conversations listlengths 3 3 |
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[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A fertilizer makes plants grow such that on the first day, the plant's height increases by one-half, on the second day by one-third of its height from the previous day, on ... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A finite sequence of three-digit integers has the property that the tens and units digits of each term are, respectively, the hundreds and tens digits of the next term, and... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A finite set $\\mathcal{S}$ of distinct real numbers has the following properties: the mean of $\\mathcal{S} \\cup \\{1\\}$ is $13$ less than the mean of $\\mathcal{S}$, an... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A firing squad is composed of three policemen, Abel, Baker, and Charlie, who have probabilities 0.6, 0.7, and 0.8, respectively, of hitting the victim. Only one of the thre... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A flat board has a circular hole with radius $1$ and a circular hole with radius $2$ such that the distance between the centers of the two holes is $7.$ Two spheres with eq... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A flat board has a circular hole with radius $1$ and a circular hole with radius $2$ such that the distance between the centers of the two holes is 7. Two spheres with equa... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A flea jumps 1/2 ft right, then 1/4 ft left, 1/8 ft right, and so on. To what point is the flea converging?"
},
{
"from": "gpt",
"value": "<think>\n<think>\nOka... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A floor tile has the shape of a regular polygon. If the tile is removed from the floor and rotated through $50^\\circ$, it will fit back exactly into its original place in ... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A fly and a blind spider are on opposite corners of a unit cube. The fly never moves, and the spider moves at random from its initial corner to another corner connected by ... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A four-digit integer \\( m \\) and the four-digit integer obtained by reversing the order of the digits of \\( m \\) are both divisible by 45. If \\( m \\) is also divisibl... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A four-digit number \\( N \\) leaves a remainder of 10 when divided by 21, a remainder of 11 when divided by 23, and a remainder of 12 when divided by 25. What is the sum o... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A freight train travels 1 mile in 1 minute 30 seconds. At this rate, how many miles will the train travel in 1 hour?"
},
{
"from": "gpt",
"value": "<think>\n<th... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A frequency distribution of the scores for Mr. Sampson's algebra class is shown. What percent of the class received a score in the $60\\%$-$69\\%$ range? \\begin{tabular}{|... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A frog can jump forward any number of lily pads but never backward. Starting on the first of a sequence of twelve lily pads, in how many ways can the frog jump to the last ... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A frog jumps on five points numbered 1, 2, 3, 4, 5 in clockwise order on a circle. The jumping rules are:\n- If the frog is at an odd-numbered point, it moves one place clo... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A frog performs a random walk on the integers, starting from zero. At each step, it jumps one integer forward or one integer backward with equal probability. Find the proba... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A frustum has regular hexagonal bases with side lengths of $5$ and $6$ and a slant height of $7$. What is the height of the frustum?"
},
{
"from": "gpt",
"value... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A function $f$ has domain $[0,2]$ and range $[0,1]$. (The notation $[a,b]$ denotes $\\{x:a \\le x \\le b \\}$.) Let\n\\[g(x) = 1 - f(x + 1).\\]Then the domain of $g(x)$ is ... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A function $g(x)$ has derivatives of all orders for all real numbers $x$. Assume $g(2) = 5$, $g'(2) = -6$, $g''(2) = 10$, $g'''(2) = -3$. Write the third-degree Taylor poly... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A function \\( f : \\mathbb{N} \\rightarrow \\mathbb{N} \\) is defined such that \\( f(1) = 1 \\) and for all \\( n \\in \\mathbb{N} \\), \\( f(2n) = f(n) \\) and \\( f(2n+... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A function \\( f : \\mathbb{N} \\to \\mathbb{N} \\) satisfies the condition \\(\\sum_{k = 1}^n \\left(\\frac{1}{f(k)f(k + 1)}\\right) = \\frac{f(f(n))}{f(n +1)}\\) for all ... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A function \\( f \\) from the integers to the integers is defined as follows:\n\\[ f(n) = \\begin{cases} \nn + 3 & \\text{if } n \\text{ is odd} \\\\\nn/2 & \\text{if } n \... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A function \\( f \\) is defined for all real numbers and satisfies \\( f(2 + x) = f(2 - x) \\) and \\( f(7 + x) = f(7 - x) \\) for all real \\( x \\). If \\( x = 0 \\) is a... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A function \\( f \\) is defined over all the positive integers and satisfies:\n\\[ f(1) = 1996 \\]\n\\[ f(1) + f(2) + f(3) + \\cdots + f(n) = n^2 f(n) \\]\nCalculate the ex... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A function \\( f \\), continuous on the positive real axis, satisfies the property\n\\[\n\\int _{ 1 }^{ xy }{ f\\left( t \\right) \\, \\text{d}t } = y \\int _{ 1 }^{ x }{ f... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A function \\( f: \\mathbb{N} \\to \\mathbb{N} \\) satisfies the following conditions:\n1. \\( f(a+b) = f(f(a) + b) \\)\n2. \\( f(a+b) = f(a) + f(b) \\) if \\( 0 \\leq a+b ... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A function \\( f: \\mathbb{N}_0 \\to \\mathbb{R} \\) satisfies \\( f(1) = 3 \\) and \\( f(m+n) + f(m-n) - m + n - 1 = \\frac{f(2m) + f(2n)}{2} \\) for any nonnegative integ... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A gambler plays a game \\( n \\) times, with the probability of winning each game being \\( p \\). If it is known that the gambler did not win \\( x \\) times, what is the ... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A game is played on the vertices of the cube $(x, y, z): 0 \\le x, y, z \\le 1$ as follows. Beginning at $(0, 0, 0)$, the player moves along one of the edges emanating from... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A gardener plants three maple trees, four oak trees, and five birch trees in a row. The trees are planted in random order, with each arrangement being equally likely. Find ... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A gardener plants three maple trees, four oak trees, and five birch trees in a row. What is the probability that no two birch trees are next to one another?"
},
{
"... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A gear with radius 34.5 rolls around the circumference of a larger gear with radius 103.5. How many revolutions around the larger gear would the smaller gear have completed... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A giant, circular and perfectly flat pizza must be shared by 22 people. What is the least number of cuts to be made on the pizza so that each person can have a piece of piz... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A glass of iced water at $3^{\\circ}C$ is left in the shade outside at $27^{\\circ}C$. After 5 minutes, the water’s temperature is $7^{\\circ}C$. Find the water’s temperatu... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A gold bar with dimensions $2 \\times 3 \\times 4$ is melted and recast into three spheres of equal volume. What is the radius of each sphere?"
},
{
"from": "gpt",
... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A grasshopper is situated at the origin (0, 0) on the Cartesian plane. If the grasshopper is at a given point (m, n), it can jump to either (m + 6, n + 9) or (m + 9, n + 6)... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A grasshopper lies on a line at position $0$. It can jump one unit left ($-1$) or one unit right ($+1$) with equal probability. What is the probability that the grasshopper... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A group consists of $2005$ members. Each member holds exactly one secret. Each member can send a letter to any other member to convey all the secrets he holds. The number o... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A group of 16 people will be partitioned into 4 indistinguishable 4-person committees. Each committee will have one chairperson and one secretary. The number of different w... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A group of 5 is to be chosen from 6 men and 4 women. In how many ways can the group be chosen if it contains a particular man X but excludes a particular woman Y?"
},
{... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A group of 6 friends went to the movies. Colin and Megan insist on sitting next to each other and Abe wants to sit next to them as well. Lee, Dani, and Libbey also want to ... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A group of \\(n \\geq 3\\) people sit at a round table, where \\(n\\) is an odd number. Some people shake hands with each other in such a way that no two pairs of people sh... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A group of women working together at the same rate can build a wall in 45 hours. They joined the work over a period of time, one by one, at equal intervals. Once at work, e... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A hardware supplier manufactures 3 kinds of clamps; types A, B, and C. Production restrictions force it to make 20 more type C clamps than the total of the other types and ... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A heavy rope, 50 ft long, weighs 0.5 lb/ft and hangs over the top of a building 120 ft high. How much work is done in pulling the rope to the top of the building?"
},
{... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A helicopter is directly above an oil tanker at a height of 5000 km. The pilot spots an oil platform with an angle of depression of 26 degrees. What is the distance between... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A hexagon inscribed in a circle has three consecutive sides of length 3 and three consecutive sides of length 5. The chord of the circle that divides the hexagon into 2 tra... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A hexagon is inscribed in a circle of radius \\( r \\). If two sides of the hexagon are 7 units long, while the other four sides are 20 units long, find \\( r \\)."
},
... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A hexagon is inscribed in a circle of radius \\( r \\). Two sides of the hexagon are 7 units long, while the other four sides are 20 units long. Find \\( r \\)."
},
{
... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A hexagon that is inscribed in a circle has side lengths $22$, $22$, $20$, $22$, $22$, and $20$ in that order. The radius of the circle can be written as $p+\\sqrt{q}$, whe... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A high school basketball game between the Raiders and Wildcats was tied at the end of the first quarter. The number of points scored by the Raiders in each of the four quar... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A home buyer can afford to spend no more than \\$N per month on mortgage payments. Suppose that the interest rate is $r$ and that the term of the mortgage is 20 years. Assu... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A hunter shoots at a fox from a distance of 100 yards with a probability of hitting the fox of $1/2$. If he misses, he reloads and shoots again after the fox has run 50 yar... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A hyperbola has the property that, if F and F' are its foci, then the four points of intersection of this hyperbola with the circle having diameter FF' are the vertices of ... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A jar contains blue, red, and yellow marbles. Yellow marbles make up 20% of the marbles in the jar. After adding 5 blue marbles, blue marbles make up 64% of the marbles. Af... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A jar contains red, yellow, green, and blue marbles. If $1/5$ of the non-yellow marbles are red, $1/3$ of the non-red marbles are blue, and $1/2$ of the non-blue marbles ar... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A jar has 10 red candies and 10 blue candies. Terry picks two candies at random, then Mary picks two of the remaining candies at random. Given that the probability that the... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A jar has 10 red candies and 10 blue candies. Terry picks two candies at random, then Mary picks two of the remaining candies at random. Given that the probability that the... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A jar has blue, red, and yellow marbles. Yellow marbles make up 20% of the marbles in the jar. Once 5 blue marbles are added, blue marbles make up 64% of the marbles. Anoth... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A judoka team was weighed before a competition. The three heaviest members weigh 41% of the total weight of the team, and the two lightest members weigh 17% of the total we... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A kite with an area of 22 is inscribed in a circle with a diameter of 10. Find the sum of the reciprocals of the side lengths of the kite."
},
{
"from": "gpt",
... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A knight starts at the origin on a chessboard. Each step, it moves to one of the 8 possible positions chosen uniformly and at random. What is the expected value of the squa... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A ladder 10 ft long rests against a vertical wall. If the bottom of the ladder slides away from the wall at a rate of 1.5 ft/s, how fast is the angle between the ladder and... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A large candle is $119$ centimeters tall. It is designed to burn down more quickly when it is first lit and more slowly as it approaches its bottom. Specifically, the can... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A large candle is $119$ centimeters tall. It is designed to burn down more quickly when it is first lit and more slowly as it approaches its bottom. Specifically, the candl... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A large cube is dipped in green paint and then divided into 216 smaller, congruent cubes. One of the smaller cubes is then randomly selected. What is the probability, expre... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A large rectangle is partitioned into four rectangles by two segments parallel to its sides. The areas of three of the resulting rectangles are 6, 14, and 35. What is the a... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A laser is placed at the point $(3,5)$. The laser beam travels in a straight line. Larry wants the beam to hit and bounce off the $y$-axis, then hit and bounce off the $x$-... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A lattice point is a point whose coordinates are both integers. How many lattice points are on the boundary or inside the region bounded by $y=|x|$ and $y=-x^2+6$?"
},
... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A lattice point is a point whose coordinates are both integers. Suppose we draw a line segment from (0, 2004) to (a, b) where a and b are integers each randomly selected fr... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A leaf 30 feet from the ground falls from a branch of a tree at a rate of 2 feet per second. There is a slight wind blowing at a rate of 2 feet per second. Describe the loc... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A lemonade stand analyzes its earning and operations. For the previous month it had a \\$50 dollar budget to divide between production and advertising. If it spent $k$ doll... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A license plate has six digits from 0 to 9 and may have leading zeros. If two plates must always differ in at least two places, what is the largest number of plates that is... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A license plate in a certain state consists of 4 digits, not necessarily distinct, and 2 letters, also not necessarily distinct. These six characters may appear in any orde... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A license plate number consists of any combination of three letters followed by any combination of three digits. The value of a license plate is determined by adding the va... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A light is mounted on a pole 15 feet high. A six-foot tall man is walking in a straight line away from the pole at 5 feet per second. Find the speed of the leading edge of ... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A light is on the ground, 15m from a wall. A woman, 1.6m tall, walks away from the light and towards the wall at 0.5 m/s. Calculate the rate of change of the length of her ... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A light source at the point $(0, 16)$ in the coordinate plane casts light in all directions. A disc (circle along with its interior) of radius $2$ with center at $(6, 10)$ ... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A lighthouse facing East sends out a fan-shaped beam of light extending from S.-E to N.-E. An observer sailing due North continues to see the light for \\(10\\sqrt{2}\\) mi... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A line $ y=x+2 $ is rotated $ 45^{\\circ} $ clockwise about the point $ (0,2) $. A second line $ y=-2x+10 $ is drawn. What is the area of the triangle formed by these three... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A line \\( l \\) is tangent to a circle of radius \\( r \\) at a point \\( P \\). From a variable point \\( R \\) on the circle, a perpendicular \\( RQ \\) is drawn to \\( ... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A line in the first quadrant is at a distance of 6 units from the origin and makes an angle of $120^\\circ$ with the positive X-axis. Find the equation of the line."
},
... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A line is expressed in the form\n\\[\\begin{pmatrix} -2 \\\\ -5 \\end{pmatrix} \\cdot \\left( \\begin{pmatrix} x \\\\ y \\end{pmatrix} - \\begin{pmatrix} 1 \\\\ 11 \\end{pm... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A line is moving so that its distance from the origin is always equal to \\( a \\). Find the equation of the locus of the image of the origin with respect to this line."
... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A line of slope $\\dfrac{1}{2}$ intersects the $x$-axis at an angle $\\theta$ from the horizontal. Compute the following sum:\n\n$$\\sum_{n = 0}^{\\infty} \\dfrac{5^n \\sin... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A line parallel to the parallel sides of a trapezium is drawn through the midpoints of the unequal sides, dividing the trapezium into two regions in the ratio of $5:2$. Fin... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A line passes through points \\( A(1,1) \\) and \\( B(100,1000) \\). How many other points with integer coordinates are on the line and strictly between \\( A \\) and \\( B... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A line whose $y$-intercept is $(0,5)$ intersects the ellipse $9x^2 + 16y^2 = 144.$ Find all possible slopes of this line."
},
{
"from": "gpt",
"value": "<think... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A list of 2018 positive integers has a unique mode that occurs exactly 10 times. What is the least number of distinct values that can occur in the list?"
},
{
"from... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A list of consecutive integers beginning with $1$ is written on the chalkboard. One of the numbers is erased. If the average of the numbers that remain is $35\\frac{6}{17}$... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A loader has a waggon that can carry up to 1000 kg and a cart that can carry up to 1 kg. There are a finite number of sacks of sand in a storehouse with a total weight exce... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A machine either works or does not work on a particular day. If the machine works one day, the probability that it will work the next day is 0.4. If the machine does not wo... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A machine outputs integers from $1$ to $10$ inclusive, with each integer having a fixed probability of being produced. The expected value of each output is $7$ and the prob... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A machine-shop cutting tool has the shape of a notched circle with a radius of $\\sqrt{50}$ cm. The length of $AB$ is 6 cm, and the length of $BC$ is 2 cm. The angle $ABC$ ... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A man 6 ft tall walks at the rate of 200 feet per minute towards a street light which is 18 ft above the ground. At what rate is the tip of his shadow moving?"
},
{
... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A man buys an item for 50 dollars using a 100-dollar bill. The clerk exchanges the 100-dollar bill for change from an adjacent shop and gives the man the item and his chang... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A man deposits $2000 on June 1st each year for 20 years. If interest at 8% is compounded annually, what is the value of his account after 20 years? Use the formula \\( S_{n... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A man in a speed boat traveling at 50 km/h dropped his spare life jacket into a river flowing at a constant speed. Twenty minutes later, he turned back and found the jacket... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A man is 6 miles east and 5 miles south of his home. He is also 3 miles north of a river which is 8 miles south of his home. What is the least number of miles he may travel... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A man is running through a train tunnel. When he is 2/5 of the way through, he hears a train approaching the tunnel from behind him at a speed of 60 mph. Whether he runs ah... |
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