conversations listlengths 3 3 |
|---|
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A quadrilateral has distinct integer side lengths. If the second-largest side has a length of 10, what is the maximum possible length of the largest side?"
},
{
"fr... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A quadrilateral has perpendicular diagonals and three of its sides have lengths of 2, 3, and 4. There are two possible lengths for the fourth side: \\( \\sqrt{m} \\) or \\(... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A quadrilateral has side lengths 3, 3, \\(x\\), and \\(y\\). The values of \\(x\\) and \\(y\\) are integers. After \\(x\\) is chosen, let \\(N\\) be the number of possible ... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A quadrilateral is inscribed in a circle. If an angle is inscribed into each of the four segments outside the quadrilateral, what is the sum of these four angles, expressed... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A quadrilateral with sides 15, 15, 15, and 20 is inscribed in a circle which is tangential to a square. Find the area of the square. Do not use Brahmagupta's formula."
},... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A racetrack is divided into two sections of equal length. Farley can drive the first section at 18 miles per hour and the second section at a faster \\( k \\) miles per hou... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A random number selector can only select one of the nine integers 1, 2,..., 9, and it makes these selections with equal probability. Determine the probability that after $n... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A ray is drawn from the origin tangent to the graph of the upper part of the hyperbola $y^2 = x^2 - x + 1$ in the first quadrant. This ray makes an angle of $\\theta$ with ... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A ray of light passing through the point $(1,2)$ reflects on the $x$-axis at point $A$ and the reflected ray passes through the point $(5,3)$. Find the coordinates of $A$."... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A real $2024 \\times 2024$ matrix $A$ is called nice if $(Av, v) = 1$ for every vector $v \\in \\mathbb{R}^{2024}$ with unit norm. Find an example of a nice non-identity ma... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A real function \\( f \\) is always positive and satisfies \\( \\frac{f(x)}{f(y)} \\leq 2^{(x-y)^2} \\) for all \\( x, y \\in \\mathbb{D} \\), where \\( \\mathbb{D} \\) is ... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A real number $x$ is chosen randomly from the interval $(0,1)$. What is the probability that $\\lfloor \\log_5(3x) \\rfloor = \\lfloor \\log_5(x) \\rfloor$?"
},
{
"... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A rectangle \\(ABCD\\) with \\(AB = 12\\) and \\(BC = 16\\) is the base of a pyramid \\(P\\) with height 24. A plane parallel to \\(ABCD\\) divides \\(P\\) into a frustum \... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A rectangle is split into 9 squares. If the side lengths of the rectangle are relatively prime, what is the perimeter of the rectangle?"
},
{
"from": "gpt",
"va... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A rectangle measures 6 meters by 10 meters. Drawn on each side of the rectangle is a semicircle that has the endpoints of its diameter on the vertices of the rectangle. Wha... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A rectangular box \\( P \\) is inscribed in a sphere of radius \\( r \\). The surface area of \\( P \\) is 384, and the sum of the lengths of its 12 edges is 112. What is \... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A rectangular box has a volume of $64$. Find the sum of the dimensions of the box with the minimum surface area."
},
{
"from": "gpt",
"value": "<think>\n<think>... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A rectangular box has a volume of 108 cubic feet. How many cubic yards are in the volume of this box?"
},
{
"from": "gpt",
"value": "<think>\n<think>\nOkay, so ... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A rectangular box has width $12$ inches, length $16$ inches, and height $\\frac{m}{n}$ inches, where $m$ and $n$ are relatively prime positive integers. Three faces of the ... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A rectangular box measures $a \\times b \\times c$, where $a$, $b$, and $c$ are integers and $1 \\le a \\le b \\le c$. The volume and the surface area of the box are numeri... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A rectangular box measures \\(a \\times b \\times c\\), where \\(a\\), \\(b\\), and \\(c\\) are integers and \\(1 \\le a \\le b \\le c\\). The volume and the surface area o... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A rectangular cow pasture is enclosed on three sides by a fence and the fourth side is part of the side of a barn that is $400$ feet long. The fence costs $\\$5$ per foot, ... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A rectangular floor that is $10$ feet wide and $17$ feet long is tiled with $170$ one-foot square tiles. A bug walks from one corner to the opposite corner in a straight li... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A rectangular parallelepiped has sides of integer lengths. The sum of the number of its vertices, the length of its edges, the area of its faces, and its volume is 2015. Wh... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A rectangular pen uses the wall of a house as one side and is enclosed by 1000 m of fencing. Describe how the area of the pen changes as the length \\( L \\) increases."
... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A rectangular prism has dimensions 8 inches by 2 inches by 32 inches. If a cube has the same volume as the prism, what is the surface area of the cube, in square inches?"
... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A rectangular prism with dimensions $p \\times q \\times r$, where $p$, $q$, and $r$ are pairwise coprime, is cut into $1 \\times 1 \\times 1$ cubes. How many of these cube... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A rectangular sheet of paper is divided by a single cut into a triangle and a pentagon. The lengths of the sides of the pentagon are $17$, $25$, $28$, $33$, and $43$, in so... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A rectangular tile measures 3 inches by 4 inches. What is the fewest number of these tiles that are needed to completely cover a rectangular region that is 2 feet by 5 feet... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A rectangular yard contains two flower beds in the shape of congruent isosceles right triangles. The remainder of the yard has a trapezoidal shape with parallel sides of le... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A red ball and a green ball are randomly and independently tossed into bins numbered with the positive integers so that for each ball, the probability that it is tossed int... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A regular 6-sided die is rolled three times. Given that one 6 was rolled, what is the probability that a 6 was rolled at least twice?"
},
{
"from": "gpt",
"valu... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A regular n-gon is in the same position after a rotation of 35 degrees about its center. What is the smallest possible value of n?"
},
{
"from": "gpt",
"value":... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A regular octagon is to be formed by cutting equal isosceles right triangles from the corners of a square with sides of one unit. What is the length of the leg of each of t... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A regular tetrahedron and a cube share an inscribed sphere. What is the ratio of the volume of the tetrahedron to the volume of the cube?"
},
{
"from": "gpt",
"... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A rental truck agency has 6 moving vans, two of which need new brakes. The vans are tested one at a time. What is the probability that the last van with brake problems is t... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A retail jeweler sells a pair of gold earrings for $213.90. The jeweler uses a 130% markup based on cost. Based on this information, what was the wholesale cost of the earr... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A revolving beacon of light sits 2 miles from a straight shoreline, revolving at 1 revolution per minute. Find the speed of the spot of light on the shore in miles per hour... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A rhombus ABCD has a perimeter of 52 meters and a diagonal AC of 24 meters. What is the area of the rhombus in square meters?"
},
{
"from": "gpt",
"value": "<th... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A rhombus is inscribed in triangle \\( \\triangle ABC \\) with vertex \\( A \\) as one of its vertices. Two sides of the rhombus lie along \\( AB \\) and \\( AC \\), where ... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A right circular cone has a base of radius $1$ and a height of $3$. A cube is inscribed in the cone so that one face of the cube is contained in the base of the cone. What ... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A right circular cone has a base of radius $1$ and height $3$. A cube is inscribed in the cone such that one face of the cube is contained in the base of the cone. What is ... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A right circular cone has base radius $r$ and height $h$. The cone lies on its side on a flat table. As the cone rolls on the surface of the table without slipping, the poi... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A right circular cone is inscribed in a sphere such that the circumference of its base lies on the interior of the sphere’s surface, as does its apex. The cone has a base r... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A right circular cone of height 2 is cut into two pieces by a plane parallel to its base. The lateral areas of the two pieces are equal. What is the distance from the cone'... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A right circular cone sits on a table, pointing up. The cross-section triangle, perpendicular to the base, has a vertex angle of 60 degrees. The diameter of the cone's base... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A right circular cylinder with its diameter equal to its height is inscribed in a right circular cone. The cone has a diameter of 10 and an altitude of 12, and the axes of ... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A right circular cylinder with its diameter equal to its height is inscribed in a right circular cone. The cone has diameter 10 and altitude 12, and the axes of the cylinde... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A right cylinder with a height of 5 inches has a radius of 2 inches. What is the area of the curved surface of the cylinder, in square inches? Express your answer in terms ... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A right cylindrical oil tank is $15$ feet tall and its circular bases have diameters of $4$ feet each. When the tank is lying flat on its side (not on one of the circular e... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A right pyramid has a regular octagon $ABCDEFGH$ with side length $1$ as its base and apex $V$. Segments $\\overline{AV}$ and $\\overline{DV}$ are perpendicular. What is th... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A right rectangular prism has integer side lengths \\(a\\), \\(b\\), and \\(c\\). Given that \\(\\text{lcm}(a,b)=72\\), \\(\\text{lcm}(a,c)=24\\), and \\(\\text{lcm}(b,c)=1... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A right square pyramid with volume $54$ has a base with side length $6.$ The five vertices of the pyramid all lie on a sphere with radius $\\frac mn$, where $m$ and $n$ are... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A right triangle has an inscribed circle that cuts the hypotenuse into lengths of 7 and 11. What is the area of the triangle?"
},
{
"from": "gpt",
"value": "<th... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A right triangle has area \\( A \\) and perimeter \\( P \\). Find the largest possible value for the positive constant \\( k \\) such that for every such triangle, \\( P^2 ... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A right triangle has legs measuring 20 inches and 21 inches. What is the length of the hypotenuse, in inches?"
},
{
"from": "gpt",
"value": "<think>\n<think>\nO... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A right triangle has side lengths 3, 4, and 5 units. Another triangle has sides of length 4, \\( x \\), and \\( y \\) units and has the same area as the right triangle. If ... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A right triangle has sides with lengths \\(4 + 2^{1/3}\\) and \\(3 + 7 \\cdot 2^{1/3}\\). What is the length of the hypotenuse? Write your answer in the form \\(a + b \\cdo... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A right triangle is inscribed in the parabola \\(y = x^2\\), with the hypotenuse parallel to the \\(Ox\\) axis. What can be the height of the triangle lowered to the hypote... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A right-angled triangle has legs \\(a\\) and \\(b\\) with \\(a > b\\). The right angle is bisected, splitting the original triangle into two smaller triangles. Find the dis... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A ring is formed by bending a cylinder and gluing the faces together. The top and bottom portions of the cylinder are cut at an angle to form the ring. Given that the insid... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A road is to be constructed to pass through a small mountain. A two-dimensional model for the shape of the mountain is the curve with equation \\( y = 24 - x^2 \\). A recta... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A rocket is fired vertically from a point 5 miles from the tracking station. For the first 20 seconds, its angle of elevation changes at $2^{\\circ}$ per second. Find the r... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A rocket is launched on a vertical trajectory and is tracked by a radar station that is 3 km from the launch pad. Find the vertical speed of the rocket at the instant when ... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A round pizza is $\\frac13$ of an inch thick and has a diameter of 12 inches. It is cut into 12 congruent pieces. What is the number of cubic inches in the volume of one pi... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A round table has room for \\( n \\) diners (\\( n \\ge 2 \\)). There are napkins in three different colors. In how many ways can the napkins be placed, one for each seat, ... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A router distributes 60 distinguishable parcels to four different channels in one second. If the order in which the parcels pass through each channel does not matter, deter... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A school day consists of eight 67-minute classes and 3-minute breaks in between classes. How long is a school day, in minutes?"
},
{
"from": "gpt",
"value": "<t... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A secretary writes letters to 8 different people and addresses 8 envelopes with the people's addresses. He randomly puts the letters in the envelopes. What is the probabili... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A sector is cut out from a circular sheet of paper, and the two edges of the cut are joined together to form a cone. The cone formed has a radius of 6 and a volume of 96π. ... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A semi-circle $ABCD$ has diameter $AD$ of length $4$. Chord $AB$ has length $1$ and chord $BC$ has length $1$. Find the length of chord $CD$."
},
{
"from": "gpt",
... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A semicircle is inscribed in a right triangle with sides 5, 12, and 13, such that the base of the semicircle is touching the side of the triangle that is 12 cm long, and th... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A semicircle of radius $r$ is divided into $n + 1$ equal parts. For each division point $k$, a triangle $A_k$ is formed with the ends of the semicircle. Calculate the limit... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A sequence of numbers $x_{1},x_{2},x_{3},\\ldots,x_{100}$ has the property that, for every integer $k$ between $1$ and $100,$ inclusive, the number $x_{k}$ is $k$ less t... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A sequence $(a_i)$ is defined by $a_1 = a_2 = 1$ and $a_n = \\dfrac{n+1}{n-1}(a_1 + a_2 + \\cdots + a_{n-1})$. Find $a_{2005}$."
},
{
"from": "gpt",
"value": "<... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A sequence $(a_{n})$ is defined by $a_{1}= 1$ and $a_{n}= a_{1}a_{2}...a_{n-1}+1$ for $n \\geq 2$. Find the smallest real number $M$ such that $\\sum_{n=1}^{m}\\frac{1}{a_{... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A sequence $(a_{n})$ of positive numbers is defined by \n\\[ a_{n} = \\frac{1}{16} \\left(1 + 4a_{n-1} + \\sqrt{1 + 24a_{n-1}}\\right) \\]\nwith $a_{0} = 1$. Find a general... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A sequence $(a_{n})$ satisfies $a_{1} = \\frac{1}{2}$ and $a_{1} + a_{2} + \\ldots + a_{n} = n^{2} a_{n}$. Determine $a_{n}$."
},
{
"from": "gpt",
"value": "<th... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A sequence $(x_n)$ is given as follows: $x_0$ and $x_1$ are arbitrary positive real numbers, and $x_{n+2} = \\frac{1 + x_{n+1}}{x_n}$ for $n \\ge 0$. Find $x_{1998}$."
},... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A sequence $\\{R_n\\}_{n \\ge 0}$ obeys the recurrence $7R_n = 64 - 2R_{n-1} + 9R_{n-2}$ for any integers $n \\ge 2$. Additionally, $R_0 = 10$ and $R_1 = -2$. Let $$S = \\s... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A sequence $\\{a_n\\}$ is defined as follows: $a_1 = 1$ and $a_{n+1} = 420a_n - 69$ for all positive integers $n$. Determine the number of squares in this sequence. If ther... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A sequence $\\{a_n\\}$ is defined as follows:\n\\[a_1 = 1\\]\nFor $n \\geq 1$,\n\\[a_{n+1} = \\frac{a_n}{1 + n \\cdot a_n}\\].\nFind an explicit formula for $a_n$, if it ex... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A sequence $\\{a_n\\}$ is defined by $a_n=\\int_0^1 x^3(1-x)^n dx$ for $n=1,2,3,\\ldots$. Find the constant number $c$ such that $\\sum_{n=1}^{\\infty} (n+c)(a_n-a_{n+1})=\... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A sequence \\(a_1, a_2, \\dots\\) of non-negative integers is defined by the rule \\(a_{n+2} = |a_{n+1} - a_n|\\) for \\(n \\geq 1\\). If \\(a_1 = 999\\), \\(a_2 < 999\\), ... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A sequence \\(a_1, a_2, a_3, \\ldots\\) satisfies the recurrence relation \\(a_{n+3} = 2a_{n+2} - 2a_{n+1} + a_n\\) for \\(n = 1, 2, 3, \\ldots\\). Given \\(a_1 = 222\\), \... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A sequence consisting of X's and O's is considered basic if every subset of consecutive symbols containing the first symbol satisfies the following conditions:\n1. The numb... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A sequence has 15 terms, each H or T. There are 14 pairs of adjacent terms: 2 are HH, 3 are HT, 4 are TH, and 5 are TT. How many sequences meet these criteria?"
},
{
... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A sequence is called an arithmetic progression of the first order if the differences of the successive terms are constant. It is called an arithmetic progression of the sec... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A sequence is defined as $a_0 = e$, $a_1 = \\pi$, and $a_{n+2} = \\frac{1 + a_{n+1}}{a_n}$ for $n \\ge 0$. Find $a_{2016}$."
},
{
"from": "gpt",
"value": "<thin... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A sequence is defined as \\( a_n = 13a_{n-1} - 30a_{n-2} \\). Let \\( a_0 = 10 \\) and \\( a_1 = 58 \\). The product of the units digits of \\( a_{2023} \\), \\( a_{2024} \... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A sequence is defined by \\( x_1 = 1 \\), \\( x_2 = 4 \\), and \\( x_{n+2} = 4x_{n+1} - x_n \\) for \\( n \\geq 1 \\). Find all natural numbers \\( m \\) such that the numb... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A sequence is defined such that $a_1=1$, $a_2=2$, and $a_{n+2}=a_{n+1}^{a_n}$ for $n\\ge 1$. What is the 9th term of this sequence?"
},
{
"from": "gpt",
"value"... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A sequence is generated as follows: if the \\( n \\)-th term is even, then the \\( (n+1) \\)-th term is half the \\( n \\)-th term; otherwise it is two more than twice the ... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A sequence of $A$s and $B$s is called antipalindromic if writing it backwards and then turning all the $A$s into $B$s and vice versa produces the original sequence. For exa... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A sequence of 0s and 1s of length $n$ is randomly generated, with each digit having an equal probability of being 0 or 1. Given that $m$ of the digits are 0s, what is the p... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A sequence of numbers is defined as follows: $x_{n}=\\frac{4}{7}$ if $n=0$, and $x_{n}=7x_{(n-1)}-3$ for all positive integers $n$. It is given that $x_{n}=\\frac{k^{n-1}+p... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A sequence of numbers is defined by the relation $a_n = -a_{n-1} + 6a_{n-2}$ with initial conditions $a_1 = 2$ and $a_2 = 1$. Find the value of $a_{100} + 3a_{99}$."
},
... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A set \\( M \\) consists of \\( n \\) elements. Find the greatest \\( k \\) for which there is a collection of \\( k \\) subsets of \\( M \\) such that for any subsets \\( ... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A set of discs is being placed in a stack. The thickness of each successive disc added to the stack is $1.5$ times the thickness of the disc immediately beneath it. What is... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A set of three numbers has both a mean and median equal to 4. If the smallest number in the set is 1, what is the range of the set of numbers?"
},
{
"from": "gpt",
... |
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