conversations listlengths 2 2 |
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[
{
"from": "human",
"value": "Find the value of $k$ such that the polynomial $x^3 + 2x^2 - kx + 6$ can be factored into the product of a linear and quadratic term."
},
{
"from": "gpt",
"value": "We are given the polynomial:\n\n$$\nP(x) = x^3 + 2x^2 - kx + 6\n$$\n\nWe are told that this polynomial... |
[
{
"from": "human",
"value": "A fair coin is flipped 6 times. What is the probability that the sequence of heads and tails obtained is not the same when its order is reversed? Express your answer as a common fraction."
},
{
"from": "gpt",
"value": "We are given that a **fair coin is flipped 6 tim... |
[
{
"from": "human",
"value": "Consider the polynomial $P(x)=ax^4+bx^3+cx^2+dx+e$. Suppose that we know that $P(2)=0$ and $P(-3)=0$. Find the value of the coefficient $b$ in terms of $a$."
},
{
"from": "gpt",
"value": "We are given a polynomial:\n\n$$\nP(x) = ax^4 + bx^3 + cx^2 + dx + e\n$$\n\nand... |
[
{
"from": "human",
"value": "Find the volume of a tetrahedron with vertices $\\mathbf{0}$, $\\mathbf{a}$, $\\mathbf{b}$, and $\\mathbf{c}$, where $\\mathbf{a} = \\begin{pmatrix} 2 \\\\ 0 \\\\ 1 \\end{pmatrix}$, $\\mathbf{b} = \\begin{pmatrix} 1 \\\\ -1 \\\\ 3 \\end{pmatrix}$, and $\\mathbf{c} = \\begin{pmat... |
[
{
"from": "human",
"value": "Find the value of $b$ such that the quadratic expression $bx^2 + 12x + 9$ can be written as the square of a binomial."
},
{
"from": "gpt",
"value": "We are given the quadratic expression:\n\n$$\nbx^2 + 12x + 9\n$$\n\nand we are told that it can be written as the **sq... |
[
{
"from": "human",
"value": "Let $a$, $b$, and $c$ be constants such that the equation\n\\[f(x) + ax^2 + f(y) + by^2 + f(z) + cz^2 = 0\\]holds for all real numbers $x,y,z$ satisfying $x+y+z=0$. Determine the value of $f(5) + f(-5)$."
},
{
"from": "gpt",
"value": "We are given the functional equ... |
[
{
"from": "human",
"value": "A group of 8 friends wants to take a photo together, but they have a peculiar requirement: each person must stand next to someone who is at least 2 inches taller or shorter than them. If the heights of the friends are 60, 62, 65, 66, 68, 70, 72, and 75 inches, in how many ways c... |
[
{
"from": "human",
"value": "How many cubic polynomials $P(x)$ with coefficients in the set $\\{-3, -2, -1, 0, 1, 2, 3\\}$ satisfy $P(2) = P(-2)$?"
},
{
"from": "gpt",
"value": "We are given that $ P(x) $ is a **cubic polynomial** with coefficients in the set $ \\{-3, -2, -1, 0, 1, 2, 3\\} $, an... |
[
{
"from": "human",
"value": "A rectangle is inscribed in a 10 x 10 square, with two of its sides parallel to the sides of the square and one of its vertices at a vertex of the square. If the side lengths of the rectangle are integers and the area of the rectangle is 48, what is the largest possible area of ... |
[
{
"from": "human",
"value": "Given the equations $ax^3 + bx^2 + cx + d = 0$ and $ex^3 + fx^2 + gx + h = 0$, find the value of $m$ such that the sum of the roots of the equation $(ax^3 + bx^2 + cx + d)(ex^3 + fx^2 + gx + h) + mx = 0$ is equal to 5."
},
{
"from": "gpt",
"value": "We are given two ... |
[
{
"from": "human",
"value": "The sum of the squares of the roots of the equation $x^2 + 2ax + b = 0$ is equal to the sum of the squares of the roots of the equation $x^2 + 3bx + a = 0$. Determine the value of $a$ and $b$."
},
{
"from": "gpt",
"value": "We are given two quadratic equations:\n\n1.... |
[
{
"from": "human",
"value": "A group of friends are playing a game where they take turns saying a positive integer. The rule is that the next person's number must be a divisor of the previous number. If someone says a number that is not a divisor of the previous number, they lose. The game starts with the n... |
[
{
"from": "human",
"value": "Determine the value of $x$ for which the graph of the function\n\n$$f(x)=\\frac{x^3-2x^2-5x+1}{x^2+2x-3}$$\n\nhas a hole."
},
{
"from": "gpt",
"value": "We are given the function:\n\n$$\nf(x) = \\frac{x^3 - 2x^2 - 5x + 1}{x^2 + 2x - 3}\n$$\n\nWe are asked to **find t... |
[
{
"from": "human",
"value": "Quadrilateral $ABCD$ is inscribed in a circle with center $O$. If $m \\angle BAD = 85^\\circ$, $m \\angle ABC = 105^\\circ$, and $m \\angle BCD = 70^\\circ$, find $m \\angle ADC$."
},
{
"from": "gpt",
"value": "We are given a **cyclic quadrilateral** $ABCD$ inscribed... |
[
{
"from": "human",
"value": "In a certain sequence the first term is $a_1 = 2007$ and the second term is $a_2 = 2008.$ Furthermore, the values of the remaining terms are chosen so that\n\\[a_n + a_{n + 1} + a_{n + 2} = n\\]for all $n \\ge 1.$ Determine $a_{1000}.$\n\nWrite another problem inspired by this... |
[
{
"from": "human",
"value": "In quadrilateral $ABCD$, $\\angle B = 45^\\circ$ and $\\angle C = 60^\\circ.$ The diagonals $AC$ and $BD$ intersect at $O$. Find the ratio of the area of triangle $AOB$ to the area of triangle $COD$."
},
{
"from": "gpt",
"value": "We are given a quadrilateral $ABCD$ ... |
[
{
"from": "human",
"value": "The graph of the line $x+y=b$ is a perpendicular bisector of the line segment from $(1,3)$ to $(5,7)$. What is the value of b?\n\nNew Problem:\n\nA circle with center $(2,4)$ and radius $3$ is drawn. The graph of the line $x+y=b$ is a tangent to this circle at the point $(5,1)$.... |
[
{
"from": "human",
"value": "Find the value of the expression\n$\\tan \\frac{\\pi}{15} \\tan \\frac{2 \\pi}{15} \\tan \\frac{3 \\pi}{15}$."
},
{
"from": "gpt",
"value": "We are asked to evaluate the expression:\n\n$$\n\\tan \\frac{\\pi}{15} \\tan \\frac{2\\pi}{15} \\tan \\frac{3\\pi}{15}\n$$\n\n... |
[
{
"from": "human",
"value": "Determine all possible values of the expression\n\n$$\\sqrt{(a - c)^2 + (b - d)^2} + \\sqrt{(b - d)^2 + (a - c)^2} + \\sqrt{(a - b)^2 + (c - d)^2} + \\sqrt{(c - d)^2 + (a - b)^2},$$\n\nwhere $a,$ $b,$ $c,$ $d$ are arbitrary real numbers."
},
{
"from": "gpt",
"value":... |
[
{
"from": "human",
"value": "A bakery sells a total of 250 loaves of bread per day. They sell a combination of whole wheat and white bread. If the ratio of whole wheat to white bread is 3:5, and the profit on a loaf of whole wheat bread is $0.50, while the profit on a loaf of white bread is $0.25, what is t... |
[
{
"from": "human",
"value": "Let $p$ and $q$ be two distinct irrational roots of the cubic equation $x^3+px^2+qx+r=0$, where $p$, $q$, and $r$ are integers. If $p$ and $q$ are also roots of the equation $x^2-6x-5=0$, find the third root of the cubic equation."
},
{
"from": "gpt",
"value": "We ar... |
[
{
"from": "human",
"value": "A hexagonal prism has a base with an area of 108 square centimeters. Its height is 6 centimeters. What is the length of the diagonal of the prism from one vertex of the base to the opposite vertex of the other base, in centimeters?"
},
{
"from": "gpt",
"value": "We a... |
[
{
"from": "human",
"value": "If $a$ and $b$ are positive integers such that $ab=72$, what is the maximum value of $a^b+b^a$?"
},
{
"from": "gpt",
"value": "We are given that $ a $ and $ b $ are **positive integers** such that:\n\n$$\nab = 72\n$$\n\nWe are to find the **maximum value** of:\n\n$$\... |
[
{
"from": "human",
"value": "Let $f(n)$ be the number of regions in the coordinate plane defined by the inequalities $1 \\le x \\le n$ and $0 \\le y \\le x \\lfloor \\log_2 x \\rfloor$, where $\\lfloor \\log_2 x \\rfloor$ is the greatest integer not exceeding $\\log_2 x$. Find the number of values of $n$ wi... |
[
{
"from": "human",
"value": "What is the remainder when $x^6+3x^4+3x^2+1$ is divided by $x^2+1$?"
},
{
"from": "gpt",
"value": "We are asked to find the **remainder** when the polynomial:\n\n$$\nP(x) = x^6 + 3x^4 + 3x^2 + 1\n$$\n\nis divided by:\n\n$$\nD(x) = x^2 + 1\n$$\n\n---\n\n### Step 1: Us... |
[
{
"from": "human",
"value": "What is the largest value of $x$ that satisfies the equation $\\sqrt{2x}=4x$?\n\nA boat travels across a river from point A to point B, with the current moving at a speed of 5 km/h. If the boat travels at a speed of 15 km/h in still water, and the distance between points A and B... |
[
{
"from": "human",
"value": "How many integers from $1$ to $200$ have at least one digit that is a prime number?"
},
{
"from": "gpt",
"value": "We are asked to find how many integers from **1 to 200** have **at least one digit that is a prime number**.\n\n---\n\n### Step 1: Identify the prime di... |
[
{
"from": "human",
"value": "Consider the equation $x^4 + ax^2 + bx + c = 0$, where $a, b, c$ are rational numbers. One of the roots of this equation is $2 + i\\sqrt{3}$, and another root is a rational number. Find the value of this rational root."
},
{
"from": "gpt",
"value": "We are given the ... |
[
{
"from": "human",
"value": "Let $x$ and $y$ be positive real numbers, $x>y$.\nFind the value of the expression\n$$x^2\\cdot\\sqrt{x^2-y^2}+y^2\\cdot\\sqrt{x^2-y^2}-xy\\cdot\\sqrt{x^2+y^2}$$"
},
{
"from": "gpt",
"value": "We are given the expression:\n\n$$\nx^2\\cdot\\sqrt{x^2 - y^2} + y^2\\cdot... |
[
{
"from": "human",
"value": "These two spinners are divided into fifths and thirds, respectively. If each of these spinners is spun once, what is the probability that the sum of the results of the two spins will be an odd number? Express your answer as a common fraction.\n\n[asy]\n\nsize(5cm,5cm);\n\ndraw(C... |
[
{
"from": "human",
"value": "Let $\\mathcal{S}$ be the set of real numbers that can be represented as repeating decimals of the form $0.\\overline{abcd}$ where $a, b, c, d$ are distinct digits. Find the number of elements in $\\mathcal{S}$ that are greater than 0.5."
},
{
"from": "gpt",
"value":... |
[
{
"from": "human",
"value": "Find the smallest positive solution to\n\\[\\sin^2 2x + \\sin^2 3x = 1 + \\sin 2x \\sin 3x\\]in radians."
},
{
"from": "gpt",
"value": "We are given the trigonometric equation:\n\n$$\n\\sin^2 2x + \\sin^2 3x = 1 + \\sin 2x \\sin 3x\n$$\n\nWe are to find the **smalles... |
[
{
"from": "human",
"value": "A bookshelf has 5 shelves, and each shelf can hold at most 3 books. If you have 11 books to arrange on the shelves, in how many ways can you distribute the books so that no shelf holds more than 3 books?"
},
{
"from": "gpt",
"value": "We are given:\n\n- A **bookshelf... |
[
{
"from": "human",
"value": "Find the value of $c$ for the smallest positive real number $d$ such that all the roots of the polynomial $x^4 - dx^3 + cx^2 - dx + 1$ are real. For this value of $d$, the value of $c$ is unique."
},
{
"from": "gpt",
"value": "We are given the polynomial:\n\n$$\nP(x)... |
[
{
"from": "human",
"value": "Find all 4-digit multiples of 11 of the form $b27,\\!c$ where $b$ and $c$ are digits, and the sum of the digits is a multiple of 3. What is the minimum value of $b$?"
},
{
"from": "gpt",
"value": "We are given a 4-digit number of the form:\n\n$$\nb27c\n$$\n\nWhere:\n... |
[
{
"from": "human",
"value": "Let $\\triangle ABC$ be a right triangle with $BC = 24$ and $AB = 30$. The point $M$ is the midpoint of $BC$. The incenter of $\\triangle ABC$ is $I$ and the inradius is $6$. What is the length of the segment $IM$?"
},
{
"from": "gpt",
"value": "We are given a right ... |
[
{
"from": "human",
"value": "The digits of a three-digit positive integer $n$ are randomly selected. What is the probability that $n$ is divisible by $11$?"
},
{
"from": "gpt",
"value": "We are given a **three-digit positive integer** $ n $, and we are told that its **digits are randomly selecte... |
[
{
"from": "human",
"value": "Find the equation of the line passing through the points (2,7) and (-1,-3) in the form y = mx + b. What is the value of 2m - b?"
},
{
"from": "gpt",
"value": "We are given two points: \n- Point 1: $ (2, 7) $ \n- Point 2: $ (-1, -3) $\n\nWe are to find the equation ... |
[
{
"from": "human",
"value": "Solve for $y$:\n\n$$\\frac{2y^2 - 3y + 2}{y + 2} = 2y - 1$$"
},
{
"from": "gpt",
"value": "We are given the equation:\n\n$$\n\\frac{2y^2 - 3y + 2}{y + 2} = 2y - 1\n$$\n\n---\n\n### **Step 1: Eliminate the denominator**\n\nMultiply both sides of the equation by $ y + ... |
[
{
"from": "human",
"value": "What is the smallest positive integer $y$ that, when multiplied by $225$, produces a product that is a multiple of $315$?"
},
{
"from": "gpt",
"value": "We are given:\n\n- A number $ y $\n- When multiplied by $ 225 $, the product is a multiple of $ 315 $\n\nWe are to... |
[
{
"from": "human",
"value": "In $\\triangle PQR$, we have $PQ = QR$ and $m\\angle PQR = 72^{\\circ}$. If $M$ is the midpoint of $\\overline{PR}$, what is the measure of $\\angle PMQ$?"
},
{
"from": "gpt",
"value": "We are given:\n\n- Triangle $ \\triangle PQR $ with $ PQ = QR $, so it is **isosc... |
[
{
"from": "human",
"value": "Vivian operates a food cart. She pays a weekly fee of $30 to rent a spot in the market. If Vivian earns $12 per day, what is her total profit every week?"
},
{
"from": "gpt",
"value": "To find Vivian's **total profit every week**, we need to calculate her **weekly ea... |
[
{
"from": "human",
"value": "Find the smallest positive integer $k$ such that the product of all the roots of the equation $x^6 - 2x^4 + 3x^2 - 1 = 0$ divides $x^k - 1$."
},
{
"from": "gpt",
"value": "We are given the polynomial:\n\n$$\nP(x) = x^6 - 2x^4 + 3x^2 - 1\n$$\n\nWe are to find the **sm... |
[
{
"from": "human",
"value": "Find all real values of x such that\n\\[\\frac{(x+i)^2}{x^2+1} = i.\\]"
},
{
"from": "gpt",
"value": "We are given the equation:\n\n$$\n\\frac{(x+i)^2}{x^2+1} = i\n$$\n\nWe are to find all **real values of $ x $** that satisfy this equation.\n\n---\n\n### Step 1: Exp... |
[
{
"from": "human",
"value": "Find the equation of the circle that passes through the points $(1,2)$, $(3,4)$, and $(5,6)$, and has its center on the line $x+y=7$.\n\nCan you solve it?"
},
{
"from": "gpt",
"value": "We are given three points: \n- $ A = (1, 2) $ \n- $ B = (3, 4) $ \n- $ C = (5,... |
[
{
"from": "human",
"value": "Find the largest positive integer whose square is a palindrome and has a length of 5 digits."
},
{
"from": "gpt",
"value": "We are asked to find the **largest positive integer** such that its **square is a 5-digit palindrome**.\n\n---\n\n### Step 1: Understand the pr... |
[
{
"from": "human",
"value": "In a triangle, the length of the median to one side is 12 cm, and the length of the median to another side is 15 cm. If the two medians intersect at a right angle, what is the length of the third side of the triangle?"
},
{
"from": "gpt",
"value": "We are given:\n\n-... |
[
{
"from": "human",
"value": "The matrix\n\\[\\begin{pmatrix} 2 & 1 & 0 \\\\ 0 & 1 & 2 \\\\ 3 & 0 & -1 \\end{pmatrix}\\]\nis an inverse of a matrix of the form\n\\[\\begin{pmatrix} a & 0 & b \\\\ c & 1 & d \\\\ e & 0 & f \\end{pmatrix}.\\]\nFind the ordered triple $(a, f, g),$ where $g$ is the value of the d... |
[
{
"from": "human",
"value": "Emily's bookshelf is overflowing, so she decides to donate some books to the local library. She takes out 5 novels, 3 biographies, 2 sets of encyclopedias (each set has 5 volumes), and 10 children's books, then asks her siblings if they have any books to donate. 2 of her sibling... |
[
{
"from": "human",
"value": "Quadrilateral $A_0A_1A_2A_3$ has its sides extended to form a new quadrilateral $B_0B_1B_2B_3$, with $B_i$ lying on the line containing the side opposite to $A_i$. Points $C_i$ are defined such that $A_iC_i$ is parallel to $B_{i+1}B_{i+2}$, where indices are taken modulo 4. Show... |
[
{
"from": "human",
"value": "A square with side length $n$ is divided into $n^2$ squares of side length $1$. How many of these squares can be removed so that the remaining ones still form a connected shape with at least one square in each row and column?"
},
{
"from": "gpt",
"value": "We are giv... |
[
{
"from": "human",
"value": "Brianna is baking a batch of 60 cupcakes. She plans to add 240 sprinkles and half as many chopped nuts to the entire batch. What are the average number of decorations in each cupcake?"
},
{
"from": "gpt",
"value": "We are given the following information:\n\n- Brianna... |
[
{
"from": "human",
"value": "Let $\\mathbf{P}$ be a $3 \\times 3$ matrix, with real entries, such that $\\mathbf{P}^2 = \\mathbf{P}.$ Find the number of different possible values that $\\det(\\mathbf{P})$ can have."
},
{
"from": "gpt",
"value": "We are given that $\\mathbf{P}$ is a $3 \\times 3... |
[
{
"from": "human",
"value": "Let $a$ and $b$ be two distinct positive real numbers. We define a sequence $(X_n)$ as follows:\n\n$X_1 = \\frac{a+b}{2}, \\quad X_2 = \\frac{2}{\\frac{1}{a} + \\frac{1}{b}}, \\quad X_n = \\frac{X_{n-1} + X_{n-2}}{2}$ for $n \\geq 3$.\n\nDetermine the behavior of the sequence $(... |
[
{
"from": "human",
"value": "The perimeters of three rectangles are 24, 42, and 70. What is the average (mean) of their diagonal lengths?"
},
{
"from": "gpt",
"value": "We are given the **perimeters** of three rectangles: \n- Rectangle 1: Perimeter = 24 \n- Rectangle 2: Perimeter = 42 \n- Rec... |
[
{
"from": "human",
"value": "The coefficients of a quadratic polynomial are such that the sum of the roots is 6, the product of the roots is 8, and the sum of the reciprocals of the roots is 4/3. Determine the constant term of the quadratic polynomial."
},
{
"from": "gpt",
"value": "We are given... |
[
{
"from": "human",
"value": "Find the maximum value of the expression\n\\[ \\sqrt{x^3 - xy^2} + \\sqrt{y^3 - yx^2} \\]\nin the domain $0 \\le x \\le 2$ and $0 \\le y \\le 3$."
},
{
"from": "gpt",
"value": "We are given the expression:\n\n$$\n\\sqrt{x^3 - xy^2} + \\sqrt{y^3 - yx^2}\n$$\n\nand the... |
[
{
"from": "human",
"value": "The length of a rectangle is 5 cm more than its width. If the area of the rectangle is 64 cm^2, what is the length of the diagonal of the rectangle?"
},
{
"from": "gpt",
"value": "We are given:\n\n- The **length** of a rectangle is **5 cm more** than its **width**.\n... |
[
{
"from": "human",
"value": "Find all functions $f(x)$ that satisfy\n$f(x-1)+f(x+1) = f(x)$ for all $x$\nand\n$f(x-1)-f(x+1) = -f(-x)$ for all $x$\nand\n$f(0) = 5.$"
},
{
"from": "gpt",
"value": "We are given a functional equation involving a function $ f(x) $, and we are to find all such functi... |
[
{
"from": "human",
"value": "Let $x,$ $y,$ and $z$ be real numbers such that\n\\begin{align*}\ne^x &= 2 + \\sin y, \\\\\ne^y &= 2 + \\sin z, \\\\\ne^z &= 2 + \\sin x.\n\\end{align*}Find the largest possible value of $e^x.$"
},
{
"from": "gpt",
"value": "We are given the system of equations:\n\n$... |
[
{
"from": "human",
"value": "Let $A_n$ be the sum of the terms of the sequence $a_1, a_2, \\dots, a_n$ and $B_n$ be the sum of the terms of the sequence $b_1, b_2, \\dots, b_n$. If $A_n = B_n$ for all positive integers $n$, and $a_{n+1} = a_n + b_n$ for all $n \\geq 1$, find $a_{100}$."
},
{
"from":... |
[
{
"from": "human",
"value": "Let $a$ and $b$ be two positive integers such that $ab + 3a + 4b + 12$ is divisible by $a + 5$. Find the least possible value of $b$ such that $\\gcd(a, b) = 1$."
},
{
"from": "gpt",
"value": "We are given that $ a $ and $ b $ are **positive integers** such that:\n\n... |
[
{
"from": "human",
"value": "Let's say we have a circle with a radius of 4 cm. We want to find the area of the largest inscribed rectangle that can be drawn inside the circle. The rectangle has a width of 3 cm. What is the length of the rectangle?"
},
{
"from": "gpt",
"value": "We are given:\n\n... |
[
{
"from": "human",
"value": "Let's create a new problem inspired by the given one:\n\nFind the largest possible value of the sum $S$ modulo 2010, where $S$ is formed by arranging the numbers 1 to 2010 in a sequence such that the sum of each pair of adjacent terms is divisible by 3.\n\nThis problem requires ... |
[
{
"from": "human",
"value": "There are 7 students in a class, each of whom is taking exactly one of 5 courses. For each course, there is a teacher who teaches that course to all students enrolled in it.\n\nSuppose there is a tournament where each student selects one of his teachers to work with for the dura... |
[
{
"from": "human",
"value": "Determine the largest possible value of $n$ such that $n!$ is divisible by $2^{11}$."
},
{
"from": "gpt",
"value": "We are asked to find the **largest possible value of $ n $** such that $ n! $ is divisible by $ 2^{11} $.\n\n---\n\n### Step 1: Understanding the Probl... |
[
{
"from": "human",
"value": "Compute the number of ways to choose a team of 3 players from a group of 8 players, where the team must include at least one player from each of the 3 different age groups (young, middle-aged, and old)."
},
{
"from": "gpt",
"value": "We are given:\n\n- A group of **8... |
[
{
"from": "human",
"value": "Let $a$ and $b$ be two complex numbers such that $|a|=2$ and $|b|=\\sqrt{5}$.\nFind the argument of $a$ if $ab=1-2i$."
},
{
"from": "gpt",
"value": "We are given:\n\n- $ |a| = 2 $\n- $ |b| = \\sqrt{5} $\n- $ ab = 1 - 2i $\n\nWe are to **find the argument of $ a $**.\... |
[
{
"from": "human",
"value": "In a box, there are five balls with different radii: 1, 2, 3, 4, and 5 cm. Three balls are randomly selected from the box. What is the probability that the three balls can be stacked on top of each other in a way that each ball is tangent to the one below it? Express your answer... |
[
{
"from": "human",
"value": "Samantha purchases 300 decorative vases for her boutique, tripling the existing stock. After selling 2/3 of the total inventory, her business partner acquires 200 more decorative vases, and a week later, sells 100. What is the total number of decorative vases remaining in the bo... |
[
{
"from": "human",
"value": "Given a regular hexagon inscribed in a circle, a point P is chosen inside the circle, not necessarily on a vertex or side of the hexagon. Three lines pass through P, each perpendicular to a side of the hexagon. Each line intersects the circle at two points. Let L be the length o... |
[
{
"from": "human",
"value": "Each of the twelve dots in this figure is to be colored red, white, blue or green. No two dots connected by a segment (with no other dots between) may be the same color. How many ways are there to color the dots of this figure?\n\n[asy]\ndraw((-75,0)--(-45,0)--(-60,26)--cycle);\... |
[
{
"from": "human",
"value": "Find the sum of the infinite series $1+3\\left(\\dfrac{1}{2002}\\right)+5\\left(\\dfrac{1}{2002}\\right)^2+7\\left(\\dfrac{1}{2002}\\right)^3+\\cdots$."
},
{
"from": "gpt",
"value": "We are given the infinite series:\n\n$$\n1 + 3\\left(\\frac{1}{2002}\\right) + 5\\le... |
[
{
"from": "human",
"value": "Find the equation of the curve that is traced by the point whose coordinates are the sum and product of the coordinates of a point on the line segment connecting $(-2,7)$ and $(3,11)$. Express the equation in terms of $x$ and $y$, where $x$ is the sum of the coordinates and $y$ ... |
[
{
"from": "human",
"value": "In quadrilateral $ABCD$, the diagonals $AC$ and $BD$ intersect at point $O$. If $AO:OC = 3:4$ and $BO:OD = 2:3$, find the ratio of the area of quadrilateral $AOBD$ to the area of quadrilateral $ABCD$."
},
{
"from": "gpt",
"value": "We are given a quadrilateral $ABCD$... |
[
{
"from": "human",
"value": "Three friends randomly select 2 cards each from a standard deck of 52 cards.\nWhat is the probability that each friend has exactly one red card and one black card?"
},
{
"from": "gpt",
"value": "We are given a standard deck of 52 cards, which contains:\n\n- **26 red ... |
[
{
"from": "human",
"value": "A parabola has a focus at $(1, 1)$ and a directrix $y = -1$. It has two $y$-intercepts, one of which is the point $(0, 2)$. What is the other one? Enter your answer as an ordered pair."
},
{
"from": "gpt",
"value": "We are given:\n\n- A **parabola** with:\n - **Focu... |
[
{
"from": "human",
"value": "The function $f(x)$ is defined for $x \\geq 0$ and satisfies $f(4x) = f(2x) + 1$ for all $x \\geq 0$. If $f(0) = 0$, determine the value of $f(16)$."
},
{
"from": "gpt",
"value": "We are given the functional equation:\n\n$$\nf(4x) = f(2x) + 1 \\quad \\text{for all } ... |
[
{
"from": "human",
"value": "What is the smallest four-digit number that is a multiple of $17$ and has the digits $2,4,5,$ and $6$?"
},
{
"from": "gpt",
"value": "We are asked to find the **smallest four-digit number** that:\n\n1. Is a **multiple of 17**\n2. Uses the **digits 2, 4, 5, and 6 exac... |
[
{
"from": "human",
"value": "What is the smallest positive integer $n$ such that the sum of its digits is equal to the number of its divisors?"
},
{
"from": "gpt",
"value": "We are asked to find the **smallest positive integer** $ n $ such that:\n\n$$\n\\text{Sum of the digits of } n = \\text{Nu... |
[
{
"from": "human",
"value": "Simplify the expression $12^{12} \\cdot 6^6 \\div 3^{24}$ using the properties of exponents.\n\nNote: This problem requires a similar approach to the original problem, but with different numbers and a slight twist. The solution involves applying the properties of exponents, such... |
[
{
"from": "human",
"value": "Starting with a positive integer, Maria repeatedly takes the square root of her number and then rounds it up to the nearest integer. If she starts with 256, how many times must she do this before she reaches 1?"
},
{
"from": "gpt",
"value": "We are given that Maria s... |
[
{
"from": "human",
"value": "We call a number a valley number if its middle digit is smaller than any other digit. For example, 849 is not a valley number, but 462 is a valley number. How many 4-digit valley numbers are there?"
},
{
"from": "gpt",
"value": "We are asked to find how many **4-di... |
[
{
"from": "human",
"value": "Let $z$ and $w$ be complex numbers such that $|z + 1 + 3i| = 1$ and $|w - 7 - 8i| = 3.$ Find the smallest possible value of $|z - w|.$\n\nNew problem:\n\nLet $f(x) = x^2 + ax + b$ and $g(x) = x^2 + cx + d$ be two quadratic functions with real coefficients. If $f(x)$ has roots $... |
[
{
"from": "human",
"value": "A street has 20 houses on each side, for a total of 40 houses. The addresses on one side of the street form an arithmetic sequence, while the addresses on the other side form a geometric sequence. The sum of the addresses on both sides of the street is 640. Determine the common ... |
[
{
"from": "human",
"value": "Given the function $f(x)$ defined on the interval $[0, 2\\pi]$, such that\n\n$$f(x) = \\begin{cases} \\sin(x) &\\quad \\lfloor x / \\pi \\rfloor\\text{ is even} \\\\ f(y) + \\cos(x) &\\quad \\text{otherwise} \\end{cases}$$\n\nwhere $y$ is the greatest even factor of $\\lfloor x ... |
[
{
"from": "human",
"value": "Determine the value of a if the polynomial $x^3 - ax^2 + (a-3)x + 1$ is divisible by $x^2 - 4x + 3$."
},
{
"from": "gpt",
"value": "We are given a polynomial:\n\n$$\nP(x) = x^3 - ax^2 + (a - 3)x + 1\n$$\n\nand told that it is divisible by:\n\n$$\nQ(x) = x^2 - 4x + 3\... |
[
{
"from": "human",
"value": "At a dinner party, five couples are seated around a circular table. If each husband wants to be seated opposite his wife, and Mrs. Smith cannot be seated next to Mrs. Johnson, in how many ways can the guests be seated?"
},
{
"from": "gpt",
"value": "We are given a pr... |
[
{
"from": "human",
"value": "A gardener is planting flowers in a row of 15 pots. She wants to plant at least one flower in each pot, but she also wants to make sure that no three consecutive pots have the same color flower. If she has 3 different colors of flowers to choose from, how many ways can she plant... |
[
{
"from": "human",
"value": "There are five switches controlling five lights in a room. Each switch corresponds to one of the lights, but none of them are labelled. You can turn the lights on and off as many times as you want, but you can only enter the room one time to observe the lights. How can you figur... |
[
{
"from": "human",
"value": "Find the value of $a$ in the function $f(x) = a \\sin (2x + \\pi) + 1$ given that the graph of $f(x)$ intersects the graph of $y = 2x^2 + 1$ at exactly two points."
},
{
"from": "gpt",
"value": "We are given the function:\n\n$$\nf(x) = a \\sin(2x + \\pi) + 1\n$$\n\na... |
[
{
"from": "human",
"value": "There are $2n$ complex numbers that satisfy both $z^{28} - z^{8} - 1 = 0$ and $|z| = 1$. These numbers have the form $z_{m} = \\cos\\theta_{m} + i\\sin\\theta_{m}$, where $0\\leq\\theta_{1} < \\theta_{2} < \\dots < \\theta_{2n} < 360$ and angles are measured in degrees. Find the... |
[
{
"from": "human",
"value": "Determine the smallest base $b$ such that the number $45_b$ can be expressed as the sum of two perfect squares."
},
{
"from": "gpt",
"value": "We are given a number in base $ b $: $ 45_b $, and we are to find the **smallest base $ b $** such that this number can be e... |
[
{
"from": "human",
"value": "In the coordinate plane, two parabolas are graphed according to the equations $y = -x^2 + 3x - 1$ and $y = x^2 - 7x + 15.$ Find the distance between the points of intersection of these two parabolas and the origin $(0,0).$"
},
{
"from": "gpt",
"value": "We are given ... |
[
{
"from": "human",
"value": "The set of all points $(x,y)$ in the coordinate plane for which the sum of the coordinates is a multiple of 3. If we were to plot all such points, into how many regions would the resulting graph split up the plane?"
},
{
"from": "gpt",
"value": "We are given a condit... |
[
{
"from": "human",
"value": "Let $x^2+bx+30$ be a quadratic equation with integer roots $r_1$ and $r_2$. How many different values of $b$ are possible?"
},
{
"from": "gpt",
"value": "We are given a quadratic equation:\n\n$$\nx^2 + bx + 30\n$$\n\nwith **integer roots** $ r_1 $ and $ r_2 $. We are... |
[
{
"from": "human",
"value": "Let a polynomial with integer coefficients $f(x)$ be $x^4-12x^3+47x^2+12x+27$. Alice evaluated $f(-1)$, $f(0)$, and $f(1)$. She was able to factorize the polynomial using these three evaluations. How did she do it?"
},
{
"from": "gpt",
"value": "We are given a poly... |
[
{
"from": "human",
"value": "Given that a polynomial of degree n has n+1 distinct coefficients, find the sum of the coefficients that divide both 14 and 105. Recall that we consider the constant term of the polynomial to be its divisor."
},
{
"from": "gpt",
"value": "We are given:\n\n- A **polyn... |
[
{
"from": "human",
"value": "A coin is tossed until a head is obtained for the first time. What is the probability that the number of tails obtained before the first head is a multiple of 3?"
},
{
"from": "gpt",
"value": "We are given a coin that is tossed repeatedly until the **first head** app... |
[
{
"from": "human",
"value": "The organizers of a conference want to distribute the participants into discussion groups, each consisting of 5 students and 1 teacher. If there are 180 participants in total, and each student has a preference for one of 4 topics, find the maximum number of discussion groups tha... |
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