conversations listlengths 2 2 |
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[
{
"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?"... |
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