messages listlengths 3 3 |
|---|
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Solve the crossword puzzle. You are presented with a clue as input and the number of letters in brackets.\n\nGreat dosh arranged for those looking after some kids? (9)",
"rol... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "d. Assuming most atoms are in the interior of the box, what is the total energy of the atoms in the box?",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Solve the crossword puzzle. You are presented with a clue as input and the number of letters in brackets.\n\nIn remix, bury a vocal that's list of words … (10)",
"role": "use... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Given the rest of reaction components:\nreactant: Cc1ccc2c(cnn2C2CCCCO2)c1B1OC(C)(C)C(C)(C)O1\nligand: CC(C)c1cc(C(C)C)c(-c2ccccc2P(C2CCCCC2)C2CCCCC2)c(C(C)C)c1\nsolvent: C1CCOC1... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A2 Find the electron number density $n_{e}$ at equilibrium when both external ionizers are switched on simultaneously.",
"role": "user"
},
{
"content": "<think>\nOkay... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "You identified several new antigens in a refractory pancreatic solid tumor and want to target them with the chimeric antigen receptor T-cell therapy. You design your chimera so t... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "The solutions to the system of equations \\begin{align*} \\log_{225}{x}+\\log_{64}{y} = 4\\\\ \\log_{x}{225}- \\log_{y}{64} = 1 \\end{align*} are $(x_1,y_1)$ and $(x_2, y_2)$ . F... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $a,$ $b,$ $c,$ and $d$ be positive real numbers such that $36a + 4b + 4c + 3d = 25.$ Find the maximum value of\n\\[a \\times \\sqrt{b} \\times \\sqrt[3]{c} \\times \\sqrt[4]... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $P$ denote the set of all ordered pairs $ \\left(p,q\\right)$ of nonnegative integers. Find all functions $f: P \\rightarrow \\mathbb{R}$ satisfying\n\\[ f(p,q) \\equal... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Given the rest of reaction components:\nreactant 1: O=S(=O)(Oc1cnc2ccccc2c1)C(F)(F)F\nreactant 2: Cc1ccc2c(cnn2C2CCCCO2)c1B(O)O\nligand: Cc1ccccc1P(c1ccccc1C)c1ccccc1C\nbase: [Li... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Consider the multiplicative group $A=\\{z\\in\\mathbb{C}|z^{2006^k}=1, 0<k\\in\\mathbb{Z}\\}$ of all the roots of unity of degree $2006^k$ for all positive integers $k$ .\n\... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Evaluate\n\n$$\n\\int_{0}^{\\frac{\\pi}{2}} \\ln \\left(9 \\sin ^{2} \\theta+121 \\cos ^{2} \\theta\\right) d \\theta\n$$",
"role": "user"
},
{
"content": "<think>\nO... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A group of clerks is assigned the task of sorting $1775$ files. Each clerk sorts at a constant rate of $30$ files per hour. At the end of the first hour, some of the clerks are r... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "There are a set of RNAs of approximately 200 or more nucleotides in length, that have no or limited coding capacity. Which one of the following statements contains an untruth ab... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "The set $S$ contains nine numbers. The mean of the numbers in $S$ is $202.$ The mean of the five smallest of the numbers in $S$ is $100.$ The mean of the five largest n... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Which one of the following is the least correct statement about the following post-transcription processes?\n\n\nAnswer Choices:\n(A) Some histone mRNAs end in a stem loop struct... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "2.2.1. Derive the expression of $\\Gamma$",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to derive the expression for the gamma function, Γ. Hmm, where do... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Which one of the following in NOT a constraint for detection of very massive black holes with microlensing technique?\nAnswer Choices:\n(A) Wavelength at which the event is obser... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A frog is positioned at the origin of the coordinate plane. From the point $(x, y)$ , the frog can jump to any of the points $(x + 1, y)$ , $(x + 2, y)$ , $(x, y + 1)$ , or $(x, ... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A hexagon that is inscribed in a circle has side lengths $22$ , $22$ , $20$ , $22$ , $22$ , and $20$ in that order. The radius of the circle can be written as $p+\\sqrt{q}$ , whe... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "For any $n \\geq 1$, let $A$ denote the $\\mathbb{C}$ algebra consisting of $n \\times n$ upper triangular complex matrices $\\left\\{\\left(\\begin{array}{ccc}* & * & * \\\\ 0 &... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "The equation of a digital filter is given by $y(n)=1 / 3(x(n)+x(n-1)+x(n-2))$, where $y(n)$ and $x(n)$ are, respectively, the nth samples of the output and input signals. Determi... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A.4 In the ultrarelativistic limit, the energy of the electron as a function of time is:\n\n$$\nE(t)=\\frac{E_{0}}{1+\\alpha E_{0} t},\n$$\n\nwhere $E_{0}$ is the initial energy ... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "(b) Any real inductor has undesired, or parasitic, resistance. We can model the real inductor as an ideal inductor $L$ in series with a parasitic resistance $R$.\n\nDue to the th... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "The numbers in the sequence $101$ , $104$ , $109$ , $116$ , $\\ldots$ are of the form $a_n=100+n^2$ , where $n=1,2,3,\\ldots$ . For each $n$ , let $d_n$ be the greatest common di... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A train with cross-sectional area $S_{t}$ is moving with speed $v_{t}$ inside a long tunnel of cross-sectional area $S_{0}\\left(S_{0}=4 S_{t}\\right)$. Assume that almost all th... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "In order to calculate the necessary beam to produce a particle X, we ran the following simulation in a High Energy software $e^{+}e^{-}\\rightarrow X$, where the electron $e^{-}$... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "c. Find the wind power $P_{w}$ in terms of $P$ and the temperature ratio $T_{2} / T_{1}$.",
"role": "user"
},
{
"content": "<think>\nAlright, so I need to find the wi... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "If $|x|$ is less than 0.7, then if we use fifth Maclaurin polynomial approximate $sin(x)$ the error is less than 0.0001. Is this correct? Answer True or False.",
"role": "use... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "1.2.1. Write down the expression giving the atmospheric pressure $p$ as a function of the altitude $Z$.",
"role": "user"
},
{
"content": "<think>\nOkay, so I have to ... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Given the rest of reaction components:\nreactant: Cc1ccc2c(cnn2C2CCCCO2)c1B1OC(C)(C)C(C)(C)O1\nligand: CN(C)Cc1ccc(P(C(C)(C)C)C(C)(C)C)cc1\nsolvent: CN(C)C=O\nbase: CCN(CC)CC \n... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "a. Consider an atom in the interior of this container of volume $V$. Suppose the potential energy of the interaction is given by\n\n\n$$\n\nu(r)= \\begin{cases}0 & r<d \\\\ -\\ep... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "How many components of a Dirac neutrino are a part of the SM?\nAnswer Choices:\n(A) 2\n(B) 4\n(C) 3\n(D) 1",
"role": "user"
},
{
"content": "<think>\nOkay, let's see.... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Suppose that $X$ and $Y$ are two metric spaces and $f:X \\longrightarrow Y$ is a continious function. Also for every compact set $K \\subseteq Y$ , it's pre-image $f^{pre... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "The value of the following series\n\n$$\n\\sum_{n=2}^{\\infty} \\frac{3 n^{2}+3 n+1}{\\left(n^{2}+n\\right)^{3}}\n$$\n\ncan be written in the form $\\frac{m}{n}$, where $m$ and $... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "What is the smallest number of standard deviations from the mean that we must go if we want to ensure that we have at least 50% of the data of a distribution?",
"role": "user... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Define $n!!$ to be $n(n-2)(n-4)\\cdots 3\\cdot 1$ for $n$ odd and $n(n-2)(n-4)\\cdots 4\\cdot 2$ for $n$ even. When $\\sum_{i=1}^{2009} \\frac{(2i-1)!!}{(2i)!!}$ is expressed as ... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "1. At a certain moment, the time experienced by the particle is $\\tau$. What reading $t_{0}$ on a stationary clock located at $x=0$ will be observed by the particle? After a lo... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "c. Assuming the radius is the value you found in part (b), how much energy is stored in the electric field of the electron? Express your answer in terms of $E_{0}=m c^{2}$ and th... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "During a recent campaign for office, a candidate made a tour of a country which we assume lies in a plane. On the first day of the tour he went east, on the second day he went no... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $f : \\mathbb{C} \\to \\mathbb{C} $ be defined by $ f(z) = z^2 + iz + 1 $. How many complex numbers $z $ are there such that $ \\text{Im}(z) > 0 $ and both the real and the i... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Which of the following products of gamma matrices is Hermitian?\nAnswer Choices:\n(A) i gamma0 * gamma0 * gamma2* gamma3\n(B) i gamma0 * gamma1 * gamma2 * gamma3\n(C) i gamma0 * ... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $v$ and $w$ be distinct, randomly chosen roots of the equation $z^{1997}-1=0.$ Find the probability that\n\\[\\sqrt{2+\\sqrt{3}}\\le\\left|v+w\\right|.\\]",
"role": "use... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $a$ and $b$ be positive integers satisfying $\\frac{ab+1}{a+b} < \\frac{3}{2}$ . The maximum possible value of $\\frac{a^3b^3+1}{a^3+b^3}$ is $\\frac{p}{q}$ , where $p$ and $... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Do either $(1)$ or $(2)$ $(1)$ $x$ and $y$ are functions of $t.$ Solve $x' = x + y - 3, y' = -2x + 3y + 1,$ given that $x(0) = y(0) = 0.$ $(2)$ A weightless r... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "The polynomial $f(z)=az^{2018}+bz^{2017}+cz^{2016}$ has real coefficients not exceeding $2019,$ and $f\\left(\\tfrac{1+\\sqrt3i}{2}\\right)=2015+2019\\sqrt3i$ . Find the remainde... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $z$ be a complex number such that $z^{23} = 1$ and $z \\neq 1.$ Find\n\\[\\sum_{n = 0}^{22} \\frac{1}{1 + z^n + z^{2n}}.\\]",
"role": "user"
},
{
"content": "<th... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "For an isosceles prism of angle $A$ and refractive index $\\mu$, it is found that the angle of minimum deviation $\\delta_{m}=A$. Which of the following options is/are correct?\n... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Compute the nearest integer to $$100 \\sum_{n=1}^{\\infty} 3^{n} \\sin ^{3}\\left(\\frac{\\pi}{3^{n}}\\right)$$",
"role": "user"
},
{
"content": "<think>\nOkay, so I'... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Every group of order $5\\cdot7\\cdot47=1645 is abelian, and cyclic. Is this true? Answer true or false.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to ... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Which one of the following methods/or/observations is NOT potentially relevant for constraining dark energy equation of state?\n\nAnswer Choices:\n(A) Measuring the change in fre... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A pyramid has a triangular base with side lengths $20$ , $20$ , and $24$ . The three edges of the pyramid from the three corners of the base to the fourth vertex of the pyramid a... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $R$ be a square region and $n \\ge 4$ an integer. A point $X$ in the interior of $R$ is called *n-ray partitional* if there are $n$ rays emanating from $X$ that d... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "What is the value of the sum\n\\[\n \\sum_z \\frac{1}{{\\left|1 - z\\right|}^2} \\, ,\n\\]where $z$ ranges over all 7 solutions (real and nonreal) of the equation $z^7 = -1$?",
... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "The CORRECT statement(s) for cubic close packed (ccp) three dimensional structure is(are)\n\n(A) The number of the nearest neighbours of an atom present in the topmost layer is 1... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $\\triangle ABC$ be a right triangle with right angle at $C.$ Let $D$ and $E$ be points on $\\overline{AB}$ with $D$ between $A$ and $E$ such that $\\overline{CD}$ and $\\ove... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "For a positive real number $x > 1,$ the Riemann zeta function $\\zeta(x)$ is defined by\n\\[\\zeta(x) = \\sum_{n = 1}^\\infty \\frac{1}{n^x}.\\]Compute\n\\[\\sum_{k = 2}^\\infty ... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A scientist studies the stress response of barley to increased temperatures and finds a protein which contributes to heat tolerance through the stabilisation of cell membrane. Th... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Consider Convolutional Neural Network D2 which takes input images of size 32x32 with 1 colour channels. The first layer of D2 uses 4 filters of size 5x5, a stride of 2, and zero-... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "The increasing sequence $3, 15, 24, 48, \\ldots\\,$ consists of those positive multiples of 3 that are one less than a perfect square. What is the remainder when the 1994th term... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Point $B$ is on $\\overline{AC}$ with $AB = 9$ and $BC = 21.$ Point $D$ is not on $\\overline{AC}$ so that $AD = CD,$ and $AD$ and $BD$ are integers. Let $s$ be the sum of all po... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Steve says to Jon, \"I am thinking of a polynomial whose roots are all positive integers. The polynomial has the form $P(x) = 2x^3-2ax^2+(a^2-81)x-c$ for some positive integers $... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Consider the integral $$ \\int_{-1}^1 x^nf(x) \\; dx $$ for every $n$ -th degree polynomial $f$ with integer coefficients. Let $\\alpha_n$ denote the smallest positive ... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find the complex numbers $ z$ for which the series \r\n\\[ 1 \\plus{} \\frac {z}{2!} \\plus{} \\frac {z(z \\plus{} 1)}{3!} \\plus{} \\frac {z(z \\plus{} 1)(z \\plus{} 2)}{4!} \... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "(a) Does $\\sum_{i=1}^{p-1} \\frac{1}{i} \\equiv 0\\left(\\bmod p^{2}\\right)$ for all odd prime numbers $p$? (Note that $\\frac{1}{i}$ denotes the number such that $\\left.i \\c... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "For every $x \\ge -\\frac{1}{e}\\,$ , there is a unique number $W(x) \\ge -1$ such that \n\\[\n W(x) e^{W(x)} = x.\n\\]\nThe function $W$ is called Lambert's $W$ function... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A train pulls out of the station at constant velocity. The received signal energy thus falls off with time as $1/i^2$. The total received signal at time $i$ is $Y_i = \\frac{1}{i... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "There is a positive real number $x$ not equal to either $\\tfrac{1}{20}$ or $\\tfrac{1}{2}$ such that \\[\\log_{20x} (22x)=\\log_{2x} (202x).\\] The value $\\log_{20x} (22x)$ can... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A particle is located on the coordinate plane at $(5,0)$ . Define a move for the particle as a counterclockwise rotation of $\\pi/4$ radians about the origin followed by a transl... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "An ellipse in the first quadrant is tangent to both the $x$-axis and $y$-axis. One focus is at $(3,7)$ and the other focus is at $(d,7).$ Compute $d.$",
"role": "user"
},... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Consider all 1000-element subsets of the set $\\{ 1, 2, 3, ... , 2015 \\}$ . From each such subset choose the least element. The arithmetic mean of all of these least elements ... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Point $P$ lies on the diagonal $AC$ of square $ABCD$ with $AP > CP$ . Let $O_1$ and $O_2$ be the circumcenters of triangles $ABP$ and $CDP$ , respectively. Given that $AB = 12$ a... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Consider the set of continuous functions $f$ , whose $n^{\\text{th}}$ derivative exists for all positive integer $n$ , satisfying $f(x)=\\tfrac{\\text{d}^3}{\\text{dx}^3}f(x... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "For $\\pi \\le \\theta < 2\\pi$ , let \\[P=\\dfrac12\\cos\\theta-\\dfrac14\\sin2\\theta-\\dfrac18\\cos3\\theta+\\dfrac1{16}\\sin4\\theta+\\dfrac1{32}\\cos5\\theta-\\dfrac1{64}\\s... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "FemtoPravis is walking on an $8 \\times 8$ chessboard that wraps around at its edges (so squares on the left edge of the chessboard are adjacent to squares on the right edge, and... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "There is a C-NOT gate where the condition is that the first qubit of control has to be in the state \\left|0\\right\\rangle.\n\\left|\\psi\\right\\rangle =\\left(\\alpha\\left|0\... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A sphere is inscribed in the tetrahedron whose vertices are $A = (6,0,0), B = (0,4,0), C = (0,0,2),$ and $D = (0,0,0).$ The radius of the sphere is $m/n,$ where $m$ and $n$ are r... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Suppose that $x$ , $y$ , and $z$ are complex numbers such that $xy = -80 - 320i$ , $yz = 60$ , and $zx = -96 + 24i$ , where $i$ $=$ $\\sqrt{-1}$ . Then there are real numbers $a$... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Complex numbers $a,$ $b,$ and $c$ are zeros of a polynomial $P(z) = z^3 + qz + r,$ and $|a|^2 + |b|^2 + |c|^2 = 250.$ The points corresponding to $a,$ $b,$ and $c$ in the complex... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "For nonnegative integers $a$ and $b$ with $a + b \\leq 6$ , let $T(a, b) = \\binom{6}{a} \\binom{6}{b} \\binom{6}{a + b}$ . Let $S$ denote the sum of all $T(a, b)$ , where $a$ an... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "In triangle $ABC$, medians $AD$ and $CE$ intersect at $P$, $PE=1.5$, $PD=2$, and $DE=2.5$. What is the area of $AEDC$?",
"role": "user"
},
{
"content": "<think>\nOka... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A4 Define $x_{\\mathrm{g}}=h v_{\\mathrm{g}} / k_{\\mathrm{B}} T_{\\mathrm{s}}$ where $E_{\\mathrm{g}}=h v_{\\mathrm{g}}$. Express the useful output power of the cell, $P_{\\text... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Enya and John are of normal phenotype but they have a son with a genetic condition that is caused by a mutation in the DMD gene and is a sex-linked recessive disorder. What are t... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Assume we are going to sell a bundle item including one unit of Item 1 and one unit of Item 2 at price p_{12}. The total cost of this item is t(c_{1} + c_{2}), where 0 < t < 1. A... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Trapezoid $ABCD$ has sides $AB=92$, $BC=50$, $CD=19$, and $AD=70$, with $AB$ parallel to $CD$. A circle with center $P$ on $AB$ is drawn tangent to $BC$ and $AD$. Given that $AP=... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "The Chi-square statistic $\\chi^2=\\sum_c\\frac{(P(x)-Q(x))^2}{Q(x)}$ is (twice) the first term in the Taylor series expansion of $D(P||Q)$ about $Q$. True or False?",
"role"... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Determine the exact value of\n\\[\\sqrt{\\left( 2 - \\sin^2 \\frac{\\pi}{7} \\right) \\left( 2 - \\sin^2 \\frac{2 \\pi}{7} \\right) \\left( 2 - \\sin^2 \\frac{3 \\pi}{7} \\right)... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Which one of the following is a correct expression for the Scharfetter-Gummel discretized drift-diffusion model equation for current? In the following expressions $J{n,j+1/2}$ i... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $P(x)$ be a nonzero polynomial such that $(x-1)P(x+1)=(x+2)P(x)$ for every real $x$ , and $\\left(P(2)\\right)^2 = P(3)$ . Then $P(\\tfrac72)=\\tfrac{m}{n}$ , where $m$ and $... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "The majority of stars in our Galaxy form and evolve in multi-stellar systems. Below are five potential multi-star systems that are presented. How many of these systems can coexis... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A simple calculator is given to you. (It contains 8 digits and only does the operations +,-,*,/, $ \\sqrt{\\mbox{}}$ ) How can you find $ 3^{\\sqrt{2}}$ with accuracy of 6 digi... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find the range of $ k$ for which the following inequality holds for $ 0\\leq x\\leq 1$ .\r\n\\[ \\int_0^x \\frac {dt}{\\sqrt {(3 \\plus{} t^2)^3}}\\geq k\\int _0^x \\frac {dt}... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let a sequence be defined as follows: $a_1 = 3,$ $a_2 = 3,$ and for $n \\ge 2,$\n\\[a_{n + 1} a_{n - 1} = a_n^2 + 2007.\\]Find the largest integer less than or equal to $\\frac{a... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "How many functions $f:\\{0,1\\}^{3} \\rightarrow\\{0,1\\}$ satisfy the property that, for all ordered triples \\left(a_{1}, a_{2}, a_{3}\\right) and \\left(b_{1}, b_{2}, b_{3}\\r... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A container in the shape of a right circular cone is 12 inches tall and its base has a 5-inch radius. The liquid that is sealed inside is 9 inches deep when the cone is held with... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Compute $\\sum_{k=1}^{1007}\\left(\\cos \\left(\\frac{\\pi k}{1007}\\right)\\right)^{2014}$.",
"role": "user"
},
{
"content": "<think>\nOkay, so I have this problem h... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "On the complex plane, the parallelogram formed by the points 0, $z,$ $\\frac{1}{z},$ and $z + \\frac{1}{z}$ has area $\\frac{35}{37}.$ If the real part of $z$ is positive, let $... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $x,$ $y,$ $z$ be real numbers such that $x + y + z = 1,$ and $x \\ge -\\frac{1}{3},$ $y \\ge -1,$ and $z \\ge -\\frac{5}{3}.$ Find the maximum value of\n\\[\\sqrt{3x + 1} + ... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Equilateral $\\triangle ABC$ is inscribed in a circle of radius 2. Extend $\\overline{AB}$ through $B$ to point $D$ so that $AD=13,$ and extend $\\overline{AC}$ through $C$ to po... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A spin-half particle is in a mixed ensemble. The spin half system is in state |\\psi1 \\rangle with probabilty (1/3) and in state in state |\\psi2 \\rangle with probability (2/3... |
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