conversations listlengths 2 2 |
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[
{
"from": "human",
"value": "Problem:\nIn a cryptographic system, a message is encrypted using the formula:\n\nC ≡ M^e (mod n)\n\nwhere C is the ciphertext, M is the plaintext, e is the public key, and n is the modulus. Given that e = 17, n = 323, and C = 179, find the original plaintext message M, which is... |
[
{
"from": "human",
"value": "Simplify the expression $\\left( \\frac{2x^2}{3y} \\right)^{-3} \\cdot (3xy^2)^2$."
},
{
"from": "gpt",
"value": "We are given the expression:\n\n$$\n\\left( \\frac{2x^2}{3y} \\right)^{-3} \\cdot (3xy^2)^2\n$$\n\nLet's simplify it **step by step**.\n\n---\n\n### **St... |
[
{
"from": "human",
"value": "What is the length of the altitude dropped from the vertex $A$ of the triangle $ABC$ with vertices $A(0,3)$, $B(4,0)$, and $C(-1,-1)$ to the line $BC$?"
},
{
"from": "gpt",
"value": "We are given the coordinates of the vertices of triangle $ ABC $:\n\n- $ A(0, 3) $\n... |
[
{
"from": "human",
"value": "Given a cube, how many different rectangular prisms (with edges parallel to the edges of the cube) can be inscribed within it, where at least one vertex of the prism coincides with a vertex of the cube?"
},
{
"from": "gpt",
"value": "We are given a **cube** and asked... |
[
{
"from": "human",
"value": "The graph of the equation $y=ax^2+bx+c$ intersects the $x$-axis at two points, $P$ and $Q$. If the vertex of the parabola is equidistant from $P$ and $Q$, what is the relationship between the coefficients $a$, $b$, and $c$?"
},
{
"from": "gpt",
"value": "We are given... |
[
{
"from": "human",
"value": "The altitude of a right-angled triangle is increased by $20\\%$, resulting in an increase in the area of the triangle by $10\\%$. Determine the original ratio of the base to the altitude."
},
{
"from": "gpt",
"value": "We are given:\n\n- A **right-angled triangle**.\... |
[
{
"from": "human",
"value": "Let n be a positive integer. We call a proper divisor d of n \"good\" if d is a divisor of at least one proper divisor of n not equal to d. For example, 1 is a good divisor of every positive integer, since 1 divides every positive integer. On the other hand, 3 is not a good d... |
[
{
"from": "human",
"value": "Maria wants to set up a flower stand with tulips. She intends to arrange 5 rows of 12 tulips each. However, she decides to add 8 more tulips to fill in some gaps. How many tulips does Maria need to purchase in total?"
},
{
"from": "gpt",
"value": "Let's break the pro... |
[
{
"from": "human",
"value": "Find the interval of all $y$ such that both $4y$ and $5y$ are in the interval $(2,3)$."
},
{
"from": "gpt",
"value": "We are given that **both** $ 4y $ and $ 5y $ are in the interval $ (2, 3) $. We are to find the **interval of all $ y $** that satisfies this conditi... |
[
{
"from": "human",
"value": "Find the sum of the smallest and largest prime numbers between 30 and 50."
},
{
"from": "gpt",
"value": "We are asked to find the **sum of the smallest and largest prime numbers between 30 and 50**.\n\n---\n\n### Step 1: List all numbers between 30 and 50\nWe are con... |
[
{
"from": "human",
"value": "Find all possible values of $k$ for which the equation $\\cos^2 x + k\\sin^2 x = 1$ has exactly two distinct real solutions in $x$ on the interval $[0, 2\\pi)$, counting multiplicity."
},
{
"from": "gpt",
"value": "We are given the equation:\n\n$$\n\\cos^2 x + k \\si... |
[
{
"from": "human",
"value": "Ava sold some of her artwork to a gallery. Ava sold each painting for $75. The gallery sells them for $10 more than 2 times that. How much does the gallery profit from each painting?"
},
{
"from": "gpt",
"value": "Let's break this down step by step.\n\n---\n\n### Ste... |
[
{
"from": "human",
"value": "Emily's dad gave her $50 to buy school supplies. The list included a backpack for $25, a set of pens for $5.50, a notebook for $3.25, and 3 folders that were $1.50 each. Her dad also gave her a discount card for 10% off the backpack. At the store, the pens were on sale for $1 of... |
[
{
"from": "human",
"value": "A dart is thrown onto a circular dartboard with a radius of 25 cm. The probability of the dart hitting the dartboard within a certain region is proportional to the area of that region. If the probability of the dart hitting within 10 cm of the center is 1/5, what is the probabil... |
[
{
"from": "human",
"value": "A group of friends wants to schedule a meeting every week. They have different work schedules and can only meet on specific days: Sarah can meet on Mondays, Wednesdays, and Fridays, John can meet on Tuesdays and Thursdays, and Emily can meet on Mondays and Fridays. If they want ... |
[
{
"from": "human",
"value": "Tom has a large container that can hold 240 liters of a mixture of water and juice, with the ratio of water to juice being 5 to 3. If he wants to fill the container completely and add a flavor enhancer that comes in 10-liter bottles, how many bottles of flavor enhancer can he ad... |
[
{
"from": "human",
"value": "What is the ratio of the numerical value of the area, in square units, of an equilateral triangle of side length 6 units to the numerical value of its perimeter, in units? Express your answer as a common fraction in simplest radical form."
},
{
"from": "gpt",
"value"... |
[
{
"from": "human",
"value": "A nurse is paid $45 per hour for a 36-hour workweek. She pays 1/4 of the money on rent, $700 on food, and $800 on transportation per month. Calculate the total amount of money the nurse has after paying her monthly expenses."
},
{
"from": "gpt",
"value": "Let's break... |
[
{
"from": "human",
"value": "Let $S$ be the set of all polynomials with real coefficients such that the derivative of each polynomial is divisible by the polynomial itself. Determine the maximum number of elements in $S$ such that no two elements have the same derivative."
},
{
"from": "gpt",
"v... |
[
{
"from": "human",
"value": "In the diagram, $F$ is the midpoint of $\\overline{CD}$, and $G$ is the midpoint of $\\overline{AB}$. Determine the sum of the $x$ and $y$ coordinates of $H$, the point of intersection of $\\overline{FG}$ and $\\overline{AE}$. [asy]\nsize(180); defaultpen(linewidth(.7pt)+fontsiz... |
[
{
"from": "human",
"value": "Find the sum of the reciprocals of the divisors of $32$."
},
{
"from": "gpt",
"value": "We are asked to find the **sum of the reciprocals of the divisors of 32**.\n\n---\n\n### Step 1: Find all the divisors of 32\n\nFirst, factorize 32:\n\n$$\n32 = 2^5\n$$\n\nThe div... |
[
{
"from": "human",
"value": "In a rectangular garden, there are 26 rows of trees, each with 33 trees. If the gardener decides to add 67 more trees to each row, how many trees will there be in total in the garden?"
},
{
"from": "gpt",
"value": "We are given the following information:\n\n- There a... |
[
{
"from": "human",
"value": "Find the largest possible degree of the polynomial $p(x)$ such that the graph of the rational function $\\frac{p(x)}{x^3-2x^2+3x+1}$ has a slant asymptote."
},
{
"from": "gpt",
"value": "We are given a rational function:\n\n$$\n\\frac{p(x)}{x^3 - 2x^2 + 3x + 1}\n$$\n... |
[
{
"from": "human",
"value": "The water level in a cylindrical tank increases over the course of a week due to rainfall and decreases due to evaporation. The level is measured on Sundays, Mondays, Wednesdays, and Fridays. On Sunday, the level was 3 meters. On Monday, it rose by 1 meter. By Wednesday, it had ... |
[
{
"from": "human",
"value": "The radius of a cone is $q$ units and the radius of a cylinder is $3q$ units. What is the ratio of the volume of the cone to the volume of the cylinder?"
},
{
"from": "gpt",
"value": "We are given:\n\n- Radius of the **cone** = $ q $ units \n- Radius of the **cylind... |
[
{
"from": "human",
"value": "A bookstore is offering a discount on bulk purchases of paperbacks. If you buy 50 paperbacks or more, the cost per book is reduced by 15%. How many paperbacks can you buy with a budget of $\\$300$if the original price per book is $\\$4.95$?"
},
{
"from": "gpt",
"valu... |
[
{
"from": "human",
"value": "An ellipse is inscribed in a rectangle with sides of length $a$ and $b$. Three consecutive points on the ellipse have distances to a vertex of the rectangle equal to $\\frac{a}{4}$, $\\frac{b}{4}$, and $\\frac{\\sqrt{a^2+b^2}}{4}$. Find the ratio of the semi-major axis to the se... |
[
{
"from": "human",
"value": "Tom pulled into a food truck festival and decided to try a few things. He ordered an 8-dollar BBQ pulled pork sandwich and his daughter asked for a 3-dollar hot dog, so he added that to the order. Tom then decided to get some sides and ordered a 5-dollar bucket of fries and two ... |
[
{
"from": "human",
"value": "In isosceles triangle $\\triangle ABC$ we have $AB=AC=4$. The altitude from $B$ meets $\\overline{AC}$ at $H$. If $AH=3(HC)$ then determine $BC$.\n\nAnother problem similar to this one:\n\nIn isosceles triangle $\\triangle ABC$ we have $AB=AC=5$. The altitude from $B$ meets $... |
[
{
"from": "human",
"value": "Determine the sum of the series\n$$S=\\sum_{n=0}^\\infty\\frac{2}{G(2n+1)},$$\nwhere the function $G(n)$ is defined recursively as $G(0)=1$, $G(1)=2$, and $G(n)=G(n-1)+G(n-2)+n$ for $n\\ge 2$."
},
{
"from": "gpt",
"value": "We are given a recursive function $ G(n) $ ... |
[
{
"from": "human",
"value": "A pair of fair dice is rolled three times. What is the probability that the sum of the numbers shown on the dice is 7 exactly two times out of the three rolls? Express your answer as a common fraction."
},
{
"from": "gpt",
"value": "We are given that a **pair of fair... |
[
{
"from": "human",
"value": "Solve for $y$ in the equation $2y + 5 = 11y - 3$. Then, find the value of $y^2 + 2y - 1$."
},
{
"from": "gpt",
"value": "We are given the equation:\n\n$$\n2y + 5 = 11y - 3\n$$\n\n### Step 1: Solve for $ y $\n\nSubtract $ 2y $ from both sides:\n\n$$\n5 = 9y - 3\n$$\n\... |
[
{
"from": "human",
"value": "The sum of three consecutive even numbers is 18. What is the middle number?"
},
{
"from": "gpt",
"value": "We are given that the **sum of three consecutive even numbers is 18**, and we are to find the **middle number**.\n\n---\n\n### Step 1: Represent the numbers\n\n... |
[
{
"from": "human",
"value": "Given the points $(x_1,y_1)$ and $(x_2,y_2)$, find the number of integer values of $n$ for which the point $(x_1+y_1n,y_2+nx_2)$ lies inside the rectangle defined by the points $(0,0)$, $(4,0)$, $(4,5)$, and $(0,5)$."
},
{
"from": "gpt",
"value": "We are given two po... |
[
{
"from": "human",
"value": "A bag contains 10 red balls and 20 blue balls. A ball is drawn at random and then replaced. This process is repeated 5 times. What is the probability that at least 3 consecutive draws result in red balls?"
},
{
"from": "gpt",
"value": "We are given:\n\n- A bag contai... |
[
{
"from": "human",
"value": "How many non-similar rectangles with integer side lengths have an area of 24 square units?"
},
{
"from": "gpt",
"value": "We are asked to find how many **non-similar rectangles** with **integer side lengths** have an **area of 24 square units**.\n\n---\n\n### Step 1:... |
[
{
"from": "human",
"value": "Find the value of $x$ such that $\\sqrt{x - 2} = 8$.\n\nAnother problem inspired by this one:\n\nA water tank can hold 1000 liters of water. Water is flowing into the tank at a rate of 10 liters per minute, and at the same time, water is leaking out of the tank at a rate of 2 li... |
[
{
"from": "human",
"value": "Four points $A,$ $B,$ $C,$ and $D$ are on a circle, as shown. The point $O$ is the center of the circle. The angle subtended by the arc $BC$ at the center is $140^\\circ$ and the angle subtended by the arc $AD$ at the center is $100^\\circ.$ If the reflex angle at $A$ is $x^\\ci... |
[
{
"from": "human",
"value": "Alex purchases 8 dozen carnations. Each carnation costs $4. As a bulk buyer, he is entitled to a 25% discount on the total price. How much will he pay?"
},
{
"from": "gpt",
"value": "Let's break the problem down step by step:\n\n---\n\n### **Step 1: Calculate the tot... |
[
{
"from": "human",
"value": "If 3 printers in an office can produce 15 documents per minute, how many printers would be needed to produce 75 documents per minute?"
},
{
"from": "gpt",
"value": "We are given:\n\n- **3 printers** can produce **15 documents per minute**.\n- We want to find out how ... |
[
{
"from": "human",
"value": "Ashley participates in a large tournament. The tournament consists of 3 rounds, with players eliminated at the end of each round. 20 players are eliminated after the first round. A quarter of the remaining players are eliminated after the second round. Half of the remaining play... |
[
{
"from": "human",
"value": "Let $D,$ $E,$ and $F$ be constants such that the equation \\[\\frac{(x+E)(Dx+42)}{(x+F)(x+11)} = 3\\]has infinitely many solutions for $x.$ For these values of $D,$ $E,$ and $F,$ it turns out that there are only finitely many values of $x$ which are not solutions to the equation... |
[
{
"from": "human",
"value": "Mr. Music has a collection of 15 vinyl records, each with a unique cover art. He wants to create a playlist by arranging the records in a specific order. However, he has a peculiar condition: he wants to alternate between records with red covers and records with blue covers. If ... |
[
{
"from": "human",
"value": "Let $f(x) = \\sin x$ and $g(f(x)) = 2\\cos^2 x + 1.$ Find $g\\left(\\frac{\\sqrt{3}}{2}\\right).$"
},
{
"from": "gpt",
"value": "We are given:\n\n- $ f(x) = \\sin x $\n- $ g(f(x)) = 2\\cos^2 x + 1 $\n- We are asked to find: $ g\\left(\\frac{\\sqrt{3}}{2}\\right) $\n\... |
[
{
"from": "human",
"value": "The equation $\\frac{x-3}{x+5} = \\frac{y-2}{y-7}$ is equivalent to the equation $ax+by+c=0$, where $a, b, c$ are nonzero constants. Find $a+b+c$."
},
{
"from": "gpt",
"value": "We are given the equation:\n\n$$\n\\frac{x - 3}{x + 5} = \\frac{y - 2}{y - 7}\n$$\n\nWe a... |
[
{
"from": "human",
"value": "A rectangle is inscribed in a circle with radius $r$, such that the length of one of the rectangle's diagonals is $2r-1$. Find the length of the other diagonal of the rectangle."
},
{
"from": "gpt",
"value": "We are given:\n\n- A **rectangle** is inscribed in a **cir... |
[
{
"from": "human",
"value": "A ship is sailing on a course that takes it from the origin (0,0) to the point (x,y) = (6,8). However, due to strong winds, the ship is being blown off course, resulting in a constant angular deviation of 20 degrees from its intended path. Determine the new point (x',y') that th... |
[
{
"from": "human",
"value": "A palindrome is a number that reads the same forwards and backwards. A three-digit palindrome in base 10, $ABA_{10}$, is expressed as $CD_C$ in base $C$. Find the smallest base $C$ for which such a palindrome exists."
},
{
"from": "gpt",
"value": "We are given a **th... |
[
{
"from": "human",
"value": "Two circles have radii $r_1$ and $r_2$ respectively. The sum of their areas is $16\\pi$ and the difference of their circumferences is $4\\pi$. What is the positive difference between the radii of the two circles?"
},
{
"from": "gpt",
"value": "We are given:\n\n- The ... |
[
{
"from": "human",
"value": "Given the triangle with vertices at (0, 0), (6, 0), and (0, 8), find the equation of the line that goes through the centroid of the triangle and is perpendicular to the line segment between the points (3, 2) and (9, 4)."
},
{
"from": "gpt",
"value": "We are given a t... |
[
{
"from": "human",
"value": "Three dice are rolled simultaneously. What is the probability that the sum of the numbers obtained on the first two dice is greater than the number obtained on the third die?"
},
{
"from": "gpt",
"value": "We are given that **three dice are rolled simultaneously**, a... |
[
{
"from": "human",
"value": "Martin eats 1/2 cup of berries every day. The grocery store is selling a package of berries (1 cup per pack) for $2.00. How much will he spend on berries in a 30 day period?"
},
{
"from": "gpt",
"value": "Let's break the problem down step by step:\n\n---\n\n### **S... |
[
{
"from": "human",
"value": "There exists a scalar $m$ such that for any vectors $\\mathbf{u},$ $\\mathbf{v},$ and $\\mathbf{w}$ such that $\\mathbf{u} + \\mathbf{v} + \\mathbf{w} = \\mathbf{0},$ the equation\n\\[m (\\mathbf{v} \\times \\mathbf{w}) + \\mathbf{w} \\times \\mathbf{u} + \\mathbf{u} \\times \\m... |
[
{
"from": "human",
"value": "Tom wants to put his books into $X$ boxes, each containing the same number of books, $Y$. Each box will have more than one book and no box will have all the books. If there are 17 possible values for $Y$ given all of the restrictions, what is the smallest number of books he coul... |
[
{
"from": "human",
"value": "In a certain geometric sequence, the sum of the first three terms is 21 and the sum of the first six terms is 63. Find the eighth term of the sequence."
},
{
"from": "gpt",
"value": "We are given a **geometric sequence** with the following information:\n\n- The **sum... |
[
{
"from": "human",
"value": "What is the probability of drawing four cards from a standard deck of 52 cards and getting four cards of different suits? Express your answer as a common fraction."
},
{
"from": "gpt",
"value": "We are asked to find the probability of drawing **four cards of differen... |
[
{
"from": "human",
"value": "What value of $x$ will give the maximum value for $x^3- 12x^2 + 36x - 1$?"
},
{
"from": "gpt",
"value": "We are given the function:\n\n$$\nf(x) = x^3 - 12x^2 + 36x - 1\n$$\n\nWe are asked to find the value of $ x $ that gives the **maximum value** of this function.\n... |
[
{
"from": "human",
"value": "A geometric series has a common ratio of 1/5 and a sum of 45. What is the first term of the series?"
},
{
"from": "gpt",
"value": "We are given:\n\n- A **geometric series** with a **common ratio** $ r = \\frac{1}{5} $\n- The **sum of the series** is $ S = 45 $\n- We... |
[
{
"from": "human",
"value": "Find the median of all solutions for $y$ when $y^4 - 2y^3 - 5y^2 = 0$."
},
{
"from": "gpt",
"value": "We are given the equation:\n\n$$\ny^4 - 2y^3 - 5y^2 = 0\n$$\n\n### Step 1: Factor the equation\n\nFirst, factor out the greatest common factor (GCF), which is $ y^2 ... |
[
{
"from": "human",
"value": "Piravena must make a trip from $A$ to $B,$ then from $B$ to $C,$ then from $C$ to $A.$ Each of these three parts of the trip is made entirely by bus or entirely by airplane. The cities form a right-angled triangle as shown, with $C$ a distance of $3000\\text{ km}$ from $A$ and w... |
[
{
"from": "human",
"value": "Let $f(x)=\\{x\\}^2$, where $\\{x\\}$ denotes the fractional part of $x$. Find the largest positive integer $n$ such that the equation $nf(x^2f(x^2))=x^2$ has at most 100 real solutions.\n\nNote: the fractional part of $x$ is a real number $y=\\{x\\}$ such that $0\\le y<1$ and $... |
[
{
"from": "human",
"value": "In triangle $ABC$, $\\angle B$ is a right angle, $AB=6$ units, and $BC=4$ units. A circle with center $D$ and radius $r$ is drawn such that it passes through points $A$ and $C$. What is the value of $r$ in simplest radical form?"
},
{
"from": "gpt",
"value": "We are ... |
[
{
"from": "human",
"value": "Six people, A, B, C, D, E, and F, are to be seated around a round table. Each person has a preference for the two people they would like to sit next to, and these preferences are given as follows:\n\nA: B, C\nB: A, D\nC: A, F\nD: B, E\nE: D, F\nF: C, E\n\nDetermine the number of... |
[
{
"from": "human",
"value": "What is the sum of the squares of the $x$-intercepts of the graph of $y = (x^2+3x+1)(x^3-x-2)$?"
},
{
"from": "gpt",
"value": "We are given the function:\n\n$$\ny = (x^2 + 3x + 1)(x^3 - x - 2)\n$$\n\nWe are asked to find the **sum of the squares of the $x$-intercepts... |
[
{
"from": "human",
"value": "Two bags contain different numbers of red and white marbles. Bag A contains 3 red marbles and 5 white marbles, while bag B contains 9 red marbles and M white marbles. A single marble is randomly drawn from each bag. The probability that both marbles are of the same color is 0.55... |
[
{
"from": "human",
"value": "How many inches are there in the perimeter of a square that has a diagonal of 24.5 feet?"
},
{
"from": "gpt",
"value": "We are given that the **diagonal of a square is 24.5 feet**, and we are asked to find the **perimeter of the square in inches**.\n\n---\n\n### Step... |
[
{
"from": "human",
"value": "Let's consider the positive integers whose decimal representation contains only the digits 1, 5 or 6, or, more precisely, all the positive integers whose decimal representation is composed only of zeros and 1s, only of zeros and 5s, only of zeros and 6s. Find the number of such ... |
[
{
"from": "human",
"value": "A group of $n$ students take a multiple-choice exam with $q$ questions. For each question, each student can choose one of two possible answers. After the exam, it turns out that for every question, exactly half of the students have chosen each answer. However, if the students' a... |
[
{
"from": "human",
"value": "Find all real numbers $b$ such that the roots of the polynomial\n$$x^3 - 4x^2 + 15x + b$$\nform an arithmetic progression and are not all real."
},
{
"from": "gpt",
"value": "We are given the cubic polynomial:\n\n$$\nP(x) = x^3 - 4x^2 + 15x + b\n$$\n\nWe are to **fin... |
[
{
"from": "human",
"value": "In a certain company, there are 15 employees, including a group of 4 siblings: Alex, Ben, Charlie, and David. The company is planning to send a team of 7 employees to a conference. However, due to budget constraints, they can only send at most 2 of the siblings. In how many ways... |
[
{
"from": "human",
"value": "A certain number is called \"influential\" if it is equal to the product of its distinct prime factors. For example, 6 is influential because 6 = 2 * 3, where 2 and 3 are distinct prime factors of 6. What is the largest influential number less than 1000?"
},
{
"from": "g... |
[
{
"from": "human",
"value": "Let $\\alpha$ and $\\beta$ be the roots of the quadratic equation $37x^2+111x+15=0$. Find the value of $(\\alpha^3+\\beta^3)/\\alpha\\beta$."
},
{
"from": "gpt",
"value": "We are given a quadratic equation:\n\n$$\n37x^2 + 111x + 15 = 0\n$$\n\nLet the roots be $\\alph... |
[
{
"from": "human",
"value": "Find the real value of x if $\\sqrt{x^4}=|x^2|+2x$."
},
{
"from": "gpt",
"value": "We are given the equation:\n\n$$\n\\sqrt{x^4} = |x^2| + 2x\n$$\n\n---\n\n### Step 1: Simplify the left-hand side\n\nWe know that:\n\n$$\n\\sqrt{x^4} = |x^2|\n$$\n\nBecause $x^4$ is alw... |
[
{
"from": "human",
"value": "Find the number of integers $k$ between 2 and 12 (inclusive) for which the decimal representation of $\\frac{1}{k}$ has a finite number of digits after the decimal point."
},
{
"from": "gpt",
"value": "We are asked to find the number of integers $ k $ between 2 and 1... |
[
{
"from": "human",
"value": "Compute $(-64)\\div (-32)$.\n\nAnother problem:\n\nA rectangular garden has an area of 24 square meters. If the length of the garden is 3 times the width, what is the width of the garden?"
},
{
"from": "gpt",
"value": "Let's solve each problem step by step.\n\n---\n\... |
[
{
"from": "human",
"value": "A 4-by-4 grid of unit squares is to be colored with either green or yellow. Each square has an equal chance of being colored with either color. What is the probability that no 3-by-3 sub-grid is entirely yellow? Express your answer as the sum of the numerator and denominator of ... |
[
{
"from": "human",
"value": "A city has 15 districts, and each district has a direct bus route to exactly four other districts. If no two districts have the same set of bus routes, what is the minimum number of bus routes required to connect all districts?"
},
{
"from": "gpt",
"value": "We are g... |
[
{
"from": "human",
"value": "Tom is filling a large tank with a pump that fills 30 gallons/hour. After 4 hours, the tank is not yet full, so he decides to add another pump that fills 60 gallons/hour. If the tank is full after 2 more hours, how many gallons can Tom's tank hold?"
},
{
"from": "gpt",
... |
[
{
"from": "human",
"value": "Let $a$ be an even multiple of $289$. Find the greatest common divisor of $a^2 + 4a + 5$ and $a + 2$."
},
{
"from": "gpt",
"value": "We are given that $ a $ is an **even multiple of 289**, and we are to find the **greatest common divisor** (GCD) of the two expression... |
[
{
"from": "human",
"value": "Mr. Thompson pays his gym membership fees biannually at $249. How much does he pay in a year for his gym membership?"
},
{
"from": "gpt",
"value": "We are told that Mr. Thompson pays his gym membership fees **biannually** at **$249**.\n\n---\n\n### Step 1: Understand... |
[
{
"from": "human",
"value": "In a binary tree, each node has at most two children (left and right).\nFind the number of distinct binary trees with 7 nodes, where each node represents a person, and the left child is always the elder sibling.\nAssume that the tree is not necessarily balanced, and the root nod... |
[
{
"from": "human",
"value": "What is the greatest prime factor of 45?"
},
{
"from": "gpt",
"value": "To find the **greatest prime factor** of **45**, we follow these steps:\n\n---\n\n### Step 1: Prime Factorization of 45\n\nWe start by breaking down 45 into its prime factors.\n\n- 45 is divisibl... |
[
{
"from": "human",
"value": "Arrange the four distinct red books and six distinct blue books on a shelf so that no two books of the same color are adjacent. How many ways are there to do this?"
},
{
"from": "gpt",
"value": "We are given:\n\n- 4 **distinct red books**\n- 6 **distinct blue books**... |
[
{
"from": "human",
"value": "The graphs of $y = x^3 - 2x^2 - x + 1$ and $x + 3y = 2$ intersect in the points $(x_1,y_1),$ $(x_2,y_2),$ and $(x_3,y_3).$ If $x_1x_2x_3 = A$ and $y_1y_2y_3 = B,$ compute the ordered pair $(A,B).$"
},
{
"from": "gpt",
"value": "We are given two equations:\n\n1. $ y ... |
[
{
"from": "human",
"value": "Find the number of integer values of $x$ such that $x^2 + 2x + 1$ is greater than 4 and less than 9."
},
{
"from": "gpt",
"value": "We are given the inequality:\n\n$$\n4 < x^2 + 2x + 1 < 9\n$$\n\n### Step 1: Simplify the expression\nNotice that:\n\n$$\nx^2 + 2x + 1 =... |
[
{
"from": "human",
"value": "Let $x$ and $y$ be real numbers such that $\\log_{10} x+\\log_{10} (x+y)=1$ and $\\log_{10} y+\\log_{10} (x+y)=1$. Find $\\log_{10} (xy)$."
},
{
"from": "gpt",
"value": "We are given two logarithmic equations involving real numbers $ x $ and $ y $:\n\n$$\n\\log_{10} ... |
[
{
"from": "human",
"value": "What is the smallest positive integer $n$ such that $2^n - 1$ is not a prime number?"
},
{
"from": "gpt",
"value": "We are asked to find the **smallest positive integer** $ n $ such that $ 2^n - 1 $ is **not a prime number**.\n\n---\n\n### Step 1: Understand the expr... |
[
{
"from": "human",
"value": "The vertices of triangle $ABC$ are $A = (3,-1,2),$ $B = (1,2,-4),$ and $C = (-1,1,2)$. Find the coordinates of the point $D$ on the line $AC$ such that $ABCD$ is a rhombus."
},
{
"from": "gpt",
"value": "We are given the coordinates of the vertices of triangle $ ABC... |
[
{
"from": "human",
"value": "Express $0.525252...$ as a fraction."
},
{
"from": "gpt",
"value": "We are given the repeating decimal:\n\n$$\n0.525252\\ldots\n$$\n\nThis is a **repeating decimal** with the repeating block **\"52\"**. Let's express this as a fraction step by step.\n\n---\n\n### Ste... |
[
{
"from": "human",
"value": "A rectangular garden measures 10 meters by 5 meters. If the length is increased by x meters and the width is decreased by the same amount, the area of the garden remains the same. What is the value of x?"
},
{
"from": "gpt",
"value": "We are given a rectangular garde... |
[
{
"from": "human",
"value": "Let \\[g(x) =\n\\begin{cases}\n\\sqrt{4-x^2} &\\text{if } x\\ge 0, \\\\\nb-x &\\text{if } x<0.\n\\end{cases}\n\\]Find $b$ if the graph of $y=g(x)$ is symmetric about the $y$-axis (which means if $(x,y)$ is on the graph, then $(-x,y)$ is also on the graph)."
},
{
"from": ... |
[
{
"from": "human",
"value": "At a certain university, the number of students who choose to major in biology is decreasing by 10% each year. If 120 students chose to major in biology this year, and 144 students chose to major in biology last year, what was the number of students who chose to major in biology... |
[
{
"from": "human",
"value": "In the diagram, $PQR$ is a straight line and $QS$ is perpendicular to $PR$. The measure of $\\angle QPR$ is $x^\\circ$, the measure of $\\angle QRS$ is $y^\\circ$, and the measure of $\\angle PSQ$ is $z^\\circ$. If $x = 2y$ and $z = 3x$, determine the measure of $\\angle RPS$.\n... |
[
{
"from": "human",
"value": "In quadrilateral $ABCD$, $AB = BC = CD$ and $\\angle ABC = \\angle BCD = 90^\\circ$. What is the measure in degrees of $\\angle DAB$?"
},
{
"from": "gpt",
"value": "We are given a quadrilateral $ABCD$ with the following properties:\n\n- $AB = BC = CD$\n- $\\angle ABC... |
[
{
"from": "human",
"value": "In the Cartesian plane, a point $P$ is chosen randomly inside the unit circle centered at the origin. Two lines are then randomly selected from the set of all lines passing through $P$ and the center of the circle. What is the probability that the angle between these two lines i... |
[
{
"from": "human",
"value": "Julian cycled 36 miles on Sunday, carrying a 40-pound bicycle bag. He is exactly 5 feet 8 inches tall. He spent 8 hours cycling. The temperature during the day was 70 degrees Fahrenheit. What was Julian's average cycling pace, in miles per hour?"
},
{
"from": "gpt",
... |
[
{
"from": "human",
"value": "A machine generates a random integer between 1 and 100 inclusive, and then checks if it is divisible by 4 or 5. If it is divisible by 4, the machine wins 4 points; if it is divisible by 5 but not 4, the machine wins 5 points; otherwise, the machine loses 2 points. What is the ex... |
[
{
"from": "human",
"value": "Rahul needs to remove soil from a rectangular garden bed in his backyard. The garden bed measures 12 feet long and 3 feet wide. If the soil is $\\frac{2}{3}$ foot deep, what volume of soil, in cubic feet, does Rahul have to remove?"
},
{
"from": "gpt",
"value": "To f... |
[
{
"from": "human",
"value": "A bookstore has 12 identical shelves, and the owner wants to display 20 identical books on the shelves. However, there is a constraint: each shelf can hold a maximum of 3 books. How many ways are there to distribute the books among the shelves, considering that the order of the ... |
[
{
"from": "human",
"value": "In the diagram, $ABCD$ is a cyclic quadrilateral where $\\angle B = 3x^\\circ$ and $\\angle D = x^\\circ.$ Find the measure of $\\angle C.$"
},
{
"from": "gpt",
"value": "We are given that $ABCD$ is a **cyclic quadrilateral**, and the angles are:\n\n- $\\angle B = 3x... |
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