conversations listlengths 2 2 |
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[
{
"from": "human",
"value": "Compute\n\\[\\prod_{n = 1}^\\infty \\left(1 + \\frac{2n^3 - n^2 - n + 1}{n^6 - n^5 + n^4 - n^3 + n^2 - n + 1}\\right).\\]"
},
{
"from": "gpt",
"value": "Compute\n\\[\\prod_{n = 1}^\\infty \\left(1 + \\frac{2n^3 - n^2 - n + 1}{n^6 - n^5 + n^4 - n^3 + n^2 - n + 1}\\rig... |
[
{
"from": "human",
"value": "Find the value of $k$ if $k$ is an integer such that $x^2 - 4x + 4$ is a factor of $kx^3 + x^2 + 1$."
},
{
"from": "gpt",
"value": "Find the value of $k$ if $k$ is an integer such that $x^2 - 4x + 4$ is a factor of $kx^3 + x^2 + 1$."
}
] |
[
{
"from": "human",
"value": "Consider a number system with base $b$ and let $N$ be the number of positive integer multiples of $b^2 + b + 1$ that can be expressed in the form $b^j - b^i$, where $i$ and $j$ are integers and $0\\leq i < j \\leq b^2 - 1$. Determine $N$ when $b = 3$."
},
{
"from": "gpt"... |
[
{
"from": "human",
"value": "Compute\n\\[\\sum_{1 \\le a < b < c} \\frac{1}{2^a 3^b 5^c}.\\](The sum is taken over all triples $(a,b,c)$ of positive integers such that $1 \\le a < b < c.$)\n\nHere is another problem:\n\nFind the value of the following infinite product:\n\\[\\prod_{n=1}^{\\infty} \\left(1 + ... |
[
{
"from": "human",
"value": "The function\n$$\\ell (x) = \\sqrt {x^2-6x+11}$$\nhas the range $[a,b]$.\nDetermine $a+b$"
},
{
"from": "gpt",
"value": "The function\n$$\\ell (x) = \\sqrt {x^2-6x+11}$$\nhas the range $[a,b]$.\nDetermine $a+b$"
}
] |
[
{
"from": "human",
"value": "A square is inscribed inside a regular hexagon. The sides of the square are tangent to the sides of the hexagon. If the length of a side of the square is 1, what is the area of the hexagon?"
},
{
"from": "gpt",
"value": "A square is inscribed inside a regular hexagon... |
[
{
"from": "human",
"value": "Line $l_1$ has equation $2x + 3y = 7$ and goes through $A = (2, 1)$. Line $l_2$ has equation $y = -2$ and meets line $l_1$ at point $B$. Line $l_3$ has positive slope, goes through point $A$, and meets $l_2$ at point $C$. The area of $\\triangle ABC$ is $2.5$. What is the slope ... |
[
{
"from": "human",
"value": "Rationalize the denominator of $\\frac{\\sqrt{x} + \\sqrt{x+1}}{\\sqrt{x+2} - \\sqrt{x}}$ and write your answer in the form $\\displaystyle \\frac{A\\sqrt{B} + C\\sqrt{D}}{E}$, where $B < D$, the fraction is in lowest terms and all radicals are in simplest radical form. What is ... |
[
{
"from": "human",
"value": "Let ABC be a triangle with side lengths AB = 8, BC = 12, and CA = 16. Let D be a point on side BC such that the circumcircles of triangles ABD and ADC have equal radii. Find the ratio of the areas of triangles ABD and ADC, expressed as a fraction p/q where p and q are relatively... |
[
{
"from": "human",
"value": "In the diagram below, points $A$, $B$, and $C$ lie on a circle with center $O$, such that $\\angle AOB = 2\\theta$ and $\\angle BOC = \\theta$. The triangle $ABC$ is inscribed in the circle, and the points $D$ and $E$ are the midpoints of sides $AC$ and $BC$, respectively.\n\n[a... |
[
{
"from": "human",
"value": "A five-digit palindrome number is divisible by 15. Find the sum of all possible values of its middle digit."
},
{
"from": "gpt",
"value": "A five-digit palindrome number is divisible by 15. Find the sum of all possible values of its middle digit."
}
] |
[
{
"from": "human",
"value": "In pentagon ABCDE, AB = 7, BC = 8, CD = 9, DE = 10, EA = 11. Calculate\n\n\\[\\frac{\\sin(B - C)}{\\cos(D - E)} + \\frac{\\cos(B - C)}{\\sin(D - E)}.\\]"
},
{
"from": "gpt",
"value": "In pentagon ABCDE, AB = 7, BC = 8, CD = 9, DE = 10, EA = 11. Calculate\n\n\\[\\frac... |
[
{
"from": "human",
"value": "Determine the number of digits in the decimal representation of the number $A$, written in base 7, if the sum of its digits is 12."
},
{
"from": "gpt",
"value": "Determine the number of digits in the decimal representation of the number $A$, written in base 7, if the... |
[
{
"from": "human",
"value": "Let $w$ be a complex number that satisfies\n\\[|w^2 - 2i| + |w^2 - 3| = 7.\\]Find the maximum value of $|w|.$"
},
{
"from": "gpt",
"value": "Let $w$ be a complex number that satisfies\n\\[|w^2 - 2i| + |w^2 - 3| = 7.\\]Find the maximum value of $|w|.$"
}
] |
[
{
"from": "human",
"value": "A polynomial $P(x)$ with integer coefficients is such that $P(2)=24$, $P(3)=54$, and $P(5)=120$.\n\nDetermine the number of positive integer solutions of the equation $P(x)=x^2$."
},
{
"from": "gpt",
"value": "A polynomial $P(x)$ with integer coefficients is such tha... |
[
{
"from": "human",
"value": "A polynomial $p(x)$ is said to be \"balanced\" if the graph of $y=p(x)$ is symmetric about the line $y=x$. Let $q(x)$ be a quadratic polynomial with a range of $[3,6]$. What is the smallest possible value of the coefficient of the $x^2$ term in a balanced polynomial of the form ... |
[
{
"from": "human",
"value": "Count the number of non-congruent convex polygons whose side lengths form a geometric progression and whose angles are positive integers not exceeding 180 degrees."
},
{
"from": "gpt",
"value": "Count the number of non-congruent convex polygons whose side lengths for... |
[
{
"from": "human",
"value": "In a regular octahedron, 12 points are placed on each edge, dividing it into 13 congruent segments. A triangle is formed by selecting three non-collinear points from these 48 points. How many distinct planes contain the centroid of this triangle?"
},
{
"from": "gpt",
... |
[
{
"from": "human",
"value": "Tom's younger sister's height is two-thirds Tom's height. If Tom was 5 inches taller than his sister when she was 2 years old, and Tom's height at that time was 3 times his sister's height at that time, how many inches tall is Tom's sister now?"
},
{
"from": "gpt",
"... |
[
{
"from": "human",
"value": "In a school play, there are 5 actors who will play the roles of 3 different characters. Each actor can play only one character, and each character can be played by only one actor. The director wants to arrange the actors in a line such that no two actors playing the same charact... |
[
{
"from": "human",
"value": "In quadrilateral $PQRS$ with right angles at $P$ and $R$, points $K$ and $L$ lie on diagonal $\\overline{PR}$. If $PK = 5$, $QK = 4$, and $KR = 3$, find the length of segment $SL$.\n\n[asy]\npair P = (0,0), Q = (5,3), R = (10, 0), S = (3, -4), K = (5, 0), L = (8,0);\ndraw(P--Q--... |
[
{
"from": "human",
"value": "Let $f(x)=x^2+2x+1$. Find the product of all $x$ that satisfy the equation $f^{-1}(x)=f(\\sqrt{x})$."
},
{
"from": "gpt",
"value": "Let $f(x)=x^2+2x+1$. Find the product of all $x$ that satisfy the equation $f^{-1}(x)=f(\\sqrt{x})$."
}
] |
[
{
"from": "human",
"value": "A semicircular strip of radius 4 cm is bounded by two parallel chords of lengths 4 cm and 12 cm. Find the length of a line segment tangent to the semicircle at its midpoint and intersecting the chords at their midpoints."
},
{
"from": "gpt",
"value": "A semicircular ... |
[
{
"from": "human",
"value": "Given that $\\log_{5}\\tan x + \\log_{5}\\cot x= 0$ and that $\\log_{5}(\\tan x-\\cot x)=\\frac{1}{2}(\\log_{5}m+1)$, find $m$."
},
{
"from": "gpt",
"value": "Given that $\\log_{5}\\tan x + \\log_{5}\\cot x= 0$ and that $\\log_{5}(\\tan x-\\cot x)=\\frac{1}{2}(\\log_... |
[
{
"from": "human",
"value": "Let $(a_n)$ be a sequence such that\n\\begin{align*}\na_1&=1,\\\\\na_2&=1+\\sqrt 3,\\\\\na_{n+2}&=\\sqrt {2a_n^2+2a_{n+1}^2}, \\qquad n=1,2,3,\\dots\n\\end{align*}\nDetermine $a_{11}$."
},
{
"from": "gpt",
"value": "Let $(a_n)$ be a sequence such that\n\\begin{align*... |
[
{
"from": "human",
"value": "Determine the values of $x$ for which the inequality\n\n$$\\frac{x^2\\sin x + x^3\\cos x - 2x^4\\sin x}{x\\sin x + x^2\\cos x - 2x^3\\sin x} \\le 1$$\n\nholds, where $x$ is a real number."
},
{
"from": "gpt",
"value": "Determine the values of $x$ for which the inequa... |
[
{
"from": "human",
"value": "Let $q(x) = x^{2009} - x^{2008} + x^{2007} - x^{2006} + \\cdots - x + 1,$\nand let $s(x)$ be the polynomial remainder when $q(x)$ is divided by $x^4-x^3+2x^2-x+1$. Find the remainder when $|s(2010)|$ is divided by $1000$."
},
{
"from": "gpt",
"value": "Let $q(x) = x^... |
[
{
"from": "human",
"value": "Determine the smallest positive integer $c$ for which $x^3 + 3x^2 + cx + 120$ can be factored into a product of three polynomials, each having integer coefficients, where one of the polynomials has a leading coefficient of 2."
},
{
"from": "gpt",
"value": "Determine ... |
[
{
"from": "human",
"value": "For a nonnegative integer $n$, let $f(n)$ be the largest real number such that for all positive real numbers $x$, we have \\[\\sin x + \\cos x \\le f(n) \\left( x + \\frac{1}{x} \\right)^n.\\]Find the smallest positive integer $m$ such that $f(m) > 1$."
},
{
"from": "gpt... |
[
{
"from": "human",
"value": "A right triangle has a hypotenuse of length 10 inches and an area of 15 square inches. What is the length, in inches, of the leg that is not adjacent to the 45-degree angle?"
},
{
"from": "gpt",
"value": "A right triangle has a hypotenuse of length 10 inches and an a... |
[
{
"from": "human",
"value": "The polynomial $cx^3 + dx^2 - 11x + 1$ has a factor of $2x^2 + x - 1.$ Find the ordered pair $(c,d)$, given that the sum of the roots of the original polynomial is equal to 2."
},
{
"from": "gpt",
"value": "The polynomial $cx^3 + dx^2 - 11x + 1$ has a factor of $2x^... |
[
{
"from": "human",
"value": "Consider a geometric sequence with first term $a_1 = 2$ and common ratio $r$. Suppose the sequence is truncated after $n$ terms, where $n$ is a positive integer. Find the smallest value of $n$ such that the sum of the remaining terms (after the first term) is at least twice the ... |
[
{
"from": "human",
"value": "Let $x$ be a real number such that\n\\[x^2 - x\\sqrt{2} + \\sqrt{3} - \\sqrt{2} + \\sqrt{4} - \\sqrt{3} + \\dots + \\sqrt{x} - \\sqrt{x-1} = 2^{m}\\]\nfor some integer $m$. Determine $m$.\n\n(Hint: Express $x^2-x\\sqrt{2}$ as $x(x-\\sqrt{2})$ and factor out $\\sqrt{2}$ from the ... |
[
{
"from": "human",
"value": "Let $f(x)$ be a monic polynomial of degree $4$ with rational coefficients such that $f(-1+\\sqrt {2})=0$ and $f(1)=10$. Determine $f(x)$."
},
{
"from": "gpt",
"value": "Let $f(x)$ be a monic polynomial of degree $4$ with rational coefficients such that $f(-1+\\sqrt {... |
[
{
"from": "human",
"value": "Find the real root of the equation $x^4 + x^3 - 6x^2 + 14x - 25 = 0$ in the form $x = \\sqrt{\\frac{a}{b}} + \\sqrt{\\frac{c}{d}}$, where $a, b, c,$ and $d$ are positive integers, and determine the value of $a + b + c + d$."
},
{
"from": "gpt",
"value": "Find the rea... |
[
{
"from": "human",
"value": "A fair 6-sided die is rolled three times. What is the probability that the sum of the numbers obtained on the first two rolls is greater than the number obtained on the third roll?"
},
{
"from": "gpt",
"value": "A fair 6-sided die is rolled three times. What is the p... |
[
{
"from": "human",
"value": "In quadrilateral $ABCD$, $AD = 12$. The length of diagonal $AC$ is 15. Let $M$ be the largest possible value of $AB^2 + BC^2 + CD^2$, and let $m$ be the smallest possible value. Find $M - m$."
},
{
"from": "gpt",
"value": "In quadrilateral $ABCD$, $AD = 12$. The ... |
[
{
"from": "human",
"value": "In a certain library, there are 4 shelves, and each shelf can hold 3 books of different genres (fiction, non-fiction, and biography). If the library wants to display exactly 6 books, with at least one book from each genre on each shelf, how many different arrangements are possib... |
[
{
"from": "human",
"value": "Determine the number of integers $n$ such that there exists an integer $a$ for which the congruence $a^2x \\equiv 1\\pmod{n}$ has a solution in $x$."
},
{
"from": "gpt",
"value": "Determine the number of integers $n$ such that there exists an integer $a$ for which th... |
[
{
"from": "human",
"value": "A cylinder with radius $r$ and height $h$ is inscribed in a sphere of radius 4, and it also contains two spheres with radius 1. The smallest possible volume of the cylinder can be expressed in the form $V\\pi.$ Find $V.$"
},
{
"from": "gpt",
"value": "A cylinder with... |
[
{
"from": "human",
"value": "The diagonals of a convex hexagon are connected to form a triangle. The triangle has an area of 30 square units. If the diagonals are also perpendicular to each other, what is the area of the hexagon?"
},
{
"from": "gpt",
"value": "The diagonals of a convex hexagon a... |
[
{
"from": "human",
"value": "Let $d(n)$ denote the number of positive integer divisors of $n$ that are also divisors of $n+1$. Find the sum of the five least positive integers $n$ that are solutions to $d(n) + d(n+1) = 5$."
},
{
"from": "gpt",
"value": "Let $d(n)$ denote the number of positive i... |
[
{
"from": "human",
"value": "Five friends - Alex, Ben, Charlie, David, and Emily - are wearing hats of different colors - blue, green, red, white, and yellow. Each person can see the colors of the other four people's hats but not their own. Starting from Alex, each person is asked to guess the color of thei... |
[
{
"from": "human",
"value": "In a triangle $PQR$ with $PR=PQ$, the bisectors of $\\angle P$ and $\\angle R$ intersect at $S$. If the circumcircle of $\\triangle PSR$ has a diameter of 14 cm, find the smallest possible area of $\\triangle PQR$."
},
{
"from": "gpt",
"value": "In a triangle $PQR$ w... |
[
{
"from": "human",
"value": "Let $P(x) = 0$ be the polynomial equation of least possible degree, with rational coefficients, having $\\sqrt[4]{10} + \\sqrt[4]{5}$ as a root. Compute the sum of all of the roots of $P(x) = 0.$"
},
{
"from": "gpt",
"value": "Let $P(x) = 0$ be the polynomial equati... |
[
{
"from": "human",
"value": "Determine the constants $a$ and $b$ such that\n$\\ds \\tan ^2 x=a \\tan 2x+b$\n\nholds for all real numbers $x$ which do not make the expressions in the equation undefined."
},
{
"from": "gpt",
"value": "Determine the constants $a$ and $b$ such that\n$\\ds \\tan ^2 x... |
[
{
"from": "human",
"value": "What is the probability that a randomly selected permutation of the digits 1 through 10 will have the property that the number formed by the first three digits is greater than the number formed by the last three digits?"
},
{
"from": "gpt",
"value": "What is the prob... |
[
{
"from": "human",
"value": "Let $a_1$, $a_2$, $\\dots,$ $a_{50}$ be the fifty points in the plane, no three collinear, and each point is colored red or blue. For each pair of these points there is a disk with the center at the midpoint between the points and the radius equal to half the distance between th... |
[
{
"from": "human",
"value": "A set of three fractions has the property that if you remove any one of the fractions, the remaining two have the same arithmetic mean. Given that one of the fractions is $\\frac{1}{2}$, and the other two have denominators of 3 and 4 respectively, what is the sum of the two unkn... |
[
{
"from": "human",
"value": "A pirate's ship is anchored at point $A$ in the Caribbean Sea. The pirate sets sail in a northwest direction, traveling $2$ miles to point $B$. For $k \\ge 1$, once the pirate reaches point $B_k$, she turns $45^{\\circ}$ clockwise and then sails $2k+1$ miles straight to point $B... |
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