conversations listlengths 2 2 |
|---|
[
{
"from": "human",
"value": "Three friends - Charlie, David, and Emily - are invited to a meeting that starts at a random time between 3:00 and 5:00. Each friend has a 30% chance of being late to the meeting, and if late, they arrive at a random time between 15 minutes and 30 minutes after the meeting start... |
[
{
"from": "human",
"value": "Let $f(x)$ be a polynomial function of degree 3 with leading coefficient 1 and the other coefficients $a$ and $b$, and let $c$ be the average value of the function on the interval $[0, 1]$. For what values of $a$ and $b$ is the value $c$ not in the range of $f(x)$? Express your ... |
[
{
"from": "human",
"value": "Convert $212121_3$ to base 5."
},
{
"from": "gpt",
"value": "We are given the number $ 212121_3 $ and we are to convert it to base 5.\n\n---\n\n### **Step 1: Convert from base 3 to base 10 (decimal)**\n\nWe will convert $ 212121_3 $ to base 10 by expanding it using p... |
[
{
"from": "human",
"value": "In the Cartesian plane, consider the circle centered at the origin with radius $r$. Three points $A$, $B$, and $C$ lie on the circle, and the line segments $AB$, $BC$, and $CA$ are initially equal in length. The points $A$, $B$, and $C$ begin moving counterclockwise along the ci... |
[
{
"from": "human",
"value": "Find the two-digit positive integer $n$ such that the base-10 number formed by the digits of $3n$ in reverse order is divisible by $n$."
},
{
"from": "gpt",
"value": "We are given a two-digit positive integer $ n $, and we are told that the **base-10 number formed by... |
[
{
"from": "human",
"value": "Consider the complex equation\n\n$$z^4+16z^2+100=0$$\n\nwhere $z$ is a complex number. Find the product of the solutions to this equation that lie in the third quadrant of the complex plane."
},
{
"from": "gpt",
"value": "We are given the complex equation:\n\n$$\nz^4... |
[
{
"from": "human",
"value": "Determine all positive integers $a, b, c, d$ such that\n\\[(\\sqrt[3]{a}+\\sqrt[3]{b}+\\sqrt[3]{c})(\\sqrt[3]{a}+\\sqrt[3]{b}-\\sqrt[3]{c})= \\sqrt[3]{4d^2}.\\]"
},
{
"from": "gpt",
"value": "We are given the equation:\n\n$$\n(\\sqrt[3]{a} + \\sqrt[3]{b} + \\sqrt[3]... |
[
{
"from": "human",
"value": "Find the smallest possible distance between a point on the parabola $y=x^2$ and a point on the circle $x^2+y^2=1$."
},
{
"from": "gpt",
"value": "We are asked to find the **smallest possible distance** between a point on the **parabola** $ y = x^2 $ and a point on th... |
[
{
"from": "human",
"value": "A group of friends want to make a road trip across the country in several cars. Each car has a different fuel tank capacity, and the friends have a limited amount of fuel available for the entire trip. If they want to divide the fuel equally among the cars, how much fuel can eac... |
[
{
"from": "human",
"value": "Let $a,$ $b,$ and $c$ be positive real numbers such that $a + b + c = 1.$ Find the maximum value of\n\\[\\frac{(a + b)(b + c)(c + a)}{a^2b^2c^2}.\\]"
},
{
"from": "gpt",
"value": "We are given that $ a, b, c $ are **positive real numbers** such that:\n\n$$\na + b + ... |
[
{
"from": "human",
"value": "Problem:\nMy friend Emily is very peculiar. She only likes numbers that are palindromes. How many different 3-digit numbers are possible that Emily likes?\n\nSolution:\nSince a palindrome reads the same forwards and backwards, we can count the number of 3-digit palindromes by co... |
[
{
"from": "human",
"value": "A square table is filled with 16 equal balls, with each ball being colored red, blue, or white. Determine the number of colorings of the table such that each row and each column contains exactly two red balls, two blue balls, and two white balls."
},
{
"from": "gpt",
... |
[
{
"from": "human",
"value": "In $\\triangle ABC$, $AB = 10$ and $AC = 15$. $D$ is a point on $\\overline{BC}$ such that $BD = 3CD$. What is the length of median $\\overline{AD}$? Express your answer as a decimal to the nearest tenth."
},
{
"from": "gpt",
"value": "We are given:\n\n- Triangle $ \... |
[
{
"from": "human",
"value": "Samantha wants to buy some books at a store. The romance novels are $8 each, the science fiction books are $12 each, the mystery novels are $6 each, and the biography books are $9 each. She wants to buy 3 of each, but she only has 80 dollars. How much money is she short?"
},
... |
[
{
"from": "human",
"value": "Emily has a 30-cookie pack and eats 2 cookies for dessert every evening this month. On Friday, she had friends over and they ate 4 cookies together. How many cookies are left from the pack Emily started with?"
},
{
"from": "gpt",
"value": "Let's break the problem dow... |
[
{
"from": "human",
"value": "The functions $f(x)$ and $g(x)$ are inverses of each other. If the points $(a, b)$, $(c, d)$, and $(e, f)$ are on the graph of $f(x)$ and the points $(b, a)$, $(d, c)$, and $(f, e)$ are on the graph of $g(x)$, what is the value of $f^{-1}(d)$?"
},
{
"from": "gpt",
"v... |
[
{
"from": "human",
"value": "Express the recurring decimal 0.27 as a fraction."
},
{
"from": "gpt",
"value": "We are given the recurring decimal:\n\n$$\n0.27\\overline{27}\n$$\n\nThis means the digits **27** repeat indefinitely. So the decimal is:\n\n$$\n0.27272727\\ldots\n$$\n\n---\n\n### Step ... |
[
{
"from": "human",
"value": "Problem:\nA ship is sailing in the direction of the vector $\\vec{v} = (3, 4)$ with a speed of 5 miles per hour. At time $t = 0$, the ship is at the origin. Find the polar coordinates of the ship's position at time $t = 2$ hours. Express your answer in the form $(r, \\theta)$, w... |
[
{
"from": "human",
"value": "A right circular cone is tangent to the planes $x+y+z=5$ and $x+y+z=-3.$ The vertex of the cone lies on the line $x-2y+3z=0.$ Find the coordinates of the vertex."
},
{
"from": "gpt",
"value": "We are given:\n\n- A **right circular cone** is **tangent** to the planes:... |
[
{
"from": "human",
"value": "Convert the complex number $5(\\cos(3\\pi/2)+i\\sin(3\\pi/2))$ into the standard form $a+bi$."
},
{
"from": "gpt",
"value": "We are given the complex number in **polar form**:\n\n$$\n5\\left(\\cos\\left(\\frac{3\\pi}{2}\\right) + i\\sin\\left(\\frac{3\\pi}{2}\\right)... |
[
{
"from": "human",
"value": "Problem:\nIn a bag, there are 18 red marbles, 21 blue marbles, and 5 green marbles. 3 marbles are chosen at random from the bag. What is the probability that at least one red marble and at least one blue marble will be chosen?"
},
{
"from": "gpt",
"value": "We are gi... |
[
{
"from": "human",
"value": "Let $g$ be a function mapping positive integers to positive integers, such that\n\n(i) $g$ is increasing (i.e. $g(n + 1) > g(n)$ for all positive integers $n$)\n(ii) $g(m + n) = g(m) + g(n)$ for all positive integers $m$ and $n,$ and\n(iii) if $m \\neq n$ and $m^2 + n^2$ is a pe... |
[
{
"from": "human",
"value": "Consider the function\n\\[g(x) = \\left\\{\n\\begin{array}{cl}\n\\sqrt{-x} & \\text{if } x \\le 0, \\\\\n1-x & \\text{if } x > 0.\n\\end{array}\n\\right.\\]\nDetermine the range of values of $k$ such that the equation $g(g(x)) = k$ has exactly three distinct real solutions."
}... |
[
{
"from": "human",
"value": "Given the complex number $z = a + bi$, find the value of $(z + i)^2 + 2(z + i)(2 - z) + (2 - z)^2$."
},
{
"from": "gpt",
"value": "We are given a complex number:\n\n$$\nz = a + bi\n$$\n\nWe are asked to evaluate the expression:\n\n$$\n(z + i)^2 + 2(z + i)(2 - z) + (2... |
[
{
"from": "human",
"value": "Solve $\\arctan x + \\arctan \\frac{1}{x} = \\arccos \\frac{x^2 - 1}{x^2 + 1}$."
},
{
"from": "gpt",
"value": "We are given the equation:\n\n$$\n\\arctan x + \\arctan \\frac{1}{x} = \\arccos \\frac{x^2 - 1}{x^2 + 1}\n$$\n\nWe are to **solve** this equation for $ x $,... |
[
{
"from": "human",
"value": "A bottle with a volume of 1.5 liters has the shape of a right circular cylinder with a height of 10 cm and an outer radius of 8 cm. Water is poured into the bottle at a rate of 0.1 liters per second until it reaches the brim. Determine the rate at which the surface area of the w... |
[
{
"from": "human",
"value": "Find the number of three-digit positive integers that are not divisible by 5 and have a sum of digits equal to 12."
},
{
"from": "gpt",
"value": "We are asked to find the number of **three-digit positive integers** that:\n\n1. Are **not divisible by 5**\n2. Have a **... |
[
{
"from": "human",
"value": "Let $a$, $b$, and $c$ be the roots of the equation $x^3-3x^2+2x-1=0$. Find the value of $\\frac{a^2}{bc} + \\frac{b^2}{ca} + \\frac{c^2}{ab}$."
},
{
"from": "gpt",
"value": "We are given that $ a, b, c $ are the roots of the cubic equation:\n\n$$\nx^3 - 3x^2 + 2x - 1... |
[
{
"from": "human",
"value": "Let $p(x)$ be a polynomial of degree 2010 such that $p(x) = x^{2010} + x^{2009} + \\ldots + x + 1$. Determine the remainder when $p(x)$ is divided by $x^3 - 1$."
},
{
"from": "gpt",
"value": "We are given a polynomial:\n\n$$\np(x) = x^{2010} + x^{2009} + \\ldots + x ... |
[
{
"from": "human",
"value": "The graph of a function $f(x)$ is shown below.\n\n[asy]\nsize(150);\nreal ticklen=3;\nreal tickspace=2;\n\nreal ticklength=0.1cm;\nreal axisarrowsize=0.14cm;\npen axispen=black+1.3bp;\nreal vectorarrowsize=0.2cm;\nreal tickdown=-0.5;\nreal tickdownlength=-0.15inch;\nreal tickdow... |
[
{
"from": "human",
"value": "A circular pizza has a diameter of 16 inches and a thickness of 1/8 inch. What is the volume of the pizza in cubic inches?"
},
{
"from": "gpt",
"value": "To find the **volume of a circular pizza**, we can treat it as a **cylinder**. The formula for the volume of a cy... |
[
{
"from": "human",
"value": "Michael was watching a TV show, which was aired for 1.5 hours. During this time, there were 3 commercials, which lasted 10 minutes each. How long (in hours) was the TV show itself, not counting commercials?"
},
{
"from": "gpt",
"value": "We are given:\n\n- The total ... |
[
{
"from": "human",
"value": "In triangle $\\triangle ABC$, $AB = 12$, $BC = 17$, and $AC = 15$. Points $D, E, F$ are the midpoints of sides $BC, AC, AB$ respectively. Three circles with centers $O_1, O_2, O_3$ pass through points $D, E, F$ and intersect the sides of the triangle at the following points: $O_... |
[
{
"from": "human",
"value": "A bookstore sold 18 novels on the first day of a promotional event. On the second day, 30 more novels were sold. On the third day, the store sold three times the number of novels it did on the day before. How many novels were sold during the promotional event?"
},
{
"fro... |
[
{
"from": "human",
"value": "What is the probability that a randomly chosen positive integer less than or equal to 100 has a prime factorization that contains at least one prime number greater than 5? Express your answer as a common fraction."
},
{
"from": "gpt",
"value": "We are asked to find t... |
[
{
"from": "human",
"value": "A library has a collection of books, each written by one or more authors. The table below shows the availability of authors to attend a book signing event on different days of the week.\n\n\\begin{tabular}{|l|c|c|c|c|c|}\n\\multicolumn{6}{c}{}\\\\\\hline\n&Mon&Tues&Wed&Thurs&Fri... |
[
{
"from": "human",
"value": "A ray of light is reflected by two mirrors. Each mirror reflects the ray at an angle that is smaller by a fixed number of degrees than the angle of incidence. Let's call this number of degrees $k$. The reflected ray then forms an angle with the incident ray. For any incident ang... |
[
{
"from": "human",
"value": "Find $XY$ in the triangle below.\n\n\n[asy]\nunitsize(1inch);\npair P,Q,R;\nP = (0,0);\nQ= (1,0);\nR = (0,1);\ndraw (P--Q--R--P,linewidth(0.9));\ndraw(rightanglemark(Q,P,R,3));\nlabel(\"$X$\",P,S);\nlabel(\"$Y$\",Q,S);\nlabel(\"$Z$\",R,N);\nlabel(\"$12\\sqrt{2}$\",(Q+R)/2,NE);\n... |
[
{
"from": "human",
"value": "A region $R$ in the complex plane is defined by \\begin{align*}\nR = \\{x + iy: - 1\\le x\\le1, - 1\\le y\\le1\\}.\n\\end{align*}A complex number $z = x + iy$ is chosen uniformly at random from $R$. What is the probability that $\\left(\\frac{1}{2} + \\frac{\\sqrt{3}}{2}i\\right... |
[
{
"from": "human",
"value": "Find the $2 \\times 2$ matrix $\\mathbf{M}$ such that $\\mathbf{M} \\begin{pmatrix} 2 & 0 \\\\ 0 & 2 \\end{pmatrix} = \\begin{pmatrix} 1 & 1 \\\\ 0 & 1 \\end{pmatrix}$ and $\\mathbf{M} \\begin{pmatrix} 1 & 1 \\\\ 1 & 1 \\end{pmatrix} = \\begin{pmatrix} 2 & 2 \\\\ 1 & 1 \\end{pma... |
[
{
"from": "human",
"value": "What is $\\dbinom{n}{2}$ for any positive integer $n$?"
},
{
"from": "gpt",
"value": "We are asked to find the value of the binomial coefficient:\n\n$$\n\\dbinom{n}{2}\n$$\n\n### Step 1: Recall the formula for binomial coefficients\n\nThe binomial coefficient is defi... |
[
{
"from": "human",
"value": "Rationalize the denominator of $\\displaystyle \\frac{1}{\\sqrt{2} - \\sqrt{3} + \\sqrt{7}}$, and write your answer in the form \\[\n\\frac{A\\sqrt{2} + B\\sqrt{3} + C\\sqrt{7} + D\\sqrt{E}}{F},\n\\]where everything is in simplest radical form and the fraction is in lowest terms... |
[
{
"from": "human",
"value": "Find the modulo $7$ remainder of the product $1\\cdot 2\\cdot 3\\cdot 4\\cdot 5\\cdot \\ldots \\cdot 120\\cdot 121\\cdot 122\\cdot 123$ after removing all the numbers that are divisible by $7$ from the product."
},
{
"from": "gpt",
"value": "We are asked to find the ... |
[
{
"from": "human",
"value": "How many ways can 99897 be factored as a product of two prime numbers?"
},
{
"from": "gpt",
"value": "We are asked to find how many ways the number **99897** can be factored as a product of **two prime numbers**.\n\n---\n\n### Step 1: Understand the Problem\n\nWe are... |
[
{
"from": "human",
"value": "A five-digit number is divisible by 16 and its digits add up to 24. The last two digits of the number are the square of the first digit. Find the number."
},
{
"from": "gpt",
"value": "We are given the following conditions for a **five-digit number**:\n\n1. It is **d... |
[
{
"from": "human",
"value": "Find the number of positive factors of 360 that are one more than a perfect square."
},
{
"from": "gpt",
"value": "We are asked to find the **number of positive factors of 360** that are **one more than a perfect square**.\n\n---\n\n### Step 1: Prime Factorization of... |
[
{
"from": "human",
"value": "The graph of $y = f(x)$ is shown below.\n\n[asy]\nunitsize(0.5 cm);\n\nreal func(real x) {\n real y;\n if (x >= -3 && x <= 0) {y = -2 - x;}\n if (x >= 0 && x <= 2) {y = sqrt(4 - (x - 2)^2) - 2;}\n if (x >= 2 && x <= 3) {y = 2*(x - 2);}\n return(y);\n}\n\nint i, n;\n\nfor (i... |
[
{
"from": "human",
"value": "Emily knits scarves for her family members as holiday gifts. Each scarf requires 50 yards of yarn. She has 8 cousins on her mother's side and 2/3 that number on her father's side. Emily also wants to make scarves for her parents. How many yards of yarn will Emily need in total?"... |
[
{
"from": "human",
"value": "A music festival is hosting eight bands on two separate stages. The first stage will feature three rock bands, two pop bands, and one hip-hop band, while the second stage will feature one jazz band and one classical band. If the organizers want to ensure that no two bands of the... |
[
{
"from": "human",
"value": "The sum of two numbers is 72. Their difference is 11. What is the greater of the two numbers?"
},
{
"from": "gpt",
"value": "We are given two pieces of information:\n\n1. The **sum** of two numbers is **72**.\n2. The **difference** between the two numbers is **11**.\... |
[
{
"from": "human",
"value": "Let $a,$ $b,$ $c,$ $d$ be real numbers such that $a + b + c + d = 0,$ and $a \\ge -1,$ $b \\ge -\\frac{1}{2},$ $c \\ge -\\frac{3}{4},$ and $d \\ge -2.$ Find the maximum value of\n\\[\\sqrt{a + 1} + \\sqrt{2b + 1} + \\sqrt{4c + 3} + \\sqrt{d + 2}.\\]"
},
{
"from": "gpt",... |
[
{
"from": "human",
"value": "The height of a building above the ground is 120m. If this height is one-fifth of the vertical distance between the bottom of the foundation and the roof of the building, how deep is the foundation below the ground?"
},
{
"from": "gpt",
"value": "We are given:\n\n- T... |
[
{
"from": "human",
"value": "Determine the remainder when $2^{17} + 2^{18}$ is divided by 27."
},
{
"from": "gpt",
"value": "We are asked to find the **remainder** when $ 2^{17} + 2^{18} $ is divided by 27.\n\n---\n\n### Step 1: Factor the expression\n\n$$\n2^{17} + 2^{18} = 2^{17}(1 + 2) = 2^{1... |
[
{
"from": "human",
"value": "Find the point on the curve defined by $x^2 + y^2 = 1$ that is closest to the point $(2,0)$."
},
{
"from": "gpt",
"value": "We are given the curve defined by the equation:\n\n$$\nx^2 + y^2 = 1\n$$\n\nThis is the **unit circle** centered at the origin $(0, 0)$ with ra... |
[
{
"from": "human",
"value": "In the trapezoid shown, the ratio of the area of triangle $ABD$ to the area of triangle $ACD$ is $5:11$. If $AD + BC = 288$ cm, how long is segment $\\overline{AD}$? [asy]\nimport olympiad; size(150); defaultpen(linewidth(0.8));\npair A = (0,0), B = (5,0), C = (3,2), D = (1,2);\... |
[
{
"from": "human",
"value": "In Texas, the total number of influenza cases was reported as 3500 on a certain day. The number of cases rose by 700 on the second day, with 100 recoveries. On the third day, the total number of new cases jumped to 2000 with 300 recoveries. What's the total number of influenza c... |
[
{
"from": "human",
"value": "In a class of 48 students, 25 are boys and the rest are girls. Two-thirds of the girls wear glasses, and the rest do not. How many girls do not wear glasses?"
},
{
"from": "gpt",
"value": "We are given the following information:\n\n- Total number of students = 48 \n... |
[
{
"from": "human",
"value": "Gary buys 4 chickens. After two years, he has 8 times as many chickens as he started with. If each chicken lays 6 eggs a day, how many eggs does Gary currently collect every week?"
},
{
"from": "gpt",
"value": "Let's break the problem down step by step.\n\n---\n\n###... |
[
{
"from": "human",
"value": "If $x$, $y$, and $z$ are positive numbers satisfying \\[\nx+\\frac{1}{y}=4,\\ \\ \\ y+\\frac{1}{z}=1,\\text{ and }z+\\frac{1}{x}=\\frac{7}{3},\n\\]find the value of $xyz$.\n\nInspired problem:\n\nIf $a$, $b$, and $c$ are positive integers satisfying \\[\na^2+b^2+c^2=50,\\ \\ \\ ... |
[
{
"from": "human",
"value": "Find the number of real solutions to the equation $\\sqrt{x} = -3\\sin x$ in the interval $[0, 2\\pi]$."
},
{
"from": "gpt",
"value": "We are asked to find the number of **real solutions** to the equation:\n\n$$\n\\sqrt{x} = -3\\sin x\n$$\n\nin the interval $[0, 2\\p... |
[
{
"from": "human",
"value": "A group of friends want to plan a dinner party where each person will be seated next to exactly two people they have never met before. If there are 12 people in total, how many different seating arrangements can be made?"
},
{
"from": "gpt",
"value": "We are given a ... |
[
{
"from": "human",
"value": "In the coordinate plane, a point $(x, y)$ is randomly chosen such that $0\\le x, y\\le 5$. What is the probability that the point is inside or on the boundary of the square with vertices $(1, 1), (1, 4), (4, 1)$, and $(4, 4)$? Express your answer as a common fraction."
},
{
... |
[
{
"from": "human",
"value": "Find the matrix $\\mathbf{R}$ such that for any vector $\\mathbf{v}$, $\\mathbf{R} \\mathbf{v}$ is the reflection of $\\mathbf{v}$ across the line $L$ defined by the equation $\\begin{pmatrix} 1 \\\\ 2 \\\\ -1 \\end{pmatrix} \\times \\begin{pmatrix} x \\\\ y \\\\ z \\end{pmatrix... |
[
{
"from": "human",
"value": "A circular cone is inscribed in a sphere with radius 5 cm. The height of the cone is 7 cm. What is the volume of the cone, in cubic centimeters?"
},
{
"from": "gpt",
"value": "We are given:\n\n- A **sphere** with radius $ R = 5 $ cm.\n- A **circular cone** is **inscr... |
[
{
"from": "human",
"value": "The matrix\n\\[\\begin{pmatrix} \\frac{4}{29} & -\\frac{10}{29} \\\\ -\\frac{10}{29} & \\frac{25}{29} \\end{pmatrix}\\]\ncorresponds to projecting onto a certain vector $\\begin{pmatrix} x \\\\ y \\end{pmatrix}.$ Find $\\frac{y}{x}.$\n\nWrite another problem inspired by this on... |
[
{
"from": "human",
"value": "What is the value of $x$ in the equation $4^{4x}+4^{4x}+4^{4x}+4^{4x}=2^{10}$?"
},
{
"from": "gpt",
"value": "We are given the equation:\n\n$$\n4^{4x} + 4^{4x} + 4^{4x} + 4^{4x} = 2^{10}\n$$\n\n### Step 1: Simplify the left-hand side\n\nThere are **four** terms of $ ... |
[
{
"from": "human",
"value": "Point $A$ is on the circumference of a circle centered at $O$, and $B$ is the midpoint of $\\overline{OA}$. The circular sector formed by points $A$, $B$, and $O$ is then rolled along a straight line. When the circle first returns to its original orientation, what is the ratio o... |
[
{
"from": "human",
"value": "After collecting all the old electronics in their house, Lauryn made $2000 from selling the items on eBay. If her friend Aurelia also made 70% of what she sold on eBay after selling her used electronics, calculate the total amount of money the two friends made on eBay."
},
{... |
[
{
"from": "human",
"value": "I have to memorize 300 phrases for my French exam, and I want to achieve a grade of at least 90%. Assuming that I will recall the phrases I learn correctly, and that guessing will not yield any points, what is the minimum number of phrases I need to learn to achieve my desired g... |
[
{
"from": "human",
"value": "Samantha walked 3 miles in the morning and then 4 times that distance in the afternoon. How many miles did Samantha walk in total?"
},
{
"from": "gpt",
"value": "Let's break the problem down step by step:\n\n1. **Distance walked in the morning:** \n Samantha walke... |
[
{
"from": "human",
"value": "Let $x$ and $y$ be two complex numbers such that\n$x^2+xy+2018y^2=0$\n$x^3+2017y^3=2018$\nand $xy\\neq 0$.\nDetermine the sum of the possible values of\n$\\ds \\left|\\frac{x}{y}\\right|$."
},
{
"from": "gpt",
"value": "We are given the following equations involving ... |
[
{
"from": "human",
"value": "Let $g(n)$ be the integer closest to $\\log_2 n.$ Find $\\sum_{k=2}^{2000} (-1)^{g(k)}.$"
},
{
"from": "gpt",
"value": "We are given a function $ g(n) $, which is the **integer closest to** $ \\log_2 n $, and we are to compute the sum:\n\n$$\n\\sum_{k=2}^{2000} (-1)^... |
[
{
"from": "human",
"value": "Determine the value of $k$ such that $x^3 + kx^2 + 5x + 6$ is the cube of a binomial."
},
{
"from": "gpt",
"value": "We are given the polynomial:\n\n$$\nx^3 + kx^2 + 5x + 6\n$$\n\nand we are told that it is the **cube of a binomial**. That is, there exists some binom... |
[
{
"from": "human",
"value": "A parabola has its vertex at $(1,0)$ and opens upwards with focus at $(1,3)$.\nA line is tangent to the parabola at a point in the first quadrant and intersects the $y$-axis at point $P$.\nA circle of radius $1$ has center at $(0,1)$.\nFind the distance between the center of the... |
[
{
"from": "human",
"value": "Carmen is selling girl scout cookies. She sells three boxes of samoas to the green house for $4 each; two boxes of thin mints for $3.50 each and one box of fudge delights for $5 to the yellow house; and nine boxes of sugar cookies for $2 each to the brown house. How much money d... |
[
{
"from": "human",
"value": "At Maria's bakery, she sells small and large cakes. The small cake requires 2 cups of flour, and the large cake requires 3 cups of flour. In the morning, 8 customers order the small cakes. In the early afternoon, 10 customers order the large cakes. Later, 5 customers order the s... |
[
{
"from": "human",
"value": "In the given figure, $AB$ is the diameter of a circle with center $O$. Point $P$ lies on the extension of $BA$ past $A$; point $Q$ lies on the circle, and $R$ is the point of intersection (distinct from $Q$) of line segment $PQ$ with the circle. If length $PR$ equals length $OB$... |
[
{
"from": "human",
"value": "Find all complex solutions to the equation $z^4 + 6z^2 + 25 = 0$, and then express them in the form $a + bi$, where $a$ and $b$ are real numbers and $i = \\sqrt{-1}$."
},
{
"from": "gpt",
"value": "We are given the equation:\n\n$$\nz^4 + 6z^2 + 25 = 0\n$$\n\nThis is ... |
[
{
"from": "human",
"value": "Let $p(x)$ be a fourth-degree polynomial with real coefficients satisfying $p(1)=p(2)=p(3)=p(4)=1$ and $p(6)=p(7)=p(8)=p(9)=81$. Find $|p(0)|$."
},
{
"from": "gpt",
"value": "We are given a **fourth-degree polynomial** $ p(x) $ with **real coefficients** such that:\n... |
[
{
"from": "human",
"value": "In convex quadrilateral $ABCD$, angles $A$ and $B$ are supplementary, and angles $C$ and $D$ are supplementary. If the measure of angle $A$ is $20$ degrees more than the measure of angle $C$, what is the measure of angle $A$?"
},
{
"from": "gpt",
"value": "We are giv... |
[
{
"from": "human",
"value": "Factor the expression $x^4+4$ in the form $(x^2+ax+b)(x^2+cx+d)$, where $a$, $b$, $c$, and $d$ are constants. What is $a+b+c+d$?"
},
{
"from": "gpt",
"value": "We are given the expression:\n\n$$\nx^4 + 4\n$$\n\nand asked to factor it in the form:\n\n$$\n(x^2 + ax + b... |
[
{
"from": "human",
"value": "Let $g$ be a function taking the positive integers to the positive integers such that\n\\[g(\\text{lcm}(m, n)) + g(\\text{gcd}(m, n)) = g(m) g(n) + 1\\]for all positive integers $m$ and $n.$\n\nLet $k$ be the number of possible values of $g(3),$ and let $t$ be the product of all... |
[
{
"from": "human",
"value": "Alex's aunt gave him $75 to spend on art supplies. He bought 8 paintbrushes that cost $3 each, a set of watercolors that cost $22, and a sketchbook that cost $10. How much money does Alex have left?"
},
{
"from": "gpt",
"value": "Let's break down the problem step by ... |
[
{
"from": "human",
"value": "Emily read seven fewer books last month than this month. Last month, she read eleven books. How many books did Emily read in the two months?"
},
{
"from": "gpt",
"value": "We are given the following information:\n\n- Emily read **seven fewer books last month than thi... |
[
{
"from": "human",
"value": "Emily read seven fewer books last month than this month. Last month, she read eleven books. How many books did Emily read in the two months?"
},
{
"from": "gpt",
"value": "We are given the following information:\n\n- Emily read **seven fewer books last month than thi... |
[
{
"from": "human",
"value": "Compute the sum of the series:\n\n\\[\\frac{1}{1^2} + \\frac{2}{2^2} + \\frac{3}{3^2} + \\frac{4}{4^2} + \\dots + \\frac{99}{99^2} + \\frac{100}{100^2}.\\]"
},
{
"from": "gpt",
"value": "We are asked to compute the sum of the series:\n\n$$\n\\frac{1}{1^2} + \\frac{2}... |
[
{
"from": "human",
"value": "Solve for $q$: $\\frac 34 = \\frac x{48} = \\frac {x+y}{60}= \\frac {q - y}{90}$."
},
{
"from": "gpt",
"value": "We are given the equation:\n\n$$\n\\frac{3}{4} = \\frac{x}{48} = \\frac{x + y}{60} = \\frac{q - y}{90}\n$$\n\nThis means all these fractions are equal to ... |
[
{
"from": "human",
"value": "Find the equation of the ellipse whose foci are at $(3,4)$ and $(-1,-2)$, and the sum of the distances from any point on the ellipse to the foci is 20."
},
{
"from": "gpt",
"value": "We are given:\n\n- **Foci** of the ellipse: $ F_1 = (3, 4) $ and $ F_2 = (-1, -2) $\... |
[
{
"from": "human",
"value": "Emily plans to walk dogs in her neighborhood and use the earnings to purchase art supplies and craft materials. She charges $8 per walk. The art supplies are $12 each, and the craft materials are $3 each. After walking 28 dogs, she wants to buy 4 art supply kits and then spend t... |
[
{
"from": "human",
"value": "In triangle $ABC$, the coordinates of $A$ and $B$ are $(0,0)$ and $(6,0)$, respectively. The perpendicular bisector of segment $AB$ intersects the circumcircle of triangle $ABC$ at points $M$ and $N$. If the coordinates of point $C$ are $(x,y)$, determine the value of $x+y$ if t... |
[
{
"from": "human",
"value": "Let $x,y,z$ be three positive real numbers such that $x,y,z$ is a geometric progression and $x,z,y$ is an arithmetic progression. Find the largest possible value of $\\frac{x+y+z}{xyz}$."
},
{
"from": "gpt",
"value": "We are given that:\n\n- $ x, y, z $ is a **geomet... |
[
{
"from": "human",
"value": "The graph of the equation $y = \\frac{x^2}{x^3 - Px^2 + Qx + R}$, where $P,Q,R$ are integers, is shown below. Find $P + Q + R$. [asy]\nimport graph; size(8.14cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-3.52,xmax=4.62,ymin=-3.... |
[
{
"from": "human",
"value": "James buys 3 dirt bikes for $150 each and 4 off-road vehicles for $300 each. It also cost him $25 to register each of these. How much did he pay for everything?"
},
{
"from": "gpt",
"value": "Let's break down the problem step by step to find the total amount James ... |
[
{
"from": "human",
"value": "A subset of the integers $1, 2, \\cdots, 120$ has the property that none of its members is twice another. What is the largest number of members such a subset can have?"
},
{
"from": "gpt",
"value": "We are given the set of integers from $ 1 $ to $ 120 $, and we are t... |
[
{
"from": "human",
"value": "Let a , b , c be the roots of x ^{3}− px ^{2}+ q − r . Determine p , if it is known that a ^{2}+ b ^{2}+ c ^{2} =252; a ^{3}+ b ^{3}+ c ^{3} =−2592."
},
{
"from": "gpt",
"value": "We are given the cubic polynomial:\n\n$$\nx^3 - px^2 + qx - r = 0\n$$\n\nLet the roo... |
[
{
"from": "human",
"value": "A spherical vase rests on a table so that the diameter of its base is 30 cm. A flower put in the vase is at a distance of 2 cm from the edge of the table. Find the radius of the ball-shaped part of the vase."
},
{
"from": "gpt",
"value": "We are given:\n\n- A **spher... |
[
{
"from": "human",
"value": "Let $f(x)=ax^2+bx+c$. Determine $\\Delta$, if $f(2)=11$, $f(-1)=-2$, and $f(0)=3$."
},
{
"from": "gpt",
"value": "We are given a quadratic function:\n\n$$\nf(x) = ax^2 + bx + c\n$$\n\nand the following values:\n\n- $ f(2) = 11 $\n- $ f(-1) = -2 $\n- $ f(0) = 3 $\n\nW... |
[
{
"from": "human",
"value": "Consider a regular polygon with $n$ sides. Suppose that $O$ is the center of the polygon and $A_1, A_2, \\ldots, A_n$ are its vertices. Define $B_k$ as the midpoint of the line segment $OA_k$, and denote by $\\theta_k$ the angle $OA_kB_{k + 1}$, where $k = 1, 2, \\ldots, n$. If\... |
[
{
"from": "human",
"value": "From Monday to Friday, Lucas sleeps 7 hours a night. On Saturday and Sunday, he sleeps for 10 hours a night. He also takes a 30-minute nap on weekdays and a 2-hour nap on weekends. How much sleep does he get in 3 weeks?"
},
{
"from": "gpt",
"value": "Let's break this... |
[
{
"from": "human",
"value": "In the plane, six congruent circles are arranged so that each circle is tangent to its two neighbors. Inside each circle, a triangle is inscribed such that the sides of the triangle are tangent to the circle and the vertices of the triangle lie on the boundary of the circle. Det... |
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