conversations listlengths 3 3 |
|---|
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A man is walking due east at 2 m.p.h. and to him, the wind appears to be blowing from the north. On doubling his speed to 4 m.p.h. and still walking due east, the wind appe... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A man may take a step forward, backward, left, or right with equal probability. Find the probability that after nine steps, he will be just one step away from his initial p... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A man stands 30 feet from the base of a tree and holds a ruler 6 inches in front of his eye. The bottom of the ruler aligns with the base of the tree, and the top of the tr... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A math contest consists of 9 objective type questions and 6 fill in the blanks questions. All students attempted exactly 14 out of 15 questions. Let \\( O_1, O_2, \\dots, O... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A math teacher has a bag containing \\( n > 3 \\) identically sized and shaped weights weighing 1 kilogram to \\( n \\) kilograms. You are to weigh them one at a time on a ... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A mathematical organization is producing a set of commemorative license plates. Each plate contains a sequence of five characters chosen from the four letters in AIME and t... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A merchant buys goods at $25\\%$ off the list price. He desires to mark the goods so that he can give a discount of $20\\%$ on the marked price and still clear a profit of ... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A mirror has an area of 1 m² and is shaped like a rectangle with a semicircle at each end. The frame around the mirror costs twice as much per square centimeter for the cur... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A monic quadratic $p(x)$ has $p(5) = 12$ and $p(p(3)) = 5$. If the quadratic $p(x)$ has the property that $p(2)$ is minimized, and $p(1)$ can be expressed as $a - \\sqrt{b}... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A monkey tries to climb a slippery 10-foot high pole. Starting at the bottom, he climbs 5 feet and slides half way to the ground. He then climbs an additional 5 feet and ag... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A month with 31 days has the same number of Mondays and Wednesdays. How many of the seven days of the week could be the first day of this month?"
},
{
"from": "gpt"... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A motorboat is 10 km South of a lighthouse and is on a course of 053 degrees. What is the shortest distance between the motorboat and the lighthouse?"
},
{
"from": ... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A mountain's height is given by the function \\( z = y^5 - 2x^2y + 1 \\) at the coordinate \\((x, y)\\). A rocket is launched from the point \\((-1, -1, 2)\\) in the direct... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A movie place charges $7 for an adult ticket and $4 for a child ticket. On a recent night, the sales of children tickets were 3 times the sales of adult tickets. If the tot... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A movie theater charges \\$5 for a ticket. The cashier starts with no change, and each customer either pays with a \\$5 bill or offers a \\$10 bill (and gets change). There... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A natural number 'a' has four digits and \\(a^2\\) ends with the same four digits of \\(a\\). Find the value of \\(10080 - a\\)."
},
{
"from": "gpt",
"value": "... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A natural number \\( N \\) has exactly two divisors, and \\( N+1 \\) has exactly three divisors. Find the number of divisors of \\( N+2 \\)."
},
{
"from": "gpt",
... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A non-degenerate triangle has sides of lengths $\\log 101$, $\\log 2018$, and $\\log n$, where $n$ is an integer. How many possible values are there for $n$?"
},
{
... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A non-negative integer $n$ is said to be squaredigital if it equals the square of the sum of its digits. Find all non-negative integers which are squaredigital."
},
{
... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A number 1 multiplies by 1.1 when a bot receives the letter A and multiplies by 1.02489 when the same bot receives the letter B. If I insert the letter C, the number is rou... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A number \\( N \\) leaves a remainder of 11 when divided by \\( M \\) and a remainder of 92 when divided by \\( 8M \\). What will be the remainder when \\( N \\) is divided... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A number \\( m \\) is randomly selected from the set \\(\\{11, 13, 15, 17, 19\\}\\), and a number \\( n \\) is randomly selected from \\(\\{1999, 2000, 2001, \\ldots, 2018\... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A number \\( n \\) is called a \"double\" if its base-7 digits form the base-10 number \\( 2n \\). For example, 51 in base-10 is 102 in base-7. What is the largest such \"d... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A number follows these conditions:\ni) It is 1 more than a multiple of 2.\nii) It is 2 more than a multiple of 3.\niii) It is 4 more than a multiple of 5.\nFind the smalles... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A number in the set $\\{50, 51, 52, 53, ... , 999\\}$ is randomly selected. What is the probability that it is a two-digit number? Express your answer as a common fraction.... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A number is drawn from the set of natural numbers less than ten thousand. What is the probability that the first and last digits of the number are the same?"
},
{
"... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A number is hole-y if at least one of its digits has a hole in them when you write them out. By this, we mean that $0$, $4$, $6$, $8$, $9$ have holes in them, and $1$, $2$,... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A number is said to be \"fourfriend\" if it and all the numbers obtained by rearranging its digits are multiples of 4. Determine how many 7-digit numbers are fourfriends. N... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A number of players form seven teams, with each pair of teams having one common member and each player being on exactly two teams. How many players are there?"
},
{
... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A number with 8 digits is a multiple of 73 and also a multiple of 137. The second digit from the left equals 7. Determine the 6th digit from the left."
},
{
"from":... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A pair of dice are rolled until the sum of either $5$ or $7$ appears. Find the probability that $5$ occurs first."
},
{
"from": "gpt",
"value": "<think>\n<think... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A palindrome is a number that reads the same forward and backward. How many three-digit perfect squares are palindromes?"
},
{
"from": "gpt",
"value": "<think>\... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A paperboy delivers newspapers to 10 houses along Main Street. Wishing to save effort, he doesn't always deliver to every house, but to avoid being fired he never misses th... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A parent particle divides into 0, 1, or 2 particles with probabilities 1/4, 1/2, and 1/4, respectively, and then disappears. Starting with one particle, let \\( X_i \\) be ... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A park is in the shape of a regular hexagon with each side measuring 2 km. Starting at a corner, Alice walks along the perimeter of the park for a distance of 5 km. How man... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A particle is moving along the curve \\( y = \\sqrt{x} \\). As the particle passes through the point \\( (4, 2) \\), its x-coordinate increases at a rate of \\( 3 \\text{cm... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A particle of unit mass moves on a straight line under the action of a force which is a function $f(v)$ of the velocity $v$ of the particle. The distance $x$ covered in tim... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A particle starts at the point \\((4,4)\\) and moves to one of the points \\((a-1,b)\\), \\((a,b-1)\\), or \\((a-1,b-1)\\) with equal probability \\(\\frac{1}{3}\\) each ti... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A pawn is on the middle square of a 9x9 blackboard. In each move, the pawn moves to an adjacent square with probability 1/4. What is the probability that after 4 moves, the... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A penny is thrown straight down off the top of the Empire State building at 30 feet/sec. If the Empire State building is 1440 feet high, and the equation for the height of ... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A pentagon is formed by cutting a triangular corner from a rectangular piece of paper. The five sides of the pentagon have lengths 13, 19, 20, 25, and 31 (not necessarily i... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A permutation \\( a_1, a_2, \\ldots, a_n \\) of \\( 1, 2, \\ldots, n \\) is said to be indecomposable if \\( n \\) is the least positive integer \\( j \\) for which \\( \\{... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A perpendicular is drawn from the midpoint of a side of a rhombus to a point above the plane of the rhombus. The upper end of this perpendicular is at a distance equal to h... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A person has 12 CDs of the following types: A (3), B (4), C (2), D (1), E (2). If 5 CDs are randomly selected, what is the probability that either 3 A's or 3 B's will be se... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A person has a sum of \\( n \\) dollars. Each day, they can buy one of the following: an ice cream for 1 dollar, a sandwich for 2 dollars, or a cake for 2 dollars. In how m... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A person reads a book. On the first day, he reads \\( n \\) pages in \\( t \\) minutes, where \\( n \\) and \\( t \\) are positive integers. The next day he reads \\( n + 1... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A person walks along a straight sidewalk at a constant speed of 6 feet/second. A spotlight is trained on the person from across the street, which is 20 feet wide. How fast ... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A pharmacist needs to prepare 15 ml of a special eye drop solution with a 2% active ingredient. The pharmacist has only 10% solution and 1% solution in stock. How much of e... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A photo is printed on an 11-inch by 13-inch piece of paper. The photo covers 80 square inches and has a uniform border. What is the width of the border?"
},
{
"from... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A physics club with 22 members needs to select 3 officers: chairman, vice chairman, and sergeant-at-arms. Each person can hold at most one office. Two members, Penelope and... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A piece of graph paper is folded once so that $(0,2)$ is matched with $(4,0)$ and $(7,3)$ is matched with $(m,n)$. Find $m + n$."
},
{
"from": "gpt",
"value": "... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A pile of gold and silver coins contains more silver coins than gold, but the total weight of each type is the same. When enough silver coins are removed to make the number... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A pirate is counting the loot he has stolen from S.S. AOPS, on which everything is counted in base 5. A treasure chest contains $3124_{5}$ dollars worth of jewelry, $3122_... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A pitcher has a $\\frac{1}{2}$ chance of pitching a strike. If he pitches a strike, the batter will hit the ball $\\frac{3}{4}$ of the time, ending the at bat. If he pitche... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A pizza shop offers 30 different toppings, and a customer can choose any number of toppings. Let $M$ be the number of possible pizzas the customer can buy. What is the rema... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A plane intersects a sphere of radius $R$ at a distance $x$ from the center of the sphere, cutting a cap-shaped piece off from the sphere. A cone is inscribed in this cap s... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A plane is uniquely determined by three non-collinear points. What is the maximum possible number of planes that can be determined by 12 points in space?"
},
{
"fro... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A player chooses one of the numbers 1 through 4. Two regular four-sided (tetrahedral) dice are rolled, with the sides numbered 1 through 4. If the chosen number appears on ... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A player flips 100 coins and scores \\( x(100-x) \\) points for flipping \\( x \\) heads. What is the expected value of the points the player gets?"
},
{
"from": "g... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A player scored 23, 14, 11, and 20 points in the sixth, seventh, eighth, and ninth basketball games of the season, respectively. Her points-per-game average was higher afte... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A point $(x,y)$ is randomly and uniformly chosen inside the square with vertices (0,0), (0,2), (2,2), and (2,0). What is the probability that $x+y < 3$?"
},
{
"fro... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A point $P$ is randomly selected from the rectangular region with vertices $(0,0),\\, (2,0),\\, (2,1),\\, (0,1)$. What is the probability that $P$ is closer to the origin t... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A point $P$ is randomly selected from the square region with vertices at $(\\pm 2, \\pm 2)$. What is the probability that $P$ is within one unit of the origin? Express your... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A point $P$ lies inside square $ABCD$ such that $PA = 11, PB = 22,$ and $PC = 33.$ The area of $ABCD$ can be expressed in the form $a + b\\sqrt{c}$, where $a, b,$ and $c$ a... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A point \\( C \\) is chosen on the line segment \\( AB \\) such that \\( \\frac{AC}{BC} = \\frac{BC}{5AB} \\). Find \\( \\frac{AC}{BC} \\)."
},
{
"from": "gpt",
... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A point \\( P \\) lies in the same plane as a given square of side 1. Let the vertices of the square, taken counterclockwise, be \\( A, B, C, \\) and \\( D \\). Also, let t... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A point \\( P \\) lies in the same plane as a given square of side \\( 1 \\). Let the vertices of the square, taken counterclockwise, be \\( A, B, C, D \\). Let the distanc... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A point \\((X, Y)\\) is picked at random, uniformly from the square with corners at \\((0,0)\\), \\((1,0)\\), \\((0,1)\\), and \\((1,1)\\).\n(a) Compute \\(Cov(X, Y)\\).\n(... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A point \\(P\\) is located in the interior of a rectangle such that the distance from \\(P\\) to one corner is 5 yards, from the opposite corner is 14 yards, and from a thi... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A point inside square \\(ABCD\\) has distances \\(5\\sqrt{2}\\), \\(5\\sqrt{10}\\), \\(15\\sqrt{2}\\), and \\(5\\sqrt{10}\\) from the vertices. Find the area of \\(ABCD\\).... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A point is located outside a square. The distance from the point to the nearest corner of the square is $5$ units; to the next nearest, $11$ units, and to the farthest, $17... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A point is moving randomly inside an equilateral triangle with a side length of 3 cm. What is the probability that the distance from the point to any corner is greater than... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A point is situated inside an angle of measure 60 degrees, at a distance 'a' and 'b' from its sides. Find the distance of the point from the vertex of the given angle."
}... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A point within an equilateral triangle is at distances of 5, 7, and 8 from each of the vertices. What is the length of a side of the triangle?"
},
{
"from": "gpt",
... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A point-sized ant is located at the left end of a rubber band $e$ meters long at time $t = 0$. The ant walks to the right at a rate of $\\frac{1}{\\ln(t + e)}$ centimeters ... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A poker hand consists of 5 cards. If the cards have distinct consecutive values and are not all of the same suit, we say that the hand is a straight. What is the probabilit... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A polynomial $P(x)$ of degree $2013$ with real coefficients satisfies $P(n) = 3^n$ for $n = 0, 1, \\ldots, 2013$. Evaluate $P(2014)$."
},
{
"from": "gpt",
"valu... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A polynomial $p(x)$ leaves a remainder of $-1$ when divided by $x - 1,$ a remainder of 3 when divided by $x - 2,$ and a remainder of 4 when divided by $x + 3.$ Let $r(x)$ ... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A polynomial \\( P(x) \\) of degree \\( n \\) satisfies \\( P(k) = 2^k \\) for \\( k = 0, 1, 2, \\ldots, n \\). Find the value of \\( P(n+1) \\)."
},
{
"from": "gpt... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A polynomial \\( P(x) \\) of degree \\( n \\) satisfies \\( P(k) = 2^k \\) for all integers \\( k \\) in the range \\([0, n]\\). Find \\( P(n+1) \\)."
},
{
"from": ... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A polynomial \\( P(x) \\) with integer coefficients returns the values \\( 4, 3, 2, 1, 0 \\) when \\( x \\) is \\( 1, 2, 3, 4, 5 \\) respectively. What is the smallest poss... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A polynomial \\( f(x) \\) has the remainder \\( x + 1 \\) when divided by \\( x^2 - 4x + 3 \\) and the remainder \\( 3x - 1 \\) when divided by \\( x^2 - 3x + 2 \\). Find t... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A polynomial \\( f(x) \\) leaves a remainder of 10 when divided by \\( (x-2) \\) and a remainder of \\( (2x-3) \\) when divided by \\( (x+1)^2 \\). Find the remainder when ... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A polynomial with integer coefficients is of the form\n\\[9x^4 + a_3 x^3 + a_2 x^2 + a_1 x + 15 = 0.\\]Find the number of different possible rational roots of this polynomi... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A pomegranite has 20 seeds, with exactly half red and half blue. If Kee Young picks and eats 4 seeds one by one, what is the probability that he ends up with 2 seeds of eac... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A pond is shaped as an inverted truncated pyramid with a square base. The depth of the pond is 6 meters. The bottom square has a side length of 2 meters, and the top square... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A positive factor of \\( 60^4 \\) is randomly chosen. The probability it has exactly \\( 6 \\) positive factors can be expressed as \\( \\frac{m}{n} \\), where \\( m \\) an... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A positive integer $N$ is divided into $n$ parts inversely proportional to the numbers $2, 6, 12, 20, \\ldots$. The smallest part is equal to $\\frac{1}{400} N$. Find the v... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A positive integer $n$ is called a good number if every integer multiple of $n$ is divisible by $n$ regardless of how its digits are rearranged. How many good numbers are t... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A positive integer $n$ is said to be good if there exists a perfect square whose sum of digits in base 10 is equal to $n$. How many good numbers are there among $1, 2, 3, \... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A positive integer \\( n \\) is called special if \\( n \\) is divisible by 4, \\( n+1 \\) is divisible by 5, and \\( n+2 \\) is divisible by 6. How many special integers s... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A positive integer \\( n \\) not exceeding 100 is chosen in such a way that if \\( n \\le 50 \\), then the probability of choosing \\( n \\) is \\( p \\), and if \\( n > 50... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A positive integer is called a [i]dragon[/i] if it can be written as the sum of four positive integers $a, b, c,$ and $d$ such that $a+4=b-4=4c=d/4$. Find the smallest drag... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A printer prints 17 pages per minute. How many minutes will it take to print 200 pages? Express your answer to the nearest whole number."
},
{
"from": "gpt",
"v... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A professor is assigning grades to a class of 10 students. As a very kind professor, he only gives out A's, B's, and C's. How many ways can the professor assign grades to a... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A projection takes $\\begin{pmatrix} 4 \\\\ 4 \\end{pmatrix}$ to $\\begin{pmatrix} \\frac{60}{13} \\\\ \\frac{12}{13} \\end{pmatrix}.$ Which vector does the projection tak... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A province consists of nine cities named $1, 2, 3, 4, 5, 6, 7, 8, 9$. From city $a$ there is a direct road to city $b$ if and only if $\\overline{ab}$ and $\\overline{ba}$ ... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A pyramid has a square base \\(ABCD\\) with an area of 196. The areas of triangles \\( \\triangle ABE \\) and \\( \\triangle CDE \\) are 105 and 91, respectively. What is t... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A pyramid has a triangular base with side lengths $20$, $20$, and $24$. The three edges of the pyramid from the three corners of the base to the fourth vertex of the pyrami... |
[
{
"from": "system",
"value": "You are a mathematical reasoning expert. Solve step-by-step."
},
{
"from": "human",
"value": "A quadrilateral has all of its vertices on the sides of a square with side length \\(a\\). What is the minimum value of the perimeter of this quadrilateral? Determine its n... |
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