conversations listlengths 2 2 | token_length int64 298 93.4k |
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"value": "1. On the table lie 54 piles of stones with $1,2,3, \\ldots, 54$ stones. In each step, we select any pile, say with $k$ stones, and remove it from the table along with $k$ stones from each pile that has at least $k$ stones. For example, after the first step, if we select the pile... | 11,051 |
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"value": "Return your final response within \\boxed{}. An acute angle at the base of a trapezoid inscribed in a circle with a radius of 13 is $30^{\\circ}$. The leg (non-parallel side) of the trapezoid is 10. Find the median (the midsegment) of the trapezoid."
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"value": "Given the function $f(x) = \\log_2 (2^{x}-b)$ where $b$ is a constant, if $x \\geqslant 1$, then $f(x) \\geqslant 0$ always holds true. Determine the range of values for $b$."
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"value": "<think>\nOkay, so I need to figure out the range of valu... | 2,733 |
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"value": "Given the polar equation of curve $C_1$ is $\\rho^2=\\frac {2}{3+\\cos2\\theta}$, establish a rectangular coordinate system with the pole O as the origin and the polar axis as the positive direction of the x-axis. After stretching all the x-coordinates of points on curve $C_1$ to... | 5,781 |
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"value": "Return your final response within \\boxed{}. \nGiven a circle \\( K \\) and a line \\( e \\), which intersects \\( K \\) at two distinct points \\( P \\) and \\( Q \\). Let \\( F \\) be the midpoint of one of the arcs \\( PQ \\). Consider the circles \\( k \\) that are tangent to... | 5,674 |
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"value": "Five times $A$'s money added to $B$'s money is more than $$51.00$. Three times $A$'s money minus $B$'s money is $$21.00$.\nIf $a$ represents $A$'s money in dollars and $b$ represents $B$'s money in dollars, then:\n$\\textbf{(A)}\\ a>9, b>6 \\qquad \\textbf{(B)}\\ a>9, b9, b=6\\qq... | 3,446 |
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"value": "Given the inequality $ax^{2}-bx-1\\geqslant 0$ has solutions $[-\\frac{1}{2},-\\frac{1}{3}]$ \n$(1)$ Find the values of $a$ and $b$; \n$(2)$ Find the solution set of the inequality $x^{2}-bx-a < 0$."
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"value": "<think>\nAlright, let's tackle... | 3,398 |
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"value": "Return your final response within \\boxed{}. The students take a bus to the Martyrs' Cemetery for the Qingming Festival. If the bus increases its speed by one-fifth after driving for 1 hour, they can arrive 20 minutes earlier than the scheduled time; if the bus first travels 72 k... | 2,725 |
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"value": "Given that $a$ and $b$ are the roots of the equation $x^{2}-3x-5=0$, then the value of the expression $\\frac{1}{a}+\\frac{1}{b}$ is ____."
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"value": "<think>\nOkay, so I have this problem here where I need to find the value of the expression ... | 2,275 |
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"value": "Let the complex number $z$ satisfy $\\frac{i}{1+i}z=1$, then $|z|=\\left(\\ \\ \\right)$\n\nA: $1$\n\nB: $5$\n\nC: $\\sqrt{2}$\n\nD: $2$"
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"value": "<think>\nOkay, so I need to solve this complex number problem. Let me read it again. It says, ... | 2,629 |
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"value": "The solution to the system of equations $\\left\\{\\begin{array}{l}{x=4y}\\\\{x+2y=-12}\\end{array}\\right.$ is ()\n\nA: $\\left\\{\\begin{array}{l}{x=-4,}\\\\{y=-1}\\end{array}\\right.$\n\nB: $\\left\\{\\begin{array}{l}{x=-8,}\\\\{y=-2}\\end{array}\\right.$\n\nC: $\\left\\{\\beg... | 1,262 |
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"value": "[u]Round 1[/u]\n\n\n[b]p1.[/b] How many powers of $2$ are greater than $3$ but less than $2013$?\n\n\n[b]p2.[/b] What number is equal to six greater than three times the answer to this question? \n\n\n[b]p3.[/b] Surya Cup-a-tea-raju goes to Starbucks Coffee to sip coffee out of a... | 10,683 |
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"value": "Given point $P(8,1)$, the line passing through $P$ intersects the hyperbola $x^{2}-4y^{2}=4$ at points $A$ and $B$. Point $P$ is the midpoint of segment $AB$. Find the equation of line $AB$."
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"value": "<think>\nOkay, so I have this problem he... | 6,802 |
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"value": "Given the function $f(x)=a\\ln x-bx^{2}$, if the graph of the function $f(x)$ is tangent to the line $y=- \\frac {1}{2}$ at $x=1$.\n(Ⅰ) Find the values of the real numbers $a$ and $b$;\n(Ⅱ) Find the maximum value of the function $f(x)$ on the interval $\\left[ \\frac {1}{e},e\\ri... | 4,360 |
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"value": "Generate an executable Python function generated from the given prompt. The function should take stdin as input and print the output. Simply call the function after the definition.Fox Ciel wants to write a task for a programming contest. The task is: \"You are given a simple undi... | 22,474 |
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"value": "The equations of two tangents of an ellipse are:\n\n$$\n4 x+5 y=25, \\quad 9 x+20 y=75\n$$\n\nWhat is the equation of the ellipse if its axes coincide with the coordinate axes?"
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"value": "<think>\nOkay, so I need to find the equation of an el... | 5,016 |
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"value": "Given that $x_{0}$ is a zero of the function $f(x)={(\\frac{1}{3})}^{x}-x+4$, if $x_{1}\\in (2$,$x_{0})$, $x_{2}\\in (x_{0}$,$+\\infty )$, then which of the following statements is true?\n\nA: $x_{0}\\in \\left(2,4\\right)$\n\nB: $f(x_{1}) \\gt f(x_{2})$\n\nC: $f(x_{1}) \\lt 0$，$... | 3,601 |
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"value": "Given the universal set $U=\\mathbb{R}$, if set $A=\\{x\\mid \\frac{x}{x-1} > 0\\}$, then $\\lnot\\_U A=$ \\_\\_\\_\\_\\_\\_."
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"value": "<think>\nOkay, so I need to find the negation of set A with respect to the universal set U, which is all ... | 2,460 |
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"value": "Given the inequality about $x$, $(a^{2}-4)x^{2}+(a+2)x-1 \\geqslant 0$, its solution set is an empty set. Find the range of the real number $a$ __ ___ ."
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"value": "<think>\nOkay, so I need to find the range of the real number \\( a \\) such t... | 5,604 |
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"value": "The graph of the power function $f(x)$ passes through the point $(2, 16)$. Therefore, $f(x) = \\boxed{x^4}$."
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"value": "<think>\nOkay, so I need to figure out why the power function f(x) that passes through the point (2, 16) is equal to x⁴. L... | 1,604 |
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"value": "4. Calculate: $672 \\times 673 \\times 674 - 671 \\times 673 \\times 675=$"
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"value": "<think>\nOkay, so I need to calculate 672 × 673 × 674 minus 671 × 673 × 675. Hmm, that looks like a big multiplication problem. Let me see if there's a way ... | 2,569 |
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"value": "5. Find the minimum distance from the point $(0,5 / 2)$ to the graph of $y=x^{4} / 8$."
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"value": "<think>\nOkay, so I need to find the minimum distance from the point (0, 5/2) to the graph of y = x⁴/8. Hmm, let me think. First, I remember tha... | 5,206 |
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"value": "Given vectors $\\overrightarrow{a}=(\\cos \\frac{3}{2}x, \\sin \\frac{3}{2}x)$ and $\\overrightarrow{b}=(\\cos \\frac{x}{2}, -\\sin \\frac{x}{2})$, where $x \\in (0, \\frac{\\pi}{2})$:\n1. Find $\\overrightarrow{a} \\cdot \\overrightarrow{b}$ and $|\\overrightarrow{a} + \\overrig... | 4,518 |
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"value": "Let's determine the value of the sum $\\sum_{n=1}^{\\infty} \\operatorname{arcctg}\\left(2 n^{2}\\right)$."
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"value": "<think>\nAlright, so I need to find the value of the infinite sum \\(\\sum_{n=1}^{\\infty} \\operatorname{arcctg}\\left(2 n^... | 5,079 |
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"value": "5. On the table, there are 40 red and 50 green stones. Two players, Petya and Vasya, take turns: on each turn, a player chooses a color and removes an arbitrary number of stones of that color, which must be a divisor of the number of stones of the other color at the moment of the... | 6,738 |
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"value": "4. Along a circle, 123 points are placed at equal distances from each other. Anya and Borya take turns painting one point either blue or red (they can paint any previously unpainted point). The player loses if, after their move, two adjacent points of the same color appear. Who w... | 7,646 |
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"value": "The equation of the directrix of the parabola $x^2=4y$ is ( ) \nA: $y=-1$ \nB: $y=-2$ \nC: $x=-1$ \nD: $x=-2$"
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"value": "<think>\nOkay, so I need to find the equation of the directrix for the parabola given by x² = 4y. The options are A ... | 1,209 |
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"value": "A certain civil aviation regulation allows passengers to carry a kilograms of goods for free. However, for goods exceeding a kilograms, a certain fee is charged. The fee is calculated as follows: the weight of the goods carried by the passenger, b kilograms (b > a), multiplied by... | 1,759 |
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"value": "Given that function $f(x)$ is an odd function defined on $\\mathbb{R}$ and satisfies $f(x+1)=f(x-1)$ for any real number $x$. When $0 < x < 1$, $f(x)=4^{x}$. Find the value of $f(- \\frac {5}{2})+f(1)=$ ___."
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"value": "<think>\nOkay, let's se... | 4,248 |
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"value": "Five children sit around a circular table. Each child chooses a positive integer and reports it to the others. Then, each child performs the following calculation: they add the numbers of the two children to their left, subtract the sum of the numbers of the other two children to... | 5,649 |
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"value": "Return your final response within \\boxed{}. \\[\n\\sum_{k=1}^{70} \\frac{k}{x-k} \\geq \\frac{5}{4}\n\\]\nis a union of disjoint intervals the sum of whose lengths is 1988."
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"value": "<think>\n\nOkay, so I need to solve this inequality: the ... | 14,825 |
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"value": "If $\\triangle + \\triangle = ※$, $○ = □ + □$, $\\triangle = ○ + ○ + ○ + ○$, then $※ ÷ □ = (\\ )$ \nA: $2$ \nB: $4$ \nC: $8$ \nD: $16$"
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"value": "<think>\nOkay, let's try to solve this problem step by step. So we have these equations invo... | 965 |
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"value": "Let $i$ be the imaginary unit. If the complex number $z$ satisfies $iz=1+2i$, then $|z-1|=\\_\\_\\_\\_\\_\\_$."
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"value": "<think>\nOkay, so I need to solve this complex number problem. Let's see. The problem says that for the complex number z... | 1,551 |
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"value": "In $\\triangle ABC$, $a$, $b$, $c$ are the sides opposite to angles $A$, $B$, $C$ respectively, and it satisfies $\\frac {2a-b}{c}= \\frac {\\cos B}{\\cos C}$, \n$(1)$ Find the size of angle $C$; \n$(2)$ Let the function $f(x)=2\\sin x\\cos x\\cos C+2\\sin ^{2}x\\sin C- \\frac ... | 5,783 |
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"value": "Given the space vectors $\\overrightarrow {a}=(0,-1,1)$ and $\\overrightarrow {b}=(1,0,1)$, find the magnitude of $|2 \\overrightarrow {a}+ \\overrightarrow {b}|$."
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"value": "<think>\nOkay, let's see. I need to find the magnitude of the vecto... | 1,235 |
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"value": "2. (10 points) Calculate:\n$$\n10 \\times 9 \\times 8 + 7 \\times 6 \\times 5 + 6 \\times 5 \\times 4 + 3 \\times 2 \\times 1 - 9 \\times 8 \\times 7 - 8 \\times 7 \\times 6 - 5 \\times 4 \\times 3 - 4 \\times 3 \\times 2\n$$"
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"value": "<thin... | 2,816 |
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"value": "5. In the cells of a $2015 \\times n$ table, non-negative numbers are arranged such that in each row and each column there is at least one positive number. It is known that if a cell contains a positive number, then the sum of all numbers in its column is equal to the sum of all ... | 5,531 |
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"value": "1. The modulus of the difference between the two roots of the equation $x^{2}+a x+4=0$ equals $a$, then $a=$"
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"value": "<think>\nOkay, let's see. I need to find the value of 'a' such that the modulus of the difference between the two roots of... | 3,600 |
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"value": "2. [5] Given right triangle $A B C$, with $A B=4, B C=3$, and $C A=5$. Circle $\\omega$ passes through $A$ and is tangent to $B C$ at $C$. What is the radius of $\\omega$ ?"
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"value": "<think>\nOkay, so I need to find the radius of circle ω th... | 1,986 |
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"value": "1. The monotonic increasing interval of the function $f(x)=\\log _{\\frac{2}{2}}\\left(x^{2}-2 x-3\\right)$ is $(\\quad)$.\nA. $(-\\infty,-1)$\nB. $(-\\infty, 1)$\nC. $(1,+\\infty)$\nD. $(3,+\\infty)$"
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"value": "<think>\nOkay, so I need to fi... | 4,085 |
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"value": "Generate an executable Python function generated from the given prompt. The function should take stdin as input and print the output. Simply call the function after the definition.N people run a marathon. There are M resting places on the way. For each resting place, the i-th run... | 19,856 |
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"value": "A five-digit natural number $\\overline{a_1a_2a_3a_4a_5}$ is considered a \"concave number\" if and only if $a_1 > a_2 > a_3$ and $a_3 < a_4 < a_5$, with each $a_i \\in \\{0,1,2,3,4,5\\}$ for $i=1,2,3,4,5$ (for example, 32014 and 53134). Find how many such \"concave numbers\" are... | 4,382 |
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"value": "Return your final response within \\boxed{}. In how many ways can the numbers from 1 to 100 be rearranged so that neighboring numbers differ by no more than 1?"
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"value": "<think>\n\nOkay, so I have this problem: \"In how many ways can the num... | 5,187 |
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"value": "(solved by Alice Héliou). The sum of twenty consecutive integers is 1030. What is the smallest of these integers?"
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"value": "<think>\nOkay, so the problem is that the sum of twenty consecutive integers is 1030, and we need to find the smalles... | 1,866 |
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"value": "Return your final response within \\boxed{}. \nFind the third derivative of the given function.\n\n\\[ y = \\frac{\\ln x}{x^5}, \\quad y''' = ? \\]"
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"value": "<think>\n\nOkay, so I need to find the third derivative of the function y = (ln x)/... | 2,535 |
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"value": "Let the function $f(x) = \\frac{1}{3}x^3 - 4x + 4$ have its maximum value $M$ and minimum value $m$ in the interval $[0, 3]$. The value of $M - m$ is ______."
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"value": "<think>\nOkay, so I need to find the maximum and minimum values of the fu... | 2,603 |
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"value": "10、If $\\frac{2^{12}+1}{2^{4}+1}=2^{a_{1}}+2^{a_{2}}+\\ldots+2^{a_{k}}$ where $a_{1} \\sim a_{\\mathrm{k}}$ are natural numbers, and $a_{1}<a_{2}<\\cdots$ $<a_{2}$, then $\\mathrm{k}=$ $\\qquad$"
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"value": "<think>\nOkay, let me see. I have th... | 5,352 |
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"value": "Given the function $f(x)=ax^{2}+bx+c$, where $a > 0$, and for any real number $x$, $f(2+x)=f(2-x)$. Then,\nA: $f(2) < f(1) < f(4)$\nB: $f(1) < f(2) < f(4)$\nC: $f(2) < f(4) < f(1)$\nD: $f(4) < f(2) < f(1)$"
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"value": "<think>\nOkay, so I have ... | 4,177 |
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"value": "Given $f(x)=\\frac{1}{4}x^3-x^2+x$. \n$(1)$ Find the local minimum value of the function $y=f(x)$. \n$(2)$ Prove that for $x\\in [-2,4]$, $x-6\\leq f(x)\\leq x$. \n$(3)$ Let $F(x)=|f(x)-(x+a)|$ $(a\\in R)$. Denote the maximum value of the function $y=F(x)$ on the interval $[-2... | 5,804 |
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"value": "Given an ellipse $\\frac{{x}^{2}}{{a}^{2}}+\\frac{{y}^{2}}{{b}^{2}}=1(a>b>0)$ with the right vertex $A$ and the top vertex $B$, and $O$ as the origin. A point $M$ inside the ellipse satisfies $\\overrightarrow{OM}=\\overrightarrow{MA}$, and $\\frac{|BM|}{|AB|}=\\frac{\\sqrt{6}}{4... | 7,164 |
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"value": "## Task A-1.3.\n\nThe lengths of the legs of a right triangle are $6 \\mathrm{~cm}$ and $8 \\mathrm{~cm}$. What is the radius of the circle circumscribed around this triangle?"
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"value": "<think>\nOkay, so I need to find the radius of the circ... | 1,124 |
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"value": "Return your final response within \\boxed{}. Let \\( x, y \\in \\mathbb{R} \\). If \\( 2x, 1, y-1 \\) form an arithmetic sequence and \\( y+3, |x+1| + |x-1|, \\cos(\\arccos x) \\) form a geometric sequence, then the value of \\( (x+1)(y+1) \\) is \\(\\qquad\\)."
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"value": "Given that the focus $F$ of the parabola $x=4y^{2}$ intersects the parabola at points $M$ and $N$, and $|MF|= \\frac{1}{8}$, find the value of $|MN|$."
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"value": "<think>\nOkay, let me try to figure out this problem step by step. So, we have t... | 4,792 |
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"value": "The equation of line $l$ that passes through point A(3, 0) and point B(0, 2) is ( )\n\nA: $2x+3y-6=0$\n\nB: $3x+2y-6=0$\n\nC: $2x+3y-1=0$\n\nD: $3x+2y-1=0$"
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"value": "<think>\nOkay, so I need to find the equation of the line that passes thro... | 2,056 |
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"value": "Observe the following equations: ①$1+6\\times 1=4^{2}-9\\times 1^{2}$; ②$1+6\\times 2=7^{2}-9\\times 2^{2}$; ③$1+6\\times 3=10^{2}-9\\times 3^{2}$; $\\ldots \\ldots $ According to the above pattern, write down the $n$-th equation you guess ______."
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"from": "assistant... | 2,155 |
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"value": "Given $\\sin \\alpha + \\cos \\alpha = \\frac{1}{5}$, and $- \\frac{\\pi}{2} \\leqslant \\alpha \\leqslant \\frac{\\pi}{2}$, then $\\tan \\alpha$ equals to ( ).\nA: $- \\frac{4}{3}$\nB: $- \\frac{3}{4}$\nC: $\\frac{3}{4}$\nD: $\\frac{4}{3}$"
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... | 2,630 |
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"value": "【Example 3.5】 $A B, A C$ are tangent to $\\odot O$ at $B, C$, point $P$ is on $\\odot O$, $P D, P E, P F$ are perpendicular to $B C, C A, A B$ at $D, E, F$ (Figure 3.12), prove: $P D^{2}=P E \\cdot P F$."
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"value": "<think>\nOkay, so I need to... | 4,706 |
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"value": "Given $a > 0$, $b > 0$, compare the magnitude of $M= \\sqrt{a} + \\sqrt{b}$ and $N= \\sqrt{a+b}$."
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"value": "<think>\nOkay, so I need to compare the magnitudes of M and N, where M is the sum of the square roots of a and b, and N is the square... | 2,770 |
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"value": "The monotonic decreasing interval of the function $y=\\log_{\\frac{1}{2}}(x^{2}-3x+2)$ is $(\\quad)$\n\nA: $(-\\infty,\\dfrac{3}{2})$\n\nB: $(\\dfrac{3}{2},+\\infty)$\n\nC: $(-\\infty,1)$\n\nD: $(2,+\\infty)$"
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"value": "<think>\nOkay, let's t... | 2,981 |
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"value": "Return your final response within \\boxed{}. Given the sequence defined as \\( a_{1}=1 \\), \\( a_{2}=3 \\), and for all positive integers \\( n \\),\n\\[ \na_{n+2}=(n+3) a_{n+1}-(n+2) a_{n},\n\\]\nfind all values of \\( n \\) such that \\( a_{n} \\) is divisible by 11."
},
{... | 6,737 |
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"value": "In the same plane rectangular coordinate system, the line $x - 2y = 2$ undergoes a stretch transformation $\\begin{cases} x' = x \\\\ y' = 2y \\end{cases}$ and becomes line $l$. Determine the equation of line $l$."
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"value": "<think>\nOkay, so... | 1,782 |
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"value": "4. For a given positive integer $n$. Try to find the smallest positive integer $d_{n}$ that cannot be expressed in the form $\\sum_{i=1}^{n}(-1)^{a_{i}} \\times 2^{b_{i}}$, where $a_{i} 、 b_{i}(i=1,2, \\cdots, n)$ are all non-negative integers."
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... | 12,350 |
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"value": "A [i]substring [/i] of a number $n$ is a number formed by removing some digits from the beginning and end of $n$ (possibly a different number of digits is removed from each side). Find the sum of all prime numbers $p$ that have the property that any substring of $p$ is also prime... | 11,144 |
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"value": "It is known that $\\frac{7}{13} + \\sin \\phi = \\cos \\phi$ for some real $\\phi$. What is sin $2\\phi$?"
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"value": "<think>\nOkay, so I need to find sin 2φ given that 7/13 + sin φ equals cos φ. Hmm, let's see. Let me write down the equation ... | 4,132 |
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"value": "Find the maximum value of the function $f(x)=|x|-|x-3|$."
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"value": "<think>\nOkay, so I need to find the maximum value of the function f(x) = |x| - |x - 3|. Hmm, absolute value functions can sometimes be tricky because of the piecewise defini... | 2,538 |
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"value": "A sequence of positive integers is defined by $a_0=1$ and $a_{n+1}=a_n^2+1$ for each $n\\ge0$. Find $\\text{gcd}(a_{999},a_{2004})$."
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"value": "<think>\nOkay, let me try to figure out how to find the greatest common divisor (gcd) of a_{999} a... | 10,369 |
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{
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"value": "Among the following specific propositions, the false one is ( )\nA: In space, there is exactly one line through a point outside a given line that is perpendicular to the given line.\nB: There exists only one real number $b_2$ such that -9, $b_1$, $b_2$, $b_3$, -1 form a geometri... | 5,510 |
[
{
"from": "user",
"value": "In $\\triangle ABC$, $a$, $b$, $c$ are the sides opposite to angles $A$, $B$, $C$ respectively, and they satisfy $\\frac{a+c}{a+b}=\\frac{b-a}{c}$.\n\n(1) Find the measure of angle $B$;\n\n(2) If the longest side of $\\triangle ABC$ has a length of $\\sqrt{14}$, and $\\sin A=2\\s... | 3,899 |
[
{
"from": "user",
"value": "Return your final response within \\boxed{}. Show that the greatest common divisor (GCD) of $(5a + 3b, 13a + 8b)$ is equal to the GCD of $(a, b)$."
},
{
"from": "assistant",
"value": "<think>\n\nOkay, so I need to show that the greatest common divisor (GCD) of (5a + 3... | 2,342 |
[
{
"from": "user",
"value": "Given the complex number $z$ that satisfies the equation $z(1 - i) = 2$, where $i$ is the imaginary unit, find the value of $z$."
},
{
"from": "assistant",
"value": "<think>\nOkay, so I need to find the complex number z that satisfies the equation z(1 - i) = 2. Hmm, c... | 2,217 |
[
{
"from": "user",
"value": "Return your final response within \\boxed{}. Given a natural number $n$, consider all fractions of the form $\\frac{1}{pq}$, where $p$ and $q$ are coprime, $0 < p < q \\leq n$, and $p + q > n$. Prove that the sum of all such fractions is equal to $\\frac{1}{2}$."
},
{
"fr... | 18,767 |
[
{
"from": "user",
"value": "Given the function $f(x)=x^{3}+3ax^{2}+3bx$, it has an extreme value at $x=2$. The tangent line of its graph at $x=1$ is parallel to the line $6x+2y+5=0$. Find the difference between the maximum and minimum values of $f(x)$."
},
{
"from": "assistant",
"value": "<think... | 3,174 |
[
{
"from": "user",
"value": "Given the complex number $\\dfrac{1+z}{1-z}=i$, find $|z|=$ ______."
},
{
"from": "assistant",
"value": "<think>\nOkay, so I need to solve for the modulus of the complex number z given that (1 + z)/(1 - z) equals i. Hmm, let's start by recalling that modulus refers to... | 2,151 |
[
{
"from": "user",
"value": "The number of sets $A$ satisfying $\\{0,1,2\\} \\subset A \\subseteq \\{0,1,2,3,4,5\\}$ is $\\boxed{7}$."
},
{
"from": "assistant",
"value": "<think>\nOkay, let me try to figure out how to solve this problem. The question is asking for the number of sets A that satisf... | 2,248 |
[
{
"from": "user",
"value": "Given that $x$ and $y$ are real numbers, and they satisfy $(2x-1) + i = -y - (3-y)i$, the values of $x$ and $y$ are ( )\n\nA: $x= \\frac {3}{2}$, $y=4$\n\nB: $x=- \\frac {3}{2}$, $y=4$\n\nC: $x=- \\frac {3}{2}$, $y=-4$\n\nD: $x= \\frac {3}{2}$, $y=-4$"
},
{
"from": "assi... | 1,381 |
[
{
"from": "user",
"value": "Problem 7.8. Given an isosceles triangle $ABC (AB = BC)$. On the ray $BA$ beyond point $A$, point $E$ is marked, and on side $BC$, point $D$ is marked. It is known that\n\n$$\n\\angle ADC = \\angle AEC = 60^{\\circ}, AD = CE = 13.\n$$\n\nFind the length of segment $AE$, if $DC = ... | 12,310 |
[
{
"from": "user",
"value": "In the sequence $\\{a\\_n\\}$, $a\\_1=1$, $a\\_{n+1}=a\\_n+1$, then $a\\_{2014}$ is equal to $(\\;\\;\\;)$.\nA: $2\\;013$\nB: $2\\;012$\nC: $2\\;011$\nD: $2\\;014$"
},
{
"from": "assistant",
"value": "<think>\nOkay, let's see. The problem is about a sequence where the... | 1,808 |
[
{
"from": "user",
"value": "The sales volume $y$ (pieces) of a product has a linear relationship with its selling price $x$ (yuan/piece). Based on a set of sample data $(x_i, y_i)$ ($i=1,2,\\ldots,n$), the regression equation established using the least squares method is $\\hat{y} = -10x + 200$. Among the f... | 1,263 |
[
{
"from": "user",
"value": "Given an arithmetic sequence $\\{a_n\\}$ with a common difference of 2, if $a_1$, $a_3$, and $a_4$ form a geometric sequence, then $a_1$ equals to."
},
{
"from": "assistant",
"value": "<think>\nAlright, let's see. So the problem is about an arithmetic sequence {a_n} w... | 2,577 |
[
{
"from": "user",
"value": "Task 2. A poultry farm delivered 720 eggs, which were packed in boxes of 6 eggs each. During transportation, 140 eggs were broken. The remaining eggs were packed in boxes of 12 eggs each. How many boxes of eggs were packed. Were there any unbroken eggs left that could not be pack... | 1,509 |
[
{
"from": "user",
"value": "1-191 Write down all natural numbers from 1 to 1988 on a blackboard. Perform operations $A$ and $B$ repeatedly on these numbers: first $A$, then $B$, followed by $A$ again, and then $B$, and so on. Operation $A$ involves subtracting the same natural number from each number writte... | 17,340 |
[
{
"from": "user",
"value": "Given a circle C with equation $(x-1)^2 + y^2 = 16$, there is a point P(2, 2) inside it. Let line l pass through point P and intersect circle C at points A and B.\n(1) When line l passes through the center of circle C, find the equation of line l;\n(2) When chord AB is bisected b... | 4,724 |
[
{
"from": "user",
"value": "# Problem 5.\n\nB-1\n\nChicks hatch on the night from Sunday to Monday. For two weeks, the chick sits with its beak open, the third week it silently grows feathers, and on the fourth week, it flies away from the nest. Last week, 20 chicks were sitting with their beaks open, and 1... | 3,682 |
[
{
"from": "user",
"value": "Given triangle $ABC$ with angles $A$, $B$, and $C$ and their respective opposite sides $a$, $b$, and $c$. It is known that $2\\cos C(a\\cos B+b\\cos A)=c$.\n\n(1) Find angle $C$;\n\n(2) If $c= \\sqrt{7}$, and the area of triangle $ABC$ is $\\dfrac{3 \\sqrt{3}}{2}$, find the perim... | 2,428 |
[
{
"from": "user",
"value": "Problem 2.4. Consider an alphabet of $n$ letters, with which we will form words. We will say that a word contains a palindrome if a segment of that word, of more than one letter, reads the same forwards as backwards. For example, the word OLIMPIADAS contains the palindrome ADA. G... | 5,979 |
[
{
"from": "user",
"value": "10.042. One of the angles of the trapezoid is $30^{\\circ}$, and the lines containing the lateral sides of the trapezoid intersect at a right angle (Fig. 10.42). Find the length of the shorter lateral side of the trapezoid if its midline is 10 cm, and one of the bases is $8 \\mat... | 8,073 |
[
{
"from": "user",
"value": "For an arithmetic sequence $\\{a_n\\}$, the sum of the first $n$ terms is $S_n$. Given that $a_1=13$ and $S_3=S_{11}$, find the value of $n$ for which $S_n$ is maximized."
},
{
"from": "assistant",
"value": "<think>\nOkay, let's see. I need to find the value of n for ... | 2,456 |
[
{
"from": "user",
"value": "Problem 7.4. In triangle $ABC$, the median $CM$ and the bisector $BL$ were drawn. Then, all segments and points were erased from the drawing, except for points $A(2 ; 8)$, $M(4 ; 11)$, and $L(6 ; 6)$. What were the coordinates of point $C$?\n\n![](https://cdn.mathpix.com/cropped/... | 4,907 |
[
{
"from": "user",
"value": "A geometric body's front (main) view, side (left) view, and top view are all identical in size and shape. This geometric body could be ( )\nA: Sphere\nB: Cylinder\nC: Triangular Prism\nD: Cone"
},
{
"from": "assistant",
"value": "<think>\nOkay, so I need to figure ou... | 1,976 |
[
{
"from": "user",
"value": "Let $O$ be the circumcenter of $\\triangle ABC$ (the intersection point of the perpendicular bisectors of the three sides), and $\\sqrt{2}\\overrightarrow{OA} + \\overrightarrow{OB} + \\overrightarrow{OC} = \\overrightarrow{0}$, then $\\angle BAC = \\_\\_\\_\\_\\_\\_$."
},
{
... | 3,213 |
[
{
"from": "user",
"value": "10. Given the function $f(x)=x^{3}-a x^{2}+\\left(a^{2}-2\\right) x+1$, if there exists $m>0$, such that $f(m) \\leqslant 0$, then the maximum value of the real number $a$ is $\\qquad$ ."
},
{
"from": "assistant",
"value": "<think>\nOkay, let's tackle this problem ste... | 11,038 |
[
{
"from": "user",
"value": "7. In the tetrahedron $S-ABC$,\n$$\nSA=SB=SC=\\sqrt{21}, BC=6 \\text{.}\n$$\n\nIf the projection of point $A$ onto the plane containing the side $SBC$ is exactly the orthocenter of $\\triangle SBC$, then the volume of the inscribed sphere of the tetrahedron $S-ABC$ is"
},
{
... | 8,119 |
[
{
"from": "user",
"value": "Given the sets $I=\\{1,2,3,4,5,6\\}$, $M=\\{1,2,6\\}$, and $N=\\{2,3,4\\}$, then $\\{1,6\\}=$ ( )\nA: $M\\cap N$\nB: $M\\cup N$\nC: $M\\cap (C_{I}N)$\nD: None of the above"
},
{
"from": "assistant",
"value": "<think>\nOkay, let's try to figure this problem out. So, we... | 1,702 |
[
{
"from": "user",
"value": "Given two operations: $a⊕b= \\sqrt {a^{2}-b^{2}}$ and $a⊗b= \\sqrt {(a-b)^{2}}$, determine the type of function $f(x)= \\dfrac {2⊕x}{x⊗2-2}$.\n\nA: Odd function\nB: Even function\nC: Both an odd and an even function\nD: Neither an odd nor an even function"
},
{
"from": "a... | 4,568 |
[
{
"from": "user",
"value": "7. The number of sets $X$ that satisfy the condition $\\{1,2,3\\} \\subseteq X \\subseteq\\{1,2,3,4,5,6\\}$ is"
},
{
"from": "assistant",
"value": "<think>\nOkay, so the problem here is asking for the number of sets X that satisfy the condition {1,2,3} is a subset of ... | 1,537 |
[
{
"from": "user",
"value": "Given point $Q(\\sqrt{5},\\frac{4}{3}$) is a point on the ellipse $C$: $\\frac{x^{2}}{a^{2}}+\\frac{y^{2}}{b^{2}}=1(a>b>0)$, $F_{1}$, $F_{2}$ are the left and right foci of the ellipse, and $\\overrightarrow{Q{F}_{2}}\\cdot \\overrightarrow{{F}_{2}{F}_{1}}=0$. Find the equation o... | 9,503 |
[
{
"from": "user",
"value": "5. Given a two-digit number $\\overline{p q}$, such that the quadratic function $y=x^{2}+q x+p$ intersects the $x$-axis at two distinct points $A$ and $B$, with the vertex at $C$, and $S_{\\triangle A B C} \\leqslant 1$. Then the number of such two-digit numbers is\n(A) 1\n(B) 2\... | 8,161 |
[
{
"from": "user",
"value": "A sequence is formed by the sum of terms from an arithmetic sequence with common difference $d \\neq 0$, denoted as $a_1, a_2, \\ldots, a_n, \\ldots$. The new sequence is $a_1 + a_3, a_2 + a_4, a_3 + a_5, \\ldots$. Which of the following statements is correct? ( )\nA: The new seq... | 2,118 |
[
{
"from": "user",
"value": "The graph of the function $y=a^{x-1}+2$ ($a>0$ and $a\\neq 1$) must pass through the point ( )\nA: $(1,3)$\nB: $(0,1)$\nC: $(1,0)$\nD: $(3,0)$"
},
{
"from": "assistant",
"value": "<think>\nOkay, so I need to figure out which of the given points the graph of the func... | 1,704 |
[
{
"from": "user",
"value": "4.40 Find $\\operatorname{ctg} 2 \\alpha$, if it is known that $\\sin \\left(\\alpha-90^{\\circ}\\right)=-\\frac{2}{3}$ and $270^{\\circ}<\\alpha<360^{\\circ}$."
},
{
"from": "assistant",
"value": "<think>\nAlright, let's tackle this problem step by step. We need to f... | 2,704 |
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