messages listlengths 3 3 | conversation_id int64 0 20k |
|---|---|
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "Given the circle $O: x^2 + y^2 = 4$ and the point $M(1, a)$.\n(1) If there is exactly one line passing through point $M$ that is tangent to circle $O$, find the value of the real number $a$ and the equation of the tangent... | 12,534 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "Find a number with $3$ digits, knowing that the sum of its digits is $9$, their product is $24$ and also the number read from right to left is $\\frac{27}{38}$ of the original. Let's think step by step and output the fina... | 16,210 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "Define the set operation: $A\\odot B = \\{z | z = xy(x+y), x \\in A, y \\in B\\}$. Given the sets $A=\\{0,1\\}$ and $B=\\{2,3\\}$, the sum of all elements of the set $A\\odot B$ is ___. Let's think step by step and output... | 11,174 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "The parabola with its vertex at the origin, its axis of symmetry along the y-axis, and its focus on the line $3x-4y-24=0$ has the standard equation of __ . Let's think step by step and output the final answer within \\box... | 11,799 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "$16 \\cdot 6$ Find a point on the side of a rectangle such that the lines connecting this point to the endpoints of the opposite side divide the rectangle into three mutually similar triangles. The number of such points:\... | 17,466 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "Let $p, q$ and $r$ be distinct [prime numbers](https://artofproblemsolving.com/wiki/index.php/Prime_number), where $1$ is not considered a prime. Which of the following is the smallest positive [perfect cube](https://arto... | 13,351 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "4A. Let $ABCD$ be a triangular pyramid where all three angles at vertex $D$ are right angles. Calculate the height of the pyramid dropped from vertex $D$, given the lengths of the edges $\\overline{AB}=\\overline{AC}=5$ a... | 13,263 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "18. (10 points) As shown in the figure, the side length of square $A B C D$ is 20 cm, and $E$ and $F$ are the midpoints of $A B$ and $B C$ respectively. What is the area of quadrilateral $B E G F$ in square centimeters? L... | 18,423 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "36 . 18 If person A's height or weight is greater than person B's in at least one aspect, then we say A is not inferior to B. Among 100 young men, if someone is not inferior to the other 99, he is called a great young man... | 3,205 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "8.394. $\\left\\{\\begin{array}{l}\\sin x+\\cos y=0, \\\\ \\sin ^{2} x+\\cos ^{2} y=\\frac{1}{2} .\\end{array}\\right.$ Let's think step by step and output the final answer within \\boxed{{}}",
"role": "user"
},
{... | 4,033 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "Three, let $M$ be a set of $n$ points in the plane, satisfying:\n(1) There exist 7 points in $M$ that are the 7 vertices of a convex heptagon;\n(2) For any 5 points in $M$, if these 5 points are the 5 vertices of a convex... | 18,797 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "(1) $\\sqrt{8} + 2\\sqrt{3} - (\\sqrt{27} - \\sqrt{2})$ \n(2) $(\\sqrt{2} - \\sqrt{3})^2 + 2\\sqrt{\\frac{1}{3}} \\times 3\\sqrt{2}$. Let's think step by step and output the final answer within \\boxed{{}}",
"role": ... | 2,359 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "Among the following statements: \n① In the residual plot, if the residual points are evenly distributed within a horizontal band, it indicates that the chosen model is appropriate; \n② The regression effect can be depic... | 17,852 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "Use the Horner's Rule to calculate the value of an $n$-th degree polynomial $f(x) = a_nx^n + a_{n-1}x^{n-1} + \\ldots + a_1x + a_0$ when $x = x_0$. The steps of the algorithm are as follows: \nStep 1, input $n$, $a_n$ an... | 11,950 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "Given that the function $f\\left(x\\right)=x^{3}(a\\cdot 2^{x}-2^{-x})$ is an even function, find the value of $a$. Let's think step by step and output the final answer within \\boxed{{}}",
"role": "user"
},
{
... | 15,901 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "(IMO 1975, Problem 6)\n\nLet $n$ be a positive integer. Find all polynomials $f$ in two variables such that:\n\n1. for all real numbers $t, x, y: f(t x, t y)=t^{n} f(x, y)$;\n2. for all real numbers $a, b, c: f(a+b, c)+f(... | 13,399 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "They ask for a two-digit number such that when it is written three times next to itself and then a 1 is added next to it, we get a 7-digit number which is a perfect cube. Let's think step by step and output the final answ... | 17,043 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "13. Salvo-control. The artillery system of a missile cruiser fires at a target. If the target is not hit, the system automatically makes a second shot at the same target. A third shot at this target is not made. The cruis... | 135 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "## Problem Statement\n\nWrite the equation of the plane passing through point $A$ and perpendicular to vector $\\overrightarrow{B C}$.\n\n$A(5, -1, 2)$\n\n$B(2, -4, 3)$\n\n$C(4, -1, 3)$ Let's think step by step and output... | 1,873 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "In $\\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively. If $a^{2}=b^{2}+ \\frac{1}{4}c^{2}$, then $\\frac{a\\cos B}{c}=$ _____ . Let's think step by step and output the final answer... | 15,051 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "## Task 4 - 310614\n\nA total of 20 people participated in an excursion. It was observed:\n\n(1) Exactly 5 of the participants were 30 years old or younger.\n\n(2) Of the participants who were older than 30, exactly 10 bo... | 16,382 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "Given that the parametric equations of $C\\_1$ in the rectangular coordinate system are $ \\begin{cases} x= \\frac{\\sqrt{5}}{5}t \\\ny= \\frac{2\\sqrt{5}}{5}t - 1 \\end{cases} (t \\text{ is the parameter}) $, and the pol... | 6,645 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "Given that $f(x)$ is an increasing function defined on the interval $[-1, 1]$ and $f(x-2) < f(1-x)$, find the range of values for $x$. Let's think step by step and output the final answer within \\boxed{{}}",
"role": ... | 17,579 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "5. Let real numbers $x, y$ satisfy:\n$$\nx^{3}=3 y^{2} x+5-\\sqrt{7}, y^{3}=3 x^{2} y+5+\\sqrt{7} \\text {. }\n$$\n\nThen the value of $x^{2}+y^{2}$ is ( ).\n(A) $25-8 \\sqrt{7}$\n(B) $2 \\sqrt{7}$\n(C) $12 \\sqrt{7}-25$\... | 14,774 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "In $\\triangle ABC$, $D$ is the midpoint of $AB$. A line through $D$ parallel to $BC$ intersects $AC$ at $E$. If $DE = 4$, then $BC = \\boxed{8}$. Let's think step by step and output the final answer within \\boxed{{}}",
... | 15,395 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "11. The lengths of the three altitudes of a triangle are in the ratio $3: 4: 6$. If the radius of the inscribed circle of the triangle is 1 , find the radius of the circumcircle of the triangle.\n某三角形三條高的長度之比為 $3: 4: 6$ 。... | 1,726 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "2. Determine the remainder of the division of the number $3^{100}$ by 13. Let's think step by step and output the final answer within \\boxed{{}}",
"role": "user"
},
{
"content": "To determine the remainder wh... | 3,781 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "Calculate: $|\\sqrt{25}-3|+\\sqrt[3]{-27}-(-2)^{3}$. Let's think step by step and output the final answer within \\boxed{{}}",
"role": "user"
},
{
"content": "First, we calculate the absolute value term \\( |\... | 8,104 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "14. One of the longest-living animals in the world is the Arctic quahog, and generally, Arctic quahogs can live for hundreds of years. Now there is a large Arctic quahog, 70 years old, and 4 smaller Arctic quahogs whose a... | 9,275 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "It is a custom of five good friends that they wear a different tie each day, each one the one they wore the longest ago. Each of them has at least two ties, but none of them has a dozen. None of them has two ties of the s... | 4,768 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "$\\underline{\\text { Folklore }}$\n\nAmong the actors of Karabas Barabas theater, a chess tournament was held. Each participant played exactly one game with each of the others. One solido was given for a win, half a soli... | 18,640 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "6. Given $\\sqrt{2009}=\\sqrt{x}+\\sqrt{y}$, and $0<x<y$. Find all integer pairs $(x, y)$ that satisfy the equation. Let's think step by step and output the final answer within \\boxed{{}}",
"role": "user"
},
{
... | 6,783 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "For a set $ P$ of five points in the plane, no three of them being collinear, let $ s(P)$ be the numbers of acute triangles formed by vertices in $ P$.\r\nFind the maximum value of $ s(P)$ over all such sets $ P$. Let's t... | 150 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "Given point $(a,b)$ is outside the circle $x^{2}+y^{2}=r^{2}$, the position relationship between the line $ax+by=r^{2}$ and the circle is $( \\ )$\nA: Separate\nB: Tangent\nC: Intersecting and not passing through the cent... | 5,200 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "The minimum positive period of the function $y=\\sin 2x \\cdot \\cos 2x$ is ( )\n\nA: $2\\pi$\n\nB: $4\\pi$\n\nC: $\\frac{\\pi}{4}$\n\nD: $\\frac{\\pi}{2}$ Let's think step by step and output the final answer within \\bo... | 8,280 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "The graph of the power function $f(x)$ passes through the point $(2,8)$. The expression for $f(x)$ is \\_\\_\\_\\_\\_\\_. Let's think step by step and output the final answer within \\boxed{{}}",
"role": "user"
},
... | 14,261 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "9.2. In a kindergarten, each child was given three cards, each of which had either \"MA\" or \"NYA\" written on it. It turned out that 20 children could form the word \"MAMA\" from their cards, 30 children could form the ... | 12,644 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "Given an odd function $f(x)$ with a domain of $R$, and $f(x+2)$ is an even function. If $f(1)=a$, then $f(1)+f(3)+f(5)+\\ldots +f(2019)=$\n\nA: $0$\n\nB: $a$\n\nC: $2a$\n\nD: $1010a$ Let's think step by step and output th... | 19,557 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "Translate the graph of the function $f(x)=\\cos 2x-\\sin 2x$ to the right by $\\varphi (\\varphi > 0)$ units, and it exactly coincides with the original graph. The value of $\\varphi$ that satisfies the condition can be _... | 16,650 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "There are $n$ cards such that for each $i=1,2, \\cdots n$, there are exactly one card labeled $i$. Initially the cards are piled with increasing order from top to bottom. There are two operations:\n\n[list]\n[*] $A$ : One... | 11,958 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "6. Given the function $f(x)=\\frac{(\\sqrt{2})^{x}-1}{(\\sqrt{2})^{x}+1}$, the smallest positive integer $n$ such that $f(n)>\\frac{n}{n+1}$ is $\\qquad$ Let's think step by step and output the final answer within \\boxed... | 14,394 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "# Problem 7. (4 points)\n\nNatural numbers $a, b, c$ are such that $\\operatorname{GCD}(\\operatorname{LCM}(a, b), c) \\cdot \\operatorname{LCM}(\\operatorname{GCD}(a, b), c)=200$.\n\nWhat is the greatest value that $\\op... | 13,092 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "5.2. Masha eats a bowl of porridge in 12 minutes, and Bear eats it twice as fast. How long will it take them to eat six bowls of porridge? Let's think step by step and output the final answer within \\boxed{{}}",
"rol... | 11,180 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "Let $f(x)=\\left\\{\\begin{array}{l}{\\frac{2}{3}x-1 (x\\geq 0)}\\\\{\\frac{1}{x} \\;\\;\\;\\; (x<0)}\\end{array}\\right.$. If $f(a)=a$, then the value of the real number $a$ is ____. Let's think step by step and output t... | 10,315 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "30. The figure below shows two circles with centres $\\mathrm{A}$ and $\\mathrm{B}$, and a line $\\mathrm{L}$ which is a tangent to the circles at $X$ and $Y$. Suppose that $X Y=40 \\mathrm{~cm}, A B=41 \\mathrm{~cm}$ and... | 14,280 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "Given the function $f(x) = ax^3 + bx + 8$, and $f(-2) = 10$, the value of $f(2)$ is ( )\nA: $-10$\nB: $-6$\nC: $6$\nD: $10$ Let's think step by step and output the final answer within \\boxed{{}}",
"role": "user"
},... | 524 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "A book has 500 pages, numbered 1, 2, 3, ..., how many times does the digit 1 appear in the page numbers? Let's think step by step and output the final answer within \\boxed{{}}",
"role": "user"
},
{
"content":... | 4,407 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "5. a square board consists of $2 n \\times 2 n$ squares. You want to mark $n$ of these squares so that no two marked squares are in the same or adjacent rows, and so that no two marked squares are in the same or adjacent ... | 19,619 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "Given the function $f\\left(x\\right)=4\\cos x\\sin (x+\\frac{π}{6})-1$. \n$(1)$ Find the value of $f(\\frac{π}{6})$. \n$(2)$ Find the maximum and minimum values of $f\\left(x\\right)$ on the interval $[-\\frac{π}{6}$,$... | 2,086 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "If point $P$ is on the ellipse $\\frac{x^2}{2} + y^2 = 1$, and $F_1$, $F_2$ are the two foci of the ellipse, and the angle $\\angle F_1PF_2 = 90^\\circ$, then the area of $\\triangle F_1PF_2$ is ( ).\n\nA: $\\frac{1}{2... | 12,634 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "G1.3 Let $n$ be the product 3659893456789325678 and 342973489379256 . Determine the number of digits of $n$. Let's think step by step and output the final answer within \\boxed{{}}",
"role": "user"
},
{
"conte... | 4,694 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "Given the function $f(x) = \\ln x + x$, if the tangent line to the function $f(x)$ at point $P(x_0, f(x_0))$ is parallel to the line $3x - y + 1 = 0$, find the value of $x_0$. Let's think step by step and output the final... | 19,793 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "4. Simplify $\\sqrt{13+4 \\sqrt{3}}+\\sqrt{13-4 \\sqrt{3}}$. Let's think step by step and output the final answer within \\boxed{{}}",
"role": "user"
},
{
"content": "To simplify the expression \\(\\sqrt{13 + ... | 7,986 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "5. The number of common terms (terms with the same value) in the arithmetic sequences $2,5,8, \\cdots, 2015$ and $4,9,14, \\cdots, 2014$ is $\\qquad$. Let's think step by step and output the final answer within \\boxed{{}... | 12,480 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "# 3. (3 points) Find the distance between the curves $y=e^{3 x+11}$ and $y=(\\ln x-11) / 3$. Let's think step by step and output the final answer within \\boxed{{}}",
"role": "user"
},
{
"content": "To find th... | 7,313 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "In the figure on the right, $O$ is the center of the circle, $OK$ and $OA$ are perpendicular to one another, $M$ is the midpoint of $OK$, $BN$ is parallel to $OK$, and $\\angle AMN=\\angle NMO$. Determine the measure of $... | 3,314 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "14. Given that $c$ is a positive integer not exceeding 2017, and there exist positive integers $a$ and $b$ such that\n$$\n2^{a} \\times 5^{b}=\\left(a^{3}+a^{2}+a+1\\right) c \\text {. }\n$$\n\nFind the maximum value of $... | 14,109 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "3. As shown in Figure 3, in isosceles $\\triangle ABC$, $AB = AC$, and $P$, $Q$ are points on sides $AC$, $AB$ respectively, such that $AP = PQ = QB = BC$. Then $\\angle PCQ =$ $\\qquad$ Let's think step by step and outpu... | 170 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "The chord length intercepted by the line $3x-4y=0$ on the circle $(x-1)^2+(y-2)^2=2$ is\nA: $4$\nB: $2\\sqrt{3}$\nC: $2\\sqrt{2}$\nD: $2$ Let's think step by step and output the final answer within \\boxed{{}}",
"role... | 19,088 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "2. In $\\square A B C D$, $\\angle B<90^{\\circ}, A B<B C$. From point $D$ draw tangents to the circumcircle $\\Gamma$ of $\\triangle A B C$, the points of tangency are $E$ and $F$. It is known that $\\angle E D A=\\angle... | 1,665 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "## Problem Statement\n\nCalculate the areas of figures bounded by lines given in polar coordinates.\n\n$$\nr=\\sqrt{3} \\cos \\phi, r=\\sin \\phi,\\left(0 \\leq \\phi \\leq \\frac{\\pi}{2}\\right)\n$$ Let's think step by ... | 19,612 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "2.12. Compute surface integrals of the first kind:\na) $\\iint_{\\sigma}|x| d S, \\quad \\sigma: x^{2}+y^{2}+z^{2}=1, \\quad z \\geqslant 0$\n\nb) $\\iint_{\\sigma}\\left(x^{2}+y^{2}\\right) d S, \\quad \\sigma: x^{2}+y^{... | 10,164 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "Find all positive integers $n$ such that $n!$ can be written as the product of two Fibonacci numbers. Let's think step by step and output the final answer within \\boxed{{}}",
"role": "user"
},
{
"content": "T... | 19,409 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "## Task 4 - 170824\n\nDieter tells his classmates:\n\n\"My brother Fritz is only half as old as I am. If you multiply the number of his years by itself, you get my father's age. My mother is three years younger than my fa... | 8,337 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "4. As shown in the figure, two national defense highways, each 40 m wide, intersect at an angle of $\\alpha$. Then, the area of the common part (the shaded area in the figure) of the road surface at the intersection is ( ... | 8,735 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "Define the function $f(x)$ on $\\mathbb{R}$ as follows: \n$$\nf(x) = \\begin{cases} \n\\log_{2}(8-x) & \\text{if } x \\leq 0 \\\\\nf(x-1) & \\text{if } x > 0\n\\end{cases}\n$$\nThen, $f(3) =$ （ ）\n\nA: 3 \nB: 2 \nC: $\... | 9,571 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "Given that $a$, $b$, and $c$ are the three side lengths of a triangle, simplify the expression: $|a+c-b|-|a+b+c|+|2b+c|$. Let's think step by step and output the final answer within \\boxed{{}}",
"role": "user"
},
... | 2,375 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "The four bottom corners of a cube are colored red, green, blue, and purple. How many ways are there to color the top four corners of the cube so that every face has four different colored corners? Prove that your answer ... | 14,781 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "7. Given $a_{n}=\\log _{n}(n+1)$, let $\\sum_{n=2}^{1023} \\frac{1}{\\log _{a_{n}} 100}=\\frac{q}{p}$, where $p, q$ are positive integers, and $(p, q)=1, p+q=$ $\\qquad$ Let's think step by step and output the final answe... | 19,813 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "182. Perform the operations: a) $(2+3 i)^{2} ;$ b) $(3-5 i)^{2}$; c) $(5+$ $+3 i)^{3}$ Let's think step by step and output the final answer within \\boxed{{}}",
"role": "user"
},
{
"content": "To solve the giv... | 14,009 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "Selected Optional 4-4: Coordinate Systems and Parametric Equations\n\nIn the polar coordinate system, the ray $l$: $θ = \\frac{π}{6}$ intersects with the circle $C$: $ρ = 2$ at point $A$. The equation of the ellipse $Γ$ i... | 5,180 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "Given that $0 < β < \\dfrac {π}{2} < α < π$, and $\\cos (α- \\dfrac {β}{2})=- \\dfrac { \\sqrt {2}}{2}$, $\\sin ( \\dfrac {α}{2}-β)= \\dfrac { \\sqrt {2}}{2}$, find the value of $\\cos (α+β)$. Let's think step by step and... | 18,899 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "16th Australian 1995 Problem B4 Find all real-valued functions f on the positive reals such that f(1) = 1/2 and f(xy) = f(x) f(3/y) + f(y) f(3/x). Let's think step by step and output the final answer within \\boxed{{}}",
... | 4,501 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "The position of the point $\\left(x,y\\right)$, which is the solution of the equation system $\\left\\{{\\begin{array}{l}{y=2x-5}\\\\{y=-x+1}\\end{array}}\\right.$, in the Cartesian coordinate system is ( ).\n\nA: First q... | 17,026 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "32. Find the remainder when the 2017-digit number $\\underbrace{7777}_{2017 \\text { 7s }}$ is divided by 30. Let's think step by step and output the final answer within \\boxed{{}}",
"role": "user"
},
{
"cont... | 19,306 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "7. As shown in the figure, in triangle $A B C$, $B E=E D=D C$, connect $A D$ and $A E$. Take a point $F$ on $A D$, and connect $C F$ and $B F$. If the areas of triangle $A C F$ and triangle $E F G$ are 28 and 12, respecti... | 19,495 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "7. Triangle $A B C$ is the right angled triangle with the vertex $C$ at the right angle. Let $P$ be the point of reflection of $C$ about $A B$. It is known that $P$ and two midpoints of two sides of $A B C$ lie on a line.... | 6,819 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "Given that the set A consists of the roots of the equation $(x-a)(x-a+1)=0$, and $2 \\in A$, then the value of the real number $a$ is \\_\\_\\_\\_\\_\\_. Let's think step by step and output the final answer within \\boxed... | 14,746 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "In the Cartesian coordinate system, with the origin as the pole and the positive half-axis of the x-axis as the polar axis, a polar coordinate system is established. In this polar coordinate system, it is known that circl... | 330 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "54. Continuing her research with fractions, Tanya added 4 to the numerator of a certain fraction and 10 to the denominator. After simplifying, to her surprise, she got the original fraction. She tried the same with anothe... | 1,280 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "1. Which single digit should be placed in all three of the boxes shown to give a correct calculation?\n$\\square$\n$\\square \\times$ $\\square$ $=176$\n\nA 3\nB 4\nC 5\nD 6\nE 8 Let's think step by step and output the fi... | 15,984 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "II OM - I - Task 10\n\nA circle is circumscribed around triangle $ ABC $. Given the radius $ R $ of this circle, calculate the radius of the circle passing through the centers of the three excircles (inscribed in the angl... | 2,353 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "Find the coordinates of the point that is symmetric to point A (1, 2) with respect to point P (3, 4). Let's think step by step and output the final answer within \\boxed{{}}",
"role": "user"
},
{
"content": "T... | 11,641 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "Given the function $f(x)=\\sin (\\omega x+\\varphi)+1$ ($\\omega > 0$, $0 \\leqslant \\varphi \\leqslant \\frac {\\pi}{2}$), the distance between two adjacent axes of symmetry of its graph is $\\pi$, and it reaches its ma... | 15,261 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "Given a complex number $z$ satisfying $z-|z|=-8+12i$, then the real part of $z$ is ( )\n\nA: $9$\n\nB: $7$\n\nC: $5$\n\nD: $3$ Let's think step by step and output the final answer within \\boxed{{}}",
"role": "user"
... | 10,743 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "5. The smallest natural number $n$ for which the equation $\\left[\\frac{10^{n}}{x}\\right]=2006$ has an integer solution $x$ lies in the interval $(\\quad)$\nA. $[1,6]$\nB. $[7,12]$\nC. $[13,18]$\nD. $[19,24]$ Let's thin... | 9,164 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "Four. (50 points) Let $n$ be a positive integer, and let the planar point set be\n$$\nS=\\{(x, y) \\mid x, y \\in\\{0,1, \\cdots, n\\}, x+y \\neq 0\\} \\text {. }\n$$\n\nQuestion: What is the minimum number of lines in th... | 7,682 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "If the abscissa of the points on the curve $x^{2}+y^{2}=1$ is changed to $\\frac{1}{2}$ times the original value, and the ordinate is changed to $2$ times the original value, the equation of the curve $C$ obtained is ( ).... | 1,413 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "Given that $\\log_{2}(\\log_{3}x)=1$, find the value of the real number $x$. Let's think step by step and output the final answer within \\boxed{{}}",
"role": "user"
},
{
"content": "Given the equation \\(\\lo... | 7,134 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "Example 3 Let $m$ be an integer, and $4<m<40$, and the equation $x^{2}-2(2 m-3) x+4 m^{2}-14 m+8=0$ has two integer roots. Find the value of $m$ and the roots of the equation. Let's think step by step and output the final... | 5,832 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "An arithmetic sequence with common difference $r$ is a sequence of numbers $s_{0}, s_{1}, s_{2}, \\ldots$ that satisfies for all indices $k, s_{k+1}=s_{k}+r$. Determine an expression for $s_{n}$ in terms of $s_{0}, r$, an... | 11,845 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "The radius $r=$ $\\qquad$ of the exsphere (i.e., the sphere that is tangent to one face and the extensions of the other three faces) of a regular tetrahedron with edge length 1. Let's think step by step and output the fin... | 17,745 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "If the equation $\\left(a-2\\right)x^{|a-1|}+3=9$ is a linear equation in $x$, then $a=\\_\\_\\_\\_\\_\\_$. Let's think step by step and output the final answer within \\boxed{{}}",
"role": "user"
},
{
"conten... | 1,370 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "Given proposition P: $x_1$ and $x_2$ are the two real roots of the equation $x^2 - mx - 2 = 0$, the inequality $a^2 - 5a - 3 \\geq |x_1 - x_2|$ holds for any real number $m \\in [-1,1]$; Proposition Q: There is only one r... | 11,580 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "If the solution to the system of equations $\\left\\{\\begin{array}{l}{2x+y=\\square}\\\\{x-2y=3}\\end{array}\\right.$ is $\\left\\{\\begin{array}{l}x=5\\\\ y=\\triangle\\end{array}\\right.$, then the numbers represented ... | 2,629 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "In the rectangular coordinate system, with the coordinate origin $O$ as the pole and the non-negative semi-axis of $x$ as the polar axis, the polar coordinates of point $M$ are $(2\\sqrt{2}, \\frac{\\pi}{4})$. The paramet... | 17,874 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "8. Given the function\n$$\nf(x)=x^{3}-\\log _{2}\\left(\\sqrt{x^{2}+1}-x\\right) \\text {. }\n$$\n\nFor any real numbers $a, b (a+b \\neq 0)$, the value of $\\frac{f(a)+f(b)}{a^{3}+b^{3}}$ is ( ).\n(A) always greater than... | 4,626 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "3. As shown in the figure, in the vertical coordinate plane $x o y$, line $l$ passes through the origin $O$, and $l$ is in the first and the I quadrant, with the angle between $l$ and the $x$-axis being $\\alpha\\left(0^{... | 12,070 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "Given the parabola $y=x^{2}+bx-2$ passes through the point $(1,3)$, intersects the $y$-axis at point $A$, and has its vertex at point $B$. Let $k$ be the abscissa of the intersection point of the parabola $y=x^{2}+bx-2$ w... | 8,666 |
[
{
"content": "You are a helpful assistant.",
"role": "system"
},
{
"content": "$23 \\cdot 34$ is the base area of a rectangular box, the areas of the side, and the front are all known, the product of these areas equals\n(A) the volume of the box.\n(B) the square root of the volume.\n(C) twice the vo... | 12,066 |
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