conversations listlengths 2 2 |
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[
{
"from": "human",
"value": "Let $f(x)=x^4+ax^3+bx^2+cx+d$ be a quartic polynomial with real coefficients and leading coefficient $1$.\nSuppose that for some real number $k,$\nthe equation $f(x)=k$ has exactly $3$ distinct real roots for any value of $k.$\nFind the maximum value of $b.$"
},
{
"from"... |
[
{
"from": "human",
"value": "Let $a$ and $b$ be real numbers such that $0 < a < \\frac{\\pi}{2}$ and $0 < b < \\frac{\\pi}{2}.$ Find the smallest possible value of $a + b$ such that the function\n\\[f(x) = \\cos (x - a) + \\cos (x - b)\\]achieves its maximum value."
},
{
"from": "gpt",
"value":... |
[
{
"from": "human",
"value": "A certain positive integer has the property that if any one of its digits is changed, the resulting number is always divisible by 3. What is the smallest number that has this property?"
},
{
"from": "gpt",
"value": "<think> Okay, so I need to find the smallest positi... |
[
{
"from": "human",
"value": "Find the function $g(x)$ defined on all positive real numbers $x$ that satisfies the functional equation\n\\[\\frac{g(x)}{g(y)} - \\frac{g(x+y)}{g(x-y)} = \\frac{x^2-y^2}{2}\\]\nfor all $x > y > 0$."
},
{
"from": "gpt",
"value": "<think> Okay, so I need to find a fun... |
[
{
"from": "human",
"value": "The Cybermen have again outwitted the Daleks! Unfortunately, this time the Daleks decided to abandon these tasks altogether, which means the Doctor has to deal with them.\n\nThe Doctor can handle the Daleks on his own, but Heidi now has to make sure that the Cybermen are kept bu... |
[
{
"from": "human",
"value": "Potassium ferrocyanide is diamagnetic (no unpaired electron) while potassium ferricyanide is paramagnetic (one unpaired electron)."
},
{
"from": "gpt",
"value": "<think> Okay, so I need to understand why potassium ferrocyanide is diamagnetic and potassium ferricyanid... |
[
{
"from": "human",
"value": "Express $5\\sqrt{6}+12\\sqrt{2}$ in the form $\\sqrt{a}+\\sqrt{b}+\\sqrt{c}$, where $a,$ $b,$ and $c$ are integers."
},
{
"from": "gpt",
"value": "<think> Okay, so I need to express \\(5\\sqrt{6} + 12\\sqrt{2}\\) in the form \\(\\sqrt{a} + \\sqrt{b} + \\sqrt{c}\\) wh... |
[
{
"from": "human",
"value": "Warning: this is a complex challenge.\nOverview\nA lightbox, in the context of computing, is a visual effect used in image sites. A list of thumbnail images is displayed; each image can be selected and will be displayed on top of the contents of the page either at full size or t... |
[
{
"from": "human",
"value": "While Grisha was celebrating New Year with Ded Moroz, Misha gifted Sasha a small rectangular pond of size n × m, divided into cells of size 1 × 1, inhabited by tiny evil fishes (no more than one fish per cell, otherwise they'll strife!).\n\nThe gift bundle also includes a square... |
[
{
"from": "human",
"value": "A jar contains a gas and a few drops of water at T K. The pressure in the jar is 830 mm of Hg. The temperature of the jar is reduced by 1%. The vapour pressure of water at two temperature are 30 and 25 mm of Hg. Calculate the new pressure in the jar."
},
{
"from": "gpt",... |
[
{
"from": "human",
"value": "Determine all real numbers $b$ such that the equation\n\n$$x^4+bx^3+b^2x^2-6x+1=0$$\n\nhas exactly two distinct real solutions in $x$."
},
{
"from": "gpt",
"value": "<think> Okay, so I have this equation: x⁴ + b x³ + b² x² -6x +1 =0. I need to find all real numbers b... |
[
{
"from": "human",
"value": "For the case of electrophilic addition of hydrogen bromide to 1,3-butadiene, the reaction produces a mixture of two regioisomeric allylic halides: 3-Bromo-1-butene (major product, formed by 1,2 addition) and 1-Bromo-2-butene (minor product, formed by 1,4 addition). \n\nWhat are ... |
[
{
"from": "human",
"value": "Draw the structures of BCl\\(_3\\)·NH\\(_3\\) and AlCl\\(_3\\) (dimer)."
},
{
"from": "gpt",
"value": "<think> Okay, I need to draw the structures of BCl3·NH3 and the AlCl3 dimer. Let me start with BCl3·NH3 first. \n\nSo, BCl3 is boron trichloride. I remember that bo... |
[
{
"from": "human",
"value": "For example, Kaku's QFT pp. 214-215:\n\nMassive vector theory with non-Abelian group is non-renormalizable.\nMassive vector Abelian theory is renormalizable.\n\nI heard about the following arguments, but I don't find them satisfactory.\n\nSomeone will say the propagator of massi... |
[
{
"from": "human",
"value": "The quartic polynomial $x^4+4x^3-10x^2-16x+5$ can be expressed as a product of two quadratic factors, one of which is of the form $x^2+px+q$. Find $p+q$ such that the other quadratic factor has no linear term."
},
{
"from": "gpt",
"value": "<think> Alright, let me ta... |
[
{
"from": "human",
"value": "This is an interactive problem.\n\nThe judge has a hidden rooted full binary tree with n leaves. A full binary tree is one where every node has either 0 or 2 children. The nodes with 0 children are called the leaves of the tree. Since this is a full binary tree, there are exactl... |
[
{
"from": "human",
"value": "Background\nSupplementary reading 1, Supplementary reading 2\nLinear lambda calculus is a limited form of lambda calculus, where every bound variable must be used exactly once. For example, \\a b c d e -> a b (d c) e is a valid term in linear lambda calculus. When embedded as a ... |
[
{
"from": "human",
"value": "Find all possible values of $a$ and $b$ such that the number $10a+b$ is a perfect square and the number $100a+b$ is a perfect cube. What is the maximum sum of $a$ and $b$?"
},
{
"from": "gpt",
"value": "<think> Okay, so I need to find all possible values of a and b s... |
[
{
"from": "human",
"value": "Fedya has a string S, initially empty, and an array W, also initially empty.\n\nThere are n queries to process, one at a time. Query i consists of a lowercase English letter c_i and a nonnegative integer w_i. First, c_i must be appended to S, and w_i must be appended to W. The a... |
[
{
"from": "human",
"value": "Consider a $(2,0)$ tensor $X^{\\mu \\nu}$ that can be represented in matrix form by:\n$$X^{\\mu \\nu} = \n\\pmatrix{\na & b & c & d \\\\\ne & f & g & h \\\\\ni & j & k & l \\\\\nm & n & o & p}\\tag{1}$$\nHere $a, b, c, ..., l \\in \\mathbb{R}$ are scalars.\nWe can obtain ${{X}_{... |
[
{
"from": "human",
"value": "Consider two functions, $p(x) = 2x^2 - 3x + 1$ and $q(x) = x^2 + 2x - 3$. Let $r(x) = p(x) + q(x)$ and $s(x) = p(x)q(x)$. Find the number of intersections between the graphs of $y = r(x)$ and $y = s(x)$ and determine the nature of these intersections."
},
{
"from": "gpt"... |
[
{
"from": "human",
"value": "Let\n\\[x=\\ds\\cos \\frac{4\\pi}{7}+\\ds\\cos \\frac{4\\pi}{7}\\cos \\frac{8\\pi}{7}+\\ds\\cos \\frac{4\\pi}{7}\\cos \\frac{8\\pi}{7}\\cos \\frac{16\\pi}{7}.\\]\nProve that \\(7x^{3}+7x^{2}-x-1=0\\)."
},
{
"from": "gpt",
"value": "<think> Okay, so I have this trigon... |
[
{
"from": "human",
"value": "Let $r$ and $s$ be the solutions of the equation\n$$5^x+7^x=12^x.$$\nDetermine $\\frac{1}{r}+\\frac{1}{s}$."
},
{
"from": "gpt",
"value": "<think> Okay, so I need to figure out the value of 1/r + 1/s where r and s are the solutions to the equation 5^x + 7^x = 12^x. H... |
[
{
"from": "human",
"value": "Let $a$, $b$, and $c$ be positive integers such that $a$ is the smallest integer that can be expressed as the product of three distinct primes, $b$ is the largest integer less than $500$ that can be expressed as the product of two distinct primes, and $c$ is the largest integer ... |
[
{
"from": "human",
"value": "(related: one, two, three)\nAn acrostic is a style of poem/writing where the beginning character of each line, when read vertically, also produces a word or message. For example,\nTogether\nEveryone\nAchieves\nMore\n\nalso spells out the word TEAM when the first column is read v... |
[
{
"from": "human",
"value": "How many quadrilaterals with positive area are there whose vertices are points in the $xy$-plane whose coordinates are integers $(x,y)$ satisfying $1\\le x\\le 5$ and $1\\le y\\le 5$?"
},
{
"from": "gpt",
"value": "<think> Okay, so I have this problem here: \"How man... |
[
{
"from": "human",
"value": "A rectangular prism is to be wrapped with wrapping paper. The rectangular prism has a length of $l$ cm, a width of $w$ cm, and a height of $h$ cm. The wrapping paper is rectangular with a width of $x$ cm and a height of $y$ cm. When the wrapping paper is wrapped around the recta... |
[
{
"from": "human",
"value": "A quadratic polynomial $P(x)=x^2+bx+c$ has two real roots $\\alpha$ and $\\beta$ such that\n$\\frac{\\alpha}{\\beta} + \\frac{\\beta}{\\alpha}=8$\nand $b - c = 4$.\nDetermine the sum of all possible values of $b$."
},
{
"from": "gpt",
"value": "<think> Okay, so I hav... |
[
{
"from": "human",
"value": "Chess960, a.k.a. Fisher random chess, is a variation of the game of chess.\nIn classic chess the starting positions are always the same. In chess960 they vary.\nBefore each game of chess960 one random number from 0 to 959 is drawn. Both players then arrange their pieces in the s... |
[
{
"from": "human",
"value": "Find the range of the function\n\n$$g(x) = \\frac{x^2(1+x)^2}{(x+1)(x^2-4)}$$\n\nif $x \\neq -1, 2, -2$. Enter your answer using interval notation."
},
{
"from": "gpt",
"value": "<think> Okay, so I need to find the range of the function g(x) = [x²(1 + x)²] / [(x + 1)... |
[
{
"from": "human",
"value": "A telephone company charges for international calls according to the following rules:\n\n* If the call is shorter than 10 minutes, the company charges $0.50 per minute.\n* If the call is 10-30 minutes long, the company charges $5 plus $0.25 per minute over 10 minutes.\n* If the ... |
[
{
"from": "human",
"value": "Let $N$ be a three-digit number that is divisible by 5 and has the property that when its digits are reversed, the resulting number is $\\frac{3}{5}N$. How many such numbers $N$ exist?"
},
{
"from": "gpt",
"value": "<think> Okay, let's try to tackle this problem step... |
[
{
"from": "human",
"value": "A positive integer $a$ is chosen so that $a$, $a+1$, and $a+2$ are all relatively prime to $2019$. Find the least possible value of $a$ modulo $2019$.\n\nNote: The goal is to find an integer $a$ such that $a$, $a+1$, and $a+2$ are all coprime with $2019$, which means that none o... |
[
{
"from": "human",
"value": "In triangle $PQR$, $\\angle P = 2\\angle Q$, and the bisector of $\\angle R$ intersects $\\overline{PQ}$ at point $S$. If the measure of $\\angle PSR$ is $25^\\circ$, find the measure of $\\angle Q$."
},
{
"from": "gpt",
"value": "<think> Okay, so I have this geometr... |
[
{
"from": "human",
"value": "A bookshelf has 5 shelves, and each shelf can hold a maximum of 8 books. If the bookshelf is currently empty, how many different ways can 12 books be arranged on the shelves such that no shelf is empty?"
},
{
"from": "gpt",
"value": "<think> Alright, so I need to fig... |
[
{
"from": "human",
"value": "In trapezoid $ABCD$, the diagonals $AC$ and $BD$ intersect at point $E$, and $AE:EC = 2:3$. If $AB = 12$, $CD = 9$, and $BC = 15$, find the area of $\\triangle ADE$."
},
{
"from": "gpt",
"value": "<think> Okay, so I need to find the area of triangle ADE in trapezoid ... |
[
{
"from": "human",
"value": "A rectangular piece of cardboard has an area of 24 square inches. If it is cut into smaller squares of the same size, either 3 or 5 squares can be obtained. Determine the dimensions of the original rectangle."
},
{
"from": "gpt",
"value": "<think> Okay, so I need to ... |
[
{
"from": "human",
"value": "A rotation of the plane takes $2 + 3i$ to $-1 + 4i.$ Find the complex number that the rotation takes $5 - 2i$ to."
},
{
"from": "gpt",
"value": "<think> Okay, so I have this problem here: A rotation of the plane takes 2 + 3i to -1 + 4i. Find the complex number that ... |
[
{
"from": "human",
"value": "How many rectangles are there whose four vertices are points on this grid? [asy]\nsize(50);\ndot((0,0));\ndot((5,0));\ndot((10,0));\ndot((0,5));\ndot((0,10));\ndot((5,5));\ndot((5,10));\ndot((10,5));\ndot((10,10));\n[/asy]"
},
{
"from": "gpt",
"value": "<think> Alrig... |
[
{
"from": "human",
"value": "The function $g(x)$ satisfies\n\\[g(x) + g\\left(\\frac{1}{x}\\right) = 2x\\]for all $x \\neq 0,$ and $g(2) = 3.$ Find $g\\left(\\frac{1}{5}\\right).$"
},
{
"from": "gpt",
"value": "<think> Okay, so I have this function g(x) that satisfies the equation g(x) + g(1/x)... |
[
{
"from": "human",
"value": "At a music festival, there are 5 stages featuring different bands playing various genres of music. If each band plays exactly once on each stage, and no two bands play at the same time on different stages, what is the maximum number of unique band pairings that can be created by... |
[
{
"from": "human",
"value": "Find the minimum value of\n\\[\\sqrt[3]{\\frac{1-x}{2x+1}} + \\sqrt[3]{\\frac{1-y}{2y+3}} + \\sqrt[3]{\\frac{1-z}{2z+5}}\\]\nfor real numbers $x$, $y$, and $z$ satisfying the conditions $x + y + z = 2$, $x \\ge -\\frac{1}{2}$, $y \\ge -\\frac{3}{2}$, and $z \\ge -\\frac{5}{2}$."... |
[
{
"from": "human",
"value": "The coefficients of the polynomial expansion of (x+y)^n can be used to form a matrix, where the entry in the i-th row and j-th column is the coefficient of x^i*y^(n-i). For example, for n=3, the matrix would be:\n\n```\n1 3 3 1\n3 6 3 0\n3 3 0 0\n1 0 0 0\n```\n\nFind the determi... |
[
{
"from": "human",
"value": "The quadratic function $f(x) = x^2 - (a + 1)x + a$ has the property that $f(x) > 0$ for all $x$ in the interval $(0, a)$ and $f(x) < 0$ for all $x$ in the interval $(a, \\infty)$. Determine the value of $a$."
},
{
"from": "gpt",
"value": "<think> Alright, let's see. ... |
[
{
"from": "human",
"value": "Background\nSKI combinator calculus, or simply SKI calculus, is a system similar to lambda calculus, except that SKI calculus uses a small set of combinators, namely S, K, and I instead of lambda abstraction. Unlike lambda calculus, beta reduction is possible only when a combina... |
[
{
"from": "human",
"value": "A rectangular prism has integer side lengths and a surface area of 150 square units. How many different rectangular prisms satisfy these conditions?"
},
{
"from": "gpt",
"value": "<think> Alright, so I have this problem here: \"A rectangular prism has integer side le... |
[
{
"from": "human",
"value": "A parabola with vertex $(h,k)$ has a focus at $(h,k+p)$ and a directrix $y=k-p$. If the parabola passes through the points $(h-4,k+2)$ and $(h+2,k+3)$, find $h+k+p$."
},
{
"from": "gpt",
"value": "<think> Okay, let me try to work through this problem step by step. So... |
[
{
"from": "human",
"value": "Treatment of [4*S*]-4-*t*-butyl-1-methylcyclohexene with borane-THF followed by oxidation with H2O2-NaOH gives a mixture of [1*S*,2*S*,5*S*]-5-*t*-butyl-2-methylcyclohexan-1-ol and [1*R*,2*R*,5*S*]-5-*t*-butyl-2-methylcyclohexan-1-ol and no other diastereomers. There are two ste... |
[
{
"from": "human",
"value": "A rotation of the plane takes $-3 + 2i$ to $-7 - i.$ Find the complex number that the rotation takes $-4 + 5i$ to."
},
{
"from": "gpt",
"value": "<think> Alright, so I need to figure out where a rotation takes the complex number -4 + 5i, given that the same rotation... |
[
{
"from": "human",
"value": "Let $S$ be a set of 12 points in the plane, no three of which are collinear. Each pair of points in $S$ is connected by a segment, colored red, white, or blue, such that any triangle formed by three points in $S$ has a red, a white, and a blue side. How many different colorings ... |
[
{
"from": "human",
"value": "Find the sum of all values of $k$ for which the graph of $y = kx^3 - 3kx^2 + 2kx - 5$ is tangent to the line $y = 3x - 2$."
},
{
"from": "gpt",
"value": "<think> Okay, so I need to find the sum of all values of \\( k \\) for which the graph of \\( y = kx^3 - 3kx^2 + ... |
[
{
"from": "human",
"value": "Consider a family of geometric figures formed by squares of increasing sizes, where each square is composed of smaller unit squares. The first figure consists of a single unit square, the second figure consists of four unit squares, the third figure consists of nine unit squares... |
[
{
"from": "human",
"value": "In triangle $ABC$, the bisectors of angles $A$ and $B$ intersect at point $D$. If $AB = 15$ and $BC = 20$, find $\\sin \\angle ADB$."
},
{
"from": "gpt",
"value": "<think> Okay, so I have this problem here: In triangle ABC, the bisectors of angles A and B intersect a... |
[
{
"from": "human",
"value": "A circle with radius $r$ is inscribed in a triangle with sides $a$, $b$, and $c$. The area of the circle is $\\pi r^2 = 16$. Find the value of the largest angle of the triangle, given that the length of the side opposite to it is $12$, while the lengths of the other two sides sa... |
[
{
"from": "human",
"value": "Let $b_n$ be the product of all positive integers less than or equal to $n$ that are not divisible by $3$, with $b_0=1$. What is the maximum possible value of the greatest common divisor of two consecutive terms of the sequence $c_n = b_n + n$, where $n \\ge 0$?"
},
{
"f... |
[
{
"from": "human",
"value": "In a certain quadrilateral $PQRS$, the midpoints of the sides $PQ$ and $RS$ coincide. If $PQ = 4y^2$ cm, $QR = 2z + 5$ cm, $RS = 3x^2 + 1$ cm, and $SP = 2z - 3$ cm, find the product of $x$, $y$, and $z$."
},
{
"from": "gpt",
"value": "<think> Okay, so there's this qu... |
[
{
"from": "human",
"value": "Solve the following equation:\n\n$\\frac{x^3+2x^2+3x+4}{x^2+5x+6}=x^2+7x+8$\n\nfor $x$."
},
{
"from": "gpt",
"value": "<think> Okay, so I have this equation to solve: (x³ + 2x² + 3x + 4)/(x² + 5x + 6) = x² + 7x + 8. Hmm, let me start by understanding what's going on ... |
[
{
"from": "human",
"value": "It's that time of year when many of us get our advent calendars! If you are unfamiliar with an advent calendar, these are special calendars used to count down advent in anticipation of Christmas. In my experience the doors are placed randomly, and part of the fun as a child was... |
[
{
"from": "human",
"value": "Triangle $ABC$ is an isosceles triangle with $AB=AC=5$. Let $D$ be the midpoint of $\\overline{BC}$ and let $E$ be the foot of the altitude from $A$ to $\\overline{BC}$. Point $F$ is on $\\overline{AD}$ such that the value of $\\tan\\angle FAB$ is the arithmetic mean of the val... |
[
{
"from": "human",
"value": "Consider a function $f$ defined by $f(x) = \\prod_{k=1}^n \\left( 1 + \\frac{x}{k} \\right)$, where $n$ is a positive integer. Given that $f(5) = f(8) \\cdot \\frac{29}{16}$, find the value of $n$."
},
{
"from": "gpt",
"value": "<think> Okay, so I have this problem h... |
[
{
"from": "human",
"value": "The energies achieved at modern particle accelerators are much higher than the all scales relevant to the electroweak unification. I was therefore wondering why what is measured at scattering experiments above these energies are still the $W^{\\pm}$, $Z$ and $\\gamma$ and not th... |
[
{
"from": "human",
"value": "Chebyshev Polynomials are a family of orthogonal polynomials that pop up in all kinds of places in math, and they have a lot of quite interesting properties. One characterization of them is that they are the unique polynomials that satisfy \\$T_n(\\cos(x)) = \\cos(nx)\\$.\nChall... |
[
{
"from": "human",
"value": "This is an interactive problem.\n\nJohn and his imaginary friend play a game. There are n lamps arranged in a circle. Lamps are numbered 1 through n in clockwise order, that is, lamps i and i + 1 are adjacent for any i = 1, …, n - 1, and also lamps n and 1 are adjacent. Initiall... |
[
{
"from": "human",
"value": "A random sample of 10 letters is drawn without replacement from the 26 letters of the alphabet. What is the probability that the sample contains at least one letter from each of the 5 vowels (A, E, I, O, U) and exactly 2 letters from the remaining 21 consonants? Express your ans... |
[
{
"from": "human",
"value": "A bookshelf has 5 shelves, and each shelf can hold a maximum of 8 books. If the books are stacked horizontally and each book has a unique title, how many different arrangements of books can be made on the bookshelf before it becomes completely full?"
},
{
"from": "gpt",
... |
[
{
"from": "human",
"value": "If $\\ds\\f{x}{y}=\\ds\\f{4}{5}$, determine $x$, knowing that the graphs of functions $f(t)=3t-x$ and $g(t)=\\ds\\f{6t+y}{t+5}$ intersect on the Y-axis."
},
{
"from": "gpt",
"value": "<think> Okay, so I need to find the value of x, given that the ratio of x to y is 4... |
[
{
"from": "human",
"value": "In a triangle, the ratio of the length of the altitude from a vertex to the length of the corresponding median is constant for all three vertices. Find the measure of the angle opposite the vertex where the ratio is 3/4."
},
{
"from": "gpt",
"value": "<think> Okay, s... |
[
{
"from": "human",
"value": "In triangle $ABC$, $AB=AC$, and $D$ is a point on side $BC$ such that $BD=2DC$. If $\\overline{AD}$ is reflected over the perpendicular bisector of $BC$ and intersects the extension of $\\overline{AC}$ at $E$, find the ratio $DE:EC$. [asy]\nimport geometry;\nsize(150);\npair A =... |
[
{
"from": "human",
"value": "Find the function $f(x)$ such that\n\\[f(f(x)) = f(x) + \\frac{1}{x}\\]\nfor all $x$ in its domain."
},
{
"from": "gpt",
"value": "<think> Okay, so I need to find a function f(x) such that when I compose it with itself, meaning f(f(x)), it equals f(x) plus 1/x. Hmm, ... |
[
{
"from": "human",
"value": "In quadrilateral $ABCD,$ $\\angle A = 90^\\circ$ and $\\angle B = 60^\\circ.$ The point $E$ divides $\\overline{CD}$ in the ratio $1:2$. If $DE = 6$, find\n\\[\\frac{\\sin \\angle DAE}{\\sin \\angle BAE}.\\]"
},
{
"from": "gpt",
"value": "<think> Okay, let me see. ... |
[
{
"from": "human",
"value": "A delivery truck is traveling from a warehouse to a store, and the driver notices that the truck's fuel tank is leaking at a constant rate. When the truck leaves the warehouse, the odometer reads 23,450 miles and the fuel level is 3/4 full. When the truck arrives at the store, t... |
[
{
"from": "human",
"value": "A regular polygon with 13 sides has its vertices on the graph of y=sinx/3.\nWhat is the area of the polygon?\n\nIt can be proven that the answer is around 0.42, however, it is highly unlikely that a person can calculate it manually without any external tools or devices."
},
... |
[
{
"from": "human",
"value": "Let $x$ and $y$ be positive integers such that their product is constant and $3x+y=324$. If $2x + 3y = 225$, what is the minimum value of $y$?"
},
{
"from": "gpt",
"value": "<think> Okay, so here's this problem: \n\nLet x and y be positive integers such that their pr... |
[
{
"from": "human",
"value": "Little Petya loves counting. He wants to count the number of ways to paint a rectangular checkered board of size n × m (n rows, m columns) in k colors. Besides, the coloring should have the following property: for any vertical line that passes along the grid lines and divides th... |
[
{
"from": "human",
"value": "Let $a$ and $b$ be positive integers such that $b>a>1$. What is the smallest possible value of $b$ such that there are at least 10 positive integers $n$ satisfying $a^n < 1990 < b^n$?"
},
{
"from": "gpt",
"value": "<think> Alright, so I have this problem here: I nee... |
[
{
"from": "human",
"value": "Find the range of values of $a$ for which the inequality $|x^2 + 2ax - 3| \\leq 4$ has a solution for all real numbers $x$. Express your answer in interval notation."
},
{
"from": "gpt",
"value": "<think> Okay, so I have this problem here: I need to find the range of... |
[
{
"from": "human",
"value": "In the triangle ABC, points D and E are chosen on sides AB and AC respectively, such that DE is parallel to BC. If angle BAC is 42 degrees, angle ABC is 65 degrees, and angle AED is 78 degrees, find angle EDC."
},
{
"from": "gpt",
"value": "<think> Okay, so I have th... |
[
{
"from": "human",
"value": "Consider the following system of equations:\n\n\\begin{align*}\nx + yz &= -2, \\\\\ny + xz &= -2, \\\\\nz + xy &= 3.\n\\end{align*}\n\nDetermine the value of $y_1^2 + y_2^2 + \\dots + y_n^2$, where $(x_i,y_i,z_i)$ are the ordered triples of complex numbers satisfying the above s... |
[
{
"from": "human",
"value": "Consider a pyramid of numbers in which each number is the sum of the two numbers directly below it, similar to Pascal's triangle. However, in this pyramid, each row starts and ends with the row number. For example, the third row would be \"3 1 3\". What is the sum of the numbers... |
[
{
"from": "human",
"value": "120. Compound (B) on reaction with \\(\\text{Ni(En)}_3[\\text{NO}_3]_2\\) gives a colored complex exhibiting:\\n(a) Optical isomerism\\n(b) Geometrical isomerism\\n(c) Linkage isomerism\\n(d) No isomerism"
},
{
"from": "gpt",
"value": "<think> Okay, so I have this qu... |
[
{
"from": "human",
"value": "Let $g(x)=x^2+3$. Find the product of all $x$ that satisfy the equation $g^{-1}(x)=g(x^2)$."
},
{
"from": "gpt",
"value": "<think> Okay, so I need to solve this problem: given the function g(x) = x² + 3, find the product of all x that satisfy the equation g⁻¹(x) = g(... |
[
{
"from": "human",
"value": "Let $\\gamma$ be a nonreal root of $z^4 = 1.$ Let $b_1,$ $b_2,$ $\\dots,$ $b_n$ be real numbers such that\n\\[\\frac{1}{b_1 + \\gamma} + \\frac{1}{b_2 + \\gamma} + \\dots + \\frac{1}{b_n + \\gamma} = 3 + 4i.\\]Compute\n\\[\\frac{3b_1 - 2}{b_1^3 - 2b_1^2 + 2b_1 - 1} + \\frac{3b_... |
[
{
"from": "human",
"value": "Simplify\n\\[\\sin^2 x + \\sin^2 (x - y) - 2 \\sin x \\sin y \\sin (x - y).\\]"
},
{
"from": "gpt",
"value": "<think> Okay, so I have this trigonometric expression to simplify: \\(\\sin^2 x + \\sin^2 (x - y) - 2 \\sin x \\sin y \\sin (x - y)\\). Hmm, trig identities ... |
[
{
"from": "human",
"value": "Find a five-digit number with different digits, such that the product of its digits is equal to the sum of their squares increased by 1. The digits should decrease monotonically from left to right."
},
{
"from": "gpt",
"value": "<think> Alright, let's tackle this pro... |
[
{
"from": "human",
"value": "Boiling stilbene bromide with alcoholic potash leads to the reaction represented by the equation: \nC6H5CHBr.CHBrC6H5 + 2KOH = C6H5C=CC6H5 + 2KBr + 2H2O. \nWhat is formed from the boiling of stilbene bromide with alcoholic potash?"
},
{
"from": "gpt",
"value": "<th... |
[
{
"from": "human",
"value": "Let's consider a function f(x) defined as follows:\n\nf(x) = \\sqrt{(cos(x) + 2)^2 - (sin(x) + 1)^2}\n\nWithout using a calculator, find the values of x, for which f(x) = 1, if possible. Consider all values of x from the interval [0, 2\\pi]. The final answer should contain all p... |
[
{
"from": "human",
"value": "Determine the maximum possible value of the product\n\\[\\frac{3a + 2b}{a + b} \\cdot \\frac{a + b}{4a + 3c} \\cdot \\frac{4c + 3a}{5c + 2a},\\]\nwhere $a,$ $b,$ and $c$ are positive real numbers."
},
{
"from": "gpt",
"value": "<think> Okay, so I have this problem he... |
[
{
"from": "human",
"value": "The Steenrod algebra is an important algebra that comes up in algebraic topology. The Steenrod algebra is generated by operators called \"Steenrod squares,\" one exists for each positive integer i. There is a basis for the Steenrod algebra consisting of \"admissible monomials\" ... |
[
{
"from": "human",
"value": "A parabola whose focus is $(1,2)$ intersects the hyperbola $3y^2-x^2=12$. Find all possible directrices of this parabola."
},
{
"from": "gpt",
"value": "<think> Okay, so I need to find all possible directrices of a parabola with focus at (1,2) that intersects the hyp... |
[
{
"from": "human",
"value": "Let's define a balanced multiset the following way. Write down the sum of all elements of the multiset in its decimal representation. For each position of that number check if the multiset includes at least one element such that the digit of the element and the digit of the sum ... |
[
{
"from": "human",
"value": "I am trying to show that the following expression for the electromagnetic tensor, given in Geroch's lecture notes on GR, is valid:\n$$\nF_{ab}=2\\xi_{[a}E_{b]}-\\epsilon_{abcd}\\xi^cB^d \\quad(*)\n$$\nFor that, the text suggests simple substitution of the electric ($E^a$) and ma... |
[
{
"from": "human",
"value": "Solve the following system of equations:\n\\begin{align*}\nx + y &= \\sqrt{x^2 + 1}\\\\\nx - y &= \\sqrt{x^2 - 1}\n\\end{align*}\n\n(Note: I tried to create a problem that is different in nature, but still has a similar flavor of substitution and algebraic manipulation as the or... |
[
{
"from": "human",
"value": "Simplify the expression\n\\[\\frac{(1 + \\sqrt{3})^{2 - \\sqrt{5}}}{(1 - \\sqrt{3})^{2 + \\sqrt{5}}},\\]writing your answer in the form $a + b \\sqrt{c},$ where $a,$ $b,$ and $c$ are positive integers, and $c$ is not divisible by the square of a prime."
},
{
"from": "gpt... |
[
{
"from": "human",
"value": "Let $y = \\sqrt{\\frac{\\sqrt{67}}{2} + \\frac{5}{2}}.$ There exist unique positive integers $p,$ $q,$ $r$ such that\n\\[y^{100} = 2y^{98} + 18y^{96} + 13y^{94} - y^{50} + py^{46} + qy^{44} + ry^{40}.\\]Find $p + q + r.$"
},
{
"from": "gpt",
"value": "<think> Alrigh... |
[
{
"from": "human",
"value": "What is the smallest integer value of $x$ for which $\\sin (3x) < \\frac{1}{2}$?"
},
{
"from": "gpt",
"value": "<think> Okay, so I need to find the smallest integer value of x where sin(3x) is less than 1/2. Hmm, let's see. First, I remember that the sine function os... |
[
{
"from": "human",
"value": "My question was inspired by trying to understand the paper Quantum Algorithms for Quantum Field Theories, by Jordan, Lee, and Preskill. The main result of that paper is that scattering experiments in one of the simplest-possible interacting quantum field theories, namely ϕ4 the... |
[
{
"from": "human",
"value": "Pangrams\n\nThe quick brown fox jumped over the lazy dog.\n\nThis is an example of a pangram - a sentence that contains every letter of the alphabet at least once.\nA self-enumerating pangram is a sentence that fulfills the criteria to be a pangram by producing an analysis of it... |
[
{
"from": "human",
"value": "I want you to make a gui window like this:\n\n* if you're thinking, the [quit] button closes the window\nin any language using minimum \"different\" characters..\nAND remember that the gui window must use native widgets instead of custom icons and buttons\nlike I have Windows 8,... |
[
{
"from": "human",
"value": "An infinite geometric series has the property that the sum of the first n terms of the series is always an integer. If the first term is 1 and the sum of the first 3 terms is 10, find the common ratio."
},
{
"from": "gpt",
"value": "<think> Okay, so I've got this pro... |
[
{
"from": "human",
"value": "Given that $x = \\frac{3}{4}$ is a solution to the equation $16x^2 + 15 = 24x - 5$, what is the other value of $x$ that will solve the equation? Express your answer as a common fraction."
},
{
"from": "gpt",
"value": "<think> Okay, so I need to find the other solutio... |
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