messages listlengths 3 3 |
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[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerator and denominator. For how many rational numbers between 0 and 1... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Let $ \\mathcal{H}$ be an infinite-dimensional Hilbert space, let $ d>0$ , and suppose that $ S$ is a set of points (not necessarily countable) in $ \\mathcal{H}$ such that... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Find the remainder when $9 \\times 99 \\times 999 \\times \\cdots \\times \\underbrace{99\\cdots9}_{\\text{999 9's}}$ is divided by $1000$ .",
"role": "user"
},
{
"con... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Compute the mean molecular speed v in the heavy gas radon (Rn) in m/s",
"role": "user"
},
{
"content": "<think>\nOkay, I need to calculate the mean molecular speed of ... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Two capacitors with capacitance values $C_{1}=2000 \\pm 10 \\mathrm{pF}$ and $C_{2}=3000 \\pm 15 \\mathrm{pF}$ are connected in series. The voltage applied across this combination... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "One base of a trapezoid is $100$ units longer than the other base. The segment that joins the midpoints of the legs divides the trapezoid into two regions whose areas are in the r... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Let's say a language $L \\subseteq \\{0,1\\}^*$ is in $\\textbf{P}_{angel}$ if there exists a polynomial $p : \\mathbb{N} \\mapsto \\mathbb{N}$ , a sequence of strings $\\{\... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "In a mathematics test number of participants is $N < 40$ . The passmark is fixed at $65$ . The test results are\nthe following: \nThe average of all participants is $66$ , that... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Kathy has $5$ red cards and $5$ green cards. She shuffles the $10$ cards and lays out $5$ of the cards in a row in a random order. She will be happy if and only if all the red car... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Square $AIME$ has sides of length $10$ units. Isosceles triangle $GEM$ has base $EM$ , and the area common to triangle $GEM$ and square $AIME$ is $80$ square units. Find the len... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Consider the following two person game. A number of pebbles are situated on the table. Two players make their moves alternately. A move consists of taking off the table $x$ pebb... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Suppose $a,\\,b,$ and $c$ are three complex numbers with product $1$ . Assume that none of $a,\\,b,$ and $c$ are real or have absolute value $1$ . Define\n\\begin{tabula... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Prove that the function \\[ f(\\nu)= \\int_1^{\\frac{1}{\\nu}} \\frac{dx}{\\sqrt{(x^2-1)(1-\\nu^2x^2)}}\\]\r\n(where the positive value of the square root is taken) is monotonical... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Joanie takes a $\\$6,\\!000$ loan to pay for her car. The annual interest rate on the loan is $12\\%$. She makes no payments for 4 years, but has to pay back all the money she o... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "A function $f:[0,\\infty)\\to[0,\\infty)$ is integrable and $$ \\int_0^\\infty f(x)^2 dx<\\infty,\\quad \\int_0^\\infty xf(x) dx <\\infty $$ Prove the following inequality. ... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Let $n\\geq1$ be a positive integer. $n$ lamps are placed in a line. At minute 0, some lamps are on (maybe all of them). Every minute the state of the lamps changes: A lamp is... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "For her zeroth project at Magic School, Emilia needs to grow six perfectly-shaped apple trees. First she plants six tree saplings at the end of Day $0$ . On each day afterwards, ... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "In quantum mechanics, when calculating the interaction between the electron with the proton in a hydrogen atom, it is necessary to compute the following volume integral (over all ... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Consider the additive group $\\mathbb{Z}^{2}$ . Let $H$ be the smallest subgroup containing $(3,8), (4,-1)$ and $(5,4)$ .\nLet $H_{xy}$ be the smallest subgroup containing... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Find the sum of all positive integers $n$ such that when $1^3+2^3+3^3+\\cdots +n^3$ is divided by $n+5$ , the remainder is $17$ .",
"role": "user"
},
{
"content": "<th... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "For any positive integer $a,$ $\\sigma(a)$ denotes the sum of the positive integer divisors of $a$ . Let $n$ be the least positive integer such that $\\sigma(a^n)-1$ is divisible ... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Carl chooses a *functional expression** $E$ which is a finite nonempty string formed from a set $x_1, x_2, \\dots$ of variables and applications of a function $f$ , together ... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Compute the mean molecular speed v in the light gas hydrogen (H2) in m/s",
"role": "user"
},
{
"content": "<think>\nAlright, so I need to calculate the mean molecular ... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Imagine a radioactive nuclei X(Z,A) can decay into Y(Z-2, A-4) by emitting an alpha particle with partial half life 3.0 minutes. X(Z,A) can also decay into Q(Z+1,A) by decaying a ... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Consider any rectangular table having finitely many rows and columns, with a real \n\n number $a(r, c)$ in the cell in row $r$ and column $c$. A pair $(R, C)$, where $R$ is a set ... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "The path of an asteroid that comes close to the Earth can be modeled as follows: neglect gravitational effects due to other bodies, and assume the asteroid comes in from far away ... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "2 mol of $\\mathrm{Hg}(g)$ is combusted in a fixed volume bomb calorimeter with excess of $\\mathrm{O}_{2}$ at $298 \\mathrm{~K}$ and 1 atm into $\\mathrm{HgO}(s)$. During the rea... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "B.4 If $\\Delta E_{r m s}=5.54 \\times 10^{-17} \\mathrm{~J}$, calculate the rms speed of the Be nuclei, $\\mathrm{V}_{\\mathrm{Be}}$, and hence estimate $T_{\\mathrm{c}}$. (Hint:... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": " $(BEL 5)$ Let $G$ be the centroid of the triangle $OAB.$ $(a)$ Prove that all conics passing through the points $O,A,B,G$ are hyperbolas. $(b)$ Find the locus of the cen... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "10) The handle of a gallon of milk is plugged by a manufacturing defect. After removing the cap and pouring out some milk, the level of milk in the main part of the jug is lower t... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "The sequences of real numbers $\\left\\{a_{i}\\right\\}_{i=1}^{\\infty}$ and $\\left\\{b_{i}\\right\\}_{i=1}^{\\infty}$ satisfy $a_{n+1}=\\left(a_{n-1}-1\\right)\\left(b_{n}+1\\ri... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Given are real numbers $x, y$. For any pair of real numbers $a_{0}, a_{1}$, define a sequence by $a_{n+2}=x a_{n+1}+y a_{n}$ for $n \\geq 0$. Suppose that there exists a fixed non... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Let $\\omega$ be a nonreal root of $x^3 = 1,$ and let\n\\[\\mathbf{M} = \\begin{pmatrix} -\\omega^2 & - \\omega \\\\ 1 & 0 \\end{pmatrix}.\\]Find the sum of the entries of $\\math... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Find the number of permutations of $1, 2, 3, 4, 5, 6$ such that for each $k$ with $1$ $\\leq$ $k$ $\\leq$ $5$ , at least one of the first $k$ terms of the permutation is greater t... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Find continuous functions $x(t),\\ y(t)$ such that \r\n\r $\\ \\ \\ \\ \\ \\ \\ \\ \\ x(t)=1+\\int_{0}^{t}e^{-2(t-s)}x(s)ds$ \r\n\r $\\ \\ \\ \\ \\ \\ \\ \\ \\ y(t)=\\int_{0}^{t... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Let $P(x)$ be a polynomial with integer coefficients that satisfies $P(17)=10$ and $P(24)=17.$ Given that $P(n)=n+3$ has two distinct integer solutions $n_1$ and $n_2,$ find the p... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Segments $\\overline{AB}, \\overline{AC},$ and $\\overline{AD}$ are edges of a cube and $\\overline{AG}$ is a diagonal through the center of the cube. Point $P$ satisfies $BP=60\\... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Let $(a_i)_{1\\le i\\le2015}$ be a sequence consisting of $2015$ integers, and let $(k_i)_{1\\le i\\le2015}$ be a sequence of $2015$ positive integers (positive integer ex... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Ana, Bob, and Cao bike at constant rates of $8.6$ meters per second, $6.2$ meters per second, and $5$ meters per second, respectively. They all begin biking at the same time from ... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Consider the integer \\[N = 9 + 99 + 999 + 9999 + \\cdots + \\underbrace{99\\ldots 99}_\\text{321 digits}.\\] Find the sum of the digits of $N$ .",
"role": "user"
},
{
... |
[
{
"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 transla... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "How many positive integers less than 10,000 have at most two different digits?",
"role": "user"
},
{
"content": "<think>\nAlright, so the problem is asking: How many p... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Identify the final product produced when cyclobutyl(cyclopropyl)methanol reacts with phosphoric acid in water.\nAnswer Choices:\n(A) [1,1'-bi(cyclobutan)]-1-ene\n(B) 1,2-dimethylc... |
[
{
"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": "An ideal gas is expanded from $\\left(\\mathrm{p}_{1}, \\mathrm{~V}_{1}, \\mathrm{~T}_{1}\\right)$ to $\\left(\\mathrm{p}_{2}, \\mathrm{~V}_{2}, \\mathrm{~T}_{2}\\right)$ under di... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Let $ f: Z \\to Z$ be such that $ f(1) \\equal{} 1, f(2) \\equal{} 20, f(\\minus{}4) \\equal{} \\minus{}4$ and $ f(x\\plus{}y) \\equal{} f(x) \\plus{}f(y)\\plus{}axy(x\\plus{... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "(d) Express $\\frac{d^{2} x}{d t^{2}}$ and $\\frac{d^{2} y}{d t^{2}}$ in terms of $U$, where $U=-\\frac{G M_{1}}{\\rho_{1}}-\\frac{G M_{2}}{\\rho_{2}}-\\frac{\\omega^{2}}{2}\\left... |
[
{
"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 the... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "An IPv4 packet contains the following data (in hexadecimal value) in the IP header: 4500 0034 B612 4000 4006 6F80 0A00 008B 5BC6 AEE0 . Does the header contains error?",
"role... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Prove that if every subspace of a Hausdorff space $X$ is $\\sigma$ -compact, then $X$ is countable.",
"role": "user"
},
{
"content": "<think>\nOkay, let me try to... |
[
{
"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": "Let $\\theta$ be a constant number such that $0\\leq \\theta \\leq \\pi.$ \r\nEvaluate \r\n\r\n\\[ \\int_0^{2\\pi} \\sin 8x|\\sin (x-\\theta)|\\ dx. \\]",
"role": "user"
... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Say that an integer $A$ is yummy if there exist several consecutive integers, including $A$, that add up to 2014. What is the smallest yummy integer?",
"role": "user"
},
... |
[
{
"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>\nOkay... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Arrange the nucleophiles (1. 4-methylcyclohexan-1-olate, 2. Hydroxide, 3. Propionate, 4. Methanol, 5. Ethanethiolate) from most to poorest reactivity in aqueous solution (A). Also... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Let\n\\[\\mathbf{A} = \\renewcommand{\\arraystretch}{1.5} \\begin{pmatrix} \\frac{\\sqrt{3}}{2} & 0 & -\\frac{1}{2} \\\\ 0 & -1 & 0 \\\\ \\frac{1}{2} & 0 & \\frac{\\sqrt{3}}{2} \\... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Wesyu is a farmer, and she's building a cao (a relative of the cow) pasture. She starts with a triangle $A_{0} A_{1} A_{2}$ where angle $A_{0}$ is $90^{\\circ}$, angle $A_{1}$ is ... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Let a neighborhood basis of a point $ x$ of the real line consist of all Lebesgue-measurable sets containing $ x$ whose density at $ x$ equals $ 1$ . Show that this require... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Patchouli is taking an exam with $k > 1$ parts, numbered Part $1, 2, \\dots, k$ . It is known that for $i = 1, 2, \\dots, k$ , Part $i$ contains $i$ multiple choice questi... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Jackie and Phil have two fair coins and a third coin that comes up heads with probability $\\frac47$ . Jackie flips the three coins, and then Phil flips the three coins. Let $\\fr... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "A stationary source emits sound of frequency $f_{0}=492 \\mathrm{~Hz}$. The sound is reflected by a large car approaching the source with a speed of $2 \\mathrm{~ms}^{-1}$. The re... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Let $S$ be the set of integers between 1 and $2^{40}$ whose binary expansions have exactly two 1's. If a number is chosen at random from $S,$ the probability that it is divisible ... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Find the least positive integer $m$ such that $m^2 - m + 11$ is a product of at least four not necessarily distinct primes.",
"role": "user"
},
{
"content": "<think>\n... |
[
{
"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": "Let $a_{1}, a_{2}, \\ldots$ be a sequence of positive integers such that for integers $n>2, a_{n}=$ $3 a_{n-1}-2 a_{n-2}$. How many such sequences $\\left\\{a_{n}\\right\\}$ are t... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "A right rectangular prism $P_{}$ (i.e., a rectangular parallelepiped) has sides of integral length $a, b, c,$ with $a\\le b\\le c.$ A plane parallel to one of the faces of $P_{}$ ... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Triangle $ABC_0$ has a right angle at $C_0$ . Its side lengths are pairwise relatively prime positive integers, and its perimeter is $p$ . Let $C_1$ be the foot of the altitude to... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Find the eighth term of the sequence $1440,$ $1716,$ $1848,\\ldots,$ whose terms are formed by multiplying the corresponding terms of two arithmetic sequences.",
"role": "user... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Find the number of integer values of $k$ in the closed interval $[-500,500]$ for which the equation $\\log(kx)=2\\log(x+2)$ has exactly one real solution.",
"role": "user"
}... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Given that positive integers $a,b$ satisfy \\[\\frac{1}{a+\\sqrt{b}}=\\sum_{i=0}^\\infty \\frac{\\sin^2\\left(\\frac{10^\\circ}{3^i}\\right)}{\\cos\\left(\\frac{30^\\circ}{3^i}\... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Call a positive integer $N$ a $\\textit{7-10 double}$ if the digits of the base-7 representation of $N$ form a base-10 number that is twice $N$ . For example, $51$ is a 7-10 doubl... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "For some integer $m$ , the polynomial $x^3 - 2011x + m$ has the three integer roots $a$ , $b$ , and $c$ . Find $|a| + |b| + |c|$ .",
"role": "user"
},
{
"content": "<... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Find all numbers $ n $ for which there exist three (not necessarily distinct) roots of unity of order $ n $ whose sum is $ \n1. $ ",
"role": "user"
},
{
"content"... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Let $S = \\{2^0,2^1,2^2,\\ldots,2^{10}\\}$ . Consider all possible positive differences of pairs of elements of $S$ . Let $N$ be the sum of all of these differences. Find the rema... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "a. Derive an expression for the deviation of the compass, $\\delta \\theta$, from north as a function of $K_{b}$, $K_{s}$, and $\\theta$.",
"role": "user"
},
{
"conten... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Prove that the sum of the squares of any three pairwise different positive odd integers can be represented as the sum of the squares of six (not necessarily different) positive i... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Each unit square of a 3-by-3 unit-square grid is to be colored either blue or red. For each square, either color is equally likely to be used. The probability of obtaining a grid ... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "The function $f$ , defined on the set of ordered pairs of positive integers, satisfies the following properties: \\[f(x, x) = x,\\; f(x, y) = f(y, x), {\\rm \\ and\\ } (x+y)f(x, y... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Jon and Steve ride their bicycles along a path that parallels two side-by-side train tracks running the east/west direction. Jon rides east at $20$ miles per hour, and Steve rides... |
[
{
"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\nNoticed minor changes in investigators' facility (8,4)",
"role": "... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Neutrinos are extremely light particles and rarely interact with matter. The Sun emits neutrinos, each with an energy of $8 \\times 10^{-14} \\mathrm{~J}$ and reaches a flux densi... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "In triangle $ABC$ , $\\tan \\angle CAB = 22/7$ , and the altitude from $A$ divides $BC$ into segments of length $3$ and $17$ . What is the area of triangle $ABC$ ?",
"role": "... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Suppose the codeword that we use to describe a random variable X always starts with a symbol chosen from the set {7, 8, 9} , followed by binary digits {0, 1}. Thus we have a terna... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "The vertices of a regular nonagon (9-sided polygon) are to be labeled with the digits 1 through 9 in such a way that the sum of the numbers on every three consecutive vertices is ... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Gary purchased a large beverage, but only drank $m/n$ of it, where $m$ and $n$ are relatively prime positive integers. If he had purchased half as much and drunk twice as much, he... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Each lattice point with nonnegative coordinates is labeled with a nonnegative integer in such a way that the point $(0,0)$ is labeled by 0 , and for every $x, y \\geq 0$, the set ... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Let $f(x)$ be the product of functions made by taking four functions from three functions $x,\\ \\sin x,\\ \\cos x$ repeatedly. Find the minimum value of $\\int_{0}^{\\frac{... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "a. What is the electric potential at a corner of the same cube? Write your answer in terms of $\\rho, a, \\epsilon_{0}$, and any necessary numerical constants.",
"role": "user... |
[
{
"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-p... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Exactly 2/5th of the children in a certain class are girls. If there are 100 boys in the class, how many girls are in the class?",
"role": "user"
},
{
"content": "<thi... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Determine the smallest positive integer $A$ with an odd number of digits and this property, that both $A$ and the number $B$ created by removing the middle digit of the numb... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "In triangle $ABC,$ if median $\\overline{AD}$ makes an angle of $45^\\circ$ with side $\\overline{BC},$ then find the value of $|\\cot B - \\cot C|.$",
"role": "user"
},
{... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Suppose a student who was farsighted wears glasses that allows him to read at a distance of 20cm from his eyes to the book. His near-point distance is 63cm. If his glasses are 1.5... |
[
{
"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 he... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Determine which set of states mentioned below are only entangled states:\n(a) (1/√30)* (|00>+ 2i|01>− 3|10>− 4i|11>)\n(b) (1/5)* (|00>+ 2i|01>− 2|10>− 4i|11>)\n(c) (1/2)(|00>+ |01... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Triangle $ABC$ has side lengths $AB = 9$ , $BC =$ $5\\sqrt{3}$ , and $AC = 12$ . Points $A = P_{0}, P_{1}, P_{2}, ... , P_{2450} = B$ are on segment $\\overline{AB}$ with $P_{k}$ ... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Let $L$ be the line with slope $\\frac{5}{12}$ that contains the point $A = (24,-1)$ , and let $M$ be the line perpendicular to line $L$ that contains the point $B = (5,6)$ . The... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "The decomposition reaction $2 \\mathrm{~N}_{2} \\mathrm{O}_{5}(g) \\stackrel{\\Delta}{\\rightarrow} 2 \\mathrm{~N}_{2} \\mathrm{O}_{4}(g)+\\mathrm{O}_{2}(g)$ is started in a close... |
[
{
"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 coexist... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Perform 2 iterations with the Müller method for the following equation: log_{10}(x) - x + 3 = 0, x_0 = 1/4, x_1 = 1/2, x_2 = 1. What's the decimal value of x_3?",
"role": "use... |
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