messages listlengths 3 3 |
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[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "For $1 \\leq i \\leq 215$ let $a_i = \\dfrac{1}{2^{i}}$ and $a_{216} = \\dfrac{1}{2^{215}}$ . Let $x_1, x_2, ..., x_{216}$ be positive real numbers such that $\\sum_{i=1}^{216} x_... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "For $1 \\leq i \\leq 215$ let $a_i = \\dfrac{1}{2^{i}}$ and $a_{216} = \\dfrac{1}{2^{215}}$. Let $x_1, x_2, \\dots, x_{216}$ be positive real numbers such that $\\sum_{i=1}^{216} ... |
[
{
"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\n... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Luke is borrowing $\\$10{,}000$ from the bank. The bank offers him a choice between two $10$-year payment plans:\n\n${\\bf Plan~1.}$ Luke's debt accumulates $10\\%$ annual intere... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Let $a_n=6^{n}+8^{n}$ . Determine the remainder on dividing $a_{83}$ by $49$ .",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the remainder when a8... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "suppose $x=\\sqrt{17}/17$. what is the value of $\\frac{2}{\\pi} \\int_0^{+\\infty} \\frac{\\sin ^2 t}{t^2} cos(2xt) dt$? Rounding it to the hundredths place and return the value.... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Suppose that $G$ is a finite group generated by the two elements $g$ and $h,$ where the order of $g$ is odd. Show that every element of $G$ can be written in the form\n\... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Let $d \\ge 2$ be an integer. Prove that there exists a constant $C(d)$ such that the following holds: For any convex polytope $K\\subset \\mathbb{R}^d$ , which is symmetric ... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "A uniform ladder of mass $m$ and length $\\mathcal{L}$ is resting on a wall. A man of mass $m$ climbs up the ladder and is in perfect equilibrium with the ladder when he is ... |
[
{
"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 di... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Suppose that a sequence $a_0, a_1, \\ldots$ of real numbers is defined by $a_0=1$ and \\[a_n=\\begin{cases}a_{n-1}a_0+a_{n-3}a_2+\\cdots+a_0a_{n-1} & \\text{if }n\\text{ odd}a... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "(d) The one-dimensional diffusion equation (also called the \"heat equation\") is given (for a free particle) by\n\n$$\n\\frac{\\partial \\psi}{\\partial t}=a \\frac{\\partial^{2}... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "A cube with side length 10 is suspended above a plane. The vertex closest to the plane is labeled $A$ . The three vertices adjacent to vertex $A$ are at heights 10, 11, and 12 ab... |
[
{
"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 f... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Consider the function $f: \\mathbb{N}_{0} \\rightarrow \\mathbb{N}_{0}$, where $\\mathbb{N}_{0}$ is the set of all non-negative integers, defined by the following conditions: (i) ... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Let $\\rho:G\\to GL(V)$ be a representation of a finite $p$ -group $G$ over a field of characteristic $p$ . Prove that if the restriction of the linear map $\\sum_{g\\in G}... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "An aluminum cylinder 10 cm long, with a cross-sectional area of 20 $cm^2$ is used as a spacer between two steel walls. At 17.2°C it just slips between the walls. Calculate the str... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Let $x,$ $y,$ and $z$ be positive real numbers such that\n\\[\\frac{1}{x^4} + \\frac{1}{y^4} + \\frac{1}{z^4} = 1.\\]Find the minimum value of\n\\[\\frac{x^4 y^4 + x^4 z^4 + y^4 z... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "\"Microbiologist inoculated Serratia Marcescens on Mueller-Hinton agar and placed a disc of erythromycin (30 μg). After incubation, he does not notice a zone of inhibition.\nThen,... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "A straight ladder $A B$ of mass $m=1 \\mathrm{~kg}$ is positioned almost vertically such that point $B$ is in contact with the ground with a coefficient of friction $\\mu=0.15$. I... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "e. Calculate the projected physical separation, $r_{p}$, between the galaxy and the Voorwerp.g. High precision measurements showed that the Voorwerp is slightly further away than ... |
[
{
"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": "For each real number $x$ , let $\\lfloor x \\rfloor$ denote the largest integer not exceeding $x$ .\nA sequence $\\{a_n \\}_{n=1}^{\\infty}$ is defined by $a_n = \\frac{1}{... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Consider a 70\\% efficient hydrogen-oxygen fuel cell working under standard conditions at 1 bar and $298 \\mathrm{~K}$. Its cell reaction is\n\n\\[\n\\mathrm{H}_{2}(g)+\\frac{1}{2... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Suppose we have a non-ideal gas, and in a certain volume range and temperature range, it is found to satisfy the state relation\n$$\np=A V^{\\alpha} T^{\\beta}\n$$\n\nwhere $A$ is... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "A hexagon inscribed in a circle has three consecutive sides, each of length 3, and three consecutive sides, each of length 5. The chord of the circle that divides the hexagon into... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "2.6 If this spherical bubble is now electrified uniformly with a total charge $q$, find an equation describing the new radius $R_{1}$ in terms of $R_{0}, P_{a}, q$ and the permitt... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Let $t$ be positive number. Draw two tangent lines to the palabola $y=x^{2}$ from the point $(t,-1).$ Denote the area of the region bounded by these tangent lines and the pa... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Assistants are needed to prepare for preparation. Each helper can make either 2 large cakes or 35 small cakes/hr. The kitchen is available for 3 hours and 20 large cakes & 700 sma... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "In a school, more than $90\\% $ of the students know both English and German, and more than $90\\%$ percent of the students know both English and French. Prove that more than ... |
[
{
"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\nSchooner finally given free rein under master seaman (7)",
"role":... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "A point whose coordinates are both integers is called a lattice point. How many lattice points lie on the hyperbola $x^2 - y^2 = 2000^2$ ?",
"role": "user"
},
{
"cont... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Astronomers are studying a star with a 1.8 solar radius and 1.4 solar masses. When the star's surface is not covered by plages, its Teff is 6000K. However, when 30% of its surface... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Let $P$ be a point chosen uniformly at random in the interior of the unit square with vertices at $(0,0), (1,0), (1,1)$ , and $(0,1)$ . The probability that the slope of the line ... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "The shortest distances between an interior diagonal of a rectangular parallelepiped , $P$ , and the edges it does not meet are $2\\sqrt{5}$ , $\\frac{30}{\\sqrt{13}}$ , and $\\fra... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "In an old coal factory, a conveyor belt will move at a constant velocity of $20.3 \\mathrm{~m} / \\mathrm{s}$ and can deliver a maximum power of $15 \\mathrm{MW}$. Each wheel in t... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Find the sum of the roots, real and non-real, of the equation $x^{2001}+\\left(\\frac 12-x\\right)^{2001}=0$ , given that there are no multiple roots.",
"role": "user"
},
... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Radford and Peter ran a race, during which they both ran at a constant speed. Radford began the race 30 m ahead of Peter. After 3 minutes, Peter was 18 m ahead of Radford. Peter w... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Area of the cross-section of a wire is measured using a screw gauge. The pitch of the main scale is $0.5 \\mathrm{~mm}$. The circular scale has 100 divisions and for one full rota... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Let $w_1$ and $w_2$ denote the circles $x^2+y^2+10x-24y-87=0$ and $x^2 +y^2-10x-24y+153=0,$ respectively. Let $m$ be the smallest positive value of $a$ for which the line $y=ax$ c... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "For positive integers $n,$ let $\\tau (n)$ denote the number of positive integer divisors of $n,$ including 1 and $n.$ For example, $\\tau (1)=1$ and $\\tau(6) =4.$ Define $S(n)$ ... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "The workers in a factory produce widgets and whoosits. For each product, production time is constant and identical for all workers, but not necessarily equal for the two products.... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "For a reaction taking place in a container in equilibrium with its surroundings, the effect of temperature on its equilibrium constant $K$ in terms of change in entropy is describ... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "In each unit square of an infinite square grid a natural number is written. The polygons of area $n$ with sides going along the gridlines are called *admissible*, where $n > 2$... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "When the vectors $\\mathbf{a} = \\begin{pmatrix} 1 \\\\ -1 \\\\ 2 \\end{pmatrix}$ and $\\mathbf{b} = \\begin{pmatrix} 0 \\\\ 3 \\\\ 0 \\end{pmatrix}$ are both projected onto the s... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Consider two identical galvanometers and two identical resistors with resistance $R$. If the internal resistance of the galvanometers $R_{\\mathrm{C}}<R / 2$, which of the followi... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "1. When the velocity of the particle is $v$, calculate the acceleration of the particle, $a$ (with respect to the rest frame).",
"role": "user"
},
{
"content": "<thin... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Given an acute triangle $ABC$ with altituties AD and BE. O circumcinter of $ABC$ .If o lies on the segment DE then find the value of $sinAsinBcosC$ ",
"role": "user"
},
... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Call a positive integer $N$ a 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 double because... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Suppose you are chosen as a technology assistant by the director of the opening ceremony for the 2022 Winter Olympics, and your job is to evaluate the program proposals. One of th... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Does there exist a finite group $ G$ with a normal subgroup $ H$ such that $ |\\text{Aut } H| > |\\text{Aut } G|$ ? Disprove or provide an example. Here the notation $ |\\te... |
[
{
"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: x^3 - 1/2 = 0, x_0 = 0, x_1 = 1, x_2 = 1/2. What's the decimal value of x_3?",
"role": "user"
},
{
... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "For any real numbers $\\alpha$ and $\\beta$, let $y_{\\alpha, \\beta}(x), x \\in \\mathbb{R}$, be the solution of the differential equation\n\n\\[\n\\frac{d y}{d x}+\\alpha y=x e^... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Let $z_1,$ $z_2,$ $z_3$ be complex numbers such that $|z_1| = 1,$ $|z_2| = 2,$ $|z_3| = 3,$ and\n\\[|9z_1 z_2 + 4z_1 z_3 + z_2 z_3| = 12.\\]Find $|z_1 + z_2 + z_3|.$",
"role":... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Let $K$ be a closed subset of the closed unit ball in $\\mathbb{R}^3$ . Suppose there exists a family of chords $\\Omega$ of the unit sphere $S^2$ , with the following prope... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "<span style=\"color:darkred\"> Let $(R,+,\\cdot)$ be a ring and let $f$ be a surjective endomorphism of $R$ such that $[x,f(x)]=0$ for any $x\\in R$ , where $[a,b]=ab-b... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "The numbers $x_1,...x_{100}$ are written on a board so that $ x_1=\\frac{1}{2}$ and for every $n$ from $1$ to $99$ , $x_{n+1}=1-x_1x_2x_3*...*x_{100}$ . Prove that $x_{... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Two Ladders Two straight ladders $A B$ and $C D$, each with length $1 \\mathrm{~m}$, are symmetrically placed on smooth ground, leaning on each other, such that they are touching ... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "List-I shows different radioactive decay processes and List-II provides possible emitted particles. Match each entry in List-I with an appropriate entry from List-II, and choose t... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "A certain planet with radius $R=$ $3 \\times 10^{4} \\mathrm{~km}$ is made of a liquid with constant density $\\rho=1.5 \\mathrm{~g} / \\mathrm{cm}^{3}$ with the exception of a ho... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "In decimal representation $$ \\text {34!=295232799039a041408476186096435b0000000}. $$ Find the numbers $a$ and $b$ .",
"role": "user"
},
{
"content": "<think>\n... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Let $\\rho:\\mathbb{R}^n\\to \\mathbb{R}$ , $\\rho(\\mathbf{x})=e^{-||\\mathbf{x}||^2}$ , and let $K\\subset \\mathbb{R}^n$ be a convex body, i.e., a compact convex set with n... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "In trapezoid $ABCD$ , $AD$ is parallel to $BC$ . Knowing that $AB=AD+BC$ , prove that the bisector of $\\angle A$ also bisects $CD$ .",
"role": "user"
},
{
"co... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Paul and Sara are playing a game with integers on a whiteboard, with Paul going first. When it is Paul's turn, he can pick any two integers on the board and replace them with thei... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "\"Scientist aims to detect two genes using multiplex PCR by using two sets of primers.\nDNA bands were visualized by putting the gel on a UV transilluminator and clear bands were ... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Find the orthogonal trajectories of the family of conics $(x+2y)^{2} = a(x+y)$ . At what angle do the curves of one family cut the curves of the other family at the origin?",
... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "For a given positive integer $m$ , the series $$ \\sum_{k=1,k\\neq m}^{\\infty}\\frac{1}{(k+m)(k-m)} $$ evaluates to $\\frac{a}{bm^2}$ , where $a$ and $b$ are positive in... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Consider the ellipse\n\n\\[\n\\frac{x^{2}}{4}+\\frac{y^{2}}{3}=1\n\\]\n\nLet $H(\\alpha, 0), 0<\\alpha<2$, be a point. A straight line drawn through $H$ parallel to the $y$-axis c... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "In the game of *Ring Mafia*, there are $2019$ counters arranged in a circle. $673$ of these counters are mafia, and the remaining $1346$ counters are town. Two players, Tony... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "If the sum of all digits of a number is $7$ , then we call it *lucky number*. Put all *lucky numbers* in order (from small to large): $a_1,a_2,\\cdots,a_n,\\cdots$ . If $a_n=20... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Determine all sets of real numbers $S$ such that:\n\n\n\n- $1$ is the smallest element of $S$ ,\n- for all $x,y\\in S$ such that $x>y$ , $\\sqrt{x^2-y^2}\\in S$\n\n\n*Adia... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "There exist $r$ unique nonnegative integers $n_1 > n_2 > \\cdots > n_r$ and $r$ integers $a_k$ ( $1\\le k\\le r$ ) with each $a_k$ either $1$ or $- 1$ such that \\[a_13^{n_1} + a_... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "In $\\triangle ABC$ , $AB= 425$ , $BC=450$ , and $AC=510$ . An interior point $P$ is then drawn, and segments are drawn through $P$ parallel to the sides of the triangle. If these... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "In triangle $ABC$ , point $D$ is on $\\overline{BC}$ with $CD=2$ and $DB=5$ , point $E$ is on $\\overline{AC}$ with $CE=1$ and $EA=3$ , $AB=8$ , and $\\overline{AD}$ and $\\overli... |
[
{
"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": "Use the Runge-Kutta method with $h=0.1$ to find approximate values of the solution of $(y-1)^2 * y' = 2x + 3$ with y(1) = 4. What is y(0)?",
"role": "user"
},
{
"conte... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Given vectors $\\overrightarrow{a}$, $\\overrightarrow{b}$, $\\overrightarrow{c}$ form equal angles with each other, and $|\\overrightarrow{a}|=1$, $|\\overrightarrow{b}|=1$, $|\\... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "2.2 Find the numerical value of $\\frac{\\rho_{i} T_{i}}{\\rho_{a} T_{a}}-1$ using $\\gamma=0.0250 \\mathrm{Nm}^{-1}, R_{0}=1.00 \\mathrm{~cm}$, and $P_{a}=1.013 \\times 10^{5} \\... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "b. Let the box start at $x=0$, and let $t=0$ be the time the box first starts moving. Find the acceleration of the box in terms of $x, t, v_{0}, s$, and the other parameters, whil... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Lucy was born on Wednesday, December 1st, 2004. That Wednesday was the first day of her life. Her parents held a party for her on the $1000$th day of her life. On which day of the... |
[
{
"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": "2.2.2. Calculate the numerical value of $\\Gamma$.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to calculate the numerical value of Γ, but wait, Γ usuall... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": " $ f\\left( x \\right) \\equal{} \\frac {x^5}{5x^4 \\minus{} 10x^3 \\plus{} 10x^2 \\minus{} 5x \\plus{} 1}$ . $ \\sum_{i \\equal{} 1}^{2009} f\\left( \\frac {i}{2009} \\right) \\... |
[
{
"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": "In a resource-limited ecological system, a population of organisms cannot keep growing forever (such as lab bacteria growing inside culture tube). The effective growth rate $g$ (i... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Three planets orbit a star circularly in the same plane. Each moves in the same direction and moves at constant speed. Their periods are $60$ , $84$ , and $140$ years. The thre... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Consider the following ratio\n\nR=\\frac{\\lambda\\sigma\\left(e^{+}e^{-}\\rightarrow q\\bar{q}\\right)}{\\sigma\\left(e^{+}e^{-}\\rightarrow\\mu^{+}\\mu^{-}\\right)}\n\nwhere lam... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Let $A$ be a finite ring and $a,b \\in A,$ such that $(ab-1)b=0.$ Prove that $b(ab-1)=0.$ ",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to prove t... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Find the number of positive integers $n$ less than $2017$ such that \\[1+n+\\frac{n^2}{2!}+\\frac{n^3}{3!}+\\frac{n^4}{4!}+\\frac{n^5}{5!}+\\frac{n^6}{6!}\\] is an integer.",
... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Let $m \\circ n = \\frac{m+n}{mn+4}$. Compute $((\\cdots((2005 \\circ 2004) \\circ 2003) \\circ \\cdots \\circ 1) \\circ 0)$.",
"role": "user"
},
{
"content": "<think>... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "A root of unity is a complex number that is a solution to $z^n = 1$ for some positive integer $n$. Determine the number of roots of unity that are also roots of $z^2 + az + b = 0$... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "In the last few decades, reverberation mapping, interferometry and various other observations across the electromagnetic spectrum, have been used to probe the inner structure of A... |
[
{
"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 freq... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Two congruent right circular cones each with base radius $3$ and height $8$ have axes of symmetry that intersect at right angles at a point in the interior of the cones a distance... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Consider an electromagnetic wave incident on an interface from a medium#1 with refractive index n1 = 1.75 to another medium#2 of refractive index n2 = 1.26. The wave is plane-pola... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Suppose a ping pong ball of radius $R$, thickness $t$, made out of a material with density $\\rho_{b}$, and Young's modulus $Y$, is hit so that it resonates in mid-air with small ... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "Rectangle $ABCD$ is given with $AB=63$ and $BC=448.$ Points $E$ and $F$ lie on $AD$ and $BC$ respectively, such that $AE=CF=84.$ The inscribed circle of triangle $BEF$ is tangent ... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "The sum of the following seven numbers is exactly 19: $a_1 = 2.56$ , $a_2 = 2.61$ , $a_3 = 2.65$ , $a_4 = 2.71$ , $a_5 = 2.79$ , $a_6 = 2.82$ , $a_7 = 2.86$ . It is desired to rep... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "A weird calculator has a numerical display and only two buttons, $\\boxed{D\\sharp}$ and $\\boxed{D\\flat}$ . The first button doubles the displayed number and then adds $1$ .... |
[
{
"content": "Please reason step by step, and put your final answer within \boxed{}.",
"role": "system"
},
{
"content": "A bored student walks down a hall that contains a row of closed lockers, numbered 1 to 1024. He opens the locker numbered 1, and then alternates between skipping and opening each ... |
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