system stringclasses 1
value | conversations listlengths 2 2 |
|---|---|
You are a helpful assistant that can think step by step to solve problems. | [
{
"from": "user",
"value": "## Task B-1.3.\n\nA ship traveling along a river has covered $24 \\mathrm{~km}$ upstream and $28 \\mathrm{~km}$ downstream. For this journey, it took half an hour less than for traveling $30 \\mathrm{~km}$ upstream and $21 \\mathrm{~km}$ downstream, or half an hour more than for ... |
You are a helpful assistant that can think step by step to solve problems. | [
{
"from": "user",
"value": "3. (6 points) A construction company was building a tunnel. When $\\frac{1}{3}$ of the tunnel was completed at the original speed, they started using new equipment, which increased the construction speed by $20 \\%$ and reduced the working hours to $80 \\%$ of the original. As a ... |
You are a helpful assistant that can think step by step to solve problems. | [
{
"from": "user",
"value": "A right-angled triangle has side lengths that are integers. What could be the last digit of the area's measure, if the length of the hypotenuse is not divisible by 5?/think"
},
{
"from": "assistant",
"value": "<think>\nOkay, so I need to figure out the possible last d... |
You are a helpful assistant that can think step by step to solve problems. | [
{
"from": "user",
"value": "Let $a_n\\ (n\\geq 1)$ be the value for which $\\int_x^{2x} e^{-t^n}dt\\ (x\\geq 0)$ is maximal. Find $\\lim_{n\\to\\infty} \\ln a_n.$/think"
},
{
"from": "assistant",
"value": "<think>\nOkay, so I need to find the limit as n approaches infinity of the natural logarit... |
You are a helpful assistant that can think step by step to solve problems. | [
{
"from": "user",
"value": "11. Given that the internal angles $A, B, C$ of $\\triangle ABC$ have opposite sides $a, b, c$ respectively, and $\\sqrt{3} b \\cos \\frac{A+B}{2}=c \\sin B$.\n(1) Find the size of $\\angle C$;\n(2) If $a+b=\\sqrt{3} c$, find $\\sin A$./think"
},
{
"from": "assistant",
... |
You are a helpful assistant that can think step by step to solve problems. | [
{
"from": "user",
"value": "Task B-4.2. Let $n$ be the number obtained by writing 2013 zeros between every two digits of the number 14641. Determine all solutions of the equation $x^{4}=n$ in the set $\\mathbb{C}$./think"
},
{
"from": "assistant",
"value": "<think>\nOkay, let's see. I need to fi... |
You are a helpful assistant that can think step by step to solve problems. | [
{
"from": "user",
"value": "6. As shown in Figure 2, let $P$ be a point inside the equilateral $\\triangle ABC$ with side length 12. Draw perpendiculars from $P$ to the sides $BC$, $CA$, and $AB$, with the feet of the perpendiculars being $D$, $E$, and $F$ respectively. Given that $PD: PE: PF = 1: 2: 3$. Th... |
You are a helpful assistant that can think step by step to solve problems. | [
{
"from": "user",
"value": "8.59 For each pair of real numbers $x, y$, the function $f$ satisfies the functional equation\n$$\nf(x)+f(y)=f(x+y)-x y-1 .\n$$\n\nIf $f(1)=1$, then the number of integers $n$ (where $n \\neq 1$) that satisfy $f(n)=n$ is\n(A) 0.\n(B) 1.\n(C) 2.\n(D) 3.\n(E) infinitely many.\n(30t... |
You are a helpful assistant that can think step by step to solve problems. | [
{
"from": "user",
"value": "9.6. Find the minimum value of the expression $(\\sqrt{2(1+\\cos 2 x)}-\\sqrt{36-4 \\sqrt{5}} \\sin x+2) \\cdot(3+2 \\sqrt{10-\\sqrt{5}} \\cos y-\\cos 2 y) \\cdot$ If the answer is not an integer, round it to the nearest integer./think"
},
{
"from": "assistant",
"valu... |
You are a helpful assistant that can think step by step to solve problems. | [
{
"from": "user",
"value": "Given $0 \\leqslant x, y, z \\leqslant 1$, solve the equation:\n$$\\frac{x}{1+y+z x}+\\frac{y}{1+z+x y}+\\frac{z}{1+x+y z}=\\frac{3}{x+y+z} .$$/think"
},
{
"from": "assistant",
"value": "<think>\nAlright, so I have this equation to solve: \n\n\\[\n\\frac{x}{1 + y + z ... |
You are a helpful assistant that can think step by step to solve problems. | [
{
"from": "user",
"value": "## Problem Statement\n\nCalculate the definite integral:\n\n$$\n\\int_{0}^{3 / 2} \\frac{x^{2} \\cdot d x}{\\sqrt{9-x^{2}}}\n$$/think"
},
{
"from": "assistant",
"value": "<think>\nOkay, so I need to calculate the definite integral of x squared over the square root of ... |
You are a helpful assistant that can think step by step to solve problems. | [
{
"from": "user",
"value": "Example 6 The rules of a \"level-up game\" stipulate: On the $n$-th level, a die must be rolled $n$ times. If the sum of the points obtained from these $n$ rolls is greater than $2^{n}$, the level is considered passed. Questions:\n(1) What is the maximum number of levels a person... |
You are a helpful assistant that can think step by step to solve problems. | [
{
"from": "user",
"value": "2. (9th Canadian Mathematical Competition) $N$ is an integer, its representation in base $b$ is 777. Find the smallest positive integer $b$ such that $N$ is a fourth power of an integer in decimal notation./think"
},
{
"from": "assistant",
"value": "<think>\nOkay, let... |
You are a helpful assistant that can think step by step to solve problems. | [
{
"from": "user",
"value": "Problem 6. (8 points) In the plane, there is a non-closed, non-self-intersecting broken line consisting of 31 segments (adjacent segments do not lie on the same straight line). For each segment, the line defined by it is constructed. It is possible for some of the 31 constructed ... |
You are a helpful assistant that can think step by step to solve problems. | [
{
"from": "user",
"value": "7.1. Solve the equation $\\frac{n!}{2}=k!+l!$ in natural numbers, where $n!=1 \\cdot 2 \\cdot \\ldots n$. If there are no solutions, write 0; if there is one solution, write $n$; if there are multiple solutions, write the sum of the values of $n$ for all solutions. Recall that a ... |
You are a helpful assistant that can think step by step to solve problems. | [
{
"from": "user",
"value": "Example 1 (Question from the 13th \"Hope Cup\" Invitational Competition) The real roots of the equations $x^{5}+x+1=0$ and $x+\\sqrt[5]{x}+1=0$ are $\\alpha, \\beta$ respectively, then $\\alpha+\\beta$ equals ( ).\nA. -1\nB. $-\\frac{1}{2}$\nC. $\\frac{1}{2}$\nD. 1/think"
},
... |
You are a helpful assistant that can think step by step to solve problems. | [
{
"from": "user",
"value": "Example 5 Given that $x_{1}, x_{2}, \\cdots, x_{10}$ are all positive integers, and $x_{1}+x_{2}+\\cdots+x_{10}=2005$, find the maximum and minimum values of $x_{1}^{2}+x_{2}^{2}+\\cdots+x_{10}^{2}$./think"
},
{
"from": "assistant",
"value": "<think>\nOkay, let's see.... |
You are a helpful assistant that can think step by step to solve problems. | [
{
"from": "user",
"value": "Four, (50 points) In an $n \\times n$ grid, fill each cell with one of the numbers 1 to $n^{2}$. If no matter how you fill it, there must be two adjacent cells where the difference between the two numbers is at least 1011, find the minimum value of $n$.\n\n---\n\nThe translation ... |
You are a helpful assistant that can think step by step to solve problems. | [
{
"from": "user",
"value": "1. If the set\n$$\nA=\\{1,2, \\cdots, 10\\}, B=\\{1,2,3,4\\},\n$$\n$C$ is a subset of $A$, and $C \\cap B \\neq \\varnothing$, then the number of such subsets $C$ is $(\\quad)$.\n(A) 256\n(B) 959\n(C) 960\n(D) 961/think"
},
{
"from": "assistant",
"value": "<think>\nOk... |
You are a helpful assistant that can think step by step to solve problems. | [
{
"from": "user",
"value": "Augusto has a wire that is $10 \\mathrm{~m}$ long. He makes a cut at a point on the wire, obtaining two pieces. One piece has a length of $x$ and the other has a length of $10-x$ as shown in the figure below:\n\n![](https://cdn.mathpix.com/cropped/2024_05_01_d02c2755ad3373bde08ag... |
You are a helpful assistant that can think step by step to solve problems. | [
{
"from": "user",
"value": "29. Choose any three numbers from $a, b, c, d, e$ and find their sum, exactly obtaining the ten different numbers $7,11,13,14,19,21,22,25,26,28$. Then $a+b+c+d+e=$ ( i). MATHEMATICE YoUTH CLUE A. 25\nB. 31\nC. 37\nD. 43/think"
},
{
"from": "assistant",
"value": "<thin... |
You are a helpful assistant that can think step by step to solve problems. | [
{
"from": "user",
"value": "12.180. A side of the triangle is equal to 15, the sum of the other two sides is 27. Find the cosine of the angle opposite the given side, if the radius of the inscribed circle in the triangle is 4./think"
},
{
"from": "assistant",
"value": "<think>\nOkay, let's see. ... |
You are a helpful assistant that can think step by step to solve problems. | [
{
"from": "user",
"value": "3. Let $AB$ be a chord of the unit circle $\\odot O$. If the area of $\\odot O$ is exactly equal to the area of the square with side $AB$, then $\\angle AOB=$ $\\qquad$ (to 0.001 degree)./think"
},
{
"from": "assistant",
"value": "<think>\nOkay, so I need to find the ... |
You are a helpful assistant that can think step by step to solve problems. | [
{
"from": "user",
"value": "83. Fill in the following squares with $0, 1, 2, 3, 4, 5, 6, 7, 8, 9$ respectively, so that the sum of the two five-digit numbers is 99999. Then the number of different addition equations is $\\qquad$. $(a+b$ and $b+a$ are considered the same equation)/think"
},
{
"from":... |
You are a helpful assistant that can think step by step to solve problems. | [
{
"from": "user",
"value": "On a board, the numbers from 1 to 2009 are written. A couple of them are erased and instead of them, on the board is written the remainder of the sum of the erased numbers divided by 13. After a couple of repetition of this erasing, only 3 numbers are left, of which two are 9 and... |
You are a helpful assistant that can think step by step to solve problems. | [
{
"from": "user",
"value": "4. The number of real solutions to the equation $\\left|x^{2}-3 x+2\\right|+\\left|x^{2}+2 x-3\\right|=11$ is ( ).\n(A) 0\n(B) 1\n(C) 2\n(D) 4/think"
},
{
"from": "assistant",
"value": "<think>\nOkay, so I need to find the number of real solutions to the equation |x² ... |
You are a helpful assistant that can think step by step to solve problems. | [
{
"from": "user",
"value": "## Problem Statement\n\nCalculate the limit of the numerical sequence:\n\n$\\lim _{n \\rightarrow \\infty} \\frac{(n+1)^{4}-(n-1)^{4}}{(n+1)^{3}+(n-1)^{3}}$/think"
},
{
"from": "assistant",
"value": "<think>\nOkay, so I need to find the limit as n approaches infinity ... |
You are a helpful assistant that can think step by step to solve problems. | [
{
"from": "user",
"value": "Condition of the problem\n\nFind the derivative.\n\n$$\ny=\\frac{1}{24}\\left(x^{2}+8\\right) \\sqrt{x^{2}-4}+\\frac{x^{2}}{16} \\arcsin \\frac{2}{x}, x>0\n$$/think"
},
{
"from": "assistant",
"value": "<think>\nOkay, so I need to find the derivative of this function y... |
You are a helpful assistant that can think step by step to solve problems. | [
{
"from": "user",
"value": "\\section*{Problem 5 - 071225}\n\nAll ordered pairs of real numbers \\((x, y)\\) are to be determined for which the system of equations\n\n\\[\n\\begin{aligned}\nx \\cdot\\left(a x^{2}+b y^{2}-a\\right) & =0 \\\\\ny \\cdot\\left(a x^{2}+b y^{2}-b\\right) & =0\n\\end{aligned}\n\\]... |
You are a helpful assistant that can think step by step to solve problems. | [
{
"from": "user",
"value": "6. Let $[x]$ denote the greatest integer not exceeding the real number $x$,\n$$\n\\begin{array}{c}\nS=\\left[\\frac{1}{1}\\right]+\\left[\\frac{2}{1}\\right]+\\left[\\frac{1}{2}\\right]+\\left[\\frac{2}{2}\\right]+\\left[\\frac{3}{2}\\right]+ \\\\\n{\\left[\\frac{4}{2}\\right]+\\... |
You are a helpful assistant that can think step by step to solve problems. | [
{
"from": "user",
"value": "40. The sum of three consecutive odd numbers is equal to the fourth power of a single-digit number. Find all such triples of numbers./think"
},
{
"from": "assistant",
"value": "<think>\nOkay, let's see. The problem says that the sum of three consecutive odd numbers is... |
You are a helpful assistant that can think step by step to solve problems. | [
{
"from": "user",
"value": "19. Given $m \\in\\{11,13,15,17,19\\}$, $n \\in\\{1999,2000, \\cdots, 2018\\}$.\nThen the probability that the unit digit of $m^{n}$ is 1 is ( ).\n(A) $\\frac{1}{5}$\n(B) $\\frac{1}{4}$\n(C) $\\frac{3}{10}$\n(D) $\\frac{7}{20}$\n(E) $\\frac{2}{5}$/think"
},
{
"from": "ass... |
You are a helpful assistant that can think step by step to solve problems. | [
{
"from": "user",
"value": "Consider a regular hexagon with an incircle. What is the ratio of the area inside the incircle to the area of the hexagon?\n/think"
},
{
"from": "assistant",
"value": "<think>\nOkay, so I need to find the ratio of the area inside the incircle of a regular hexagon to t... |
You are a helpful assistant that can think step by step to solve problems. | [
{
"from": "user",
"value": "Let $ABC$ be a triangle with centroid $G$. Determine, with proof, the position of the point $P$ in the plane of $ABC$ such that $AP{\\cdot}AG + BP{\\cdot}BG + CP{\\cdot}CG$ is a minimum, and express this minimum value in terms of the side lengths of $ABC$./think"
},
{
"fr... |
You are a helpful assistant that can think step by step to solve problems. | [
{
"from": "user",
"value": "1. Four points $A, B, C, D$ in space are pairwise 1 unit apart, and points $P, Q$ move on line segments $AB, CD$ respectively. The minimum distance between point $P$ and $Q$ is/think"
},
{
"from": "assistant",
"value": "<think>\nOkay, so I need to figure out the minim... |
You are a helpful assistant that can think step by step to solve problems. | [
{
"from": "user",
"value": "The function $f$ maps the set of positive integers into itself, and satisfies the equation\n\n$$\nf(f(n))+f(n)=2 n+6\n$$\n\nWhat could this function be?/think"
},
{
"from": "assistant",
"value": "<think>\nOkay, so I need to figure out what the function f could be. The... |
You are a helpful assistant that can think step by step to solve problems. | [
{
"from": "user",
"value": "30. Find the remainder when the 2018-digit number $\\underbrace{\\overline{55 \\cdots}}_{2018 \\text { 555 }}$ is divided by 13./think"
},
{
"from": "assistant",
"value": "<think>\nOkay, so I need to find the remainder when this huge 2018-digit number made up entirely... |
You are a helpful assistant that can think step by step to solve problems. | [
{
"from": "user",
"value": "1. A line $l$ intersects a hyperbola $c$, then the maximum number of intersection points is ( ).\nA. 1\nB. 2\nC. 3\nD. 4/think"
},
{
"from": "assistant",
"value": "<think>\nOkay, so I need to figure out the maximum number of points where a line can intersect a hyperbo... |
You are a helpful assistant that can think step by step to solve problems. | [
{
"from": "user",
"value": "4. As shown in Figure 1, in the right triangular prism $A B C-A_{1} B_{1} C_{1}$, $A A_{1}=A B=A C$, and $M$ and $Q$ are the midpoints of $C C_{1}$ and $B C$ respectively. If for any point $P$ on the line segment $A_{1} B_{1}$, $P Q \\perp A M$, then $\\angle B A C$ equals ( ).\n... |
You are a helpful assistant that can think step by step to solve problems. | [
{
"from": "user",
"value": "7.242. $\\left(16 \\cdot 5^{2 x-1}-2 \\cdot 5^{x-1}-0.048\\right) \\lg \\left(x^{3}+2 x+1\\right)=0$./think"
},
{
"from": "assistant",
"value": "<think>\nOkay, let's see. I need to solve the equation: \n\n\\[16 \\cdot 5^{2x-1} - 2 \\cdot 5^{x-1} - 0.048 = 0\\]\nmultip... |
You are a helpful assistant that can think step by step to solve problems. | [
{
"from": "user",
"value": "Example 1 In $\\triangle ABC$, it is known that $x \\sin A + y \\sin B + z \\sin C = 0$. Find the value of $(y + z \\cos A)(z + x \\cos B)(x + y \\cos C) + (y \\cos A + z)(z \\cos B + x)(x \\cos C + y)$./think"
},
{
"from": "assistant",
"value": "<think>\nAlright, so ... |
You are a helpful assistant that can think step by step to solve problems. | [
{
"from": "user",
"value": "11.005. The plane angle at the vertex of a regular triangular pyramid is $90^{\\circ}$. Find the ratio of the lateral surface area of the pyramid to the area of its base./think"
},
{
"from": "assistant",
"value": "<think>\nOkay, let's see. I need to find the ratio of ... |
You are a helpful assistant that can think step by step to solve problems. | [
{
"from": "user",
"value": "A father wants to divide his property among his children: first, he gives 1000 yuan and one-tenth of the remaining property to the eldest child, then 2000 yuan and one-tenth of the remaining property to the second child, then 3000 yuan and one-tenth of the remaining property to t... |
You are a helpful assistant that can think step by step to solve problems. | [
{
"from": "user",
"value": "9. Given is a regular tetrahedron of volume 1 . We obtain a second regular tetrahedron by reflecting the given one through its center. What is the volume of their intersection?/think"
},
{
"from": "assistant",
"value": "<think>\nOkay, let's see. I have a regular tetra... |
You are a helpful assistant that can think step by step to solve problems. | [
{
"from": "user",
"value": "## Zadatak B-1.2.\n\nNa slici su prikazani pravilni peterokut. $A B C D E$ i kvadrat $A B F G$. Odredite mjeru kuta $F A D$.\n\n\n\n/think"
},
{
... |
You are a helpful assistant that can think step by step to solve problems. | [
{
"from": "user",
"value": "The equation of line $\\ell_1$ is $24x-7y = 319$ and the equation of line $\\ell_2$ is $12x-5y = 125$. Let $a$ be the number of positive integer values $n$ less than $2023$ such that for both $\\ell_1$ and $\\ell_2$ there exists a lattice point on that line that is a distance of ... |
You are a helpful assistant that can think step by step to solve problems. | [
{
"from": "user",
"value": "Let's determine $m$ such that the expression\n\n$$\nm x^{2}+(m-1) x+m-1\n$$\n\nis negative for all values of $x$.\n\n---\n\nDetermine $m$ so that the expression\n\n$$\nm x^{2}+(m-1) x+m-1\n$$\n\nis negative for all values of $x$./think"
},
{
"from": "assistant",
"valu... |
You are a helpful assistant that can think step by step to solve problems. | [
{
"from": "user",
"value": "2. How many integers $b$ exist such that the equation $x^{2}+b x-9600=0$ has an integer solution that is a multiple of both 10 and 12? Specify the largest possible $b$./think"
},
{
"from": "assistant",
"value": "<think>\nOkay, let's see. The problem is asking how many... |
You are a helpful assistant that can think step by step to solve problems. | [
{
"from": "user",
"value": "4.206 There are two forces $f_{1}$ and $f_{2}$ acting on the origin $O$ of the coordinate axis,\n$$\\begin{array}{l}\n\\vec{f}_{1}=\\overrightarrow{O A}=\\sqrt{2}\\left(\\cos 45^{\\circ}+i \\sin 45^{\\circ}\\right) \\\\\n\\vec{f}_{2}=\\overrightarrow{O B}=2\\left[\\cos \\left(-30... |
You are a helpful assistant that can think step by step to solve problems. | [
{
"from": "user",
"value": "6. Given that $\\overrightarrow{O A} \\perp \\overrightarrow{O B}$, and $|\\overrightarrow{O A}|=|\\overrightarrow{O B}|=24$. If $t \\in[0,1]$, then\n$$\n|t \\overrightarrow{A B}-\\overrightarrow{A O}|+\\left|\\frac{5}{12} \\overrightarrow{B O}-(1-t) \\overrightarrow{B A}\\right|... |
You are a helpful assistant that can think step by step to solve problems. | [
{
"from": "user",
"value": "1B. If for the non-zero real numbers $a, b$ and $c$ the equalities $a^{2}+a=b^{2}, b^{2}+b=c^{2}$ and $c^{2}+c=a^{2}$ hold, determine the value of the expression $(a-b)(b-c)(c-a)$./think"
},
{
"from": "assistant",
"value": "<think>\nOkay, so I have this problem here w... |
You are a helpful assistant that can think step by step to solve problems. | [
{
"from": "user",
"value": "2. As shown in Figure 1, the side length of rhombus $A B C D$ is $a$, and $O$ is a point on the diagonal $A C$, with $O A=a, O B=$ $O C=O D=1$. Then $a$ equals ( ).\n(A) $\\frac{\\sqrt{5}+1}{2}$\n(B) $\\frac{\\sqrt{5}-1}{2}$\n(C) 1\n(D) 2/think"
},
{
"from": "assistant",
... |
You are a helpful assistant that can think step by step to solve problems. | [
{
"from": "user",
"value": "V-2 If one side of the rectangle is reduced by $3 \\mathrm{~cm}$, and the other side is reduced by $2 \\mathrm{~cm}$, we get a square whose area is $21 \\mathrm{~cm}^{2}$ less than the area of the rectangle. Calculate the dimensions of the rectangle.\n\n![](https://cdn.mathpix.co... |
You are a helpful assistant that can think step by step to solve problems. | [
{
"from": "user",
"value": "1. Given $a, b>0, a \\neq 1$, and $a^{b}=\\log _{a} b$, then the value of $a^{a^{b}}-\\log _{a} \\log _{a} b^{a}$ is/think"
},
{
"from": "assistant",
"value": "<think>\nOkay, so I have this problem here: Given that a and b are both positive numbers, a isn't equal to 1... |
You are a helpful assistant that can think step by step to solve problems. | [
{
"from": "user",
"value": "4. Let the sequence $\\left\\{a_{n}\\right\\}$ satisfy $a_{1}=0, a_{2}=1$, and for all $n \\geqslant 3, a_{n}$ is the smallest positive integer greater than $a_{n-1}$ such that there is no subsequence of $a_{1}, a_{2}, \\cdots, a_{n}$ that forms an arithmetic sequence. Find $a_{2... |
You are a helpful assistant that can think step by step to solve problems. | [
{
"from": "user",
"value": "In a stairwell, there are 10 mailboxes. One distributor drops a flyer into 5 mailboxes. Later, another distributor also drops a flyer into 5 mailboxes. What is the probability that this way, at least 8 mailboxes will contain a flyer?/think"
},
{
"from": "assistant",
"... |
You are a helpful assistant that can think step by step to solve problems. | [
{
"from": "user",
"value": "18. (3 points) Li Shuang rides a bike at a speed of 320 meters per minute from location $A$ to location $B$. On the way, due to a bicycle malfunction, he pushes the bike and walks for 5 minutes to a place 1800 meters from $B$ to repair the bike. After 15 minutes, he continues tow... |
You are a helpful assistant that can think step by step to solve problems. | [
{
"from": "user",
"value": "53. How many four-digit numbers contain at least one even digit?/think"
},
{
"from": "assistant",
"value": "<think>\nOkay, so I need to figure out how many four-digit numbers contain at least one even digit. Hmm, let's see. First, let me make sure I understand the pro... |
You are a helpful assistant that can think step by step to solve problems. | [
{
"from": "user",
"value": "1. The range of the function $f(x)=\\sin x+\\cos x+\\tan x+$ $\\arcsin x+\\arccos x+\\arctan x$ is $\\qquad$ ./think"
},
{
"from": "assistant",
"value": "<think>\nOkay, so I need to find the range of the function f(x) = sin x + cos x + tan x + arcsin x + arccos x + ar... |
You are a helpful assistant that can think step by step to solve problems. | [
{
"from": "user",
"value": "10.319. The diagonals of an isosceles trapezoid are perpendicular to each other, and its area is $a^{2}$. Determine the height of the trapezoid./think"
},
{
"from": "assistant",
"value": "<think>\nOkay, let's try to figure out this problem. So, we have an isosceles tr... |
You are a helpful assistant that can think step by step to solve problems. | [
{
"from": "user",
"value": "8. Find the last four digits of $7^{7^{-7}}$ (100 sevens).\n\nTranslate the above text into English, please keep the original text's line breaks and format, and output the translation result directly./think"
},
{
"from": "assistant",
"value": "<think>\nOkay, let's see... |
You are a helpful assistant that can think step by step to solve problems. | [
{
"from": "user",
"value": "1. Let $S=\\{1,2, \\cdots, n\\}, A$ be an arithmetic sequence with at least two terms, a positive common difference, all of whose terms are in $S$, and such that adding any other element of $S$ does not form an arithmetic sequence with the same common difference as $A$. Find the ... |
You are a helpful assistant that can think step by step to solve problems. | [
{
"from": "user",
"value": "9.171. $0.6^{\\lg ^{2}(-x)+3} \\leq\\left(\\frac{5}{3}\\right)^{2 \\lg x^{2}}$./think"
},
{
"from": "assistant",
"value": "<think>\nAlright, so I need to solve this inequality: \\(0.6^{\\lg^{2}(-x) + 3} \\leq \\left(\\frac{5}{3}\\right)^{2 \\lg x^{2}}\\). Hmm, okay. L... |
You are a helpful assistant that can think step by step to solve problems. | [
{
"from": "user",
"value": "2. Find all integer solutions of the inequality\n\n$$\nx^{2} y^{2}+y^{2} z^{2}+x^{2}+z^{2}-38(x y+z)-40(y z+x)+4 x y z+761 \\leq 0\n$$/think"
},
{
"from": "assistant",
"value": "<think>\nOkay, so I have this inequality to solve for integer solutions:\n\nx²y² + y²z² + ... |
You are a helpful assistant that can think step by step to solve problems. | [
{
"from": "user",
"value": "Solve the triangle whose area $t=357.18 \\mathrm{~cm}^{2}$, where the ratio of the sides is $a: b: c=4: 5: 6$.\n\n/think"
},
{
"from": "assistant",
"value": "<think>\nOkay, let me try to solve this triangle problem. So, we know the area is 357.18 cm², and the sides ar... |
You are a helpful assistant that can think step by step to solve problems. | [
{
"from": "user",
"value": "Find all triples $(m,p,q)$ where $ m $ is a positive integer and $ p , q $ are primes.\n\\[ 2^m p^2 + 1 = q^5 \\]/think"
},
{
"from": "assistant",
"value": "<think>\nAlright, so I need to find all triples (m, p, q) where m is a positive integer and p, q are primes. Th... |
You are a helpful assistant that can think step by step to solve problems. | [
{
"from": "user",
"value": "15. As shown in the figure, the area of square $\\mathrm{ABCD}$ is 196 square centimeters, and it contains two partially overlapping smaller squares. The larger of the two smaller squares has an area that is 4 times the area of the smaller one, and the overlapping area of the two... |
You are a helpful assistant that can think step by step to solve problems. | [
{
"from": "user",
"value": "25. Anna, Bridgit and Carol run in a $100 \\mathrm{~m}$ race. When Anna finishes, Bridgit is $16 \\mathrm{~m}$ behind her and when Bridgit finishes, Carol is $25 \\mathrm{~m}$ behind her. The girls run at constant speeds throughout the race. How far behind was Carol when Anna fin... |
You are a helpful assistant that can think step by step to solve problems. | [
{
"from": "user",
"value": "Given the complex number $z$ has a modulus of 1. Find\n$$\nu=\\frac{(z+4)^{2}-(\\bar{z}+4)^{2}}{4 i} \\text {. }\n$$\n\nthe maximum value./think"
},
{
"from": "assistant",
"value": "<think>\nOkay, let's try to figure out how to find the maximum value of this expressio... |
You are a helpful assistant that can think step by step to solve problems. | [
{
"from": "user",
"value": "## Problem 1\n\nPerform the calculations:\n\na) $7 \\cdot 147 - 7 \\cdot 47$ (1p)\n\nb) $(2+4+6+8+\\cdots+50)-(1+3+5+7+\\cdots+49)$ (2p)\n\nc) $10 \\cdot 9^{2} : 3^{2} - 3^{4} \\quad(2 \\text{p})$\n\nd) $(\\overline{a b} + \\overline{b c} + \\overline{c a}) : (a + b + c) \\quad$ ... |
You are a helpful assistant that can think step by step to solve problems. | [
{
"from": "user",
"value": "9.27 In the metro train at the initial stop, 100 passengers entered. How many ways are there to distribute the exit of all these passengers at the next 16 stops of the train?/think"
},
{
"from": "assistant",
"value": "<think>\nOkay, so the problem is: At the initial s... |
You are a helpful assistant that can think step by step to solve problems. | [
{
"from": "user",
"value": "2. In the complex plane, there are 7 points corresponding to the 7 roots of the equation $x^{7}=$ $-1+\\sqrt{3} i$. Among the four quadrants where these 7 points are located, only 1 point is in ( ).\n(A) the I quadrant\n(B) the II quadrant\n(C) the III quadrant\n(D) the IV quadra... |
You are a helpful assistant that can think step by step to solve problems. | [
{
"from": "user",
"value": "## Problem 4\n\nGiven the numbers $1,2,3, \\ldots, 1000$. Find the largest number $m$ with the property that by removing any $m$ numbers from these 1000 numbers, among the $1000-m$ remaining numbers, there exist two such that one divides the other.\n\nSelected problems by Prof. C... |
You are a helpful assistant that can think step by step to solve problems. | [
{
"from": "user",
"value": "Problem 5.3. In five of the nine circles in the picture, the numbers 1, 2, 3, 4, 5 are written. Replace the digits $6, 7, 8, 9$ in the remaining circles $A, B, C, D$ so that the sums of the four numbers along each of the three sides of the triangle are the same.\n\n![](https://cd... |
You are a helpful assistant that can think step by step to solve problems. | [
{
"from": "user",
"value": "The function $f: \\mathbb{R}\\rightarrow \\mathbb{R}$ is such that $f(x+1)=2f(x)$ for $\\forall$ $x\\in \\mathbb{R}$ and $f(x)=x(x-1)$ for $\\forall$ $x\\in (0,1]$. Find the greatest real number $m$, for which the inequality $f(x)\\geq -\\frac{8}{9}$ is true for \n$\\forall$ $x\\... |
You are a helpful assistant that can think step by step to solve problems. | [
{
"from": "user",
"value": "18.3.19 $\\star \\star$ Find all positive integer triples $(a, b, c)$ that satisfy $a^{2}+b^{2}+c^{2}=2005$ and $a \\leqslant b \\leqslant c$./think"
},
{
"from": "assistant",
"value": "<think>\nAlright, so I need to find all positive integer triples (a, b, c) such th... |
You are a helpful assistant that can think step by step to solve problems. | [
{
"from": "user",
"value": "5. Through the vertex $M$ of some angle, a circle is drawn, intersecting the sides of the angle at points $N$ and $K$, and the bisector of this angle at point $L$. Find the sum of the lengths of segments $M N$ and $M K$, if the area of $M N L K$ is 49, and the angle $L M N$ is $3... |
You are a helpful assistant that can think step by step to solve problems. | [
{
"from": "user",
"value": "4. Let $A$ and $B$ be $n$-digit numbers, where $n$ is odd, which give the same remainder $r \\neq 0$ when divided by $k$. Find at least one number $k$, which does not depend on $n$, such that the number $C$, obtained by appending the digits of $A$ and $B$, is divisible by $k$./th... |
You are a helpful assistant that can think step by step to solve problems. | [
{
"from": "user",
"value": "7. Given that $z$ is a complex number, and $|z|=1$. When $\\mid 1+z+$ $3 z^{2}+z^{3}+z^{4}$ | takes the minimum value, the complex number $z=$ $\\qquad$ or . $\\qquad$/think"
},
{
"from": "assistant",
"value": "<think>\nAlright, so I have this problem here: Given that... |
You are a helpful assistant that can think step by step to solve problems. | [
{
"from": "user",
"value": "Example: Given the radii of the upper and lower bases of a frustum are 3 and 6, respectively, and the height is $3 \\sqrt{3}$, the radii $O A$ and $O B$ of the lower base are perpendicular, and $C$ is a point on the generatrix $B B^{\\prime}$ such that $B^{\\prime} C: C B$ $=1: 2... |
You are a helpful assistant that can think step by step to solve problems. | [
{
"from": "user",
"value": "5. Given positive real numbers $a$ and $b$ satisfy $a+b=1$, then $M=$ $\\sqrt{1+a^{2}}+\\sqrt{1+2 b}$ the integer part is/think"
},
{
"from": "assistant",
"value": "<think>\nOkay, so I need to find the integer part of M, which is defined as the sum of two square roots... |
You are a helpful assistant that can think step by step to solve problems. | [
{
"from": "user",
"value": "1. Arrange the consecutive natural numbers from 1 to 99 in sequence to form a large number:\n1234567891011…979899,\nBy extracting four consecutive digits, you can get a four-digit number, such as 5678, 1011, etc. Among the four-digit numbers obtained by this method, the largest i... |
You are a helpful assistant that can think step by step to solve problems. | [
{
"from": "user",
"value": "1. Simplify\n$$\n\\sqrt{1+2 \\sin \\alpha \\cdot \\cos \\alpha}+\\sqrt{1-2 \\sin \\alpha \\cdot \\cos \\alpha}\n$$\n$\\left(0^{\\circ}<\\alpha \\leqslant 90^{\\circ}\\right)$ The result is $\\qquad$ ./think"
},
{
"from": "assistant",
"value": "<think>\nOkay, so I need... |
You are a helpful assistant that can think step by step to solve problems. | [
{
"from": "user",
"value": "8. (15 points) A thin ray of light falls on a thin converging lens at a distance of $x=10$ cm from its optical center. The angle between the incident ray and the plane of the lens $\\alpha=45^{\\circ}$, and the angle between the refracted ray and the plane of the lens $\\beta=30^... |
You are a helpful assistant that can think step by step to solve problems. | [
{
"from": "user",
"value": "8. If $n$ is a natural number less than 50, find all values of $n$ such that the values of the algebraic expressions $4n+5$ and $7n+6$ have a common divisor greater than 1./think"
},
{
"from": "assistant",
"value": "<think>\nOkay, so I need to find all natural numbers... |
You are a helpful assistant that can think step by step to solve problems. | [
{
"from": "user",
"value": "6. Let set $A=\\{1,2,3,4,5,6\\}$, and a one-to-one mapping $f: A \\rightarrow A$ satisfies that for any $x \\in A$, $f(f(f(x)))$ $=x$. Then the number of mappings $f$ that satisfy the above condition is ( ).\n(A) 40\n(B) 41\n(C) 80\n(D) 81/think"
},
{
"from": "assistant",... |
You are a helpful assistant that can think step by step to solve problems. | [
{
"from": "user",
"value": "9. (16 points) Let the real number $t \\in [0, \\pi]$. If the equation $\\cos (x+t)=1-\\cos x$ has a solution for $x$, find the range of values for $t$.\n\n/think"
},
{
"from": "assistant",
"value": "<think>\nOkay, so I need to find the range of values for \\( t \\) i... |
You are a helpful assistant that can think step by step to solve problems. | [
{
"from": "user",
"value": "5. 5.1. Find the largest natural number in which all digits are different and any two adjacent digits differ by 6 or 7./think"
},
{
"from": "assistant",
"value": "<think>\nOkay, so I need to find the largest natural number where all the digits are different, and any t... |
You are a helpful assistant that can think step by step to solve problems. | [
{
"from": "user",
"value": "5. For a convex $n$-sided polygon, if circles are constructed with each side as the diameter, the convex $n$-sided polygon must be covered by these $n$ circles. Then the maximum value of $n$ is: ( ).\n(A) 3 .\n(B) 4 .\n(C) 5 .\n(D) Greater than 5 ./think"
},
{
"from": "as... |
You are a helpful assistant that can think step by step to solve problems. | [
{
"from": "user",
"value": "15. Lucia and Carla are playing bingo against each other (with no other opponents). Each has a card with 15 numbers; the two cards have exactly 4 numbers in common. What is the probability that, after 89 numbers have been drawn, neither of them has yet completed a bingo?\n(A) $2 ... |
You are a helpful assistant that can think step by step to solve problems. | [
{
"from": "user",
"value": "# Problem 6. (3 points)\n\nThree runners are moving along a circular track at constant equal speeds. When two runners meet, they instantly turn around and start running in opposite directions.\n\nAt some point, the first runner meets the second. After 15 minutes, the second runne... |
You are a helpful assistant that can think step by step to solve problems. | [
{
"from": "user",
"value": "$17 \\cdot 27$ As shown in the figure, in $\\triangle A B C$, $\\angle A=42^{\\circ}$. If the trisectors of $\\angle B$ and $\\angle C$ intersect at $D, E$. Then the degree measure of $\\angle B D C$ is\n(A) $67^{\\circ}$.\n(B) $84^{\\circ}$.\n(C) $88^{\\circ}$.\n(D) $110^{\\circ... |
You are a helpful assistant that can think step by step to solve problems. | [
{
"from": "user",
"value": "Example 5. Expand the function\n\n$$\nf(z)=\\frac{z}{z^{2}-2 z-3}\n$$\n\ninto a Taylor series in the neighborhood of the point $z_{0}=0$ using expansion (12), and find the radius of convergence of the series./think"
},
{
"from": "assistant",
"value": "<think>\nOkay, s... |
You are a helpful assistant that can think step by step to solve problems. | [
{
"from": "user",
"value": "7. The set \n$\\left\\{x \\left\\lvert\\,-1 \\leqslant \\log _{\\frac{1}{x}} 10<-\\frac{1}{2}\\right., x \\in \\mathbf{N}\\right\\}$ has $\\qquad$ proper subsets./think"
},
{
"from": "assistant",
"value": "<think>\nOkay, let me try to figure out this problem. So, we n... |
You are a helpful assistant that can think step by step to solve problems. | [
{
"from": "user",
"value": "2. Given $0<x<1$. Simplify\n$$\n=\\quad \\sqrt{\\left(x-\\frac{1}{x}\\right)^{2}+4}-\\sqrt{\\left(x+\\frac{1}{x}\\right)^{2}-4}\n$$/think"
},
{
"from": "assistant",
"value": "<think>\nOkay, let's see. I need to simplify the expression √[(x - 1/x)² + 4] - √[(x + 1/x)² ... |
You are a helpful assistant that can think step by step to solve problems. | [
{
"from": "user",
"value": "7. Given the equality\n\n$$\n\\left(a^{m_{1}}-1\\right) \\ldots\\left(a^{m_{n}}-1\\right)=\\left(a^{k_{1}}+1\\right) \\ldots\\left(a^{k_{l}}+1\\right)\n$$\n\nwhere $a, n, l$ and all exponents are natural numbers, and $a>1$. Find all possible values of the number $a$./think"
},
... |
You are a helpful assistant that can think step by step to solve problems. | [
{
"from": "user",
"value": "Find two such common fractions - one with a denominator of 8, the other with a denominator of 13 - so that they are not equal, but the difference between the larger and the smaller of them is as small as possible.\n\n#/think"
},
{
"from": "assistant",
"value": "<think... |
You are a helpful assistant that can think step by step to solve problems. | [
{
"from": "user",
"value": "1. As shown in Figure 1, in the Cartesian coordinate system, the graph of the quadratic function $y=a x^{2}+m c(a \\neq$ $0)$ passes through three vertices $A$, $B$, and $C$ of the square $A B O C$, and $a c=-2$. Then the value of $m$ is ( ).\n(A) 1\n(B) -1\n(C) 2\n(D) -2/think"
... |
You are a helpful assistant that can think step by step to solve problems. | [
{
"from": "user",
"value": "Let $a_1,a_2,a_3,a_4,a_5$ be distinct real numbers. Consider all sums of the form $a_i + a_j$ where $i,j \\in \\{1,2,3,4,5\\}$ and $i \\neq j$. Let $m$ be the number of distinct numbers among these sums. What is the smallest possible value of $m$?/think"
},
{
"from": "ass... |
You are a helpful assistant that can think step by step to solve problems. | [
{
"from": "user",
"value": "3. There are ten small balls of the same size, five of which are red and five are white. Now, these ten balls are arranged in a row arbitrarily, and numbered from left to right as $1,2, \\cdots, 10$. Then the number of arrangements where the sum of the numbers of the red balls is... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.