conversations listlengths 2 2 |
|---|
[
{
"content": "Given $$\\log_{a} \\frac {3}{5} < 1$$, the range of values for $a$ is \\_\\_\\_\\_\\_\\_. Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
},
{
"content": "Given the inequality $\\log_{a} \\frac{3}{5} < 1$, we need to determine the values of $a$ th... |
[
{
"content": "Given $\\sin \\left(\\alpha +\\frac{\\pi }{6}\\right)=\\frac{1}{3}$, then the value of $\\cos \\left(\\frac{\\pi }{3}-\\alpha \\right)$ is _______. Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
},
{
"content": "We need to find the value of $\\co... |
[
{
"content": "Distribute 7 doctors, including Doctors A and B, into three medical teams with one team having 3 members and the other two teams having 2 members each. In how many ways can this be done so that Doctors A and B are not in the same team? Let's think step by step and output the final answer within \\... |
[
{
"content": "A student was determining the interval $D$ where the root $x_0$ of the equation $x^3 = -x + 1$ lies. They set the function $f(x) = x^3 + x - 1$, calculated $f(-1) < 0$, and $f(1) > 0$. In the process, they used the \"bisection method\" and chose 3 more values, which are $x_1$, $x_2$, $x_3$, to fin... |
[
{
"content": "Given the sets $S=\\{ x\\mid | x-2 | > 3 \\}$ and $T=\\{ x\\mid a < x < a+8 \\}$, and their union $S\\cup T=R$, find the range of values for $a$. Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
},
{
"content": "To find the range of values for $a$ ... |
[
{
"content": "What is the largest integer \\( y \\) such that \\( \\frac{y}{4} + \\frac{3}{7} < \\frac{2}{3} \\)? Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
},
{
"content": "To solve the problem \\(\\frac{y}{4} + \\frac{3}{7} < \\frac{2}{3}\\), we will fir... |
[
{
"content": "Calculate the positive difference between the two solutions of the equation $|x + 8| = \\sqrt{256}$. Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
},
{
"content": "To solve the equation \\( |x + 8| = \\sqrt{256} \\), we need to consider the posi... |
[
{
"content": "What is the least prime factor of $5^4 - 5^2$? Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
},
{
"content": "First, we need to simplify the expression \\(5^4 - 5^2\\). We can factor out \\(5^2\\) from the expression:\n\n\\[5^4 - 5^2 = 5^2(5^2 -... |
[
{
"content": "What is the distance between the points $(1, 2)$ and $(6, 5)$? Express your answer in simplest radical form. Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
},
{
"content": "To find the distance between the points $(1, 2)$ and $(6, 5)$, we can use... |
[
{
"content": "Given the universal set U = {x ∈ Z | 0 ≤ x ≤ 10}, and sets A = {1, 2, 4, 5, 9}, B = {4, 6, 7, 8, 10}, and C = {3, 5, 7}. Find:\n1. A ∪ B\n2. (A ∩ B) ∩ C\n3. (∁<sub>U</sub>A) ∩ (∁<sub>U</sub>B) Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
},
{
"... |
[
{
"content": "Find the sum of all positive real solutions $x$ to the equation \\[2\\cos2x \\left(\\cos2x - \\cos{\\left( \\frac{1008\\pi^2}{x} \\right)}\\right) = \\cos4x - 1,\\] where $x$ is measured in radians. Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
},
{... |
[
{
"content": "Solve $\\log_8 x + \\log_2 x^3 = 9$. Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
},
{
"content": "To solve \\(\\log_8 x + \\log_2 x^3 = 9\\), let's start by expressing all logarithms with the same base.\n\n1. Change \\(\\log_8 x\\) to base 2.\... |
[
{
"content": "If $a$, $b\\in \\{-1,1,2\\}$, then the probability that the function $f\\left(x\\right)=ax^{2}+2x+b$ has a zero point is ______. Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
},
{
"content": "To solve the problem, we need to first understand whe... |
[
{
"content": "In parallelogram $PQRS$, angle $Q$ is trisected by $\\overline{QT}$ and $\\overline{QU}$, where $T$ is on $\\overline{PS}$, $U$ is on $\\overline{PR}$, $ST=3$, and $PU=9$. $PQRS$ is a rhombus, and the angle trisected is $120^\\circ$. Calculate the area of $PQRS$. Let's think step by step and outpu... |
[
{
"content": "Simplify first, then evaluate: $\\left(\\dfrac{a^{2}+1}{a}-2\\right)\\div \\dfrac{\\left(a+2\\right)\\left(a-1\\right)}{a^{2}+2a}$, where $-2\\leqslant a\\leqslant 2$. Choose an integer you like and substitute it to find the value. Let's think step by step and output the final answer within \\boxe... |
[
{
"content": "The sum of the interior angles of a polygon is $1800^{\\circ}$, calculate the number of diagonals drawn from a vertex of this polygon. Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
},
{
"content": "The sum of the interior angles of a polygon is ... |
[
{
"content": "Given lines $l_{1}$: $\\sqrt {3}x-y+2=0$ and $l_{2}$: $3x+ \\sqrt {3}y-5=0$, determine the angle between lines $l_{1}$ and $l_{2}$. Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
},
{
"content": "To determine the angle between the lines \\( l_1 \... |
[
{
"content": "A store owner bought 2000 pencils at $0.20 each. He incurs an additional cost of $0.05 per pencil for packaging. If he sells the pencils for $0.50 each, how many pencils must he sell to make a profit of exactly $200.00?\nA) 1250\nB) 1300\nC) 1350\nD) 1400\nE) 1450 Let's think step by step and outp... |
[
{
"content": "Given that $2^{12} \\cdot 3^3 = a^b,$ where both $a$ and $b$ are positive integers, find the smallest possible value for $a+b$. Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
},
{
"content": "To find the smallest possible value of \\( a + b \\), ... |
[
{
"content": "Solve the inequality\n\n$$\n\\left(\\sqrt{x^{2}-4x+3}+1\\right) \\log_{2} \\frac{x}{5} + \\frac{1}{x}\\left(\\sqrt{8x-2x^{2}-6}+1\\right) \\leq 0\n$$ Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
},
{
"content": "To solve the inequality \\((\\sq... |
[
{
"content": "Given the original triangle, if each of its two including sides is doubled, determine the factor by which the area is multiplied. Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
},
{
"content": "Given the original triangle, if each of its two incl... |
[
{
"content": "Find the smallest constant M, so that for any real numbers $a_1, a_2, \\dots a_{2023} \\in [4, 6]$ and $b_1, b_2, \\dots b_{2023} \\in [9, 12] $ following inequality holds: $$ \\sqrt{a_1^2 + a_2^2 + \\dots + a_{2023}^2} \\cdot \\sqrt{b_1^2 + b_2^2 + \\dots + b_{2023}^2} \\leq M \\cdot \\lef... |
[
{
"content": "The solution set of the inequality $\\dfrac{x}{x+1} < 0$ is . Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
},
{
"content": "The solution set of the inequality $\\dfrac{x}{x+1} < 0$ is, the sign changes every time we go past a zero:\n - The n... |
[
{
"content": "An uncrossed belt runs without slack around two circular pulleys with radii of $10$ inches and $6$ inches. If the distance between the points of contact of the belt with the pulleys is $26$ inches, calculate the distance between the centers of the pulleys in inches. Let's think step by step and ou... |
[
{
"content": "How many nonempty subsets of {1, 2, ..., 10} have the property that the sum of their largest element and smallest element is 11? Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
},
{
"content": " To find the number of nonempty subsets \\( S \\) of ... |
[
{
"content": "How many positive divisors of $180$ are not divisible by 3? Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
},
{
"content": "First, factorise $180$ into its prime factors. \n\n \\begin{align*}\n 180 &= 2 \\times 90 \\\\\n &= 2 \\times 2 \\time... |
[
{
"content": "What are the two smallest prime factors of $462$? Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
},
{
"content": "The only correct answer is \\boxed{\\dfrac{\\begin{tikzpicture}\n \\node at (1,1) [draw, circle] {\\textbf{(C)}};\n \\node (A)... |
[
{
"content": "As shown in the diagram, two lines intersect with two circles at 9 points. Choose 4 points from these 9 points such that any 3 points among these 4 are neither collinear nor lie on the same circle. How many different selections can be made? Let's think step by step and output the final answer with... |
[
{
"content": "Every day, Patrick the dog chews up one slipper from the available stock at home. There is a strict 0.5 probability that Patrick wants to chew on the left slipper and a 0.5 probability that he wants to chew on the right slipper. If the desired slipper is not available, Patrick gets upset. How many... |
[
{
"content": "There is a uniformly growing grassland. If 20 cows are grazed, they will just finish eating all the grass in 60 days. If 30 cows are grazed, they will just finish eating all the grass in 35 days. Now, 6 cows are grazing on the grassland. After a month, 10 more cows are added. How many more days wi... |
[
{
"content": "Calculate the value of $\\left(2^{0}-1+5^{2}-0\\right)^{-1} \\times 5$. Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
},
{
"content": "First, let's simplify the expression inside the parentheses:\n\n\\[\n2^0 - 1 + 5^2 - 0\n\\]\n\nWe know that \\... |
[
{
"content": "For how many integer values of $n$ between 1 and 419 inclusive does the decimal representation of $\\frac{n}{420}$ terminate? Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
},
{
"content": "The decimal representation of a fraction terminates if a... |
[
{
"content": "Let \\(m\\) be a positive integer such that \\((m+2)! + (m+3)! = m! \\cdot 1080\\). Calculate the sum of the digits of \\(m\\). Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
},
{
"content": "To solve the problem, we need to start by expressing t... |
[
{
"content": "In a right-angled triangle $ABC$ with $\\angle ABC = 90^\\circ$, two of the altitudes have lengths $6$ and $18$. If the length of the third altitude is also an integer, what is the largest it can be? Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
},
... |
[
{
"content": "In a right triangle, the altitude from a vertex to the hypotenuse splits the hypotenuse into two segments of lengths $a$ and $b$ . If the right triangle has area $T$ and is inscribed in a circle of area $C$ , find $ab$ in terms of $T$ and $C$ . Let's think step by step and output the fi... |
[
{
"content": "In a right triangle \\( ABC \\) with a right angle at vertex \\( B \\), median \\( BM \\) is drawn. The incircle of triangle \\( ABM \\) touches sides \\( AB \\) and \\( AM \\) at points \\( K \\) and \\( L \\), respectively. It is known that lines \\( KL \\) and \\( BM \\) are parallel. Find the ... |
[
{
"content": "Find all values of the parameter \\( a \\) for which there exists a value of the parameter \\( b \\) such that the system\n\n$$\n\\left\\{\\begin{array}{l}\n\\arcsin \\left(\\frac{a-y}{3}\\right)=\\arcsin \\left(\\frac{4-x}{4}\\right) \\\\\nx^{2}+y^{2}-8 x-8 y=b\n\\end{array}\\right.\n$$\n\nhas ex... |
[
{
"content": "Given the function $f(x) = x^{\\alpha}$ whose graph passes through the point $(2, \\frac{\\sqrt{2}}{2})$, calculate the value of $f(4)$. Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
},
{
"content": "First, we need to determine the value of $\\a... |
[
{
"content": "In the geometric sequence $\\{a\\_n\\}$, if $a\\_4$ and $a\\_8$ are the two roots of the equation $x^{2}+11x+9=0$, then the value of $a\\_6$ is $\\_\\_\\_\\_\\_\\_$. Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
},
{
"content": "In the geometric... |
[
{
"content": "In a circle with a radius of 1.4, determine the distance from the center to a chord that subtends an arc of $120^{\\circ}$. Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
},
{
"content": "In a circle with a radius of 1.4, let's determine the dist... |
[
{
"content": "Given the expectation \\( m_{x}(t) = 3t^2 + 1 \\) of the random function \\( X(t) \\), find the expectation of the integral \\( Y(t) = \\int_{0}^t X(s) \\, \\mathrm{d}s \\). Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
},
{
"content": "The expe... |
[
{
"content": "Find the greatest common factor of $7!$ and $8!$. Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
},
{
"content": "Let's begin by calculating $7!$ and $8!$.\n\n\\[\n7! = 7 \\times 6 \\times 5 \\times 4 \\times 3 \\times 2 \\times 1 = 5040\n\\]\n\n... |
[
{
"content": "Determine how many integer values of $x$ satisfy $-100 < x^4 < 100$. Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
},
{
"content": "To determine how many integer values of \\(x\\) satisfy the inequality \\(-100 < x^4 < 100\\), we need to analyze... |
[
{
"content": "A line is defined by passing through vectors $\\mathbf{a}$ and $\\mathbf{b}.$ For a certain scalar \\(k\\), the vector\n\\[k \\mathbf{a} + \\frac{5}{7} \\mathbf{b}\\] is also on the line. Determine the value of \\(k\\). Let's think step by step and output the final answer within \\boxed{}",
"r... |
[
{
"content": "Given the function $f(x)=|x+1|-|x-2|$.\n\n$(1)$ Find the solution set of the inequality $f(x) \\geqslant 1$;\n\n$(2)$ If the solution set of the inequality $f(x) \\geqslant x^2-x+m$ is non-empty, find the range of values for $m$. Let's think step by step and output the final answer within \\boxed{... |
[
{
"content": "The maximum value of the function $f(x) = -x^2 + 4x - 1$ on the interval $[t, t+1]$ is denoted as $g(t)$. Find the maximum value of $g(t)$. Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
},
{
"content": "To find the maximum value of the function ... |
[
{
"content": "Given a rectangle with vertices at $(0,0)$, $(4,0)$, $(0,2)$, and $(4,2)$, and a slanted line extending from $(2,0)$ to $(4,4)$, determine the value of $d$ if the line $(d,0)$ to $(4,4)$ divides the entire region into two regions of equal area. Let's think step by step and output the final answer ... |
[
{
"content": "Given that there are 10 screws in the box, 3 of which are defective, calculate the probability of drawing exactly 1 defective screw when 4 screws are randomly drawn. Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
},
{
"content": "To solve this pr... |
[
{
"content": "Given that $\\alpha$ and $\\beta$ are acute angles, $\\sin\\alpha=\\frac{1}{2}$, $\\cos\\beta=\\frac{1}{2}$, calculate the value of $\\sin(\\alpha+\\beta)$. Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
},
{
"content": "Firstly, we need to find ... |
[
{
"content": "Find the largest value of $c$ such that $3$ is in the range of $f(x) = x^2 - 6x + c$. Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
},
{
"content": "To determine the largest value of $c$ such that $3$ is in the range of the function $f(x) = x^2 ... |
[
{
"content": "The equation of an ellipse with foci on the x-axis is given by $\\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1 (a > b > 0)$. A triangle is formed by connecting one endpoint of the minor axis and both foci. The inscribed circle of this triangle has a radius of $\\frac{b}{3}$. Determine the eccentricity of... |
[
{
"content": "In $\\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are respectively $a$, $b$, and $c$. It is given that $\\frac{-b + \\sqrt{2}c}{\\cos B} = \\frac{a}{\\cos A}$, \n$(I)$ Find the size of angle $A$; \n$(II)$ If $a=2$, find the maximum value of the area $S$. Let's think step by ste... |
[
{
"content": "A right-angled triangle has an area of 150 cm² and a perimeter of 60 cm. What are the lengths of its sides? Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
},
{
"content": "To solve this problem, we need to find the lengths of the sides of a right... |
[
{
"content": "Given that a ticket to a concert costs $x$ dollars, where $x$ is a whole number, a group of 11th graders buys tickets costing a total of $60, and a group of 12th graders buys tickets costing a total of $90, determine the number of values for $x$ that are possible. Let's think step by step and outp... |
[
{
"content": "Given a Cartesian coordinate system in space with points O(0, 0, 0), A(1, 0, 1), B(1, 1, 0), and C(0, 1, 1), determine the volume of the tetrahedron O-ABC. Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
},
{
"content": "To find the volume of the ... |
[
{
"content": "Define an operation: $|\\begin{array}{l}{a}&{c}\\\\{b}&{d}\\end{array}|=ad-bc$. For example, if $|\\begin{array}{l}{1}&{3}\\\\{-2}&{0}\\end{array}|=1\\times 0-\\left(-2\\right)\\times 3=6$, then when $a=-1^{2}$, $b=\\left(-2\\right)^{2}-1$, $c=-3^{2}+5$, $d=\\frac{3}{4}÷(-\\frac{1}{4})$, find the ... |
[
{
"content": "Given that $α∈(0， \\frac {π}{2}), β∈(0， \\frac {π}{2}), \\text{and }sin(2α+β)= \\frac {3}{2}sinβ, \\text{ find the value of } \\frac {tan(α+β)}{tanα }$. Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
},
{
"content": "Given the equation:\n\\[ \\si... |
[
{
"content": "Simplify\n\\[\\left( \\frac{2 + i}{2 - i} \\right)^{200}.\\] Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
},
{
"content": "To simplify \\(\\left( \\frac{2 + i}{2 - i} \\right)^{200}\\), we can use properties of complex numbers. Here's the step-... |
[
{
"content": "Find the maximum value of the function \\( f(x)=\\sqrt{x-5}+\\sqrt{24-3x} \\). Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
},
{
"content": "Reasoning\nTo find the maximum value of the function \\( f(x) = \\sqrt{x-5} + \\sqrt{24-3x} \\), we nee... |
[
{
"content": "Let $Q$ be the product of the first $150$ positive odd integers. Find the largest integer $k'$ such that $Q$ is divisible by $3^{k'}$. Let's think step by step and output the final answer within \\boxed{}",
"role": "user"
},
{
"content": "To solve this problem, we need to determine the... |
[
{
"content": "Vanya received three sets of candies for New Year. Each set contains three types of candies: hard candies, chocolates, and gummy candies. The total number of hard candies in all three sets is equal to the total number of chocolates in all three sets, and also to the total number of gummy candies i... |
[
{
"content": "Synthesis and Exploration \n[Reading Comprehension] \"Holistic Thinking\" is an important mathematical thinking method, widely used in the simplification and evaluation of polynomials. \nFor example, $4x-2x+x=\\left(4-2+1\\right)x=3x$. Similarly, if we consider $\\left(a-b\\right)$ as a whole, t... |
End of preview. Expand
in Data Studio
README.md exists but content is empty.
- Downloads last month
- 4