messages listlengths 3 3 |
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[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Consider one mole of helium gas enclosed in a container at initial pressure $P_{1}$ and volume $V_{1}$. It expands isothermally to volume $4 V_{1}$. After this, the gas expands a... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $A B C$ be a solid right triangle $(A B=5 s, A C=12 s$, and $B C=13 s$ ) with uniform charge density $\\sigma$. Let $D$ be the midpoint of $B C$. We denote the electric poten... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A block of cheese in the shape of a rectangular solid measures $10$ cm by $13$ cm by $14$ cm. Ten slices are cut from the cheese. Each slice has a width of $1$ cm and is cut para... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $x(t)$ be a solution to the differential equation \\[\\left(x+x^\\prime\\right)^2+x\\cdot x^{\\prime\\prime}=\\cos t\\] with $x(0)=x^\\prime(0)=\\sqrt{\\frac{2}{5}}$ . Co... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Ten adults enter a room, remove their shoes, and toss their shoes into a pile. Later, a child randomly pairs each left shoe with a right shoe without regard to which shoes belong... |
[
{
"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 $A$ be a $2n \\times 2n$ matrix, with entries chosen independently at random. Every entry is chosen to be 0 or 1, each with probability $1/2$. Find the expected value of $\\d... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A frog begins at $P_0 = (0,0)$ and makes a sequence of jumps according to the following rule: from $P_n = (x_n, y_n),$ the frog jumps to $P_{n+1},$ which may be any of the points... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Points $A(0,0), B(9,6)$ and $C(6,12)$ are vertices of triangle $ABC$. Point $D$ is on segment $AB$ such that $2(AD) = DB$, point $E$ is on segment $BC$ such that $2(BE) = EC$ and... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Alice Czarina is bored and is playing a game with a pile of rocks. The pile initially contains 2015 rocks. At each round, if the pile has $N$ rocks, she removes $k$ of them, wher... |
[
{
"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, h... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find the number of ordered triples $(a,b,c)$ where $a$ , $b$ , and $c$ are positive integers , $a$ is a factor of $b$ , $a$ is a factor of $c$ , and $a+b+c=100$ .",
"role": "... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Suppose that $X$ is a compact metric space and $T: X\\rightarrow X$ is a continous function. Prove that $T$ has a returning point. It means there is a strictly increasing s... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Count the number of sequences $a_{1}, a_{2}, a_{3}, a_{4}, a_{5}$ of integers such that $a_{i} \\leq 1$ for all $i$ and all partial sums $\\left(a_{1}, a_{1}+a_{2}\\right.$, etc.... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $n\\ge 1$ be a fixed integer. Calculate the distance $\\inf_{p,f}\\, \\max_{0\\le x\\le 1} |f(x)-p(x)|$ , where $p$ runs over polynomials of degree less than $n$ ... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A8 The band gap of pure silicon is $E_{\\mathrm{g}}=1.11 \\mathrm{eV}$. Calculate the efficiency, $\\eta_{\\mathrm{Si}}$, of a silicon solar cell using this <br> value.",
"ro... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $z_1$, $z_2$, $z_3$, $\\dots$, $z_{12}$ be the 12 zeroes of the polynomial $z^{12} - 2^{36}$. For each $j$, let $w_j$ be one of $z_j$ or $iz_j$. Find the maximum possible va... |
[
{
"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 $\\mat... |
[
{
"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": "Find the surface generated by the solutions of \\[ \\frac {dx}{yz} = \\frac {dy}{zx} = \\frac{dz}{xy}, \\] which intersects the circle $y^2+ z^2 = 1, x = 0.$ ",
"role": "use... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Is the Fourier transform of the signal $x_1(t)=\\left\\{\\begin{array}{cc}\\sin \\omega_0 t, & -\\frac{2 \\pi}{\\omega_0} \\leq t \\leq \\frac{2 \\pi}{\\omega_0} \\\\ 0, & \\text... |
[
{
"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": "Denote $g(k)$ as the greatest odd divisor of $k$ . Put $f(k) = \\dfrac{k}{2} + \\dfrac{k}{g(k)}$ for $k$ even, and $2^{(k+1)/2}$ for $k$ odd. Define the sequence $x_1... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Suppose 2012 reals are selected independently and at random from the unit interval $[0,1]$ , and then written in nondecreasing order as $x_1\\le x_2\\le\\cdots\\le x_{2012}$ . ... |
[
{
"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": "At a certain university, the division of mathematical sciences consists of the departments of mathematics, statistics, and computer science. There are two male and two female pro... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Determine the smallest positive value of $x,$ in degrees, for which\n\\[\\tan (x + 100^{\\circ}) = \\tan (x + 50^{\\circ}) \\tan x \\tan (x - 50^{\\circ}).\\]",
"role": "user... |
[
{
"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\nCourse one's taken in river rising somewhere in Canada (4,6)",
"r... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "For a permutation $\\sigma$ of $1,2, \\ldots, 7$, a transposition is a swapping of two elements. Let $f(\\sigma)$ be the minimum number of transpositions necessary to turn $\\sig... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $S$ be a set with six elements . Let $\\mathcal{P}$ be the set of all subsets of $S.$ Subsets $A$ and $B$ of $S$ , not necessarily distinct, are chosen independently and at r... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "On square $ABCD$ , point $E$ lies on side $AD$ and point $F$ lies on side $BC$ , so that $BE=EF=FD=30$ . Find the area of the square $ABCD$ .",
"role": "user"
},
{
"c... |
[
{
"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>+ |0... |
[
{
"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 $\\f... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find all twice differentiable functions $f : \\mathbb{R} \\to \\mathbb{R}$ such that $$ f''(x) \\cos(f(x))\\geq(f'(x))^2 \\sin(f(x)) $$ for every $x\\in \\mathbb{R}$ .\n\... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Fisica and Ritmo discovered a piece of Notalium shaped like a rectangular box, and wanted to find its volume. To do so, Fisica measured its three dimensions using a ruler with in... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "(a) Knowing that Legendre polynomials are orthogonal $\\left(\\left\\langle P_{m}(x), P_{n}(x)\\right\\rangle=0\\right.$ if $m \\neq n)$ and $\\operatorname{deg} P_{n}(x)=n$, obt... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Given vector $\\mathbf{u}=\\left(\\frac{1}{3}, \\frac{1}{3}, \\frac{1}{3} \\right)\\in\\mathbb{R}^3$ and recursively defined sequence of vectors $\\{\\mathbf{v}_n\\}_{n\\geq 0... |
[
{
"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": "Find the number of positive integers $n \\le 1000$ that can be expressed in the form\n\\[\\lfloor x \\rfloor + \\lfloor 2x \\rfloor + \\lfloor 3x \\rfloor = n\\]for some real num... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "a. Find the final temperature $T_{f}$ of the two objects, and the total work $W$ done by the engine.",
"role": "user"
},
{
"content": "<think>\nOkay, so I have this p... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $w,$ $x,$ $y,$ and $z,$ be positive real numbers. Find the maximum value of\n\\[\\frac{wx + xy + yz}{w^2 + x^2 + y^2 + z^2}.\\]",
"role": "user"
},
{
"content": ... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find the number of distinct numbers in the list\n\\[\\left\\lfloor \\frac{1^2}{1000} \\right\\rfloor, \\ \\left\\lfloor \\frac{2^2}{1000} \\right\\rfloor, \\ \\left\\lfloor \\fra... |
[
{
"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": "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 becaus... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "For $\\{1, 2, 3, \\ldots, n\\}$ and each of its non-empty subsets a unique alternating sum is defined as follows. Arrange the numbers in the subset in decreasing order and then, ... |
[
{
"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": "\"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": "An angle $x$ is chosen at random from the interval $0^{\\circ} < x < 90^{\\circ}$. Let $p$ be the probability that the numbers $\\sin^2 x$, $\\cos^2 x$, and $\\sin x \\cos x$ ar... |
[
{
"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": "An infinite sequence of real numbers $a_{1}, a_{2}, \\ldots$ satisfies the recurrence $$a_{n+3}=a_{n+2}-2 a_{n+1}+a_{n}$$ for every positive integer $n$. Given that $a_{1}=a_{3}=... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "We define a sequence of natural numbers by the initial values $a_0 = a_1 = a_2 = 1$ and the recursion $$ a_n = \\bigg \\lfloor \\frac{n}{a_{n-1}a_{n-2}a_{n-3}} \\bigg \\rflo... |
[
{
"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": "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": "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$.\nWhat is the operator form of this on the basis $\\... |
[
{
"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 ... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Microbiologist found that the minimum inhibitory concentration and mutant prevention concentration of erythromycin for Klebsiella pneumoniae were 4 mg/L and 64 μg/mL respectively... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "What is the minimum kinetic energy in MeV of a proton in a medium-sized nucleus having a diameter of 8.0 x 10^-15 m?",
"role": "user"
},
{
"content": "<think>\nOkay, ... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $z$ be a non-real complex number with $z^{23}=1$. Compute $$ \\sum_{k=0}^{22} \\frac{1}{1+z^{k}+z^{2 k}} $$",
"role": "user"
},
{
"content": "<think>\nAlright, so... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "One hundred concentric circles with radii $1, 2, 3, \\dots, 100$ are drawn in a plane. The interior of the circle of radius 1 is colored red, and each region bounded by consecuti... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Anca and Bruce left Mathville at the same time. They drove along a straight highway towards Staton. Bruce drove at $50 \\mathrm{~km} / \\mathrm{h}$. Anca drove at $60 \\mathrm{~k... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Given that $O$ is a regular octahedron, that $C$ is the cube whose vertices are the centers of the faces of $O,$ and that the ratio of the volume of $O$ to that of $C$ is $\\frac... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "How many positive integers have exactly three proper divisors (positive integral divisors excluding itself), each of which is less than 50?",
"role": "user"
},
{
"con... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A3 Express the number of photons, $n_{\\gamma}(v)$, per unit time per unit frequency interval incident on the surface of <br> the solar cell in terms of $A, R_{\\odot}, d_{\\odot... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Caroline wants to plant 10 trees in her orchard. Planting $n$ apple trees requires $n^2$ square meters, planting $n$ apricot trees requires $5n$ square meters, and plant... |
[
{
"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": "Find the least positive integer $n$ such that when $3^n$ is written in base $143$ , its two right-most digits in base $143$ are $01$ .",
"role": "user"
},
{
"content"... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $n \\ge 2$ be a natural number and suppose that positive numbers $a_0,a_1,...,a_n$ satisfy the equality $(a_{k-1}+a_{k})(a_{k}+a_{k+1})=a_{k-1}-a_{k+1}$ for each $k =1... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Alpha and Beta both took part in a two-day problem-solving competition. At the end of the second day, each had attempted questions worth a total of 500 points. Alpha scored 160 p... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Consider the sequence of numbers defined recursively by $t_1=1$ and for $n>1$ by $t_n=1+t_{(n/2)}$ when $n$ is even and by $t_n=\\frac{1}{t_{(n-1)}}$ when $n$ is odd.... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find the least odd prime factor of $2019^8+1$.",
"role": "user"
},
{
"content": "<think>\nOkay, so I need to find the least odd prime factor of \\(2019^8 + 1\\). Alri... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "The Proca equation is given by\n\n\\partial_{\\mu}F^{\\mu\\nu}+m^{2}A^{\\nu}=0\n\nif \\mu,\\nu=0,…,n, what is the general solution of the motion equations A^{\\nu}?\nAnswer Choic... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "(a) Let $n$ is a positive integer. Calculate $\\displaystyle \\int_0^1 x^{n-1}\\ln x\\,dx$ .\n\n(b) Calculate $\\displaystyle \\sum_{n=0}^{\\infty}(-1)^n\\left(\\frac{1}{(n+1... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Rhombus $PQRS^{}_{}$ is inscribed in rectangle $ABCD^{}_{}$ so that vertices $P^{}_{}$ , $Q^{}_{}$ , $R^{}_{}$ , and $S^{}_{}$ are interior points on sides $\\overline{AB}$ , $\\... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $a_1,a_2,\\dots,a_{100}$ be a sequence of integers. Initially, $a_1=1$ , $a_2=-1$ and the remaining numbers are $0$ . After every second, we perform the following proce... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Two electromagnetic (em) fields are applied on the 1s->2p transition of a Hydrogen atomic vapor sample. The field#1 is polarized in z-direction, propagating in x-direction, with ... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "The real root of the equation $8x^3-3x^2-3x-1=0$ can be written in the form $\\frac{\\sqrt[3]{a}+\\sqrt[3]{b}+1}{c}$ , where $a$ , $b$ , and $c$ are positive integers. Find $a+b+... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $\\overline{CH}$ be an altitude of $\\triangle ABC$ . Let $R\\,$ and $S\\,$ be the points where the circles inscribed in the triangles $ACH\\,$ and $BCH^{}_{}$ are tangent to... |
[
{
"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": "<t... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Misha rolls a standard, fair six-sided die until she rolls 1-2-3 in that order on three consecutive rolls. The probability that she will roll the die an odd number of times is $\... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find the number of solutions in positive integers $(k ; a_{1}, a_{2}, \\ldots, a_{k} ; b_{1}, b_{2}, \\ldots, b_{k})$ to the equation $$a_{1}(b_{1})+a_{2}(b_{1}+b_{2})+\\cdots+a_... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $\\mathbf{a},$ $\\mathbf{b},$ $\\mathbf{c},$ $\\mathbf{d}$ be four distinct unit vectors in space such that\n\\[\\mathbf{a} \\cdot \\mathbf{b} = \\mathbf{a} \\cdot \\mathbf{c... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Consider a linear harmonic oscillator and let, Ψ0 and Ψ 1 be its real, normalized ground and first excited state energy eigenfunctions respectively.\nLet Ψ = AΨ0 + BΨ1 with A and... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "For real numbers $a,\\ b$ , define a point $P_n(x_n,\\ y_n)$ by \n\\[(x_0,\\ y_0)=(1,\\ 0)\\]\n\n\\[(x_{n+1},\\ y_{n+1})=(ax_n-by_n,\\ bx_n+ay_n)\\ \\ (n=0,\\ 1,\\ 2,\\ \\cdot... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "An ellipse has foci at $(9, 20)$ and $(49, 55)$ in the $xy$ -plane and is tangent to the $x$ -axis. What is the length of its major axis?",
"role": "user"
},
{
"conte... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Welcome to the USAYNO, where each question has a yes/no answer. Choose any subset of the following six problems to answer. If you answer $n$ problems and get them all correct, yo... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "How many positive integer multiples of 1001 can be expressed in the form $10^{j} - 10^{i}$ , where $i$ and $j$ are integers and $0\\leq i < j \\leq 99$ ?",
"role": "user"
}... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Call a three-term strictly increasing arithmetic sequence of integers special if the sum of the squares of the three terms equals the product of the middle term and the square of... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "An $m\\times n\\times p$ rectangular box has half the volume of an $(m + 2)\\times(n + 2)\\times(p + 2)$ rectangular box, where $m, n,$ and $p$ are integers, and $m\\le n\\le p.$... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Adults made up $\\frac5{12}$ of the crowd of people at a concert. After a bus carrying $50$ more people arrived, adults made up $\\frac{11}{25}$ of the people at the concert. Fin... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "In the Cartesian plane let $A = (1,0)$ and $B = \\left( 2, 2\\sqrt{3} \\right)$ . Equilateral triangle $ABC$ is constructed so that $C$ lies in the first quadrant. Let $P=(x,y)... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A gas is enclosed in a cylinder with a movable frictionless piston. Its initial thermodynamic state at pressure $P_{i}=10^{5} \\mathrm{~Pa}$ and volume $V_{i}=10^{-3} \\mathrm{~m... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $P(z)=z^3+az^2+bz+c$ , where $a$ , $b$ , and $c$ are real. There exists a complex number $w$ such that the three roots of $P(z)$ are $w+3i$ , $w+9i$ , and $2w-4$ , where $i^2... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Planet X orbits a star (X) in a circular orbit, and receives a certain amount of heat from the star. Planet Y orbits some other star (Y) in a circular orbit, in some other star s... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "The increasing sequence $2,3,5,6,7,10,11,\\ldots$ consists of all positive integers that are neither the square nor the cube of a positive integer. Find the 500th term of this se... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "It is given that $\\log_{6}a + \\log_{6}b + \\log_{6}c = 6$ , where $a$ , $b$ , and $c$ are positive integers that form an increasing geometric sequence and $b - a$ is the square... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A bee starts flying from point $P_0$. She flies $1$ inch due east to point $P_1$. For $j \\ge 1$, once the bee reaches point $P_j$, she turns $30^{\\circ}$ counterclockwise and t... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "In triangle $ABC, AB=13, BC=15,$ and $CA = 14.$ Point $D$ is on $\\overline{BC}$ with $CD=6.$ Point $E$ is on $\\overline{BC}$ such that $\\angle BAE\\cong \\angle CAD.$ Given th... |
[
{
"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 $(a,b,c)$ be a real solution of the system of equations $x^3 - xyz = 2$ , $y^3 - xyz = 6$ , $z^3 - xyz = 20$ . The greatest possible value of $a^3 + b^3 + c^3$ can be written... |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $m \\circ n=(m+n) /(m n+4)$. Compute $((\\cdots((2005 \\circ 2004) \\circ 2003) \\circ \\cdots \\circ 1) \\circ 0)$.",
"role": "user"
},
{
"content": "<think>\nAl... |
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