conversations listlengths 2 2 |
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[
{
"from": "human",
"value": "A company offers two investment plans:\n\n**Plan 1:** A 5-year investment with an annual interest rate of 6%, compounded monthly. At the end of each year, the investor receives 20% of the interest earned, and the rest is reinvested.\n\n**Plan 2:** A 5-year investment with an ann... |
[
{
"from": "human",
"value": "An isosceles trapezoid has parallel sides of length 10 and 25, with legs of length 15. Find the area of the trapezoid when the angle between one leg and the shorter side is 45 degrees."
},
{
"from": "gpt",
"value": "An isosceles trapezoid has parallel sides of length... |
[
{
"from": "human",
"value": "A bookshelf contains an infinite number of books. The titles of the books are written using only the letters A and B. One day, a librarian starts to list the titles of the books in alphabetical order, starting with the books whose titles contain only the letter A, followed by th... |
[
{
"from": "human",
"value": "A bookstore has 5 shelves, each with a different genre of books: fiction, non-fiction, biographies, self-help, and children's books. The store owner wants to display 3 books from each genre on a special display table. If the order of the books on the table matters, how many diff... |
[
{
"from": "human",
"value": "How many different circles pass through at least three points in this 4-by-4 grid of lattice points? \n\n[asy]\nsize(30);\ndot((0,0));\ndot((1,0));\ndot((2,0));\ndot((3,0));\ndot((0,1));\ndot((1,1));\ndot((2,1));\ndot((3,1));\ndot((0,2));\ndot((1,2));\ndot((2,2));\ndot((3,2));\n... |
[
{
"from": "human",
"value": "Let $S$ denote the sum of the areas of the squares formed by joining the consecutive points $(n^2, n)$ in the coordinate plane, for all positive integers $n$. Determine the value of $S$."
},
{
"from": "gpt",
"value": "Let $S$ denote the sum of the areas of the square... |
[
{
"from": "human",
"value": "A circle centered at (1, 2) has radius r, where r is a positive integer. The circle partitions the plane into two regions. What is the smallest value of r such that the area of the region inside the circle is at least half the area of the region outside the circle?"
},
{
... |
[
{
"from": "human",
"value": "Let f(x) be a quadratic function with a leading coefficient 1 and the equation f(f(x))=0 having exactly three distinct real roots, all of them being integers. What is the largest possible value of the product of the roots of the equation f(x)=0?"
},
{
"from": "gpt",
... |
[
{
"from": "human",
"value": "Consider a sequence of points on the complex plane defined by the recurrence relation $z_n=z_{n-1}+|z_{n-2}-z_{n-3}|z_{n-2}$ for all $n\\ge 4$, where $z_1, z_2, z_3$ are randomly chosen complex numbers with integer real and imaginary parts in the range $[-10, 10]$. Find the numb... |
[
{
"from": "human",
"value": "Let $x$, $y$, $z$ be real numbers such that $x + y + z = 6$ and $x^2 + y^2 + z^2 = 24$. Find the possible values of $\\frac{x^3 + y^3 + z^3}{xyz}$."
},
{
"from": "gpt",
"value": "Let $x$, $y$, $z$ be real numbers such that $x + y + z = 6$ and $x^2 + y^2 + z^2 = 24$. ... |
[
{
"from": "human",
"value": "Six distinct integers are to be assigned to the faces of a cube so that the sum of the three integers on each pair of opposite faces is equal. How many different ways are there to do this?"
},
{
"from": "gpt",
"value": "Six distinct integers are to be assigned to the... |
[
{
"from": "human",
"value": "In a certain base $b$, the cube of $321_b$ is $1954321_b$. What is $b$?"
},
{
"from": "gpt",
"value": "In a certain base $b$, the cube of $321_b$ is $1954321_b$. What is $b$?"
}
] |
[
{
"from": "human",
"value": "Consider the following equation:\n$$x^2 + \\frac{1}{4} x + 1 = 0.$$\nDetermine the value of\n$$2x^3 + 4x^6 + 6x^9 + 8x^{12} + \\dotsb.$$\n\n(Note: This problem requires a different approach, as the equation has complex roots, and the expression to be found involves even powers o... |
[
{
"from": "human",
"value": "Simplify the expression $\\sqrt[3]{2-3\\sqrt[3]{-20}}$, and express it in the form $a+b\\sqrt[3]{c}$, where $a,$ $b,$ and $c$ are integers. Determine the value of $a+b+c$."
},
{
"from": "gpt",
"value": "Simplify the expression $\\sqrt[3]{2-3\\sqrt[3]{-20}}$, and expr... |
[
{
"from": "human",
"value": "Five cards are randomly selected from a standard deck of 52 cards, with replacement. What is the probability that exactly three of them are of different suits?"
},
{
"from": "gpt",
"value": "Five cards are randomly selected from a standard deck of 52 cards, with repl... |
[
{
"from": "human",
"value": "The polynomials $p(x)$ and $q(x)$ satisfy the equation $p(x) \\cdot q(x) = x^2 + 1$. If $p(x) = x^2 - x + 1$, what is the value of $q(x)$?"
},
{
"from": "gpt",
"value": "The polynomials $p(x)$ and $q(x)$ satisfy the equation $p(x) \\cdot q(x) = x^2 + 1$. If $p(x) = x... |
[
{
"from": "human",
"value": "Let $f(x)$ be a function defined as $f(x) = x^2 + 2x$. Find $a$ so that $f(f(f(-2))) = f(f(f(a)))$, where $a$ is a positive number."
},
{
"from": "gpt",
"value": "Let $f(x)$ be a function defined as $f(x) = x^2 + 2x$. Find $a$ so that $f(f(f(-2))) = f(f(f(a)))$, wher... |
[
{
"from": "human",
"value": "In a triangular prism, the lateral faces meet at a right angle. The area of the base is 72 square units, the area of one of the lateral faces is 48 square units, and the perimeter of the base is 30 units. Find the volume of the prism."
},
{
"from": "gpt",
"value": "I... |
[
{
"from": "human",
"value": "A right circular cone is inscribed in a sphere of radius 10. The cone's vertex is at the center of the sphere and its base is tangent to the sphere. Find the height of the cone."
},
{
"from": "gpt",
"value": "A right circular cone is inscribed in a sphere of radius 1... |
[
{
"from": "human",
"value": "A triangle with vertices (a, 0), (0, b), and (0, 0) is cut by a line through the point (c, c) into two regions of equal area. Find the slope of this line, given that a and b are relatively prime positive integers."
},
{
"from": "gpt",
"value": "A triangle with vertic... |
[
{
"from": "human",
"value": "Determine the equation of a parabola in the form $y = ax^2 + bx + c$ with $a \\neq 0$, knowing that its vertex is located at the point $(2,-3)$, it passes through the point $(0,5)$, and is tangent to the line $y = 3x - 4$."
},
{
"from": "gpt",
"value": "Determine the... |
[
{
"from": "human",
"value": "In triangle $ABC$, points $D$, $E$, and $F$ are chosen on sides $BC$, $AC$, and $AB$ respectively, so that $BD:DC = CE:EA = AF:FB = 1:2$. If $\\angle ABC = 60^\\circ$, find the measure in degrees of $\\angle DEF$."
},
{
"from": "gpt",
"value": "In triangle $ABC$, poi... |
[
{
"from": "human",
"value": "A ship is 5 km from the shore, and the angle between the ship and a nearby lighthouse, measured from the shore, is 30 degrees. If the lighthouse is 30 meters tall, how far is the ship from the lighthouse?"
},
{
"from": "gpt",
"value": "A ship is 5 km from the shore, ... |
[
{
"from": "human",
"value": "If\n$$\\frac{f(x)}{g(x)}=\\frac{ax^2-2x+1}{bx^2+3x-2},$$\nwhere $f(x)$ and $g(x)$ are linear functions and $a$ and $b$ are constants, and if $f(1)=g(-2)=0,$ find $f(x)/g(x)$."
},
{
"from": "gpt",
"value": "If\n$$\\frac{f(x)}{g(x)}=\\frac{ax^2-2x+1}{bx^2+3x-2},$$\nwhe... |
[
{
"from": "human",
"value": "A circle is inscribed inside a regular octagon (an 8-sided polygon). If two sides of the octagon are randomly chosen, what is the probability that the center of the circle lies within the quadrilateral formed by those two sides and the remaining two adjacent sides of the octagon... |
[
{
"from": "human",
"value": "Given that $x^3 + \\frac{1}{x^3} = 10$, find the value of $x + \\frac{1}{x}$."
},
{
"from": "gpt",
"value": "Given that $x^3 + \\frac{1}{x^3} = 10$, find the value of $x + \\frac{1}{x}$."
}
] |
[
{
"from": "human",
"value": "A standard deck of 52 cards is shuffled and 5 cards are drawn at random. What is the probability that the drawn cards can be arranged in a sequence such that the suits of the cards alternate?"
},
{
"from": "gpt",
"value": "A standard deck of 52 cards is shuffled and ... |
[
{
"from": "human",
"value": "In the diagram below, $AB$ is a diameter of the circle. If $\\angle BAC = 60^\\circ$ and $AC = 6$, what is the length of arc $BC$?\n\n[asy]\n\npair A,B,C,O;\n\nO = (0,0);\n\nA = (-6,0);\n\nB = (6,0);\n\nC = (-3,3*sqrt(3));\n\ndraw(Circle(O,6));\n\ndraw(A--B);\n\ndraw(A--C);\n\nl... |
[
{
"from": "human",
"value": "A magic trick involves shuffling a standard deck of 52 cards and then drawing a hand of 5 cards. What is the probability that exactly two of the drawn cards are of the same suit as the top card on the remaining deck? Express your answer as a common fraction."
},
{
"from"... |
[
{
"from": "human",
"value": "A curve defined by the equation $y = \\sin x$ has a rectangle inscribed in it, with the vertices of the rectangle touching the curve at four distinct points. What is the ratio of the width of the rectangle to its height?"
},
{
"from": "gpt",
"value": "A curve define... |
[
{
"from": "human",
"value": "In a quadrilateral $ABCD$, the midpoints of the diagonals $AC$ and $BD$ are connected to form a segment. What is the ratio of the area of triangle formed by this segment and one of the quadrilateral's sides to the area of the quadrilateral? Express your answer as a common fracti... |
[
{
"from": "human",
"value": "Problem:\nSolve for $x$ in the equation\n\\[2 \\arcsin \\frac{1}{3} + \\arcsin \\frac{1}{5} + \\arcsin \\frac{1}{x} = \\frac{\\pi}{2}.\\]"
},
{
"from": "gpt",
"value": "Problem:\nSolve for $x$ in the equation\n\\[2 \\arcsin \\frac{1}{3} + \\arcsin \\frac{1}{5} + \\ar... |
[
{
"from": "human",
"value": "In a research laboratory, two types of experiments, A and B, are conducted. The time required to complete each experiment is constant, but the time for A is not necessarily equal to the time for B. Three researchers can complete 12 experiments of type A and 15 experiments of typ... |
[
{
"from": "human",
"value": "Find the smallest positive integer that has exactly 10 positive divisors, with the additional constraint that none of these divisors are prime numbers."
},
{
"from": "gpt",
"value": "Find the smallest positive integer that has exactly 10 positive divisors, with the a... |
[
{
"from": "human",
"value": "In the Cartesian plane, a circle centered at $(2,1)$ has radius $r$ such that it intersects the circles centered at $(0,0)$ and $(4,2)$ with radii $1$ and $2$, respectively, but contains no points that lie inside either of these two circles. What is the largest possible value of... |
[
{
"from": "human",
"value": "Find the remainder when the polynomial $p(x)$ is divided by $(x^2 + 2x + 3)$, given that the remainder when $p(x)$ is divided by $(x + 1)$ is $2x + 1$, and the remainder when $p(x)$ is divided by $(x + 2)$ is $x - 3$."
},
{
"from": "gpt",
"value": "Find the remainder... |
[
{
"from": "human",
"value": "The geometric progressions $\\{2, 6, 18, 54, \\ldots\\}$ and $\\{3, 9, 27, 81, \\ldots \\}$ have some common values. What is the smallest value greater than 1000 that they have in common?"
},
{
"from": "gpt",
"value": "The geometric progressions $\\{2, 6, 18, 54, \\l... |
[
{
"from": "human",
"value": "A polyhedron has 9 faces, each of which is an isosceles triangle. How many vertices does the polyhedron have?"
},
{
"from": "gpt",
"value": "A polyhedron has 9 faces, each of which is an isosceles triangle. How many vertices does the polyhedron have?"
}
] |
[
{
"from": "human",
"value": "In a triangle $ABC$, $\\sin A = \\frac{3}{5}$ and $BC = 24$. Find the length of the altitude from $A$ to $BC$."
},
{
"from": "gpt",
"value": "In a triangle $ABC$, $\\sin A = \\frac{3}{5}$ and $BC = 24$. Find the length of the altitude from $A$ to $BC$."
}
] |
[
{
"from": "human",
"value": "Let\nf(x)=\\left\\{\n\\begin{array}{ll}\n\\ds\\frac{12}{9^x}& \\text{if } x<0\\\\\nx+3& \\text{if } x \\geqslant 0\n\\end{array}\n\\right. .\n\nFind the smallest positive value of $x$ for which there exists a nonnegative integer $n$ such that $\\displaystyle \\sum_{k=1}^{n} f\\l... |
[
{
"from": "human",
"value": "The price of a certain stock is initially $\\$p$ and increases by a fixed percentage, $r\\%$, every day. If the stock's price after $n$ days is $\\$1000$, find the smallest value of $n$ for which the stock's price exceeds $\\$2000$, given that $p$ and $r$ are integers."
},
{... |
[
{
"from": "human",
"value": "A prism with a square base has a height of $h$ units and a side length of the square base equal to $s$ units. Its volume is equal to $V$ cubic units. The midpoints of the three faces meeting at one of its vertices form a triangle. The area of the triangle formed by the midpoints... |
[
{
"from": "human",
"value": "Problem:\nWe have a deck of 52 cards, and we draw 5 cards at random. What is the probability that we get at least 3 cards of the same suit?\n\nThis problem requires a different approach than the original coin flip problem, as it involves combinations and probability of drawing s... |
[
{
"from": "human",
"value": "A right circular cone is tangent to the x-y plane at the point $(0,0)$. Its vertex is $h$ units above this plane, and it is also tangent to the sphere $x^2 + y^2 + z^2 = 25$. What is the radius of the base of the cone?"
},
{
"from": "gpt",
"value": "A right circular ... |
[
{
"from": "human",
"value": "In a regular hexagon $ABCDEF$, the length of each side is $1$ unit. A point $P$ is located inside the hexagon such that $\\triangle PAB$ is an equilateral triangle. Determine the perimeter of the pentagon $PCDEF$."
},
{
"from": "gpt",
"value": "In a regular hexagon $... |
[
{
"from": "human",
"value": "Simplify the expression $\\dfrac{1000}{11111} \\cdot \\dfrac{1001}{111}.$"
},
{
"from": "gpt",
"value": "Simplify the expression $\\dfrac{1000}{11111} \\cdot \\dfrac{1001}{111}.$"
}
] |
[
{
"from": "human",
"value": "Let $p$ and $q$ be two positive integers. Bob's favorite number $n$ has the property that the greatest common divisor of $n$ and $p$ is greater than $1$, while the greatest common divisor of $n$ and $q$ is equal to $1$. Furthermore, $n$ is a three-digit number that has exactly... |
[
{
"from": "human",
"value": "In the triangle $APB$, the point $P$ lies on the angle bisector of angle $BAP$. The sides $AP$ and $BP$ are extended to meet the circumcircle of $APB$ at points $X$ and $Y$ respectively. If $AX = 11$ and $BY = 13$, find $\\frac{AP}{PB}$."
},
{
"from": "gpt",
"value":... |
[
{
"from": "human",
"value": "A fair coin is flipped 10 times. What is the probability that there are at least 3 consecutive heads in the sequence of flips? Express your answer as a common fraction."
},
{
"from": "gpt",
"value": "A fair coin is flipped 10 times. What is the probability that there... |
[
{
"from": "human",
"value": "Let $f(x)$ and $g(x)$ be two functions such that $f(g(x)) = x^2 + 2x + 1$ and $g(f(x)) = x^3 - 3x^2 - 2x + 1$. If $f(x) = x^2 - 2x + 1$, find the expression for $g(x)$."
},
{
"from": "gpt",
"value": "Let $f(x)$ and $g(x)$ be two functions such that $f(g(x)) = x^2 + 2... |
[
{
"from": "human",
"value": "If $\\ds \\sqrt {3x-4}-\\sqrt {2x-3}=1$,\nfind $\\ds \\sqrt {x+5}+\\sqrt {x+6}$."
},
{
"from": "gpt",
"value": "If $\\ds \\sqrt {3x-4}-\\sqrt {2x-3}=1$,\nfind $\\ds \\sqrt {x+5}+\\sqrt {x+6}$."
}
] |
[
{
"from": "human",
"value": "Let $P(x)$ be a polynomial with integer coefficients such that $P(1) = P(2) = P(3) = 0$, $P(4) = 2$, and $P(5) = -2$. What is the smallest possible degree of $P(x)$?"
},
{
"from": "gpt",
"value": "Let $P(x)$ be a polynomial with integer coefficients such that $P(1) =... |
[
{
"from": "human",
"value": "What four-digit integer is equal to the product of the cubes of its digits?"
},
{
"from": "gpt",
"value": "What four-digit integer is equal to the product of the cubes of its digits?"
}
] |
[
{
"from": "human",
"value": "Suppose $f(x) = x^2 - 1$, and $g(x)$ is a polynomial such that $f(g(x)) = x^3 + 2x^2 + x + 1$. Determine the possible polynomials $g(x)$ that satisfy the equation.\n\nNote: The new problem changes the function f(x) and the result of f(g(x)), which requires a different approach t... |
[
{
"from": "human",
"value": "The integers $A, B, C,$ and $D$ satisfy the following conditions:\n$A+B+C+D=40,\nA-4=B-4=C \\times 4=D \\div 4,\nB=2A,\nD=2C$.\nWhat is the value of $B$?"
},
{
"from": "gpt",
"value": "The integers $A, B, C,$ and $D$ satisfy the following conditions:\n$A+B+C+D=40,\nA... |
[
{
"from": "human",
"value": "Find the positive root of the equation\n\n$$x^3 - 3x^2 - x - \\sqrt{2} = 0$$\n\nsuch that $x^2 + 1 = \\sqrt{2}.$"
},
{
"from": "gpt",
"value": "Find the positive root of the equation\n\n$$x^3 - 3x^2 - x - \\sqrt{2} = 0$$\n\nsuch that $x^2 + 1 = \\sqrt{2}.$"
}
] |
[
{
"from": "human",
"value": "Consider the functions $g(x)=x^2-x-1$ and $h(x)=x^2-2x+1$. Find all the values of $x$ for which the compositions $g(g(x))$ and $h(h(x))$ are equal."
},
{
"from": "gpt",
"value": "Consider the functions $g(x)=x^2-x-1$ and $h(x)=x^2-2x+1$. Find all the values of $x$ fo... |
[
{
"from": "human",
"value": "Let $A$, $B$, $C$ be three-digit numbers such that $A$ is obtained by reversing the digits of $B$, and $C$ is obtained by reversing the digits of $A$. If $A+C=1203$, determine $B$.\n\n $1203 = A+C = A + \\bar{A} = 100a + 10b + c + 100c + 10b + a$ $a+c=12, b=0$\nA is divisible b... |
[
{
"from": "human",
"value": "Let $f(x) = x^3 + ax^2 + bx + c$ be a polynomial with integer coefficients. Determine the number of quadratic polynomials $g(x) = x^2 + px + q$ with integer coefficients such that $g(x)$ is a factor of $f(x)$ for some values of $a$, $b$, and $c$."
},
{
"from": "gpt",
... |
[
{
"from": "human",
"value": "An equilateral triangle ABC is inscribed in a circle with center O. The line segment MO is perpendicular to AB, where M is the midpoint of AB. The circle intersects MO at point P, and MP is 4 times the length of PO. What is the measure of angle CPM?"
},
{
"from": "gpt",
... |
[
{
"from": "human",
"value": "Problem:\nFind the greatest common divisor of two polynomials: $x^3 + 2x^2 - 3x - 4$ and $x^2 - 5x + 6$."
},
{
"from": "gpt",
"value": "Problem:\nFind the greatest common divisor of two polynomials: $x^3 + 2x^2 - 3x - 4$ and $x^2 - 5x + 6$."
}
] |
[
{
"from": "human",
"value": "Let $\\mathbf{u},$ $\\mathbf{v},$ $\\mathbf{w}$ be vectors, and let $E$ be the determinant of the matrix whose row vectors are $\\mathbf{u},$ $\\mathbf{v},$ and $\\mathbf{w}.$ Then the determinant of the matrix whose row vectors are $\\mathbf{u} \\times \\mathbf{v},$ $\\mathbf{... |
[
{
"from": "human",
"value": "Solve the equation $\\sqrt{x^2-4x+4} + \\frac{1}{\\sqrt{x^2-4x+4}} = \\sqrt{2x-3}.$"
},
{
"from": "gpt",
"value": "Solve the equation $\\sqrt{x^2-4x+4} + \\frac{1}{\\sqrt{x^2-4x+4}} = \\sqrt{2x-3}.$"
}
] |
[
{
"from": "human",
"value": "Determine the largest possible value of $x$ such that the polynomial $12x^3+3x^2-7x+15$ is divisible by 5."
},
{
"from": "gpt",
"value": "Determine the largest possible value of $x$ such that the polynomial $12x^3+3x^2-7x+15$ is divisible by 5."
}
] |
[
{
"from": "human",
"value": "In pentagon $ABCDE$, $\\angle B$ and $\\angle D$ are right angles, diagonal $\\overline{AC}$ is perpendicular to $\\overline{CD}$, diagonal $\\overline{AD}$ is perpendicular to $\\overline{DE}$, $AB=15$, $BC=20$, and $DE=12$. Find the length of $\\overline{AE}$."
},
{
"f... |
[
{
"from": "human",
"value": "Consider a function $f(x)$ defined on the real numbers by $f(x) = \\alpha x + \\beta,$ where $\\alpha$ and $\\beta$ are constants, and $\\alpha > 0.$ This function has the property that for each real number $x$, $f(x)$ is twice as far from $x$ as from $\\beta.$ Given that $\\alp... |
[
{
"from": "human",
"value": "What is the value of the expression $|z_1| + |z_2|$ given that $z_1$ and $z_2$ are complex numbers satisfying $z_1^2 + z_2^2 = 8$ and $z_1z_2 = 6$?"
},
{
"from": "gpt",
"value": "What is the value of the expression $|z_1| + |z_2|$ given that $z_1$ and $z_2$ are compl... |
[
{
"from": "human",
"value": "Let $g(x) = x^2|x|.$ Find $g^{-1}(8) + g^{-1}(-27)$."
},
{
"from": "gpt",
"value": "Let $g(x) = x^2|x|.$ Find $g^{-1}(8) + g^{-1}(-27)$."
}
] |
[
{
"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": "Let $\\mathbf{A}$ be a $2\\times2$ matrix such that $\\mathbf{A}^2=\\begin{pmatrix} 1 & 2 \\\\ 3 & 4 \\end{pmatrix}$. Find $\\mathbf{A}\\begin{pmatrix} 5 \\\\ 6 \\end{pmatrix}$."
},
{
"from": "gpt",
"value": "Let $\\mathbf{A}$ be a $2\\times2$ matrix such that $\\mat... |
[
{
"from": "human",
"value": "Compute the value of $\\arccos(\\cos 70^\\circ + 2 \\sin 35^\\circ)$ in degrees as an angle between $0^\\circ$ and $180^\\circ$."
},
{
"from": "gpt",
"value": "Compute the value of $\\arccos(\\cos 70^\\circ + 2 \\sin 35^\\circ)$ in degrees as an angle between $0^\\ci... |
[
{
"from": "human",
"value": "What is the largest positive integer $n$ such that $\\tan(n^2+1)$ is an integer multiple of $\\tan(n+1)$?"
},
{
"from": "gpt",
"value": "What is the largest positive integer $n$ such that $\\tan(n^2+1)$ is an integer multiple of $\\tan(n+1)$?"
}
] |
[
{
"from": "human",
"value": "In triangle $ABC$, the angle bisector of $\\angle A$ intersects the circumcircle at point $D$. If $\\angle B$ measures $60^\\circ$, what is the measure of $\\angle C$?"
},
{
"from": "gpt",
"value": "In triangle $ABC$, the angle bisector of $\\angle A$ intersects the ... |
[
{
"from": "human",
"value": "Find the product of the constants $a_1$ and $a_2$ if the polynomial $$x^9 + 3x^6 + 7x^3 + 2$$ can be factored as $$(x^3 + a_1x^2 + a_2x + 1)(x^6 + bx^4 + cx^2 + 2).$$"
},
{
"from": "gpt",
"value": "Find the product of the constants $a_1$ and $a_2$ if the polynomial $... |
[
{
"from": "human",
"value": "Let $ABC$ be a triangle with $\\angle ABC = \\angle BAC = 70^{\\circ}.$ A circle $\\Gamma$ tangent to sides $AB$ and $AC$ intersects line $BC$ at $D,$ and the angle bisector of $\\angle ABC$ meets $AC$ at $E.$ If $AB = 3,$ $AC = 4,$ and $BE = BD,$ find $BC.$"
},
{
"from"... |
[
{
"from": "human",
"value": "A circle is inscribed in a triangle whose vertices are the roots of the equation $x^3-6x^2+11x-6=0$.\nFind the radius of this circle."
},
{
"from": "gpt",
"value": "A circle is inscribed in a triangle whose vertices are the roots of the equation $x^3-6x^2+11x-6=0$.\n... |
[
{
"from": "human",
"value": "Given a polynomial $Q(x)$ of degree $6$ such that its roots are the reciprocals of the roots of $P(x)$, determine the sum of the coefficients of $Q(x)$."
},
{
"from": "gpt",
"value": "Given a polynomial $Q(x)$ of degree $6$ such that its roots are the reciprocals of ... |
[
{
"from": "human",
"value": "In tetrahedron $ABCD$, the midpoints of edges $\\overline{AB}$, $\\overline{AC}$, $\\overline{BC}$, $\\overline{BD}$, $\\overline{CD}$, and $\\overline{AD}$ are $E$, $F$, $G$, $H$, $I$, and $J$, respectively.\n\nLine segments $\\overline{EF}$, $\\overline{GI}$, and $\\overline{H... |
[
{
"from": "human",
"value": "Consider a set of 1024 logical statements, each of which can be either true or false. The statements are labeled with the 1024 different subsets of a 10-element set. At step i of a 1024-step process, the i-th statement is negated, and so are all the other statements that are log... |
[
{
"from": "human",
"value": "Consider a three-dimensional Cartesian space. Three friends, Alice, Bob, and Chris, are standing at points $A(-3, 1, -2),$ $B(4, 0, -3),$ and $C(-1, 4, 1),$ respectively. Their goal is to meet at a point on the $z$-axis that has the least total distance from their current positi... |
[
{
"from": "human",
"value": "Let A' be a point on the circle x^2 + y^2 - 12x + 31 = 0. Find the smallest possible distance between A' and the line segment connecting points (0,0) and (4,2)."
},
{
"from": "gpt",
"value": "Let A' be a point on the circle x^2 + y^2 - 12x + 31 = 0. Find the smallest... |
[
{
"from": "human",
"value": "Let $\\alpha$ and $\\beta$ be the roots of the equation $x^2+px+q=0$, where $p$ and $q$ are integers. Given that $\\left\\lceil \\alpha^2 \\right\\rceil - \\left\\lceil \\beta^2 \\right\\rceil = 5$, find all possible values of $p$."
},
{
"from": "gpt",
"value": "Let ... |
[
{
"from": "human",
"value": "How many four-digit numbers are composed of four distinct digits such that the sum of the two middle digits is equal to the sum of the first and last digits?"
},
{
"from": "gpt",
"value": "How many four-digit numbers are composed of four distinct digits such that the... |
[
{
"from": "human",
"value": "The sum of the reciprocals of the first $n$ positive integers is $S_n = \\frac{1}{1} + \\frac{1}{2} + \\frac{1}{3} +... + \\frac{1}{n}$. Express $S_n$ in terms of the sum of the reciprocals of the consecutive pairs of positive integers, i.e., in terms of $\\frac{1}{1\\times2} + ... |
[
{
"from": "human",
"value": "Let $0 \\le p,$ $q,$ $r,$ $s \\le 2.$ Find the possible values of the expression\n\\[\\sqrt[3]{p^3 + (2 - q)^3} + \\sqrt[3]{q^3 + (2 - r)^3} + \\sqrt[3]{r^3 + (2 - s)^3} + \\sqrt[3]{s^3 + (2 - p)^3}.\\]"
},
{
"from": "gpt",
"value": "Let $0 \\le p,$ $q,$ $r,$ $s \\l... |
[
{
"from": "human",
"value": "A sequence of three positive integers forms an arithmetic progression. If the middle term is multiplied by the sum of its base-3 digits, the product is equal to the sum of the squares of the first and last terms. Find the base-3 representation of the middle term."
},
{
"... |
[
{
"from": "human",
"value": "Evaluate the following expression:\n\n$$\\sqrt{\\left|2-\\sqrt{2+\\sqrt{2-\\sqrt{2+\\sqrt{2}}}}\\right|}.$$"
},
{
"from": "gpt",
"value": "Evaluate the following expression:\n\n$$\\sqrt{\\left|2-\\sqrt{2+\\sqrt{2-\\sqrt{2+\\sqrt{2}}}}\\right|}.$$"
}
] |
[
{
"from": "human",
"value": "A farmer has a farm with two types of animals: chickens and cows. One day, he counted a total of 27 animals and 70 feet. If the number of chickens is 3 more than the number of cows, how many chickens does the farmer have?"
},
{
"from": "gpt",
"value": "A farmer has a... |
[
{
"from": "human",
"value": "The function $f$ is not defined for $x=0$, but for all non-zero real numbers $x$, \n$$f(x)-\\frac{1}{x}f(1/x)=x^3$$\nFind the real solutions to $f(x)=-f(-x)$."
},
{
"from": "gpt",
"value": "The function $f$ is not defined for $x=0$, but for all non-zero real numbers ... |
[
{
"from": "human",
"value": "A semicircle with endpoints on the positive x-axis and centre on the y-axis is tangent to a square with one vertex at the origin. The length of the side of the square is 4. What is the radius of the semicircle?"
},
{
"from": "gpt",
"value": "A semicircle with endpoin... |
[
{
"from": "human",
"value": "Determine all real $p$ and $q$ such that the equations $\\frac{qy}{y^2-1} = \\frac{y}{y-1} - \\frac{q}{y+1}$ and $\\frac{py}{y^2-4} = \\frac{2}{y-2} - \\frac{py}{y+2}$ have the same roots."
},
{
"from": "gpt",
"value": "Determine all real $p$ and $q$ such that the eq... |
[
{
"from": "human",
"value": "Let p,q be positive integers such that p and q are not coprime. \nLet A(p) denote a set of p consecutive integers and B(q) denote a set of q consecutive integers.\nIt is known that A(p)∩B(q)={1}.\nFind the smallest possible value of min{max A(p),max B(q)}."
},
{
"from": ... |
[
{
"from": "human",
"value": "The 3rd and 5th terms of an arithmetic sequence are 17 and 39, respectively. What is the 7th term of the same sequence? \n\nAnother problem inspired by this one:\n\nIn a geometric sequence, the product of the 2nd and 5th terms is 72, and the product of the 3rd and 4th terms is 5... |
[
{
"from": "human",
"value": "In a small town, there are 5 hotels, each with a different rating (1 to 5 stars). A travel agency wants to assign 3 inspectors to review these hotels, with each inspector visiting exactly one hotel. However, due to a conflict of interest, inspectors from the same country cannot ... |
[
{
"from": "human",
"value": "Find the greatest possible value of $x$ such that $$\\frac{(2x+5)(x^2+7)}{(x-3)(x^2+2x-15)}=x^2+3x+8.$$\n\nNote: I've tried to create a new problem that requires a different approach to solve. The original problem involved manipulating and solving a quadratic equation, while the... |
[
{
"from": "human",
"value": "Given that $$x^3+y^3+z^3=125$$and that $$x^2y+x^2z+y^2x+y^2z+z^2x+z^2y=35$$for real numbers $x$, $y$, and $z$, what is the value of $x+y+z$?"
},
{
"from": "gpt",
"value": "Given that $$x^3+y^3+z^3=125$$and that $$x^2y+x^2z+y^2x+y^2z+z^2x+z^2y=35$$for real numbers $x$... |
[
{
"from": "human",
"value": "The projection of $\\begin{pmatrix} 0 \\\\ 1 \\\\ 4 \\end{pmatrix}$ onto a certain vector $\\mathbf{w}$ is $\\begin{pmatrix} 1 \\\\ -1/2 \\\\ 1/2 \\end{pmatrix}.$ Find the projection of $\\begin{pmatrix} 3 \\\\ 3 \\\\ -2 \\end{pmatrix}$ onto $\\mathbf{w}$.\n\nFind the vector $\... |
[
{
"from": "human",
"value": "Let A, B, C, D, and E be points on a circle with AB = BC = CD = DE = 1 and AC = 2, and let F be a point on the circle, different from A, B, C, D and E, such that ∠CEB = ∠BAD and ∠CBD = ∠ADE.\nFind the smallest possible value of EF."
},
{
"from": "gpt",
"value": "Let ... |
[
{
"from": "human",
"value": "A treasure chest contains $123_{10}$ gold coins. A group of pirates want to divide the treasure equally among themselves, but they only have an octal (base-8) scale to weigh the coins. How many pirates can there be if each pirate gets an equal number of coins that is a multiple ... |
[
{
"from": "human",
"value": "The sequences of positive integers $1,a_2, a_3, \\dots$ and $1,b_2, b_3, \\dots$ are a decreasing arithmetic sequence and a decreasing geometric sequence, respectively. Let $c_n=a_n+b_n$. There is an integer $k$ such that $c_{k-1}=200$ and $c_{k+1}=800$. Find the largest possibl... |
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