Alice`, `Bob`, `Arnold`\n - The people are of nationalities: `norwegian`, `brit`, `swede`, `dane`, `german`\n - Each person prefers a unique type of vacation: `cruise`, `mountain`, `camping`, `beach`, `city`\n - Each person has a unique level of education: `bachelor`, `master`, `associate`, `doctorate`, `high school`\n - Each person has an occupation: `artist`, `doctor`, `engineer`, `teacher`, `lawyer`\n\n## Clues:\n1. The person who likes going on cruises is the person who is a lawyer.\n2. The person who loves beach vacations is directly left of Arnold.\n3. The person with a doctorate is somewhere to the left of Bob.\n4. The person with an associate's degree is the person who likes going on cruises.\n5. Peter is not in the first house.\n6. The person who is an artist is Peter.\n7. The person who enjoys camping trips is the person with a master's degree.\n8. The Dane is somewhere to the right of the person who is a doctor.\n9. The person with an associate's degree is directly left of the person who is an engineer.\n10. The person who enjoys camping trips is the British person.\n11. The Norwegian and the person with a bachelor's degree are next to each other.\n12. The person who is an artist is the Swedish person.\n13. Bob is not in the fourth house.\n14. The person who enjoys camping trips is Eric.\n15. Alice is the German.\n16. The person who loves beach vacations is somewhere to the left of the person who prefers city breaks.\n17. The person who enjoys mountain retreats is in the fifth house.\n18. The person who likes going on cruises is somewhere to the right of the person who loves beach vacations.\n19. The person with a bachelor's degree is in the third house.\n What is the value of attribute Occupation for the person whose attribute Name is Eric? Please reason step by step, and put your final answer within \\boxed{}"}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "Let \\( n \\) be a positive integer, and let \\( b_0, b_1, \\dots, b_n \\) be a sequence of real numbers such that \\( b_0 = 48 \\), \\( b_1 = 81 \\), \\( b_n=0 \\), and\n\\[ \nb_{k+1} = b_{k-1} - \\frac{4}{b_k} \n\\]\nfor \\( k = 1, 2, \\dots, n-1 \\). Find \\( n \\)."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "How many zeros does the function f(x) = (-2(1(-1(-3|x - 1| - 2)^2 - 1(-3|x - 1| - 2) + 3) + 2))/(-3(1(-1(-3|x - 1| - 2)^2 - 1(-3|x - 1| - 2) + 3) + 2) + 2) have in the interval [-10, 10]?"}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "A billiard table has the shape of a $2 \\times 1$ rectangle, with pockets located at the corners and the midpoints of the longer sides. What is the minimum number of balls that need to be placed inside the rectangle so that each pocket is aligned on the same line with some two balls?"}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "Two distinct numbers are selected from the set $\\{1,2,3,4,\\dots,38\\}$ so that the sum of the remaining $36$ numbers equals the product of these two selected numbers plus one. Find the difference of these two numbers."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "For the function f(x) = -1|(5(-3((-2x - 3)/(-3x + 4))^2 + 2((-2x - 3)/(-3x + 4)) + 3))| + 2, find the number of maximal connected intervals in [-10, 10] where f'(x) > 0."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "Calculate the rounded difference between the largest and smallest singular values from the SVD of:\n[[-2, 3, -3, -1],\n [3, -3, 0, 4],\n [-3, -1, 3, -3],\n [1, 0, 3, -2]]"}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "A travel blogger visits 4 different hotels to review their scents. At the first hotel, the lobby has 15 lavender-scented candles, while the rooms have 3 times as many. The second hotel features a citrus scent, with 12 candles in the lobby and twice as many in the rooms. At the third hotel, the blogger finds a rose scent with 10 candles in the lobby and 5 more than that in the rooms. Finally, the fourth hotel has a sandalwood scent with 20 candles in the lobby and half as many in the rooms. How many candles, in total, does the travel blogger encounter during these visits?"}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "Let \\( a \\) be an integer such that \\( |a| \\leq 2005 \\). Find the number of values of \\( a \\) for which the system of equations \n\\[\n\\begin{cases}\nx^2 = y + a, \\\\\ny^2 = x + a\n\\end{cases}\n\\]\nhas integer solutions."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "Given the expression $(xy-z)^2 + (x+y+z)^2$, find the least possible value for this expression for real numbers $x$, $y$, and $z$."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "Suppose that $P(x, y, z)$ is a homogeneous degree 4 polynomial in three variables such that $P(a, b, c) = P(b, c, a)$ and $P(a, a, b) = 0$ for all real $a$, $b$, and $c$. If $P(1, 2, 3) = 1$, compute $P(2, 4, 8)$.\n\nNote: $P(x, y, z)$ is a homogeneous degree $4$ polynomial if it satisfies $P(ka, kb, kc) = k^4P(a, b, c)$ for all real $k, a, b, c$."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "Evaluate\n\\[\n\\begin{vmatrix} \n\\cos (\\alpha + \\phi) \\cos (\\beta + \\phi) & \\cos (\\alpha + \\phi) \\sin (\\beta + \\phi) & -\\sin (\\alpha + \\phi) \\\\ \n-\\sin (\\beta + \\phi) & \\cos (\\beta + \\phi) & 0 \\\\ \n\\sin (\\alpha + \\phi) \\cos (\\beta + \\phi) & \\sin (\\alpha + \\phi) \\sin (\\beta + \\phi) & \\cos (\\alpha + \\phi) \n\\end{vmatrix}.\n\\]"}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "Define M to be the number of ordered triples of positive integers (a, b, c) where a, b, c <= 4^2 and 0a^4 + -1b^4 + 1c^4 = 0 (mod 4^2). Find the remainder of M when divided by 1000."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "A week ago, Sandy’s seasonal Little League batting average was $360$. After five more at bats this week, Sandy’s batting average is up to $400$. What is the smallest number of hits that Sandy could have had this season?"}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "Find the integer value closest to the x-coordinate where f(x) = 1((1x - 1)/(-1x - 2)) + 1 and g(x) = -2x^2 - 4x - 2 intersect in the range -10 ≤ x ≤ 10."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "Let \\(p\\) and \\(q\\) be relatively prime positive integers such that \\(\\dfrac pq = \\dfrac1{2^1} + \\dfrac2{4^2} + \\dfrac3{2^3} + \\dfrac4{4^4} + \\dfrac5{2^5} + \\dfrac6{4^6} + \\cdots\\), where the numerators always increase by 1, and the denominators alternate between powers of 2 and 4, with exponents also increasing by 1 for each subsequent term. Compute \\(p+q\\)."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "Let $O$ be a point with three other points $A,B,C$ and $\\angle AOB=\\angle BOC=\\angle AOC=2\\pi/3$. Consider the average area of the set of triangles $ABC$ where $OA,OB,OC\\in\\{3,4,5\\}$. The average area can be written in the form $m\\sqrt n$ where $m,n$ are integers and $n$ is not divisible by a perfect square greater than $1$. Find $m+n$."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "10. Let the focus of the parabola $y^{2}=4 x$ be $F$, and draw any line through point $F$ intersecting the parabola at points $A$ and $B$. Does there exist a line $l$ such that for any point $M$ on line $l$, the slopes of lines $M A$, $M F$, and $M B$ always form an arithmetic sequence? If it exists, find the equation of line $l$; if not, explain the reason."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "Define M to be the number of ordered triples of positive integers (a, b, c) where a, b, c <= 4^2 and 0a^4 + -2b^4 + 1c^4 = 0 (mod 4^2). Find the remainder of M when divided by 1000."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "In a country there are $15$ cities, some pairs of which are connected by a single two-way airline of a company. There are $3$ companies and if any of them cancels all its flights, then it would still be possible to reach every city from every other city using the other two companies. At least how many two-way airlines are there?"}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "A semipro baseball league mandates that each team consists of 25 players. The league rule specifies that every player must receive a salary of at least $18,000, while the total salaries for all players on a team cannot exceed $1,000,000. Determine the maximum possible salary, in dollars, that a single player can attain under these rules."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "14. A construction company\n\nhas contracted two projects, each to be constructed by two different teams. According to the progress of the projects, the construction company can adjust the number of people in the two teams at any time. If 70 people are transferred from Team A to Team B, then the number of people in Team B will be twice that of Team A; if a certain number of people are transferred from Team B to Team A, then the number of people in Team A will be three times that of Team B. How many people are there in Team A at least?"}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "Calculate the lengths of the arcs of the curves given by the equations in polar coordinates.\n\n$$\n\\rho = 8(1 - \\cos \\varphi), \\quad -\\frac{2 \\pi}{3} \\leq \\varphi \\leq 0\n$$"}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "Define the function $g(x) = \\frac{k}{3x - 4}$. If $g(3) = g^{-1}(k+2)$, find the product of all possible values of $k$."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "How many sequences of $5$ positive integers $(a,b,c,d,e)$ satisfy $abcde\\leq a+b+c+d+e\\leq10$?"}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "Let $g(x)$ be the function defined on $-2 \\leq x \\leq 2$ by the formula $$g(x) = 2 - \\sqrt{4-x^2}.$$ This is a vertically stretched version of the previously given function. If a graph of $x=g(y)$ is overlaid on the graph of $y=g(x)$, then one fully enclosed region is formed by the two graphs. What is the area of that region, rounded to the nearest hundredth?"}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "ABCD is a square with center N and circumradius 4. Line M passes through point D perpendicular to segment CB. Point U is the reflection of point N across point D. Line E is the reflection of line M across point N. Point J is the reflection of point N across line E. What is the distance between points J and U? Express your answer as a decimal number rounded to 2 decimal places."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "Given a right triangle $PQR$ with $\\angle PQR = 90^\\circ$, suppose $\\cos Q = 0.6$ and $PQ = 15$. What is the length of $QR$?"}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "We want to paint some identically-sized cubes so that each face of each cube is painted a solid color, and each cube is painted with six different colors. If we have seven different colors to choose from, how many distinguishable cubes can we produce?"}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "What is the sum of all integer solutions to $|n| < |n-5| < 5$?"}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "There are 4 houses, numbered 1 to 4 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:\n - Each person has a unique name: `Eric`, `Alice`, `Arnold`, `Peter`\n - Everyone has a favorite smoothie: `cherry`, `dragonfruit`, `desert`, `watermelon`\n - Each person has a favorite color: `green`, `red`, `yellow`, `white`\n - They all have a unique favorite flower: `roses`, `carnations`, `daffodils`, `lilies`\n - People have unique favorite book genres: `fantasy`, `science fiction`, `mystery`, `romance`\n\n## Clues:\n1. The person who loves romance books is the person who loves white.\n2. The person who loves fantasy books is the person who loves a bouquet of daffodils.\n3. The person who likes Cherry smoothies is somewhere to the left of the person who loves romance books.\n4. The person who loves science fiction books is somewhere to the right of Peter.\n5. The person who loves the boquet of lilies is the person who loves science fiction books.\n6. The person who loves romance books is not in the fourth house.\n7. The person who loves fantasy books is the Dragonfruit smoothie lover.\n8. The person who loves white is the Desert smoothie lover.\n9. The person who loves a carnations arrangement is not in the third house.\n10. Alice is the person who likes Cherry smoothies.\n11. Eric is the person who loves mystery books.\n12. The person whose favorite color is green is in the second house.\n13. The person who loves yellow is somewhere to the left of Eric.\n What is the value of attribute House for the person whose attribute Smoothie is cherry? Please reason step by step, and put your final answer within \\boxed{}"}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "Calculate $\\tan A \\tan C$ given that in triangle $ABC$, the orthocenter divides altitude $\\overline{BD}$ from vertex $B$ to side $AC$ into segments with lengths $HD = 8$ and $HB = 24$."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "Two interior angles $A$ and $B$ of pentagon $ABCDE$ are $60^{\\circ}$ and $85^{\\circ}$. Two of the remaining angles, $C$ and $D$, are equal and the fifth angle $E$ is $15^{\\circ}$ more than twice $C$. Find the measure of the largest angle."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "What is the highest common divisor of 1340 and 3216?"}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "Let N denote the number of ordered pairs of positive integers (a, b) such that a, b <= 3^3 and a^3 + b^3 is a multiple of 3^4. Find N mod 1000."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "Commander Krel from Planet Verx has received 30 resumes for four specialized roles on his starship: Chief Science Officer, Head Engineer, Navigation Expert, and Communications Officer. Unfortunately, Commander Krel finds only 10 of these candidates appropriate for any role. Additionally, for the position of Chief Science Officer, only 4 of the suitable candidates have the specific scientific expertise required. In how many ways can Commander Krel staff his starship, considering these constraints?"}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "Determine the area of the region enclosed by the graphs of f(x) = (-3(-3e^(1(3|(-1x^2 - 5x + 3) + 3| - 2)) - 1))/(-2(-3e^(1(3|(-1x^2 - 5x + 3) + 3| - 2)) - 1) + 3) and g(x) = -3sin(3π(-4((-1x - 3)/(-3x + 2)) + 4) - 1) between the vertical lines x = 0.2 and x = 0.30000000000000004. Round to 1 decimal place."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "Given that the sequence $\\{a\\_n\\}$ is a positive arithmetic sequence satisfying $\\frac{1}{a\\_1} + \\frac{4}{a\\_{2k-1}} \\leqslant 1$ (where $k \\in \\mathbb{N}^*$, and $k \\geqslant 2$), find the minimum value of $a\\_k$."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "Define the determinant $D_1$ = $|1|$ , the determinant $D_2$ =\f \f \f \f \f \f $|1 1|$ $|1 3|$ \f \f \f \f \f \f\n, and the determinant $D_3=$ \f \f \f \f \f \f \f \f \f\n|1 1 1|\n|1 3 3|\n|1 3 5| \f \f \f \f \f \f \f \f \f\n.\nIn general, for positive integer n, let the determinant $D_n$ have 1s in every position of its first row and first\ncolumn, 3s in the remaining positions of the second row and second column, 5s in the remaining positions of the third row and third column, and so forth. Find the least n so that $D_n$ $\\geq$ 2015."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "There are 1993 boxes containing balls arranged in a line from left to right. If the leftmost box contains 7 balls, and every four consecutive boxes contain a total of 30 balls, then how many balls are in the rightmost box?"}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "Let $(1-x+x^2)^n = b_0 + b_1x + b_2x^2 + \\cdots + b_{2n}x^{2n}$ be an identity in $x$. Find the sum $t = b_0 + b_2 + b_4 + \\cdots + b_{2n}$."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "If $a, b, x$ and $y$ are real numbers such that $ax + by = 3,$ $ax^2+by^2=7,$ $ax^3+bx^3=16$, and $ax^4+by^4=42,$ find $ax^5+by^5$."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "If $y+4 = (x-2)^2, x+4 = (y-2)^2$, and $x \\neq y$, find the value of $x^2+y^2$."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "For all positive integers $n$ less than $2002$, let\n\\[\na_n = \\begin{cases} \n11 & \\text{if } n \\text{ is divisible by } 13 \\text{ and } 14 \\\\\n13 & \\text{if } n \\text{ is divisible by } 11 \\text{ and } 14 \\\\\n14 & \\text{if } n \\text{ is divisible by } 11 \\text{ and } 13 \\\\\n0 & \\text{otherwise} \\end{cases}\n\\]\nCalculate $\\sum_{n = 1}^{2001} a_n$."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "(4) Let the set $S=\\{1,2,3, \\cdots, 12\\}, A=\\left\\{a_{1}, a_{2}, a_{3}\\right\\}$ be a subset of $S$, and satisfy $a_{1}<a_{2}<a_{3}, a_{3}-a_{2} \\leqslant 5$. Then the number of subsets $A$ that meet the conditions is $\\qquad$."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "Let \\(ABCD\\) be a trapezoid with \\(AB \\parallel CD\\). Let \\(a = AB\\) and \\(b = CD\\). For \\(MN \\parallel AB\\) such that \\(M\\) lies on \\(AD\\), \\(N\\) lies on \\(BC\\), and the trapezoids \\(ABNM\\) and \\(MNCD\\) have the same area, find the length of \\(MN\\). The original answer is in the form \\(\\sqrt{\\frac{ka^2 + mb^2}{n}}\\). Please find the value of \\(k + m + n\\)."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "Find the x-coordinate of the local minimum of the function f(x) = (3(-2(-3|x - 1| + 2) + 4) + 3)/(-3(-2(-3|x - 1| + 2) + 4) + 4) in the interval [-10, 10]. Round to the nearest integer."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "$[$ [Arithmetic. Mental calculation, etc.] $]$\n\nAuthors: Gaityerri G.A., Grierenko D.:\n\n2002 is a palindrome year, which means it reads the same backward as forward. The previous palindrome year was 11 years earlier (1991). What is the maximum number of non-palindrome years that can occur consecutively (between 1000 and 9999 years)?"}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "In the number \\(2 \\_ 0 \\_ 1 \\_ 6 \\_ 0 \\_ 2 \\_\\), each of the 6 stars needs to be replaced with any of the digits \\(1, 2, 3, 4, 5, 6, 7, 8, 9\\) (digits can be repeated) so that the resulting 12-digit number is divisible by 18. How many ways can this be done?"}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "Given the variables $a$ and $b$ satisfying $b=-\\frac{1}{2}a^2+3\\ln(a)$ (with $a>0$), if point $Q(m,n)$ is on the line $y=2x+\\frac{1}{2}$, calculate the minimum value of $(a-m)^2+(b-n)^2$."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "The cards of a standard 52-card deck are dealt out in a circle. What is the expected number of pairs of adjacent cards such that one is black and the other is red? Express your answer as a common fraction."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "Consider the set of all triangles $OPQ$ where $O$ is the origin and $P$ and $Q$ are distinct points in the plane with nonnegative integer coordinates $(x,y)$ such that $41x + y = 2009$. Find the number of such distinct triangles whose area is a positive integer."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "On a $4 \\times 4$ board, numbers from 1 to 16 must be placed in the squares, without repetition, so that the sum of the numbers in each row, column, and diagonal is the same. We call this sum the Magic Sum.\n\na) What is the Magic Sum for this board?\n\nb) If the sum of the squares marked with $X$ on the board below is 34, what is the sum of the squares marked with $Y$?\n\n| $Y$ | | | $Y$ |\n| :--- | :--- | :--- | :--- |\n| | $X$ | $X$ | |\n| | $X$ | $X$ | |\n| $Y$ | | | $Y$ |\n\nc) If we fill the board with consecutive natural numbers from $k$ to $(k+15)$, so that the Magic Sum is 50, what is the value of $k$?"}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "Determine the largest eigenvalue by absolute value of the matrix:\n[[7, 4, -9],\n [0, -3, -4],\n [0, 0, -9]]"}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "Define $p(n)$ to be th product of all non-zero digits of $n$ . For instance $p(5)=5$ , $p(27)=14$ , $p(101)=1$ and so on. Find the greatest prime divisor of the following expression:\n\\[p(1)+p(2)+p(3)+...+p(999).\\]"}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "Determine the rank of the following matrix:\n[[-2, -18, 26, -10],\n [14, -5, 2, 2],\n [11, 10, -17, 9],\n [2, -9, 2, -6]]"}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "\\[\\left(1 + \\frac{1}{1+2^1}\\right)\\left(1+\\frac{1}{1+2^2}\\right)\\left(1 + \\frac{1}{1+2^3}\\right)\\cdots\\left(1 + \\frac{1}{1+2^{10}}\\right)= \\frac{m}{n},\\] where $m$ and $n$ are relatively prime positive integers. Find $m + n$."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "17. A young man allocates his monthly salary as follows: half of his monthly salary is deposited in the bank, half of the remaining money minus 300 yuan is used to pay the mortgage, and half of the remaining money plus 300 yuan is spent on meals. This leaves him with 800 yuan. What is the young man's monthly salary in yuan?"}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "Six couples are at a gathering. Each person shakes hands exactly once with everyone else except their own spouse and except one person who is not their spouse. How many handshakes are exchanged?"}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "A light flashes in one of three different colors: red, green, and blue. Every $3$ seconds, it flashes green. Every $5$ seconds, it flashes red. Every $7$ seconds, it flashes blue. If it is supposed to flash in two colors at once, it flashes the more infrequent color. How many times has the light flashed green after $671$ seconds?"}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "Define M to be the number of ordered triples of positive integers (a, b, c) where a, b, c <= 2^3 and 1a^2 + 2b^2 + 2c^2 = 0 (mod 2^4). Find the remainder of M when divided by 1000."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "4. In $\\triangle A B C$, $\\angle A B C=90^{\\circ}, A C=\\sqrt[3]{2}, D$ is a point on the extension of $A C$, $C D=A B=1$. Then the degree measure of $\\angle C B D$ is $\\qquad$"}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "What is the probability of such event happening: Form a word by randomly choosing 2 letters from the multiset {z: 2, v: 2, a: 1}, shuffle the letters in the word, what is the probability of no letter 'v' occupy any of their original positions?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "Compute the rounded difference between the largest and smallest singular values from the singular value decomposition of the matrix:\n[[1, -6, -3, -2, 1],\n [-3, 0, -5, 0, -6],\n [5, 2, -4, 4, 0],\n [2, 6, -2, -3, -6],\n [0, 0, 5, 6, 4]]"}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "A container contains five red balls. On each turn, one of the balls is selected at random, painted blue, and returned to the container. The expected number of turns it will take before all five balls are colored blue is $\\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "How many non-empty subsets $S$ of $\\{1,2,3,\\ldots,20\\}$ satisfy the following properties?\n1. No two consecutive integers belong to $S$.\n2. If $S$ contains $k$ elements, then $S$ contains no number less than $k$."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "If you add up all the x-coordinates where the functions f(x) = -2x - 4 and g(x) = -4x^2 - 4x + 2 meet in the interval [-10, 10] and round to the nearest integer, what do you get?"}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "Let $\\ell_1$ and $\\ell_2$ be two parallel lines, a distance of 15 apart. Points $A$ and $B$ lie on $\\ell_1$ while points $C$ and $D$ lie on $\\ell_2$ such that $\\angle BAC = 30^\\circ$ and $\\angle ABD = 60^\\circ$. The minimum value of $AD + BC$ is $a\\sqrt b$, where $a$ and $b$ are integers and $b$ is squarefree. Find $a + b$."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "You are given two line segments of length \\(2^{n}\\) for each integer \\(0 \\leq n \\leq 10\\). How many distinct nondegenerate triangles can you form with three of the segments? Two triangles are considered distinct if they are not congruent."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "Calculate the highest common factor of 35249 and 26159."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "A cube with side length $10$ is divided into two cuboids with integral side lengths by a straight cut. Afterwards, one of these two cuboids is divided into two cuboids with integral side lengths by another straight cut. What is the smallest possible volume of the largest of the three cuboids?"}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "A point in the plane, both of whose rectangular coordinates are integers with absolute values less than or equal to four, is chosen at random, with all such points having an equal probability of being chosen. Find the probability that the distance from the point to the origin is at most two units. Express your answer in the form \\frac{k}{m} where k and m are coprime, and give the value of k + m."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "How many ways are there to write $91$ as the sum of at least $2$ consecutive positive integers?"}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "Let \\( f(m) \\) be the product of the digits of the positive integer \\( m \\). Find the positive integer solutions to the equation \\( f(m) = m^2 - 10m - 36 \\)."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "Samantha has 10 green marbles and 5 purple marbles in a bag. She removes a marble at random, records the color, puts it back, and then repeats this process until she has withdrawn 7 marbles. What is the probability that exactly four of the marbles that she removes are green? Express your answer as a decimal."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "Determine the number of values x in [-10, 10] where f(x) = (2(1|(5(-4x^2 + 1x - 1) + 5) + 2| - 2))/(3(1|(5(-4x^2 + 1x - 1) + 5) + 2| - 2) - 2) equals zero."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "Find the sum of all integer values of \\( c \\) for which the equation \\( 15|p-1| + |3p - |p + c|| = 4 \\) has at least one solution for \\( p \\)."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "A line segment with endpoints \\(A(-3, 1)\\) and \\(B(9, 7)\\) is extended from \\(B\\) to a point \\(C\\) such that \\(BC = \\frac{1}{2} \\cdot AB\\). What are the coordinates of point \\(C\\)?"}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "Solve the equation $$x^2 - 13x + 40 = 0$$ and specify whether the roots are positive, negative, or of different signs."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "A 7-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is uncolored, vertex 1 is blue, vertex 2 is uncolored, vertex 3 is uncolored, vertex 4 is blue, vertex 5 is blue, vertex 6 is blue. How many valid colorings of the uncolored vertices exist such that the 7-gon can then be rotated in such a way that all of the red vertices end up at positions where there were originally blue vertices?"}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "What is the highest common factor of 367914 and 918?"}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "A quadrilateral $ABCD$ has a right angle at $\\angle ABC$ and satisfies $AB = 12$ , $BC = 9$ , $CD = 20$ , and $DA = 25$ . Determine $BD^2$ .\n.\n"}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "Given a triangle with an area of $24$, one side of length $8$, and the median to that side of length $7.5$, calculate the sine of the angle $\\theta$ between that side and the median."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "What is the probability of such event happening: Form a word by randomly choosing 2 letters from the multiset {v: 1, s: 1, n: 1, f: 2}, shuffle the letters in the word, what is the probability of no letter 'v' occupy any of their original positions?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "Authors: Dumanov I., Krumovsky R.\n\nDiscussing winter holidays in class, Sasha said: \"Now, after I flew to Addis Ababa, I have celebrated New Year's in all possible hemispheres of the Earth, except one!\n\nWhat is the minimum number of places where Sasha has celebrated New Year's?\n\nConsider the places where Sasha celebrated New Year's as points on a sphere. Points on the boundary of a hemisphere do not count as belonging to that hemisphere."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "Which are more: three-digit numbers where all digits have the same parity (all even or all odd), or three-digit numbers where adjacent digits have different parity?"}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "Find the sum of all entries of the inverse of matrix 1/60 * A where A =\n[[1, 2, -9],\n [2, 7, -3],\n [-9, -3, -4]]\nRound your answer to the nearest integer."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "A point \\((a,b)\\) in the plane is called sparkling if it also lies on the line \\(ax+by=1\\). Find the maximum possible distance between two sparkling points."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "Every week, Ben goes to the supermarket and buys the following: $7$ apples at $\\$2$ each, $4$ bottles of milk at $\\$4$ each, $3$ loaves of bread at $\\$3$ each, and $3$ bags of sugar at $\\$6$ each. This week the store has a $25\\%$ discount on all dairy products. Ben also has a coupon for $\\$10$ off any order of $\\$50$ or over. How much money does Ben spend on this shopping trip?"}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "This sequence lists the perfect squares in increasing order: $0, 1, 4, 9, 16, \\cdots, a, 10^8, b, \\cdots$. Determine the value of $b-a$."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "Find the number of x-intercepts of the function f(x) = -2|((-3(2x^2 - 4x - 3) - 2)/(1(2x^2 - 4x - 3) + 3)) - 3| + 1 where -10 ≤ x ≤ 10."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "A boy named Tolya loves to swim. When he visits his grandmother's dacha, he swims in the Volkhov River, going downstream from one beach to another in 18 minutes. It takes him exactly 1 hour to swim back upstream. When he visits another grandmother's dacha, he swims the same distance downstream in the Luga River in 20 minutes. How much time will it take him to swim back upstream?"}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "The sequence of numbers \\(a_{1}, a_{2}, a_{3}, \\ldots, a_{n}\\) is defined such that\n\n\\[a_{1}=0, \\left|a_{2}\\right| = \\left|a_{1}+1\\right|, \\left|a_{3}\\right| = \\left|a_{2}+1\\right|, \\ldots, \\left|a_{n}\\right| = \\left|a_{n-1}+1\\right|.\\]\n\nProve that the arithmetic mean \\(\\frac{a_{1} + a_{2} + \\ldots + a_{n}}{n}\\) of all the numbers is not less than \\(-\\frac{1}{2}\\)."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "Find the largest number among the following expressions: \\(30^{30}\\), \\(50^{10}\\), \\(40^{20}\\), \\(45^{15}\\), and \\(5^{60}\\). The original answer is in the form a^b, where a is the base and b is the exponent. Please provide the sum of a and b."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "Let $n$ be a fixed positive integer and fix a point $O$ in the plane. There are $n$ lines drawn passing through the point $O$ . Determine the largest $k$ (depending on $n$ ) such that we can always color $k$ of the $n$ lines red in such a way that no two red lines are perpendicular to each other.\n\n*Proposed by Nikola Velov*"}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "Find the modulo 9 value of $$1+21+333+4444+55555+666666+7777777+88888888+999999999$$ where each number's digits are summed and their sum is then reduced modulo 9."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "Triangle $ABC$ is scalene. Points $P$ and $Q$ are on segment $BC$ with $P$ between $B$ and $Q$ such that $BP=21$, $PQ=35$, and $QC=100$. If $AP$ and $AQ$ trisect $\\angle A$, then $\\tfrac{AB}{AC}$ can be written uniquely as $\\tfrac{p\\sqrt q}r$, where $p$ and $r$ are relatively prime positive integers and $q$ is a positive integer not divisible by the square of any prime. Determine $p+q+r$."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "What is the greatest common divisor of 163312 and 305266?"}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "Determine the number of $x$-intercepts on the graph of the function $y = \\cos \\frac{1}{x}$ in the interval $(0.00005, 0.0005)$ in radians."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "If $x$ and $y$ are positive integers such that $7x + 5y = 200$, what is the greatest possible value of $xy$?"}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "What is the probability of such event happening: Form a word by randomly choosing 3 letters from the multiset {s: 3, z: 4}, shuffle the letters in the word, what is the probability of no letter 's' occupy any of their original positions?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "Given the function $y=\\cos({2x+\\frac{π}{3}})$, determine the horizontal shift of the graph of the function $y=\\sin 2x$."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "If for any real number \\( x \\), the function\n\\[ f(x)=x^{2}-2x-|x-1-a|-|x-2|+4 \\]\nalways yields a non-negative real number, then the minimum value of the real number \\( a \\) is ."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "What is the area enclosed by the geoboard quadrilateral below?\n[asy] unitsize(3mm); defaultpen(linewidth(.8pt)); dotfactor=2; for(int a=0; a<=10; ++a) for(int b=0; b<=10; ++b) { dot((a,b)); }; draw((4,0)--(0,5)--(3,4)--(10,10)--cycle); [/asy]\n$\\textbf{(A)}\\ 15\\qquad \\textbf{(B)}\\ 18\\frac12 \\qquad \\textbf{(C)}\\ 22\\frac12 \\qquad \\textbf{(D)}\\ 27 \\qquad \\textbf{(E)}\\ 41$"}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "A semicircle with radius 2021 has diameter \\( AB \\) and center \\( O \\). Points \\( C \\) and \\( D \\) lie on the semicircle such that \\(\\angle AOC < \\angle AOD = 90^{\\circ} \\). A circle of radius \\( r \\) is inscribed in the sector bounded by \\( OA \\) and \\( OC \\) and is tangent to the semicircle at \\( E \\). If \\( CD = CE \\), compute \\(\\lfloor r \\rfloor\\)."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "How many local maxima does the function f(x) = -2sin(1π(-5x - 2) + 3) + 3 have in the interval [-10, 10]?"}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "In a 6x3 grid, how many different paths are there from the bottom left (0, 0) to the top right (5, 2), if you can only move right or up at each step, subject to the constraint that you cannot move through the following cells: (3, 2), (3, 0), (4, 2), (5, 1)?"}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "Let $A B C D$ be a rhombus. Let $F$ be a point on the segment $[A D]$ and $E$ a point on the segment $[A B]$. The lines $(F C)$ and $(B D)$ intersect at $L$, the lines $(E C)$ and $(B D)$ intersect at $K$. The lines $(F K)$ and $(B C)$ intersect at $Q$ and the lines $(E L)$ and $(D C)$ intersect at $P$. Show that $C P = C Q$."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "Find the smallest natural number $k$ for which there exist natural numbers $m$ and $n$ such that $1324 + 279m + 5^n$ is the $k$-th power of some natural number."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "What is the remainder when $5x^9 - 3x^7 + 4x^6 - 8x^4 + 3x^3 - 6x + 5$ is divided by $3x - 6$?"}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "Find the number of times the graphs of f(x) = -1|(1((-2x + 3)/(-2x - 1)) - 1) + 2| and g(x) = -2(1|x + 3| + 1) + 5 intersect in the interval [-10, 10]."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "## Task B-3.4.\n\nA restaurant offers its guests a selection of cold meat cuts, fish pâté, and vegetable salad for appetizers. For soup, guests can choose from chicken soup, beef soup, fish soup, broccoli soup, and asparagus soup. For the main course, the restaurant offers a fish, meat, or vegetarian plate, and for dessert, a vegan cake or vegan ice cream. If a guest chooses one dish from each course, what is the probability that they will randomly select a completely vegetarian meal (a meal without meat and fish)?"}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "Let AK = 3. Line I is the perpendicular bisector of segment KA. Let G be the rotation of point A by angle 11pi/12 around point K. Let V be the reflection of point G across point K. Point W is the reflection of point G across line I. Let P be the circumcircle of triangle AVW. Point T is the inversion of point G with respect to circle P. Let L be the midpoint of segment VW. What is the inradius of triangle LTK? Express your answer as a decimal number rounded to 2 decimal places."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "Determine the area of the region enclosed by the graphs of f(x) = -1e^(1(4x + 1)) + 3 and g(x) = -2x - 3 between the vertical lines x = -3.0 and x = 0.2. Round to 1 decimal place."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "What is the greatest common factor of 3256 and 2183?"}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "The endpoints of a line segment AB, which has a fixed length of 3, move on the parabola $y^2=x$. If M is the midpoint of the line segment AB, then the minimum distance from M to the y-axis is ______."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "Suppose you have a 10-gon, with vertices numbered 1 through 10 in counterclockwise order. Draw the diagonal from vertex 4 to vertex 8, from vertex 6 to vertex 3, from vertex 5 to vertex 9, from vertex 2 to vertex 10, from vertex 4 to vertex 6, from vertex 1 to vertex 3, and from vertex 9 to vertex 2. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?"}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "Find the determinant of the following matrix. Round to the nearest integer, then give your answer modulo 1000:\n[[-2, 4, 2],\n [4, 1, -2],\n [2, -2, 0]]"}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "Given that the positive integer \\( m \\) satisfies that \\( m^{4} + 16m + 8 \\) can be expressed as the product of two or more consecutive integers, determine the maximum value of \\( m \\)."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "A computer can perform 15,000 additions per second, calculate the total number of additions it can perform in 2 hours and 30 minutes."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": " Let $n$ be a positive integer. Mariano divides a rectangle into $n^2$ smaller rectangles by drawing $n-1$ vertical lines and $n-1$ horizontal lines, parallel to the sides of the larger rectangle. On every step, Emilio picks one of the smaller rectangles and Mariano tells him its area. Find the least positive integer $k$ for which it is possible that Emilio can do $k$ conveniently thought steps in such a way that with the received information, he can determine the area of each one of the $n^2$ smaller rectangles.\n"}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "Compute the smallest positive integer $a$ for which $$\\sqrt{a +\\sqrt{a +...}} - \\frac{1}{a +\\frac{1}{a+...}}> 7$$"}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "In a positive geometric sequence {a_n}, it is known that a_1a_2a_3 = 4, a_4a_5a_6 = 12, and a_{n-1}a_na_{n+1} = 324. Find the value of n. Express your answer as a whole number."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "Let \\(a_1, a_2, \\ldots\\) and \\(b_1, b_2, \\ldots\\) be arithmetic progressions where \\(a_1 = 15\\), \\(b_1 = 45\\), and \\(a_{50} + b_{50} = 150\\). Find the sum of the first fifty terms of the progression \\(a_1 + b_1, a_2 + b_2, \\ldots\\) ."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "In rectangle \\(ABCD\\), \\(DC = 2 \\cdot CB\\) and points \\(E\\) and \\(F\\) lie on \\(\\overline{AB}\\) so that \\(\\overline{ED}\\) and \\(\\overline{FD}\\) trisect \\(\\angle ADC\\) as shown. Find the ratio of the area of \\(\\triangle DEF\\) to the area of rectangle \\(ABCD\\). The original answer is in the format \\(\\frac{\\sqrt{k}}{m}\\), please give the value of k + m."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "What is the sum of all entries of the matrix A^3 where A is:\n[[-3, 0, -2, 0],\n [0, -1, -2, 0],\n [-2, -2, 2, 2],\n [0, 0, 2, 4]]?"}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "1. How many irreducible fractions exist where the sum of the numerator and denominator is 100?"}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "A region $S$ in the complex plane is defined by:\n\\[\nS = \\{x + iy: -1 \\leq x \\leq 1, -1 \\leq y \\leq 1\\}.\n\\]\nA complex number $z = x + iy$ is chosen uniformly at random from $S$. What is the probability that $\\left(\\frac{1}{2} + \\frac{1}{2}i\\right)z$ is also in $S$?"}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "Ava is a fashion-conscious retailer who specializes in selling high-quality designer jackets. She believes in the value of quality over quantity and has recently received a shipment of 30 jackets. Each high-quality jacket costs her $150. Ava decides to price each jacket at $210 to maintain her philosophy of offering only the best to her customers. \n\nAva's boutique sells 20 jackets in the first month. How much profit does Ava make from selling these jackets, considering only the cost and selling price?"}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "Define M to be the number of ordered triples of positive integers (a, b, c) where a, b, c <= 2^3 and -1a^2 + 2b^2 + -2c^2 = 0 (mod 2^2). Find the remainder of M when divided by 1000."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "There is an unlimited supply of congruent equilateral triangles made of colored paper. Each triangle is a solid color with the same color on both sides of the paper. A large equilateral triangle is constructed from four of these paper triangles. Two large triangles are considered distinguishable if it is not possible to place one on the other, using translations, rotations, and/or reflections, so that their corresponding small triangles are of the same color. \nGiven that there are six different colors of triangles from which to choose, how many distinguishable large equilateral triangles may be formed?"}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "The geometric series \\(a+ar+ar^2+\\ldots\\) has a sum of \\(7\\), and the terms involving odd powers of \\(r\\) have a sum of \\(3\\). The original answer is in \\(\\frac{k}{m}\\) format, please give the value of \\(k + m\\)."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "7. If real numbers $a, b, c$ make the quadratic function $f(x) = a x^{2} + b x + c$ such that when $0 \\leqslant x \\leqslant 1$, always $|f(x)| \\leqslant 1$. Then the maximum value of $|a| + |b| + |c|$ is $\\qquad$"}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "What is the greatest common divisor of $987654$ and $876543$?"}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "2. In tetrahedron $ABCD$, $\\triangle ABC$ is an equilateral triangle, $AD=BD=2, AD \\perp BD, AD \\perp CD$, then the distance from $D$ to plane $ABC$ is $\\qquad$ ."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "Chips, colored either black or white, are placed in the cells of a 4x4 grid such that: a) each cell contains at most one chip, b) all chips in the same row and all chips in the same column have the same colour, c) any additional chip placed on the grid would violate one or more of the previous two conditions. Furthermore, we have the following constraints (with the cells 0-indexed): cell (0, 2) is white, cell (3, 3) is black, cell (3, 0) is black, cell (1, 2) is white, cell (2, 1) is black. How many chips are placed on the grid?"}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "Two different digits from 1 to 9 are chosen. One digit is placed in each box to complete the two 2-digit numbers shown. The result of subtracting the bottom number from the top number is calculated. How many of the possible results are positive?"}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "Find the domain of the rational function $g(x) = \\frac{x^3-2x^2+4x+3}{x^2-4x+3}$. Express your answer as a union of intervals."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "Compute the sum of all entries of the product matrix AB given:\nA =\n[[6, -1, 7],\n [1, 5, 0],\n [0, -1, 1],\n [-2, 4, -4]]\nB =\n[[8, -3, 2],\n [4, -3, 7],\n [5, -7, -8]]"}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "Suppose you have a 10-gon, with vertices numbered 1 through 10 in counterclockwise order. Draw the diagonal from vertex 2 to vertex 7 and from vertex 1 to vertex 4. Then, rotate the entire setup, including the constructed diagonals, 4 vertices counterclockwise (so that vertex 1 ends up where vertex 5 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 10-gon will be partitioned into a collection of smaller polygons. (Only count the \"smallest\" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be with 6 sides?"}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "For a certain positive integer $n,$ there exist real numbers $x_1,$ $x_2,$ $\\dots,$ $x_n$ such that\n\\begin{align*}\nx_1 + x_2 + x_3 + \\dots + x_n &= 1000, \\\\\nx_1^4 + x_2^4 + x_3^4 + \\dots + x_n^4 &= 512000.\n\\end{align*}Find the smallest positive integer $n$ for which this is possible."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "For a positive real number $ [x] $ be its integer part. For example, $[2.711] = 2, [7] = 7, [6.9] = 6$ . $z$ is the maximum real number such that [ $\\frac{5}{z}$ ] + [ $\\frac{6}{z}$ ] = 7. Find the value of $ 20z$ ."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "Find $x$ if \\[ \\frac{1}{\\frac{1}{\\frac{1}{\\frac{1}{x} + \\frac{1}{2}} + \\frac{1}{\\frac{1}{x} + \\frac{1}{2}}} + \\frac{1}{\\frac{1}{\\frac{1}{x} + \\frac{1}{2}} + \\frac{1}{\\frac{1}{x} + \\frac{1}{2}}}} = \\frac{x}{36}. \\]"}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "Point \\( K \\) lies on side \\( BC \\) of parallelogram \\( ABCD \\), and point \\( M \\) lies on its side \\( AD \\). Segments \\( CM \\) and \\( DK \\) intersect at point \\( L \\), and segments \\( AK \\) and \\( BM \\) intersect at point \\( N \\). Find the maximum value of the ratio of the areas of quadrilateral \\( KLMN \\) to \\( ABCD \\)."}], "membership_label": 1} |
| {"messages": [{"role": "user", "content": "If you form a word of length 3 by randomly selecting letters from the multiset {v: 2, n: 1, f: 1, o: 1}, what is the expected number of times the pattern 'on*' appears? Round to 2 decimal places."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "If $x, y, z$ satisfy $x+y+z = 12$, $\\frac{1}{x} + \\frac{1}{y} + \\frac{1}{z} = 2$, and $x^3+y^3+z^3 = -480$, find $x^2 y + xy^2 + x^2 z + xz^2 + y^2 z + yz^2$."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Ellina has twelve blocks, two each of red ($\\textbf{R}$), blue ($\\textbf{B}$), yellow ($\\textbf{Y}$), green ($\\textbf{G}$), orange ($\\textbf{O}$), and purple ($\\textbf{P}$). Call an arrangement of blocks $\\textit{even}$ if there is an even number of blocks between each pair of blocks of the same color. For example, the arrangement\n\\[\\textbf{R B B Y G G Y R O P P O}\\]\nis even. Ellina arranges her blocks in a row in random order. The probability that her arrangement is even is $\\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$"}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "What is the probability of such event happening: Form a word by randomly choosing 3 letters from the multiset {s: 5, v: 1}, shuffle the letters in the word, what is the probability of no letter 'v' occupy any of their original positions?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "5. If the odd function $y=f(x)$ defined on $\\mathbf{R}$ is symmetric about the line $x=1$, and when $0<x \\leqslant 1$, $f(x)=\\log _{3} x$, then the sum of all real roots of the equation $f(x)=-\\frac{1}{3}+f(0)$ in the interval $(0,10)$ is . $\\qquad$"}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "A triangle has a side of length 6 cm, a side of length 8 cm and a right angle. What is the shortest possible length of the remaining side of the triangle? Express your answer in centimeters as a decimal to the nearest hundredth."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "let N be the maximum 2-digit positive integer with the property that whenever one of its digits changes to 1, the resulting number is a multiple of 3. Let Q and R be the quotient and remainder, respectively, when N is divided by 10. Find Q + R."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Given \\( f(x) = x^2 + 3x + 2 \\) and \\( S = \\{0, 1, 2, 3, \\cdots, 100\\} \\), if \\( a \\in S \\) and \\( f(a) \\) is divisible by 6, how many such \\( a \\) exist?"}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Suppose you have a 6-gon, with vertices numbered 1 through 6 in counterclockwise order. Draw the diagonal from vertex 3 to vertex 6, from vertex 1 to vertex 3, and from vertex 5 to vertex 2. Then, rotate the entire setup, including the constructed diagonals, 5 vertices counterclockwise (so that vertex 1 ends up where vertex 6 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 6-gon will be partitioned into a collection of smaller polygons. (Only count the \"smallest\" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be with 3 sides?"}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Let VW = 7. Point F is the midpoint of segment WV. Let B be the reflection of point V across point W. Let L be the circle with center W passing through point F. Circle I is the reflection of circle L across point V. Let T be the inversion of point W in circle I. Point P is the midpoint of segment TB. What is the distance between points P and B? Express your answer as a decimal number rounded to 2 decimal places."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Find\n\\[\\sum_{N = 1}^{1024} \\lfloor \\log_2 N \\rfloor.\\]"}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "If Alice is cycling east at a speed of 10 miles per hour and Sam is also cycling east but at a faster speed of 16 miles per hour, with Sam being 3 miles west of Alice, calculate the time in minutes it will take for Sam to catch up to Alice."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Let $ S \\equal{} \\{1,2,\\ldots,2008\\}$ . For any nonempty subset $ A\\in S$ , define $ m(A)$ to be the median of $ A$ (when $ A$ has an even number of elements, $ m(A)$ is the average of the middle two elements). Determine the average of $ m(A)$ , when $ A$ is taken over all nonempty subsets of $ S$ ."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "9.4. Vasya cut out a triangle from cardboard and numbered its vertices with the digits 1, 2, and 3. It turned out that if Vasya's triangle is rotated 15 times clockwise around its vertex numbered 1 by the angle at this vertex, the triangle will return to its original position. If Vasya's triangle is rotated 6 times clockwise around its vertex numbered 2 by the angle at this vertex, the triangle will return to its original position. Vasya claims that if his triangle is rotated $n$ times around the vertex numbered 3 by the angle at this vertex, the triangle will return to its original position. What is the smallest $n$ that Vasya could have named so that his claim is true for at least some cardboard triangle?"}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "In the expansion of $(x^2+ \\frac{4}{x^2}-4)^3(x+3)$, find the constant term."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Define M to be the number of ordered triples of positive integers (a, b, c) where a, b, c <= 3^2 and 2a^3 + -2b^3 + -1c^3 = 0 (mod 3^3). Find the remainder of M when divided by 1000."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Exercise 14. Consider a grid of size $2019 \\times 2019$. On this grid, pebbles are placed. A configuration is said to be beautiful if there does not exist a parallelogram formed by four pebbles $A B C D$, such that $A, B, C$, and $D$ are not all aligned.\n\nWhat is the maximum number of pebbles that can be placed on the grid?"}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Let $ABCDEF$ be a regular hexagon with side length 10 inscribed in a circle $\\omega$ . $X$ , $Y$ , and $Z$ are points on $\\omega$ such that $X$ is on minor arc $AB$ , $Y$ is on minor arc $CD$ , and $Z$ is on minor arc $EF$ , where $X$ may coincide with $A$ or $B$ (and similarly for $Y$ and $Z$ ). Compute the square of the smallest possible area of $XYZ$ .\n\n*Proposed by Michael Ren*"}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "We successively write all the numbers from 1 to one million. Then, we replace each number by the sum of its digits. We repeat this process until only single-digit numbers remain. Which digit appears the most frequently?"}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Let \\(x\\), \\(y\\), \\(z\\), and \\(r\\) be real numbers. Find the minimum value of the following algebraic expressions:\n1. \\(x^{2}+y^{2}+xy+x+y\\)\n2. \\(x^{2}+y^{2}+z^{2}+xy+yz+zx+x+y+z\\)\n3. \\(x^{2}+y^{2}+z^{2}+r^{2}+xy+xz+xr+yz+yr+zr+x+y+z+r\\)"}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "## Task A-1.3.\n\nFor real numbers $x_{1}, x_{2}, \\ldots, x_{30}$, the following holds:\n\n$$\n\\begin{aligned}\n& 20^{3} x_{1}+21^{3} x_{2}+\\cdots+49^{3} x_{30}=13 \\\\\n& 21^{3} x_{1}+22^{3} x_{2}+\\cdots+50^{3} x_{30}=1 \\\\\n& 22^{3} x_{1}+23^{3} x_{2}+\\cdots+51^{3} x_{30}=19\n\\end{aligned}\n$$\n\nWhat is the value of $21 x_{1}+22 x_{2}+\\cdots+50 x_{30}$?"}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Let $a,$ $b,$ and $c$ be the roots of $x^3 - 7x^2 + 5x + 2 = 0.$ Find\n\\[\\frac{a}{bc + 1} + \\frac{b}{ac + 1} + \\frac{c}{ab + 1}.\\]"}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Determine the probability that two edges selected at random from the twelve edges of a cube with side length 1 are skew lines (i.e., non-intersecting and not in the same plane)."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Let $d_1$ and $d_2$ be parallel lines in the plane. We are marking $11$ black points on $d_1$, and $16$ white points on $d_2$. We are drawing the segments connecting black points with white points. What is the maximum number of points of intersection of these segments that lies between the parallel lines (excluding the intersection points on the lines)?"}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Determine the largest eigenvalue by absolute value of the matrix:\n[[4, 0, 6, 2],\n [0, -8, -1, 4],\n [0, 0, -8, -11],\n [0, 0, 0, 2]]"}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Let \\( f(x) \\) be a function defined on \\(\\mathbf{R}\\), and for any real number \\( x \\), \\( f(x+3) \\cdot f(x-4) = -1 \\). Also, for \\( 0 \\leq x < 7 \\), \\( f(x) = \\log _{2}(9-x) \\). Determine the value of \\( f(-100) \\)."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "All the numbers $2, 3, 4, 5, 6, 7$ are assigned to the six faces of a cube, one number to each face. For each of the eight vertices of the cube, a product of three numbers is computed, where the three numbers are the numbers assigned to the three faces that include that vertex. What is the greatest possible value of the sum of these eight products?"}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "For the inverse of 1/96 times the matrix below, what is the sum of all entries rounded to the nearest integer?\n[[6, 2, -10, 9],\n [0, -9, 5, 3],\n [0, 0, -5, 10],\n [0, 0, 0, -4]]"}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "6. (10 points) The students in the youth summer camp come from 4 municipalities, with $\\frac{1}{6}$ of the students from Shanghai, 24 students from Tianjin, the number of students from Beijing is $\\frac{3}{2}$ times the sum of the number of students from Shanghai and Tianjin, and $\\frac{1}{4}$ of the students are from Chongqing. How many students are there in the youth summer camp?"}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Calculate the rank of matrix A where A =\n[[-2, -5, 29, 17, 10],\n [1, -5, 18, 14, 5],\n [3, 6, -37, -21, -13],\n [-6, -3, 35, 15, 14],\n [-4, 11, -33, -29, -8]]"}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Victor has $3$ piles of $3$ cards each. He draws all of the cards, but cannot draw a card until all the cards above it have been drawn. For example, for his first card, Victor must draw the top card from one of the $3$ piles. In how many orders can Victor draw the cards?"}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "What is the greatest common factor of 288 and 69965376?"}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Each of two sides of a triangle is divided into six equal parts; the corresponding division points are connected by line segments. Find the sum of the lengths of these segments if the third side of the triangle is equal to 12."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Find the sum of absolute values of all entries of the product of the following matrices:\nA =\n[[7, 6, 7, -4, -6],\n [3, -8, 5, -7, 0],\n [-8, 5, -3, 8, -6]]\nB =\n[[1, -3, -2, -1],\n [8, 0, 8, 2],\n [6, 0, 4, 7],\n [-6, -2, -3, -5],\n [3, 7, -5, 5]]"}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Compute the rank of the given 5x5 matrix:\n[[36, -26, 19, -29, 5],\n [-28, 30, -8, 30, -13],\n [8, -38, -19, -23, 40],\n [-8, -4, -9, -19, -12],\n [8, -8, 1, 7, 20]]"}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Determine the number of 3-digit numbers in base 10 having at least one 5 and at most one 3."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "NI = 9. Let P be the reflection of point I across point N. Let O be the reflection of point P across point I. Let M be the midpoint of segment NO. Let B be the rotation of point M by angle 11pi/12 around point O. Line C passes through points B and O. Point H is the reflection of point N across line C. Point E is the reflection of point N across point I. What is the circumradius of triangle MEH? Express your answer as a decimal number rounded to 2 decimal places."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "7. Three parallel lines touch a sphere with center at point O at points $K$, $L$, and $M$. It is known that the radius of the sphere is 5, the area of triangle $O K L$ is 12, and the area of triangle $K L M$ is greater than 30. Find the angle $K M L$.\n\nAnswer. $\\angle K M L=\\arccos \\frac{3}{5}$."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Suppose -100*d - r = -99*d + 8, -5*d + 3*r - 8 = 0. Let b be (-5)/(2/2*4/d). Solve c + 5*l - 3 = 0, -4*t - 9 = -b*l, 4*t - 3*t + 2*c = -5 for t."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Let \\( ABC \\) be a triangle such that \\(\\angle CAB = 20^\\circ\\). Let \\( D \\) be the midpoint of segment \\([AB]\\). It is given that \\(\\angle CDB = 40^\\circ\\). What is the measure of \\(\\angle ABC\\)?"}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Which three-digit numbers are equal to the sum of the factorials of their digits?"}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "In trapezoid $ABCD$ with $\\overline{BC} \\parallel \\overline{AD}$, let $BC = 900$ and $AD = 1800$. Let $\\angle A = 45^\\circ$, $\\angle D = 45^\\circ$, and $M$ and $N$ be the midpoints of $\\overline{BC}$ and $\\overline{AD}$, respectively. Find the length $MN$."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Let c = -4/30843 + 13601771/61686. Find the value of n + m, where n/m is the second largest root of c*f - 15400/3*f**4 - 126 + 3213/2*f**2 - 4795/3*f**3 = 0 in its simplest form."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "What is the probability of such event happening: Form a word by randomly choosing 4 letters from the multiset {v: 3, u: 2, p: 3}, shuffle the letters in the word, what is the probability of no same letter remains in its original position?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "The product of three prime numbers. There is a number that is the product of three prime factors whose sum of squares is equal to 2331. There are 7560 numbers (including 1) less than this number and coprime with it. The sum of all the divisors of this number (including 1 and the number itself) is 10560. Find this number."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Chips, colored either black or white, are placed in the cells of a 4x4 grid such that: a) each cell contains at most one chip, b) all chips in the same row and all chips in the same column have the same colour, c) any additional chip placed on the grid would violate one or more of the previous two conditions. Furthermore, we have the following constraints (with the cells 0-indexed): cell (0, 3) is white, cell (1, 0) is white, cell (2, 1) is white, cell (3, 2) is black. How many chips are placed on the grid?"}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "The lines containing the altitudes of the scalene triangle \\( ABC \\) intersect at point \\( H \\). Let \\( I \\) be the incenter of triangle \\( ABC \\), and \\( O \\) be the circumcenter of triangle \\( BHC \\). It is known that point \\( I \\) lies on the segment \\( OA \\). Find the angle \\( BAC \\)."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "What is the sum of absolute values of all entries when matrix A is multiplied by matrix B?\nA =\n[[0, 5, -5, 5],\n [-6, 9, 8, -6],\n [5, -4, 6, 6],\n [1, 10, 2, -10]]\nB =\n[[6, 5, -5, -1, 5, -2],\n [2, -2, 0, -6, 10, 8],\n [0, -3, 7, -10, -1, -1],\n [0, 8, 4, 1, 10, -2]]"}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Problem 2. Along the path between the houses of Nезнайка (Nезнayka) and Синеглазка (Sineglazka), there were 15 peonies and 15 tulips growing in a row, mixed together.\n\nSetting out from home to visit Nезнайка, Синеглазка watered all the flowers in a row. After the 10th tulip, the water ran out, and 10 flowers remained unwatered.\n\nThe next day, setting out from home to visit Синеглазка, Незнайка picked flowers for her one by one. After picking the 6th tulip, he decided that it was enough for the bouquet. How many flowers remained growing along the path?"}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Let $x = (2 + \\sqrt{3})^{1000},$ let $n = \\lfloor x \\rfloor,$ and let $f = x - n.$ Find\n\\[x(1 - f).\\]"}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Square $ABCD$ has side length $13$, and points $E$ and $F$ are exterior to the square such that $BE=DF=5$ and $AE=CF=12$. Find $EF^{2}$."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "In a 4x4 grid, how many different paths are there from the bottom left (0, 0) to the top right (3, 3), if you can only move right or up at each step, subject to the constraint that you cannot move through the following cells: (1, 2), (1, 3), (1, 0), (2, 2)?"}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Determine the last two digits of the sum: $$3! + 6! + 9! + 12! + 15!$$"}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Consider a right cylinder with height $5\\sqrt{3}$. A plane intersects each of the bases of the cylinder at exactly one point, and the cylindric section (the intersection of the plane and the cylinder) forms an ellipse. Find the product of the sum and the difference of the lengths of the major and minor axes of this ellipse."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Among the first 10,000 positive integers \\(1, 2, \\ldots, 10000\\), how many of them have at least two of the digits 2, 0, 1, 7? (The same digit appearing multiple times, such as in the number 2222, counts as containing only one of that digit)."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "A basketball is dropped from 150 feet and rebounds two-fifths of the distance it falls each time it bounces. How many feet will the basketball have traveled when it hits the ground the sixth time?"}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "When randomly selecting 3 letters from the multiset {v: 1, a: 2, w: 1, c: 2} to form a word, what is the expected number of matches of the pattern 'v*a'? Round to 2 decimal places."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Circle $I$ passes through the center of, and is tangent to, circle $II$. The area of circle $I$ is $4$ square inches. Find the area of circle $II$. Express your answer in square inches."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "The terms of the sequence $(b_i)$ defined by $b_{n + 2} = \\frac {b_n + 4030} {1 + b_{n + 1}}$ for $n \\ge 1$ are positive integers. Find the minimum possible value of $b_1 + b_2$."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Express the repeating decimal $7.171717\\ldots$ as a fraction and compute the sum of its numerator and denominator in their lowest terms."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Let $ABCD$ be a tetrahedron with edges $AB$, $AC$, and $AD$ mutually perpendicular at $A$. Let the lengths of $AB=a$, $AC=b$, and $AD=c$. Given that the areas of triangles $ABC$, $ACD$, and $ADB$ are $s$, $t$, and $u$ respectively, find the area of triangle $BCD$."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Given the function\n$$\nf(x)=\\sin ^{4} \\frac{k x}{10}+\\cos ^{4} \\frac{k x}{10}\\left(k \\in \\mathbf{Z}_{+}\\right),\n$$\n\nif for any real number \\(a\\), \n$$\n\\{f(x) \\mid a<x<a+1\\}=\\{f(x) \\mid x \\in \\mathbf{R}\\},\n$$\n\nfind the smallest value of \\( k \\)."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "The positive four-digit integers that use each of the four digits $2,$ $3,$ $5,$ and $6$ exactly once are ordered from least to greatest. What is the $17^{\\text{th}}$ integer in the list?"}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "22. Let the function $f : \\mathbb{Z} \\to \\mathbb{Z}$ take only integer inputs and have integer outputs. For any integers $x$ and $y$ , f satisfies $$ f(x) + f(y) = f(x + 1) + f(y - 1) $$ If $f(2016) = 6102$ and $f(6102) = 2016$ , what is $f(1)$ ?\n\n23. Let $d$ be a randomly chosen divisor of $2016$ . Find the expected value of $$ \\frac{d^2}{d^2 + 2016} $$ 24. Consider an infinite grid of equilateral triangles. Each edge (that is, each side of a small triangle) is colored one of $N$ colors. The coloring is done in such a way that any path between any two nonadjecent vertices consists of edges with at least two different colors. What is the smallest possible value of $N$ ?"}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Each letter represents a non-zero digit. What is the value of $t?$ \\begin{align*}\nc + o &= u \\\\\nu + n &= t + 1 \\\\\nt + c &= s \\\\\no + n + s &= 15\n\\end{align*}"}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Define M to be the number of ordered triples of positive integers (a, b, c) where a, b, c <= 3^3 and 1a^3 + -1b^3 + 0c^3 = 0 (mod 3^2). Find the remainder of M when divided by 1000."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Let's consider a set of distinct positive integers with a sum equal to 2023. Among these integers, there are a total of $d$ even numbers and $m$ odd numbers. Determine the maximum possible value of $2d + 4m$."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Let $S=\\left\\{ \\frac{p}{q} \\mid q \\leq 2009, \\frac{p}{q} < \\frac{1257}{2009}, p,q \\in \\mathbb{N} \\right\\}$. If the maximum element of $S$ is $\\frac{p_0}{q_0}$ in reduced form, find $p_0+q_0$."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "How many two-digit numbers have digits whose sum is a perfect square?"}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "When randomly selecting 4 letters from the multiset {h: 4, y: 2, w: 2} to form a word, what is the expected number of matches of the pattern 'y+w'? Round to 2 decimal places."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "What is the probability of such event happening: Form a word by randomly choosing 2 letters from the multiset {a: 4, k: 1, l: 2}, shuffle the letters in the word, what is the probability of no letter 'l' occupy any of their original positions?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "In $\\triangle{ABC}, AB=10, \\angle{A}=30^\\circ$ , and $\\angle{C=45^\\circ}$. Let $H, D,$ and $M$ be points on the line $BC$ such that $AH\\perp{BC}$, $\\angle{BAD}=\\angle{CAD}$, and $BM=CM$. Point $N$ is the midpoint of the segment $HM$, and point $P$ is on ray $AD$ such that $PN\\perp{BC}$. Then $AP^2=\\dfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Let $z_1 = 18 + 83i$ , $z_2 = 18 + 39i,$ and $z_3 = 78 + 99i,$ where $i = \\sqrt{-1}$ . Let $z$ be the unique complex number with the properties that $\\frac{z_3 - z_1}{z_2 - z_1} \\cdot \\frac{z - z_2}{z - z_3}$ is a real number and the imaginary part of $z$ is the greatest possible. Find the real part of $z$ .\nPlease give the answer directly without any intermediate steps."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "A point $(x,y)$ is randomly picked from inside the rectangle with vertices $(0,0)$, $(4,0)$, $(4,3)$, and $(0,3)$. What is the probability that $x < y$?"}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Five students wrote a quiz with a maximum score of 50. The scores of four of the students were 42, 43, 46, and 49. The score of the fifth student was N. The average (mean) of the five students' scores was the same as the median of the five students' scores. Calculate the number of possible values of N that satisfy this condition."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Given the planar vectors $\\overrightarrow{a}$ and $\\overrightarrow{b}$ that satisfy $\\overrightarrow{a}(\\overrightarrow{a}+ \\overrightarrow{b})=5$, and $|\\overrightarrow{a}|=2$, $|\\overrightarrow{b}|=1$, find the angle between vectors $\\overrightarrow{a}$ and $\\overrightarrow{b}$."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Let $S$ be the set of all nonzero real numbers. Let $f : S \\to S$ be a function such that\n\\[f(x) + f(y) = f(xyf(x + y))\\]for all $x,$ $y \\in S$ such that $x + y \\neq 0.$\n\nLet $n$ be the number of possible values of $f(4),$ and let $s$ be the sum of all possible values of $f(4).$ Find $n \\times s.$The answer is in the form \frac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Find the greatest common divisor of 9486, 13524, and 36582."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "What is the probability of such event happening: Form a word by randomly choosing 3 letters from the multiset {a: 3, h: 1, o: 2}, shuffle the letters in the word, what is the probability of no letter 'o' occupy any of their original positions?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$."}], "membership_label": 0} |
| {"messages": [{"role": "user", "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 the hundreds digit, the second outcome is the tens digit and the third outcome is the units digit, what is the probability that you will end up with a three-digit number that is divisible by 4? Express your answer as a common fraction.\n\n[asy]\ndraw(Circle((0,0),10));\ndraw((0,0)--(8.7,-5));\ndraw((0,0)--(-8.7,-5));\ndraw((0,0)--(0,10));\nlabel(\"1\",(7,7),SW);\nlabel(\"3\",(-7,7),SE);\nlabel(\"2\",(0,-2),S);\ndraw((0,0)--(3,5),Arrow);\n[/asy]The answer is in the form \frac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Round to the nearest integer: what is the x-coordinate where the graphs of f(x) = -2|(2sin(1πx + 2) - 1) - 2| + 2 and g(x) = 3x + 1 meet in the interval [-10, 10]?"}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "A 6-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is red, vertex 1 is red, vertex 2 is blue, vertex 3 is blue, vertex 4 is red, vertex 5 is red. What is the maximum number of red vertices that can be made to occupy a position where there were originally blue vertices by rotating the 6-gon?"}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Suppose you started at the origin on the number line in a coin-flipping game. Every time you\nflip a heads, you move forward one step, otherwise you move back one step. However, there\nare walls at positions 8 and -8; if you are at these positions and your coin flip dictates that\nyou should move past them, instead you must stay. What is the expected number of coin\nflips needed to have visited both walls?"}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Let B be the circle with center K and radius 12. Let R be the circle with center K and radius 5. Point E is on circle R. Line A passes through points K and E. Point I is the reflection of point E across point K. Point M is obtained by rotating point I by angle pi/2 about point K. Let Q be the reflection of point M across point K. Point L is the inversion of point M with respect to circle B. Let V be the midpoint of segment LQ. Point O is the reflection of point L across line A. Find the distance between points O and V. Express your answer as a decimal number rounded to 2 decimal places."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "[Set](https://artofproblemsolving.com/wiki/index.php/Set) $A$ consists of $m$ consecutive integers whose sum is $2m$, and set $B$ consists of $2m$ consecutive integers whose sum is $m.$ The absolute value of the difference between the greatest element of $A$ and the greatest element of $B$ is $99$. Find $m.$"}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Alex, a 3D graphics designer, is working on a space-themed animation project. She is designing a scene that features a fleet of alien spaceships orbiting a distant planet. Each spaceship is made up of 3 spherical modules connected by cylindrical tubes. Alex wants to create a total of 8 spaceships for her scene. If each spherical module takes 2 hours to design and each cylindrical tube takes 1 hour to design, how many total hours will Alex spend designing all the modules and tubes for the 8 spaceships?"}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Suppose you have a 6-gon, with vertices numbered 1 through 6 in counterclockwise order. Draw the diagonal from vertex 4 to vertex 2, from vertex 6 to vertex 3, from vertex 2 to vertex 5, from vertex 3 to vertex 5, from vertex 1 to vertex 5, from vertex 6 to vertex 4, from vertex 4 to vertex 1, and from vertex 1 to vertex 3. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?"}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Define M to be the number of ordered triples of positive integers (a, b, c) where a, b, c <= 2^4 and 2a^2 + -1b^2 + -1c^2 = 0 (mod 2^4). Find the remainder of M when divided by 1000."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Define M to be the number of ordered triples of positive integers (a, b, c) where a, b, c <= 3^3 and 0a^3 + 1b^3 + 1c^3 = 0 (mod 3^3). Find the remainder of M when divided by 1000."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Find the number of times the graphs of f(x) = -5(-2sin(3π(3x^2 + 1x) - 2)) + 2 and g(x) = -3|(4x^2 + 1x + 1) - 2| intersect in the interval [-10, 10]."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "What is $\\frac{1}{6}$ of $\\frac{1}{4}$ of $\\frac{1}{5}$ of 120? Express the result as the number of halves."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "The function $f: \\mathbb{N}\\to\\mathbb{N}_{0}$ satisfies for all $m,n\\in\\mathbb{N}$ : \\[f(m+n)-f(m)-f(n)=0\\text{ or }1, \\; f(2)=0, \\; f(3)>0, \\; \\text{ and }f(9999)=3333.\\] Determine $f(1982)$ ."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "A prism has 36 edges. How many faces does the prism have?"}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "The cross-section of a prism with an index of refraction $1.5$ is an equilateral triangle, as shown in the diagram. A ray of light enters horizontally from the air into the prism and exits at an angle $\\theta$ with respect to the surface of the triangle. Find $\\theta$ in degrees and round to the nearest whole number."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Alice has two pennies, three nickels, four dimes, and two quarters in her purse. Calculate the percentage of one dollar that is in her purse."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "After walking \\( \\frac{2}{5} \\) of the length of a narrow bridge, a pedestrian noticed that a car was approaching the bridge from behind. He then turned back and met the car at the beginning of the bridge. If the pedestrian had continued walking forward, the car would have caught up with him at the end of the bridge. Find the ratio of the car's speed to the pedestrian's speed."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "If you form a word of length 3 by randomly selecting letters from the multiset {r: 2, m: 3, h: 1}, what is the expected number of times the pattern 'r.*h' appears? Round to 2 decimal places."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Calculate the highest common divisor of 41020 and 344400."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "We have 1985 sets. Each of the sets has 45 elements, the union of any two sets has exactly 89 elements. How many elements has the union of all these 1985 sets? Please provide the final integer value of the total number of elements in the union."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "What is the maximum number of sides of a convex polygon that has no two adjacent obtuse angles?"}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Let $f$ be an infinitely differentiable real-valued function defined on the real numbers. If\n$$\nf\\left(\\frac{1}{n}\\right)=\\frac{n^{2}}{n^{2}+1}, \\quad n=1,2,3, \\ldots\n$$\ncompute the values of the derivatives $f^{(k)}(0), k=1,2,3, \\ldots$."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Find all polynomials with integer coefficients $P$ such that for all integers $a$ and $b$, the following holds:\n\n$$\na + 2b \\mid P(a) + 2P(b)\n$$"}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "In the following diagram, \\( AB = 60 \\). Find \\( AX \\).\n\n[asy]\nimport markers;\n\nreal t=.67;\npair A=(0,0);\npair B=(3,-2);\npair C=(1.5,1.5);\npair X=t*A+(1-t)*B;\n\ndraw(C--A--B--C--X);\n\nlabel(\"$A$\",A,SW);\nlabel(\"$B$\",B,E);\nlabel(\"$C$\",C,N);\nlabel(\"$X$\",X,SW);\n\nmarkangle(n=1,radius=15,A,C,X,marker(markinterval(stickframe(n=1),true)));\nmarkangle(n=1,radius=15,X,C,B,marker(markinterval(stickframe(n=1),true)));\n\nlabel(\"$72$\",.5*(B+C),E);\nlabel(\"$36$\",.5*(A+C),NW);\n[/asy]"}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "What is the probability of such event happening: Form a word by randomly choosing 4 letters from the multiset {i: 1, w: 2, g: 2, p: 2}, shuffle the letters in the word, what is the probability of at least 1 letter 'g' remains in the same position?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "The lengths of the sides of the pentagon \\(ABCDE\\) are equal to 1. Let points \\(P, Q, R,\\) and \\(S\\) be the midpoints of the sides \\(AB, BC, CD,\\) and \\(DE\\) respectively, and let points \\(K\\) and \\(L\\) be the midpoints of segments \\(PR\\) and \\(QS\\) respectively. Find the length of segment \\(KL\\)."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "A trapezoid inscribed in a circle with a radius of $13 \\mathrm{~cm}$ has its diagonals located $5 \\mathrm{~cm}$ away from the center of the circle. What is the maximum possible area of the trapezoid?"}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 2|((-2(-2x - 1) + 3)/(2(-2x - 1) - 1)) - 3| - 2 is strictly decreasing."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "What is the probability of such event happening: Form a word by randomly choosing 4 letters from the multiset {c: 2, a: 6}, shuffle the letters in the word, what is the probability of no letter 'c' occupy any of their original positions?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Consider the polynomial \\(x^5 + bx^4 + cx^3 + dx^2 + ex + f = 0\\) with integer coefficients. Determine all possible values for \\(m\\), which represents the exact number of integer roots of the polynomial, counting multiplicity."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "What is the highest common factor of 14280 and 2107476?"}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "What is the second-largest prime factor of 17465607?"}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "An 8-sided die has numbers from 1 to 8 on its faces (each number appears on one face). How many ways can I paint two faces of the die red, such that the numbers on these two faces do not add up to a sum greater than 7?"}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Let m(x) be the second derivative of -35*x**5/4 + 305*x**4/32 + 15*x**3/4 - 135*x**2/16 - 14*x. For the polynomial m(q), find the second largest root in the form n/m where n and m are coprime integers. What is n + m?"}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Given Laura is adding two distinct three-digit positive integers, where all six digits in these numbers are different, and the sum, $S$, is a four-digit number, calculate the smallest possible value for the sum of the digits of $S$."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Given that the volume of the circumscribed sphere of the prism $ABCD-A_1B_1C_1D_1$ is $\\frac{4}{3}\\pi$, and the base $ABCD$ is a square with side length 1, calculate the lateral area of the prism $ABCD-A_1B_1C_1D_1$. Express your answer as a numerical value."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "2. A right cylinder with radius $r=\\sqrt{39}$ and a right cone with radius $R=9$ have the same height $H=15$. At what non-zero height $x$, should the cylinder and the cone be intersected so that the newly formed cylinder with height $x$ and the intersected cone with height $x$ have equal volumes?"}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "How many solutions does the equation (-3(2cos(3π(-3|(1x^2 - 3x - 2) - 2|) - 3) + 1) - 1)/(-1(2cos(3π(-3|(1x^2 - 3x - 2) - 2|) - 3) + 1) + 2) = 0 have in the range [-10, 10]?"}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "The radius $r$ of a circle with center at the origin is an odd integer. \nThere is a point ( $p^m, q^n$ ) on the circle, with $p,q$ prime numbers and $m,n$ positive integers. \nDetermine $r$ ."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "let N be the biggest 3-digit positive integer with the property that whenever two of its digits changes to 8, the resulting number is a multiple of 8. Let Q and R be the quotient and remainder, respectively, when N is divided by 100. Find Q + R."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Circle $\\omega_1$ has radius $7$ and center $C_1$. Circle $\\omega_2$ has radius $23$ and center $C_2$ with $C_1C_2 = 34$. Let a common internal tangent of $\\omega_1$ and $\\omega_2$ pass through $A_1$ on $\\omega_1$ and $A_2$ on $\\omega_2$, and let a common external tangent of $\\omega_1$ and $\\omega_2$ pass through $B_1$ on $\\omega_1$ and $B_2$ on $\\omega_2$ such that $A_1$ and $B_1$ lie on the same side of the line $C_1C_2$. Let $P$ be the intersection of lines $A_1A_2$ and $B_1B_2$. Find the area of quadrilateral $PC_1A_2C_2$."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Find the sum of all four-digit natural numbers composed of the digits 3, 6, and 9."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Find the x-coordinate of the rightmost local extremum of the function f(x) = 1sin(1π((-2(-4(-4x - 2)^2 - 4(-4x - 2) - 4) + 1)/(1(-4(-4x - 2)^2 - 4(-4x - 2) - 4))) + 2) in the interval [-10, 10]. Round to the nearest integer."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Each letter represents a non-zero digit. Determine the value of \\( t \\) given the following equations:\n\\begin{align*}\na + b &= x \\\\\nx + d &= t \\\\\nt + a &= y \\\\\nb + d + y &= 16\n\\end{align*}"}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "1. Determine all pairs $(p, q)$ of real numbers such that the equation $x^{2}+p x+q=0$ has a solution $v$ in the set of real numbers, and if $t$ is a root of this equation, then $|2 t-15|$ is also a root."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "For the matrix power A^2 where A =\n[[-3, -3, 1],\n [0, 3, -1],\n [0, 0, -1]], determine the sum of all entries."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Suppose you have a 12-gon, with vertices numbered 1 through 12 in counterclockwise order. Draw the diagonal from vertex 10 to vertex 4, from vertex 4 to vertex 7, from vertex 2 to vertex 6, from vertex 2 to vertex 8, from vertex 2 to vertex 11, from vertex 9 to vertex 4, from vertex 2 to vertex 4, from vertex 3 to vertex 7, and from vertex 11 to vertex 3. How many distinct pairs of diagonals meet at a perpendicular angle (within the polygon's interior or on its boundary)?"}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Calculate the rounded sum of all singular values from the SVD of:\n[[-4, 0, 4, -4],\n [1, 1, -3, 3],\n [-2, -3, -1, -1],\n [4, -3, -4, -1]]"}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "The length of a rectangular playground exceeds twice its width by 25 feet, and the perimeter of the playground is 650 feet. What is the area of the playground in square feet?"}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Two circles touch each other internally. It is known that two radii of the larger circle, which form an angle of $60^\\circ$ between them, are tangent to the smaller circle. Find the ratio of the radii of the circles."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Chips, colored either black or white, are placed in a 5x5 grid such that: a) each cell contains at most one chip, b) all chips in the same row and all chips in the same column have the same colour. Furthermore, we have the following constraints (with the cells 0-indexed): cell (4, 1) is black, cell (2, 3) is black, cell (3, 0) is black. What is the maximum number of white chips that can be placed on the grid?"}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "In each cell of a $5 \\times 5$ board, there is either an X or an O, and no three Xs are consecutive horizontally, vertically, or diagonally. What is the maximum number of Xs that can be on the board?"}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Let $k$ be a positive integer. Find the maximum value of \\[a^{3k-1}b+b^{3k-1}c+c^{3k-1}a+k^2a^kb^kc^k,\\] where $a$ , $b$ , $c$ are non-negative reals such that $a+b+c=3k$ ."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "An $n$-sided regular polygon with side length $1$ is rotated by $\\frac{180^\\circ}{n}$ about its center. The intersection points of the original polygon and the rotated polygon are the vertices of a $2n$-sided regular polygon with side length $\\frac{1-\\tan^2 10^\\circ}{2}$. What is the value of $n$?"}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Example 2 Let $F$ be the focus of the parabola $y^{2}=4 x$, and $A, B$ be two points on the parabola different from the origin $O$, satisfying $\\overrightarrow{F A} \\cdot \\overrightarrow{F B}=0$. Extend $A F, B F$ to intersect the parabola at points $C, D$ respectively. Find the minimum value of the area of quadrilateral $A B C D$."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "In the diagram, $BP$ and $BQ$ trisect $\\angle ABC$. $BM$ bisects $\\angle ABP$. Find the ratio of the measure of $\\angle MBQ$ to the measure of $\\angle ABQ$."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "What is the probability of such event happening: Form a word by randomly choosing 3 letters from the multiset {y: 2, a: 2, w: 1, t: 2}, shuffle the letters in the word, what is the probability of no letter 'y' occupy any of their original positions?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "What is the probability of such event happening: Form a word by randomly choosing 2 letters from the multiset {i: 1, c: 1, q: 1, e: 3}, shuffle the letters in the word, what is the probability of no letter 'c' occupy any of their original positions?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "What is the smallest number of tetrahedrons into which a cube can be partitioned?"}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "A 6-gon is colored so that in clockwise order, the vertices are colored as follows: vertex 0 is red, vertex 1 is red, vertex 2 is blue, vertex 3 is blue, vertex 4 is red, vertex 5 is red. What is the maximum number of blue vertices that can be made to occupy a position where there were originally red vertices by rotating the 6-gon?"}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Form a word by randomly choosing 3 letters from the multiset {e: 1, u: 1, s: 2, l: 2, h: 2}. What is the expected number of occurrences of the pattern 'hl+' in each word? Round your answer to 2 decimal places."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Assume that $x_1,x_2,\\ldots,x_7$ are real numbers such that \n\\[ \\begin{array}{r} x_1+4x_2+9x_3+16x_4+25x_5+36x_6+49x_7=1\\,\\,\\,\\,\\,\\,\\,\\, 4x_1+9x_2+16x_3+25x_4+36x_5+49x_6+64x_7=12\\,\\,\\,\\,\\, 9x_1+16x_2+25x_3+36x_4+49x_5+64x_6+81x_7=123. \\end{array} \\] Find the value of \\[16x_1+25x_2+36x_3+49x_4+64x_5+81x_6+100x_7.\\]"}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "A stationery store is running a promotion: each notebook has a sticker, and for every 5 stickers, a customer can receive another notebook (also with a sticker). Fifth-grader Katya believes she needs to buy as many notebooks as possible before the new semester. Each notebook costs 4 rubles, and Katya has 150 rubles. How many notebooks will Katya get?"}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Suppose $g(x) = x^2 + 2x + 1$. Calculate $g^{-1}(g^{-1}(10))$."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "There are 6 houses, numbered 1 to 6 from left to right, as seen from across the street. Each house is occupied by a different person. Each house has a unique attribute for each of the following characteristics:\n - Each person has a unique name: `Alice`, `Carol`, `Bob`, `Peter`, `Eric`, `Arnold`\n - They all have a unique favorite flower: `lilies`, `tulips`, `roses`, `daffodils`, `iris`, `carnations`\n - The people are of nationalities: `norwegian`, `dane`, `swede`, `german`, `brit`, `chinese`\n - Everyone has a favorite smoothie: `cherry`, `watermelon`, `desert`, `dragonfruit`, `blueberry`, `lime`\n - Each person has a unique hobby: `photography`, `cooking`, `knitting`, `woodworking`, `painting`, `gardening`\n - Each person has a unique type of pet: `rabbit`, `hamster`, `fish`, `cat`, `bird`, `dog`\n\n## Clues:\n1. The Swedish person is the person who loves cooking.\n2. The person who drinks Blueberry smoothies is Alice.\n3. The person who loves the rose bouquet is not in the sixth house.\n4. The photography enthusiast is somewhere to the right of the Dragonfruit smoothie lover.\n5. The person who loves cooking is the person who loves a bouquet of daffodils.\n6. The person who loves the boquet of lilies and the person who loves the boquet of iris are next to each other.\n7. The person who paints as a hobby is Alice.\n8. Arnold is somewhere to the right of the Watermelon smoothie lover.\n9. The woodworking hobbyist is somewhere to the right of the German.\n10. The woodworking hobbyist is somewhere to the left of the person who loves a bouquet of daffodils.\n11. The Dane is the Watermelon smoothie lover.\n12. The Dragonfruit smoothie lover is Peter.\n13. The person who drinks Lime smoothies is the woodworking hobbyist.\n14. There is one house between the Watermelon smoothie lover and the person who likes Cherry smoothies.\n15. The person who keeps a pet bird is the person who drinks Blueberry smoothies.\n16. The Desert smoothie lover is the person who enjoys gardening.\n17. The person with a pet hamster is in the fifth house.\n18. The person who owns a dog is the Norwegian.\n19. The person who has a cat is somewhere to the right of Bob.\n20. The British person is directly left of Carol.\n21. The person who loves the boquet of iris is Arnold.\n22. The German is somewhere to the right of the person who loves a carnations arrangement.\n23. Bob is the photography enthusiast.\n24. The person with an aquarium of fish is Arnold.\n What is the value of attribute Pet for the person whose attribute House is 4? Please reason step by step, and put your final answer within \\boxed{}"}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "In triangle $\\triangle ABC$, $∠B≠\\frac{π}{2}$, $cos2B=\\sqrt{3}cosB-1$. \n$(1)$ Find $\\angle B$; \n$(2)$ Choose one of the three conditions, condition 1, condition 2, or condition 3, as known to ensure the existence and uniqueness of $\\triangle ABC$, and find the area of $\\triangle ABC$. \nCondition 1: $sinA=\\sqrt{3}sinC$, $b=2$; \nCondition 2: $2b=3a$, $b\\sin A=1$; \nCondition 3: $AC=\\sqrt{6}$, the height on side $BC$ is $2$. \nNote: If the chosen condition does not meet the requirements, the score for the second question is 0; if multiple suitable conditions are chosen and answered separately, the score will be based on the first answer."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Find all rational roots of\n\\[4x^4 - 3x^3 - 13x^2 + 5x + 2 = 0\\]Enter all the rational roots, separated by commas."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "What is the greatest common factor of 416 and 110344?"}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "How many points of intersection exist between the functions f(x) = (1(1(-1|x|)^2 - 5(-1|x|) - 2) - 2)/(3(1(-1|x|)^2 - 5(-1|x|) - 2) + 3) and g(x) = -2|(-1x - 3) - 1| + 1 in the range -10 ≤ x ≤ 10?"}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "What is the maximum number of numbers that can be selected from the set \\( 1, 2, \\ldots, 1963 \\) such that the sum of no two numbers is divisible by their difference?"}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Determine the number of maximal connected intervals in [-10, 10] where the function f(x) = 1cos(1π(-2|(5((-2x + 3)/(2x - 1))^2 - 5((-2x + 3)/(2x - 1))) + 1| - 2) - 1) + 2 is strictly increasing."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "What is the probability of such event happening: Form a word by randomly choosing 2 letters from the multiset {s: 1, m: 4, z: 2}, shuffle the letters in the word, what is the probability of no letter 's' occupy any of their original positions?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Express the second largest root of 183*t + 3070*t**3 - 1760 - 14634*t**3 - 2688*t**4 + 3449*t + 8810*t**2 - 7014*t**4 = 0 as n/m where n and m are coprime and calculate n + m."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Let the set $I = \\{1,2,3,4,5\\}$. Choose two non-empty subsets $A$ and $B$ from $I$. How many different ways are there to choose $A$ and $B$ such that the smallest number in $B$ is greater than the largest number in $A$?"}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Asli will distribute $100$ candies among her brother and $18$ friends of his. Asli splits the friends of her brother into several groups and distributes all the candies into these groups. In each group, the candies are shared in a fair way such that each child in a group takes the same number of candies, and this number is the largest possible. Then, Asli's brother takes the remaining candies of each group. At most, how many candies can Asli's brother have?"}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "What is the probability of such event happening: Form a word by randomly choosing 4 letters from the multiset {h: 3, y: 5}, shuffle the letters in the word, what is the probability of no letter 'h' occupy any of their original positions?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "What is the modular inverse of $13$, modulo $2000$?\n\nExpress your answer as an integer from $0$ to $1999$, inclusive."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "If $5(2y + 3\\sqrt{3}) = S$, calculate the value of $10(4y + 6\\sqrt{3})$."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Determine the values of $n$ for which an $n\\times n$ square can be tiled with pieces of the type ."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Find the smallest positive integer \\( n \\) that satisfies the following condition: For any \\( n \\) points \\( A_1, A_2, \\dots, A_n \\) chosen on the circumference of a circle \\( O \\), among the \\(\\binom{n}{2}\\) angles \\(\\angle A_i O A_j \\) (where \\( 1 \\leq i < j \\leq n \\)), there are at least 2007 angles that do not exceed \\( 120^\\circ \\)."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "9. (5 points) Xiaohong brought 5 RMB notes of 50 yuan, 20 yuan, and 10 yuan denominations, 6 notes of 20 yuan, and 7 notes of 10 yuan. She bought a commodity worth 230 yuan. Therefore, there are $\\qquad$ ways to pay."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Let \\( \\triangle ABC \\) have centroid \\( S \\), midpoint of segment \\( AS \\) be \\( H \\), and midpoint of side \\( AB \\) be \\( Q \\). Let the line parallel to \\( BC \\) through \\( H \\) intersect \\( AB \\) at \\( P \\) and line \\( CQ \\) at \\( R \\). What is the ratio of the areas of triangles \\( PQR \\) and \\( APH \\)?"}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Example 7 Given $a=\\sqrt{5}-1$. Then the value of $2 a^{3}+7 a^{2}-2 a$ -12 is $\\qquad$ [7]\n(2010) \"Mathematics Weekly\" Cup National Junior High School Mathematics Competition)"}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Given an inscribed quadrilateral \\(ABCD\\). The extensions of its opposite sides intersect at points \\(P\\) and \\(Q\\). Let \\(K\\) and \\(N\\) be the midpoints of the diagonals. Prove that the sum of the angles \\(PKQ\\) and \\(PNQ\\) is \\(180^\\circ\\)."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Let o(a) = -56*a**5 - 44*a**4 + 18*a**3 + 30*a**2 - 5*a. Let h(p) = 168*p**5 + 133*p**4 - 53*p**3 - 89*p**2 + 16*p. Let z(f) = 3*h(f) + 8*o(f). Let j(y) be the second derivative of z(y). Suppose j(x) = 0. Express the second largest root as n/m where n and m are coprime and return n + m."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "In a 5x4 grid, how many different paths are there from the bottom left (0, 0) to the top right (4, 3), provided you can move right or up by any number of cells (like a chess rook) at each step, subject to the constraint that you cannot move through the following cells: (1, 0), (3, 2), (4, 0), (2, 2)? If it's impossible, answer with -1."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "What is the value of (26417 - 27522)/((-3)/((-18)/(-39)))? Express your answer as an integer or in the form p/q where p and q are coprime."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Suppose you have a 11-gon, with vertices numbered 1 through 11 in counterclockwise order. Draw the diagonal from vertex 10 to vertex 1 and from vertex 4 to vertex 6. Then, rotate the entire setup, including the constructed diagonals, 4 vertices counterclockwise (so that vertex 1 ends up where vertex 5 was), and superimpose it on the original (so that the resulting diagram contains both the original diagonals and the rotated versions of the diagonals). The original 11-gon will be partitioned into a collection of smaller polygons. (Only count the \"smallest\" polygons, i.e, those that do not have any of the diagonals running through them; this is not a combinatorics problem.) How many such polygons will there be with 8 sides?"}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "\nProblem 5. Ann and Max play a game on a $100 \\times 100$ board.\n\nFirst, Ann writes an integer from 1 to 10000 in each square of the board so that each number is used exactly once.\n\nThen Max chooses a square in the leftmost column and places a token on this square. He makes a number of moves in order to reach the rightmost column. In each move the token is moved to a square adjacent by side or by vertex. For each visited square (including the starting one) Max pays Ann the number of coins equal to the number written in that square.\n\nMax wants to pay as little as possible, whereas Ann wants to write the numbers in such a way to maximise the amount she will receive. How much money will Max pay Ann if both players follow their best strategies?\n\n(Lev Shabanov)\n\nAnswer: 500000 coins.\n"}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "For the matrix power A^4 where A =\n[[2, 2, -1],\n [2, 3, -2],\n [-1, -2, 1]], determine the value at position (2, 2)."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Starting with the number 150, Emily repeatedly divides her number by two and then takes the greatest integer less than or equal to that number. How many times must she do this before she reaches the number 1?"}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Rose fills each of the rectangular regions of her rectangular flower bed with a different type of flower. The lengths, in feet, of the rectangular regions in her flower bed are as shown in the figure. She plants one flower per square foot in each region. Asters cost $\\$1$ each, begonias $\\$1.50$ each, cannas $\\$2$ each, dahlias $\\$2.50$ each, and Easter lilies $\\$3$ each. What is the least possible cost, in dollars, for her garden? [asy]\ndraw((0,0)--(11,0)--(11,6)--(0,6)--cycle,linewidth(0.7));\ndraw((0,1)--(6,1),linewidth(0.7));\ndraw((4,1)--(4,6),linewidth(0.7));\ndraw((6,0)--(6,3),linewidth(0.7));\ndraw((4,3)--(11,3),linewidth(0.7));\nlabel(\"6\",(3,0),S);\nlabel(\"5\",(8.5,0),S);\nlabel(\"1\",(0,0.5),W);\nlabel(\"5\",(0,3.5),W);\nlabel(\"4\",(2,6),N);\nlabel(\"7\",(7.5,6),N);\nlabel(\"3\",(11,1.5),E);\nlabel(\"3\",(11,4.5),E);\n[/asy]"}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "10. (14 points) In $\\triangle A B C$, $B C=a, C A=$ $b$, a square $A B E F$ is constructed outward with side $A B$, $O$ is the center of the square $A B E F$, $M$ and $N$ are the midpoints of sides $B C$ and $C A$, respectively. When $\\angle B C A$ varies, find the maximum value of $O M+O N$.\n\n---\n\nPlease note that the translation preserves the original text's formatting and structure."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "12.444 The base of the pyramid is an equilateral triangle. One of the lateral faces of the pyramid is perpendicular to the plane of the base. Find the cosine of the angle between the other two lateral faces, if they form an angle \\(\\alpha\\) with the plane of the base."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "There are 5 people standing in a line numbered 1 through 5 in a left to right order.\nEach person has the following attributes: Beverage, Food, Movie-Genre, Music-Genre, Transport.\nThe attributes have the following possible values:\n- Beverage: coffee, lemonade, soy-milk, sprite, tea\n- Food: avocado, cauliflower, orange, peach, peas\n- Movie-Genre: disaster, fantasy, horror, satire, sports\n- Music-Genre: d&b, disco, dubstep, jazz, punk\n- Transport: airplane, bus, scooter, skateboard, van\n\nGiven the following premises about the line of people:\n- Movie-Genre:satire is somewhere to the right of Movie-Genre:sports\n- Food:peas is on the left or right of Music-Genre:d&b\n- Movie-Genre:fantasy is on the left of Beverage:coffee\n- Music-Genre:dubstep is on the right of Food:peach\n- Movie-Genre:horror is on the far left\n- Transport:airplane is on the right of Music-Genre:jazz\n- Transport:van is somewhere to the left of Beverage:lemonade\n- Food:orange is on the far left\n- Food:cauliflower is on the left or right of Beverage:soy-milk\n- Transport:scooter is on the left or right of Music-Genre:punk\n- Music-Genre:dubstep == Transport:airplane\n- Transport:bus is on the far left or far right\n- Music-Genre:d&b is in an odd position\n- Transport:skateboard is in an odd position\n- Beverage:sprite is on the far right\n- Beverage:coffee is on the left of Movie-Genre:disaster\n- Music-Genre:punk is in an odd position\n- Transport:scooter is on the left or right of Transport:airplane\n- Movie-Genre:fantasy is in an odd position\n\nAnswer the following question:\n\nQuestion: Question: What is the Food of the person who has/is horror??\n Please provide your answer in the format: \\boxed{X}, where X is the answer."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Given sets $A=\\{2,3,4\\}$ and $B=\\{a+2,a\\}$, if $A \\cap B = B$, find $A^cB$ ___."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "What is the probability of such event happening: Form a word by randomly choosing 3 letters from the multiset {u: 4, m: 3}, shuffle the letters in the word, what is the probability of no letter 'u' occupy any of their original positions?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "A binary operation $\\diamondsuit$ has the properties that $a\\,\\diamondsuit\\, (b\\,\\diamondsuit \\,c) = (a\\,\\diamondsuit \\,b)\\cdot c$ and that $a\\,\\diamondsuit \\,a=1$ for all nonzero real numbers $a, b,$ and $c$. (Here $\\cdot$ represents multiplication). The solution to the equation $2016 \\,\\diamondsuit\\, (6\\,\\diamondsuit\\, x)=100$ can be written as $\\tfrac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find the value of $p+q.$"}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "In a triangle with sides \\(a\\), \\(b\\), and \\(c\\), angle bisectors are drawn. The points where these bisectors intersect the opposite sides become the vertices of a second triangle. Prove that the ratio of the areas of these triangles is equal to \\(\\frac{2 a b c}{(a+b)(a+c)(b+c)}\\)."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Let F be the circle with center O and radius 9. Point Z lies on circle F. Let I be the rotation of point Z by angle 11pi/12 around point O. Let Y be the midpoint of segment IO. Let C be the midpoint of segment YI. Let L be the line through points Z and I. Point P is the reflection of point C across line L. Circle D passes through points O, C, and P. Find the radius of circle D. Express your answer as a decimal number rounded to 2 decimal places."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "$10 \\cdot 44$ What is the maximum difference between a three-digit number and the sum of the cubes of its digits? What kind of three-digit number can achieve this maximum difference? What is the smallest positive difference?\n\n(10th All-Russian Mathematical Olympiad, 1984)"}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "29.34. Calculate the length of the arc of the parabola $2 y=x^{2}$ from the point $(0,0)$ to the point $\\left(x_{0}, \\frac{x_{0}^{2}}{2}\\right)$."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "5. At $17-00$ the speed of the racing car was 30 km/h. Every 5 minutes thereafter, the speed increased by 6 km/h. Determine the distance traveled by the car from $17-00$ to $20-00$ of the same day."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Amir is 8 kg heavier than Ilnur, and Daniyar is 4 kg heavier than Bulat. The sum of the weights of the heaviest and lightest boys is 2 kg less than the sum of the weights of the other two boys. All four boys together weigh 250 kg. How many kilograms does Amir weigh?"}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Given that the scores of the test are 65, 70, 75, 85, and 95, and the class average is always an integer after each score is entered, determine the last score Mrs. Johnson entered."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Let $\\sigma_k(n)$ be the sum of the $k^{th}$ powers of the divisors of $n$. For all $k \\ge 2$ and all $n \\ge 3$, we have that \\[\\frac{\\sigma_k(n)}{n^{k+2}} (2020n + 2019)^2 > m.\\] Find the largest possible value of $m$."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "What is the probability of such event happening: Form a word by randomly choosing 3 letters from the multiset {g: 2, t: 2, u: 1, s: 2}, shuffle the letters in the word, what is the probability of no letter 'g' occupy any of their original positions?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "How many consecutive \"0\"s are there at the end of the product \\(5 \\times 10 \\times 15 \\times 20 \\times \\cdots \\times 2010 \\times 2015\\)?"}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Given the function f(x) = x(x+k)(x+2k)(x-3k), and f'(0) = 6, calculate the value of k. Express your answer as a single integer."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "What is the probability of such event happening: Form a word by randomly choosing 3 letters from the multiset {u: 2, n: 1, s: 2}, shuffle the letters in the word, what is the probability of exact 1 letter 's' remains in the same position?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Given the equations 60^a = 3 and 60^b = 5, calculate the value of 12^((1-a-b)/2(1-b)). Express your answer as a single number."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Express the second largest root of 486/11 - 513/11*j - 3195/11*j**2 - 150*j**3 = 0 as n/m where n and m are coprime and calculate n + m."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "A particle is located on the coordinate plane at $ (5,0)$ . Define a *move* for the particle as a counterclockwise rotation of $ \\pi/4$ radians about the origin followed by a translation of $ 10$ units in the positive $ x$ -direction. Given that the particle's position after $ 150$ moves is $ (p,q)$ , find the greatest integer less than or equal to $ |p|\\plus{}|q|$ ."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Someones age is equal to the sum of the digits of his year of birth. How old is he and when was he born, if it is known that he is older than $11$ .\nP.s. the current year in the problem is $2010$ ."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Let $x_{k} \\in[-2,2](k=1,2, \\cdots, 2013)$,\nand\n$x_{1}+x_{2}+\\cdots+x_{2013}=0$. Try to find\n$$\nM=x_{1}^{3}+x_{2}^{3}+\\cdots+x_{2013}^{3}\n$$\n\nthe maximum value.\n(Liu Kangning)"}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Define M to be the number of ordered triples of positive integers (a, b, c) where a, b, c <= 3^2 and 2a^3 + 1b^3 + -2c^3 = 0 (mod 3^3). Find the remainder of M when divided by 1000."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "As shown in the figure, square \\(ABCD\\) and rectangle \\(BEFG\\) share a common vertex \\(B\\). The side length of square \\(ABCD\\) is \\(6 \\, \\text{cm}\\) and the length of rectangle \\(BEFG\\) is \\(9 \\, \\text{cm}\\). Find the width of the rectangle in centimeters."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Let $a,$ $b,$ $c,$ $d$ be real numbers, none of which are equal to $-1,$ and let $\\omega$ be a complex number such that $\\omega^3 = 1$ and $\\omega \\neq 1.$ If\n\\[\n\\frac{1}{a + \\omega} + \\frac{1}{b + \\omega} + \\frac{1}{c + \\omega} + \\frac{1}{d + \\omega} = \\frac{3}{\\omega},\n\\]\nthen find\n\\[\n\\frac{1}{a + 1} + \\frac{1}{b + 1} + \\frac{1}{c +1} + \\frac{1}{d + 1}.\n\\]"}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Find the sum of all entries of the inverse of matrix 1/96 * A where A =\n[[-10, 5, -8, -8],\n [0, -10, 8, 5],\n [0, 0, 7, 12],\n [0, 0, 0, -12]]\nRound your answer to the nearest integer."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "For any positive integer \\( n \\), let \\( a_{n} \\) denote the number of triangles with integer side lengths whose longest side is \\( 2n \\).\n\n(1) Find an expression for \\( a_{n} \\) in terms of \\( n \\);\n\n(2) Given the sequence \\( \\{b_{n}\\} \\) satisfies\n$$\n\\sum_{k=1}^{n}(-1)^{n-k} \\binom{n}{k} b_{k}=a_{n}\\quad (n \\in \\mathbf{Z}_{+}),\n$$\nfind the number of positive integers \\( n \\) such that \\( b_{n} \\leq 2019 a_{n} \\)."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Given square ABCD has side length s, calculate the ratio of the area of square EFGH to the area of square ABCD, where E, F, G, and H are the centers respectively of isosceles right-angled triangles with bases AB, BC, CD, and DA, each exterior to the square with the right angle at the base vertex."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "If an odd square not divisible by 9 is decreased by one, the result is divisible by 24."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Find the rank of the matrix:\n[[60, 66, -56, 18, 20],\n [-36, -9, 0, -18, -72],\n [15, -6, 9, -3, 15],\n [-40, -21, 12, -18, -60],\n [25, 45, -41, 15, 5]]"}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Find the smallest, positive five-digit multiple of $18.$"}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Find the number of all integer solutions of the inequality \\( \\sqrt{1-\\sin \\frac{\\pi x}{4}-3 \\cos \\frac{\\pi x}{2}}-\\sqrt{6} \\cdot \\sin \\frac{\\pi x}{4} \\geq 0 \\) that belong to the interval [1991; 2013]."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Determine ∫(-1((3x - 3)/(2x - 1))^2 - 4((3x - 3)/(2x - 1)) + 4) dx. Do not include the constant of integration C."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "What is the probability of such event happening: Form a word by randomly choosing 4 letters from the multiset {w: 1, v: 2, t: 1, n: 4}, shuffle the letters in the word, what is the probability of no letter 'n' occupy any of their original positions?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Let the following system of equations hold for positive numbers \\(x, y, z\\):\n\n\\[ \\left\\{\\begin{array}{l}\nx^{2}+x y+y^{2}=48 \\\\\ny^{2}+y z+z^{2}=25 \\\\\nz^{2}+x z+x^{2}=73\n\\end{array}\\right. \\]\n\nFind the value of the expression \\(x y + y z + x z\\)."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Let the set $\\mathcal{S} = \\{8, 5, 1, 13, 34, 3, 21, 2\\}.$ Susan makes a list as follows: for each two-element subset of $\\mathcal{S},$ she writes on her list the greater of the set's two elements. Find the sum of the numbers on the list."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "ABCDEGHIJ is a regular 9-gon with center F and circumradius 10. Let T be the reflection of point B across point I. Line L is the angle bisector of angle CGD. Let N be the reflection of point F across point T. Point Z is the reflection of point N across line L. Find the distance between points Z and G. Express your answer as a decimal number rounded to 2 decimal places."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "The Planar National Park is an undirected 3-regular planar graph (i.e., all vertices have degree 3). A visitor walks through the park as follows: she begins at a vertex and starts walking along an edge. When she reaches the other endpoint, she turns left. On the next vertex, she turns right, and so on, alternating left and right turns at each vertex. She does this until she gets back to the vertex where she started. What is the largest possible number of times she could have entered any vertex during her walk, over all possible layouts of the park?"}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "1. Solve the inequality\n\n$$\n\\frac{4+\\sqrt{-3-x}}{3-|x+4|} \\leqslant 1\n$$\n\nIn your answer, specify the sum of all integer values of $x$ that satisfy the given inequality and do not exceed 12 in absolute value."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Find the rank of the matrix:\n[[-6, 0, 1, -11, -26],\n [0, -4, 2, 8, 20],\n [0, -8, 4, 16, 40],\n [42, 30, -22, 17, 32],\n [18, 16, -11, 1, -2]]"}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Compute without using a calculator: $9!/8!$"}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "In the trapezoid \\(ABCD (BC \\parallel AD)\\), the diagonals intersect at point \\(M\\). Given \\(BC = b\\) and \\(AD = a\\), find the ratio of the area of triangle \\(ABM\\) to the area of trapezoid \\(ABCD\\)."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "设实数 $k, l, m$ 满足：函数 $y=(x+1)\\left(x^{2}+k x+l\\right)$ 的图像有对称中心 $(1,0)$ ，且与函数 $y=x^{3}+m$ 的图像有公共点。已知 $k+l+m$ 的取值范围为一个区间。请给出区间上限的分子与分母之和。"}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "In a given regular $n$-gon with $n \\geq 10$, select $k$ points from the $n$ vertices such that there exist 4 points among the $k$ points which form the vertices of a quadrilateral. This quadrilateral must have three of its sides as the sides of the given regular $n$-gon. Find the minimum value of $k$."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Calculate the inverse matrix of 1/96 times the following and determine the sum of all entries of the resulting matrix. Your answer should be rounded to the nearest integer:\n[[-6, -11, 8, -8],\n [0, 6, -7, -10],\n [0, 0, 2, -5],\n [0, 0, 0, 6]]"}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Find the second-largest prime factor of 65860."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "10.64 On the plane, what is the minimum number of points needed so that the distances between each pair of points can take on each of the values $1,2,4,8,16,32,64$?"}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Determine the largest eigenvalue by absolute value of the matrix:\n[[-3, -12, -10, -4],\n [-12, -9, -6, 5],\n [-10, -6, -2, -6],\n [-4, 5, -6, -10]]"}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Define the function $\\xi : \\mathbb Z^2 \\to \\mathbb Z$ by $\\xi(n,k) = 1$ when $n \\le k$ and $\\xi(n,k) = -1$ when $n > k$ , and construct the polynomial \\[ P(x_1, \\dots, x_{1000}) = \\prod_{n=1}^{1000} \\left( \\sum_{k=1}^{1000} \\xi(n,k)x_k \\right). \\]\n(a) Determine the coefficient of $x_1x_2 \\dots x_{1000}$ in $P$ .\n(b) Show that if $x_1, x_2, \\dots, x_{1000} \\in \\left\\{ -1,1 \\right\\}$ then $P(x_1,x_2,\\dots,x_{1000}) = 0$ .\n\n*Proposed by Evan Chen*"}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "The side length of square $C$ is 24 cm. The side length of square $D$ is 54 cm. What is the ratio of the area of square $C$ to the area of square $D$? Express your answer as a common fraction."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "For the equation in terms of $x$, $x^{2} + a|x| + a^{2} - 3 = 0 \\ (a \\in \\mathbf{R})$ to have a unique real solution, find the value of $a$."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Let $A B C$ be a triangle with $A B=13, B C=14, C A=15$. The altitude from $A$ intersects $B C$ at $D$. Let $\\omega_{1}$ and $\\omega_{2}$ be the incircles of $A B D$ and $A C D$, and let the common external tangent of $\\omega_{1}$ and $\\omega_{2}$ (other than $B C$) intersect $A D$ at $E$. Compute the length of $A E$."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "How many points of intersection exist between the functions f(x) = -4((1(-1|(-4x + 5)| - 1) - 3)/(3(-1|(-4x + 5)| - 1) - 2))^2 and g(x) = -2(3((-3x + 2)/(2x - 1)) + 2)^2 - 3(3((-3x + 2)/(2x - 1)) + 2) + 1 in the range -10 ≤ x ≤ 10?"}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Find the sum of all positive rational numbers that are less than $10$ and that have denominator $30$ when written in lowest terms ."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "What is the probability of such event happening: Form a word by randomly choosing 2 letters from the multiset {t: 3, f: 3, e: 1}, shuffle the letters in the word, what is the probability of no letter 'e' occupy any of their original positions?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Ivan and Mirka were sharing pears from a plate. Ivan always took two pears, and Mirka took half of what was left on the plate. They proceeded in this manner: Ivan, Mirka, Ivan, Mirka, and finally Ivan, who took the last two pears.\n\nDetermine who ended up with more pears and by how many.\n\n(M. Dillingerová)\n\nHint. How many pears did Mirka take the second time?"}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Define M to be the number of ordered triples of positive integers (a, b, c) where a, b, c <= 2^3 and -1a^2 + -1b^2 + 0c^2 = 0 (mod 2^2). Find the remainder of M when divided by 1000."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "$\\begin{array}{l}3 \\cdot 7 \\text { if } 0<x<1 \\text {, simplify: } \\\\ \\left(\\frac{\\sqrt{1+x}}{\\sqrt{1+x}-\\sqrt{1-x}}+\\frac{1-x}{\\sqrt{1-x^{2}}+x-1}\\right) \\times\\left(\\sqrt{\\frac{1}{x^{2}}-1}-\\frac{1}{x}\\right) .\\end{array}$"}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "For how many numbers $n$ does $2017$ divided by $n$ have a remainder of either $1$ or $2$?"}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "What is the remainder when $2003 \\cdot 2004 \\cdot 2005 \\cdot 2006 \\cdot 2007$ is divided by 17?"}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "The sequences of positive integers $1,a_2, a_3, \\dots$ and $1,b_2, b_3, \\dots$ are an increasing arithmetic sequence and an increasing geometric sequence, respectively. Let $c_n=a_n+b_n$. There is an integer $k$ such that $c_{k-1}=100$ and $c_{k+1}=1000$. Find $c_k$."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "What is the probability of such event happening: Form a word by randomly choosing 3 letters from the multiset {x: 4, u: 2, h: 1}, shuffle the letters in the word, what is the probability of no letter 'x' occupy any of their original positions?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Given that $g(x)=1-x^2$ and $f(g(x))=\\frac{1-x^2}{x^2}$ when $x\\not=0$, find $f(1/3)$."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Pedro wants to paint a cube-shaped box in such a way that faces sharing a common edge are painted in different colors. Calculate the minimum number of colors needed to paint the box in this manner."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Let $\\mathcal{F}$ be the set of all the functions $f : \\mathcal{P}(S) \\longrightarrow \\mathbb{R}$ such that for all $X, Y \\subseteq S$, we have $f(X \\cap Y) = \\min (f(X), f(Y))$, where $S$ is a finite set (and $\\mathcal{P}(S)$ is the set of its subsets). Find\n\\[\\max_{f \\in \\mathcal{F}}| \\textrm{Im}(f) |. \\]"}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "A set of sample data $11$, $13$, $15$, $a$, $19$ has an average of $15$. Calculate the standard deviation of this data set."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "How many points of intersection exist between the functions f(x) = -3sin(2π(-1|((-1(-3x) - 1)/(-1(-3x) + 2)) + 3|) - 2) + 2 and g(x) = 3sin(1π(5((-3x + 2)/(-1x + 3))^2 + 4((-3x + 2)/(-1x + 3)) + 3) - 2) + 3 in the range -10 ≤ x ≤ 10?"}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Let $x, y, z \\in \\mathbb{R}$ and the equation $7x^2 + 7y^2 + 7z^2 + 9xyz = 12$ holds. Find the minimum value of $x^2 + y^2 + z^2$, which can be expressed as $\\frac{a}{b}$, where $a, b \\in \\mathbb{Z}$ and $\\gcd(a, b) = 1$. Determine the value of $a + b$."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Calculate the sum of all integer solutions to $4 < (x-3)^2 < 36$."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "ABCDEFGHIJKL is a regular dodecagon with center S and circumradius 12. Let P be the line through points C and F. Point N is the reflection of point H across point J. Line U is the perpendicular bisector of segment HE. Let Y be the reflection of line P across line U. Point R is the reflection of point F across line Y. Line V passes through point N perpendicular to line P. Let Q be the line through point S parallel to line V. Point T is the intersection of lines Q and P. Point M is the midpoint of segment TC. Find the distance between points M and R. Express your answer as a decimal number rounded to 2 decimal places."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Given the function f(x) = x^2 - x + sqrt(2x^4 - 6x^2 + 8x + 16), find the minimum value of f(x). Express your answer as a single number."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Find the Singular Value Decomposition (SVD) of the matrix and determine the rounded sum of all singular values:\n[[-5, -1, -2, -1, 0],\n [-3, -6, -3, 6, -3],\n [6, -5, -1, -5, 6],\n [-1, -5, 2, 6, 3],\n [-3, 0, 4, 4, 3]]"}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "What is the probability of such event happening: Form a word by randomly choosing 3 letters from the multiset {f: 1, i: 1, g: 1, q: 2}, shuffle the letters in the word, what is the probability of no letter 'i' occupy any of their original positions?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Determine the largest constant \\( C \\) such that for all real numbers \\( x_1, x_2, \\ldots, x_6 \\), the inequality\n\n\\[ \n\\left(x_1 + x_2 + \\cdots + x_6\\right)^2 \\geq C \\cdot \\left(x_1 \\left(x_2 + x_3\\right) + x_2 \\left(x_3 + x_4\\right) + \\cdots + x_6 \\left(x_1 + x_2\\right) \\right)\n\\]\n\nholds.\n\nAlso, find for this \\( C \\) all \\( x_1, x_2, \\ldots, x_6 \\) for which equality holds."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "An equilateral triangle and a rectangle both have perimeters of 60 inches. The rectangle's length is twice its width. What is the ratio of the length of the side of the triangle to the width of the rectangle? Express your answer as a common fraction."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Determine the number of intersection points between the functions f(x) = (-2(-2x - 2) - 3)/(-2(-2x - 2) + 3) and g(x) = 1x - 2 where -10 ≤ x ≤ 10."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "What is the probability of such event happening: Form a word by randomly choosing 2 letters from the multiset {c: 2, r: 2, z: 1}, shuffle the letters in the word, what is the probability of no letter 'r' occupy any of their original positions?\nIf the probability can be written as the form $\\frac{m}{n}$, where $m$ and $n$ are relatively prime integers, find $m + n$."}], "membership_label": 0} |
| {"messages": [{"role": "user", "content": "Let l be (9 - 18/4)*(-3)/108*-16. Solve -4*b - 11 = -8*b + 3*s - l*y, -4*b - 4*s - 14 = -2*y, 2*b - 12 = 4*s for b."}], "membership_label": 0} |
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