messages
listlengths 1
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stringlengths 0
830
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[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.\nLet $U=\\{2,3,4,5,6\\}$, $A=\\{x|x^2-6x+8=0\\}$, $B=\\{2,5,6\\}$. Then, find $(C_UA) \\cup B$.",
"role": "user"
}
] | \{2,3,5,6\} |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.\nfind the length of the wire required to go 10 times round a square field containing 53824 m 2 .",
"role": "user"
}
] | 9280 |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.\nIf a number x is greater than a certain number and less than 8, and another number y is greater than 8 and less than 13, then the greatest possible positive integer difference of x and y is 7. What is the lower limit for x?",
"role": "user"
}
] | 2 |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.\nGiven the function $f(x)=x+a\\sin x$ is monotonically increasing on $(-\\infty,+\\infty)$, find the range of values for the real number $a$.",
"role": "user"
}
] | [-1,1] |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.\nFill in the appropriate “ “, \"-\", \",\", or \"宀\" operator in the squares below (parentheses can also be used in the equation) to make the following equation hold true.",
"role": "user"
}
] | 12 \times 12 \div (12 + 12) = 6, (12 \times 12 + 12) \div 12 = 13 |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.\nIn an $n \\times n \\, (n \\geq 2)$ chessboard, a piece on a certain square can move one step to the right, up, or to the lower left. For which positive integers $n$ can the piece start from the bottom left square $A$ and traverse each square on the board exactly once?",
"role": "user"
}
] | 3k \text{ and } 3k+1, \text{ where } k \in \mathbb{N}_{+} |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.\nOliver is learning words starting with an alphabet, focusing on vowels first, before moving to others. He takes 5 days to learn words of one alphabet thoroughly but decides to set a unique challenge for himself. For each vowel, the number of rest days Oliver takes before starting to learn the next vowel is equal to the index of the vowel in the alphabet. Also, Oliver plans to learn the last two vowels simultaneously, while incorporating the appropriate rest days between them. Considering these factors, how many days does Oliver need to finish learning all vowels?",
"role": "user"
}
] | 40 |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.\nThere are three fill pipes - Pipe A, Pipe B, and Pipe C. Pipeline A can fill 2/3 of a cistern in 40 minutes. Pipe B can fill the entire cistern in 45 minutes, and Pipe C can fill half of the cistern in 30 minutes. If all three pipes are working together, how many minutes does it take to fill the cistern completely?",
"role": "user"
}
] | 11.25 |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.\nIn the figure below, $ABCD$ is a parallelogram, and $M$ is the midpoint of segment $AD$.\n\nIf the area of quadrilateral $MOCD$ is $5 \\text{ cm}^2$, calculate the area of triangle $AOM$.",
"role": "user"
}
] | 1 \text{ cm}^2 |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.\nIn a right-angled triangle $ABC$, the hypotenuse $AB$ is divided in the ratio $m: n$ such that the smaller segment lies adjacent to the shorter leg. What should be the ratio of the sides of the triangle $ABC$ so that the lines connecting the division point with the midpoints of the legs form a right angle?",
"role": "user"
}
] | a : b : c = \sqrt{n} : \sqrt{m} : \sqrt{m+n} |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.\nThere are 15 million people employed in a city. The unemployment rate in this city is 8%. Find the number of unemployed people (in millions). Round your answer to the nearest thousandth.",
"role": "user"
}
] | 1.304 \text{ million} |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.\nIf a factory plans to have its output not less than twice this year's output the year after next, then the minimum average annual growth rate for the next two years is \\_\\_\\_\\_\\_.",
"role": "user"
}
] | 41.4\% |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.\nGiven $f(x)= \\frac {x}{1+x}(x\\geqslant 0)$, the sequence ${a_{n}}$ satisfies $a_{1}=f(1)$, and $a_{n+1}=f(a_{n})(n∈N_{+})$, find the value of $a_{2015}$.",
"role": "user"
}
] | a_{2015}= \frac {1}{2016} |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.\nThe method to draw a line chart is to first plot points on the grid, and then connect the points in sequence.",
"role": "user"
}
] | first plot points on the grid, and then connect the points in sequence. |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.\nGiven the sets $A=\\{x \\mid x^2+5x+6=0\\}$ and $B=\\{x \\mid mx+1=0\\}$, with the condition that $A \\cup B = A$, find the set comprising the values of the real number $m$.",
"role": "user"
}
] | \left\{0, \frac{1}{2}, \frac{1}{3}\right\} |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.\nFor a function $f(x)$ defined on $\\mathbb{R}$, if a real number $x$ satisfies $f(x) = x$, then $x$ is called a fixed point of the function $f(x)$. If the quadratic function $f(x) = x^2 + ax + 1$ has no fixed points, then the range of values for the real number $a$ is.",
"role": "user"
}
] | -1 < a < 3 |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.\nWhen the perimeter of a regular polygon is divided by 5, the length of a side is 25. Find the area of the polygon.",
"role": "user"
}
] | 1078.125 |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.\nIn triangle \\(ABC\\) with angle \\(A\\) equal to \\(60^\\circ\\), a bisector \\(AD\\) is drawn. The radius of the circumcircle of triangle \\(ADC\\) with its center at point \\(D\\) is \\( \\frac{2\\sqrt{3}}{3} \\). Find the length of segment \\(BM\\), where \\(M\\) is the intersection point of segments \\(AD\\) and \\(BO\\), if \\(AB=1\\).",
"role": "user"
}
] | \frac{\sqrt{21}}{9} |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.\nGiven that in parallelogram $ABCD$, $AB=14$ cm and $BC=16$ cm, the perimeter of this parallelogram is ______ cm.",
"role": "user"
}
] | 60\, \text{cm} |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.\nThe coordinates of the vertex of the parabola $y=2(x-3)^{2}+1$ are $\\boxed{\\text{answer}}$.",
"role": "user"
}
] | (3,1) |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.\nA store purchased a pair of pants for some amount and was selling it at a price that equaled the purchase price of the pants plus a markup that was 25 percent of the selling price. After some time, the store owner decided to decrease the selling price by 20 percent. The store's gross profit on this sale was $14. What was the original purchase price of the pants?",
"role": "user"
}
] | \$210 |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.\nThe minimum value of the function $y=\\log(x^2-2x+3)$ is.",
"role": "user"
}
] | \log2 |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.\nA sequence of positive integers \\( a_1, a_2, a_3, \\ldots \\) satisfies\n\\[ a_{n+1} = n \\left\\lfloor \\frac{a_n}{n} \\right\\rfloor + 1 \\]\nfor all positive integers \\( n \\). If \\( a_{30} = 30 \\), how many possible values can \\( a_1 \\) take? (For a real number \\( x \\), \\( \\lfloor x \\rfloor \\) denotes the largest integer that is not greater than \\( x \\).)",
"role": "user"
}
] | 274 |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.\nTen kilograms of basmati rice costing rs. 20 per kg is mixed with 5 kilograms of Long-grain rice costing rs . 25 per kg, and 15 kilograms of Short-grain rice costing rs. 18 per kg. Also, 8 kilograms of medium grain rice costing rs. 22 per kg is added to the mixture. What is the average price per kg of the resulting mixture of rice?",
"role": "user"
}
] | 20.29 |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.\nA store owner purchases 500 pens at the marked price of 450 pens from a wholesaler. In addition, the store owner must pay a 5% tax on the total cost of the pens and a shipping fee of $20. If he sells these pens offering a discount of 3%, what is the profit percentage?",
"role": "user"
}
] | 1.52\% |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.\nGiven $p$: \"The line $x+y-m=0$ intersects with the circle $(x-1)^{2}+y^{2}=1$\"; $q$: \"The two roots of the equation $x^{2}-x+m-4=0$ have opposite signs\". If $p∨q$ is true and $￢p$ is true, find the range of values for the real number $m$.",
"role": "user"
}
] | m\leqslant 1-\sqrt{2} \text{ or } 1+\sqrt{2}\leqslant m < 4 |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.\nA cashier at Apex Parts received a check for 2380 dollars for 59 parts. Some parts cost a certain amount per part and other parts cost 50 dollars per part. There are 40 50-dollar parts in the order. What is the cost of the other parts?",
"role": "user"
}
] | $20 |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.\nA trader made a profit of $960 after a week of sales. To raise $610 to pay for her next shipment of wares, she splits the profit in half and solicits donations from her family and friends. If she received a total donation of $310 and adds it to one-half of her profit, how much money did she make above her goal?",
"role": "user"
}
] | $180 |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.\nIf the sales tax is reduced from 7 3/5% to 6 2/3%, and there is an additional 5% discount applied before tax along with a handling fee of Rs. 200, what difference does it make to a person who purchases an article with a market price of Rs. 15,400?",
"role": "user"
}
] | 964.60 |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.\nIn a school, there are 402 more girls than boys. If there are 739 girls and 127 teachers in the school, what is the total number of students in the school?",
"role": "user"
}
] | 1076 |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.\nIn a class, there are three groups of children: Group A, Group B, and Group C. Each child in Group A has 4 pencils, 2 erasers, and 13 skittles. Each child in Group B has 6 pencils, 1 ruler, and 8 skittles. Each child in Group C has 3 pencils, 1 sharpener, and 15 skittles. There are 7 children in Group A, 9 children in Group B, and 5 children in Group C. Calculate the total number of pencils, erasers, rulers, sharpeners, and skittles for the whole class.",
"role": "user"
}
] | 238 |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.\nGiven vectors $\\overrightarrow{a}=(-1,3)$ and $\\overrightarrow{b}=(3,4)$, find the projection of $\\overrightarrow{a}$ in the direction of $\\overrightarrow{b}$.",
"role": "user"
}
] | \frac{9}{5} |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.\n\nThree pirates divided the diamonds they acquired during the day in the evening: Bill and Sam each received twelve diamonds, and the rest were given to John, who did not know how to count. During the night, Bill stole one diamond from Sam, Sam stole one diamond from John, and John stole one diamond from Bill. As a result, the average mass of the diamonds Bill had decreased by one carat, Sam’s average mass decreased by two carats, but John’s average mass increased by four carats. How many diamonds did John originally have?",
"role": "user"
}
] | 9 \text{ diamonds} |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.\nA chemical plant needs to study the relationship between the recovery rate of a product ($y$) and the content of effective components in the raw material. After taking $8$ pairs of observations, the following sums were calculated: $\\sum\\limits^{8}_{i=1}x_i=52$, $\\sum\\limits^{8}_{i=1}y_i=228$, $\\sum\\limits^{8}_{i=1}x_i^2=478$, $\\sum\\limits^{8}_{i=1}x_iy_i=1,849$. Find the linear regression equation between $y$ and $x$.",
"role": "user"
}
] | \hat y=11.47+2.62x |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.\nThe length of a rectangular field is some ratio of its width. The perimeter of the field is 288 meters. The width of the field is 60 meters. What is the ratio of the length to the width of the field?",
"role": "user"
}
] | 7:5 |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.\nIf the function $f(x)=\\begin{cases} x^{2}-3x+1, & x\\geqslant 1, \\\\ (\\dfrac{1}{2})^{x}+\\dfrac{1}{2}, & x < 1, \\end{cases}$, then $f(f(2))=$ \\_\\_\\_\\_\\_\\_.",
"role": "user"
}
] | \dfrac{5}{2} |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.\n$A$ makes a bet with $B$ that they will roll a sum less than 8 with two dice in a single throw. $A$ bets 10 forints, while $B$ stakes 8 forints. Is this bet fair?\n\n(Note: Refer to Prékopa András's article on \"Expected Value\" in this issue.)",
"role": "user"
}
] | 7 \frac{1}{7} \text{ forints} |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.\nGiven the point \\( P \\) inside the triangle \\( \\triangle ABC \\), satisfying \\( \\overrightarrow{AP} = \\frac{1}{3} \\overrightarrow{AB} + \\frac{1}{4} \\overrightarrow{AC} \\), let the areas of triangles \\( \\triangle PBC \\), \\( \\triangle PCA \\), and \\( \\triangle PAB \\) be \\( S_1 \\), \\( S_2 \\), and \\( S_3 \\) respectively. Determine the ratio \\( S_1 : S_2 : S_3 = \\quad \\).",
"role": "user"
}
] | 5:4:3 |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.\nIf the general term of the sequence $\\{a_{n}\\}$ is $a_{n}=2n-12$, then when $n=$______, the sum of the first $n$ terms of the sequence $\\{a_{n}\\}$, denoted as $S_{n}$, is minimized.",
"role": "user"
}
] | 5 \text{ or } 6 |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.\nObserve the pattern of the following numbers and fill in the blank: 0, -3, 8, -15, 24, …, then the 2010th number is ___.",
"role": "user"
}
] | 4040099 |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.\nGiven that point $M$ is a moving point on the parabola $y^2 = 8x$, with $F$ being the focus of the parabola, and point $A$ lies on the circle $C: (x-3)^2 + (y+1)^2 = 1$, find the minimum value of $|AM| + |MF|$.",
"role": "user"
}
] | 4 |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.\nThe dihedral angle at the lateral edge of a regular triangular pyramid is $2 \\alpha$. The height of the pyramid is $h$. Find the volume of the cone circumscribed around the pyramid.",
"role": "user"
}
] | \frac{\pi h^3}{3 - \cot^2(\alpha)} |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.\nA natural number $A$ has $d$ as its largest proper divisor. For the natural number $A+2$, its largest proper divisor is $d+2$. Find the value of $A$.",
"role": "user"
}
] | 7 |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.\nFactorize: $3x^{2}y-3y=$____.",
"role": "user"
}
] | 3y\left(x+1\right)\left(x-1\right) |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.\nIf $x \\in (0, +\\infty)$, observe the following expressions: $x+\\dfrac{1}{x} \\geqslant 2$, $x+\\dfrac{4}{{x}^{2}}=\\dfrac{x}{2}+\\dfrac{x}{2}+\\dfrac{4}{{x}^{2}} \\geqslant 3$, $x+\\dfrac{27}{{x}^{3}}=\\dfrac{x}{3}+\\dfrac{x}{3}+\\dfrac{x}{3}+\\dfrac{27}{{x}^{3}} \\geqslant 4$, $…$, by analogy, we have $x+\\dfrac{a}{{x}^{n}} \\geqslant n+1 (n \\in \\mathbb{N}^{*})$, then the value of $a$ is ______.",
"role": "user"
}
] | n^{n} |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.\nGiven the sets $A=\\left\\{ x\\in\\mathbb{R} \\mid |x+3|+|x-4|\\leqslant 9\\right\\}$ and $B=\\left\\{ x\\in\\mathbb{R} \\mid x=4t+ \\frac{1}{t}-6, t\\in(0,+\\infty)\\right\\}$, then the intersection of sets $A$ and $B$ is $\\boxed{\\{x|-2\\leqslant x\\leqslant 5\\}}$.",
"role": "user"
}
] | \{x|-2\leqslant x\leqslant 5\} |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.\nGiven the matrix $$M= \\begin{bmatrix} 1 & 2 \\\\ a & 1\\end{bmatrix}$$ with one of its eigenvalues being λ=3, and its corresponding eigenvector being $$\\overrightarrow {α}= \\begin{bmatrix} \\overset{1}{1}\\end{bmatrix}$$, find the equation of the curve l<sub>2</sub> resulting from the transformation of the line l<sub>1</sub>: x+2y+1=0 under matrix M.",
"role": "user"
}
] | x+1=0 |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.\nIn a school with 8 classes, the number of students in each class decreases by 3 students compared to the previous class. There are 3 subject groups (A, B, and C), and the number of students in each subject group is equal for all classes. Additionally, the attendance rate for class A is 90%, for class B is 85%, and for class C is 80%. If the total number of students at school is 348, how many students are there in the largest class of subject group A?",
"role": "user"
}
] | 16 |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.\nTanner saved $17 in September, $48 in October, $25 in November, and $55 in December. He then spent $49 on a video game and $65 on a new pair of shoes. Tanner received a 10% discount on the video game and had to pay 5% sales tax on the total cost of his purchases. How much money does Tanner have left?",
"role": "user"
}
] | \$30.44 |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.\nConnect all the black and white dots with horizontal or vertical lines to form a closed loop that does not intersect itself. The loop must turn at a right angle at the black dots, passing straight through the previous and subsequent cells; it must pass straight through the white dots, turning at a right angle either before or after the dot (at least at one location). For instance, the drawing in Figure 2 is the only solution for Figure 1. Following this rule, how many times does the loop turn in total in Figure 3?",
"role": "user"
}
] | 20 |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.\nIn her bakery, Teal sells pumpkin pie and custard pie by the slice. The pumpkin pie is cut into 8 pieces. The custard pie is cut into 6 pieces. Pumpkin pie is sold at a certain price per slice. Custard pie is $6 a slice. Teal sold 4 pumpkin pies and 5 custard pies, and she made $340 from her sales. How much does a slice of pumpkin pie cost?",
"role": "user"
}
] | $5 |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.\nGiven that \\( a \\) is a positive real number and \\( f(x) = \\log_{2}\\left(a x + \\sqrt{2 x^{2} + 1}\\right) \\) is an odd function, find the solution set for the inequality \\( f(x) > 0 \\).",
"role": "user"
}
] | \left( \frac{7}{8}, +\infty \right) |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.\nGiven two sets $M=\\{x \\mid \\frac{1}{x} > 1\\}$ and $N=\\{x \\mid x^2 + 2x - 3 < 0\\}$, find the union of $M$ and $N$, $M \\cup N$.",
"role": "user"
}
] | M \cup N=(-3, 1) |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.\nFrancis' family has some ukuleles, 4 guitars, and a certain number of violins. Each ukulele has 4 strings. Each guitar has 6 strings. Each violin has 4 strings. The total number of strings among the ukuleles, guitars, and violins is 40 strings. How many ukuleles and violins does Francis' family have?",
"role": "user"
}
] | 4 |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.\nThe area of a sector of a circle with a certain radius and an angle at the center of 39° is 49.02857142857143 square meters. What is the radius of the circle?",
"role": "user"
}
] | 12.03 |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.\nGiven a sequence $\\{a_{n}\\}$ where $a_{2}=3a_{1}$, let $S_{n}$ denote the sum of the first $n$ terms of the sequence. It is known that ${S_{n+1}}+{S_n}+{S_{n-1}}=3{n^2}+2$ for all $n\\geq 2, n\\in \\mathbb{N}^{*}$. If for any $n\\in \\mathbb{N}^{*}$, $a_{n} \\lt a_{n+1}$, then the range of possible values for the first term $a_{1}$ is ______.",
"role": "user"
}
] | \left(\frac{13}{15}, \frac{7}{6}\right) |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.\nFind the equation of the tangent line to the curve $y=e^{x}$ that passes through the origin.",
"role": "user"
}
] | y=ex |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.\nWhen a number is divided by 6 and then multiplied by 12, the answer is 13. What is the number?",
"role": "user"
}
] | 6.5 |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.\nA family went out to see a movie. The regular ticket costs $9 and the ticket for children is $2 less. They gave the cashier two $20 bills. There are 2 adults and 3 children in the family. How much change did they receive?",
"role": "user"
}
] | \$1 |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.\nGiven vectors $\\overset{→}{a}=(1,−x)$ and $\\overset{→}{b}=(x,−16)$, find the value of $x$ such that the two vectors have opposite directions.",
"role": "user"
}
] | x = -5 |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.\nA notebook costs three times as much as a pen. Lyle paid $18 for 4 notebooks. How much does a pen cost?",
"role": "user"
}
] | \$1.50 |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.\nFind the minimum value of the distance $MN$ between a point $M$ on the curve $y=2\\ln x$ and a point $N$ on the line $y=2x+3$.",
"role": "user"
}
] | \sqrt{5} |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.\nThere are 15 stations between Hyderabad and Bangalore. How many second class tickets have to be printed, so that a passenger can travel from any station to any other station?",
"role": "user"
}
] | 68 |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.\nGiven the circle $(x-4)^{2}+(y-5)^{2}=169$, a line passing through the point $(1,1)$ intersects the circle at points $A$ and $B$. Find the range of $|AB|$.",
"role": "user"
}
] | \left[24,26\right] |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.\nThere is a train and a car. The speed of the train and the car have a certain ratio. A bus covered a distance of 480 km in 8 hours. The speed of the bus is 3/4th of the speed of the train. The car will cover 450 km in 6 hours. What is the ratio between the speed of the train and the car?",
"role": "user"
}
] | 16:15 |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.\nOn a circle, \\( n > 1 \\) points are marked, called positions, dividing it into equal arcs. The positions are numbered clockwise from 0 to \\( n-1 \\). Vasya places a token on one of these positions. The following actions, called moves, are repeated an unlimited number of times: Petya names a natural number, and Vasya moves the token clockwise or counterclockwise by the number of positions specified by Petya. If at any point after a move the token lands on position number 0, Vasya loses, and Petya wins. For which values of \\( n \\) can Petya always win, regardless of Vasya's moves?",
"role": "user"
}
] | n = 2^k |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.\nJoseph has a refrigerator, a water heater in his house, and an electric oven that consumes power at different rates. The total amount of money that Joseph pays for the energy used by the refrigerator is three times the amount he pays for the power used by the water heater. The electric oven uses power worth some amount in a month, twice what Joseph pays for the power the water heater uses. Joseph pays a total of $1500 for the power used by these gadgets. How much does Joseph pay for the power used by the electric oven in a month?",
"role": "user"
}
] | \$500 |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.\nHow should the positive numbers $a, b, c$ be chosen, given that their sum is constant, to maximize the expression $a b^{2} c^{3}$?",
"role": "user"
}
] | i'. |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.\nLeonardo is reading an article which says temperature is increasing a certain amount every century. The change in temperature in 700 years will be 21 units. How many units does the temperature increase every century?",
"role": "user"
}
] | 3 |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.\nWe consider \\( n \\) distinct points in a plane. For each segment formed by a pair of these points, we color the midpoint of the segment blue. How many blue points are needed, at minimum?\n\nIf two midpoints coincide, they count as a single point.",
"role": "user"
}
] | 2n - 3 |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.\nGiven a tetrahedron \\(K L M N\\) with \\(K L = 5\\), \\(N M = 6\\), \\(\\angle L M N = \\angle K N M = 35^\\circ\\), and \\(\\angle L N M = \\angle K M N = 55^\\circ\\). A sphere is circumscribed around the tetrahedron. Consider the set of all points on this sphere such that the sum of spherical distances from these points to \\(K\\), \\(L\\), \\(M\\), and \\(N\\) is no more than \\(6 \\pi\\). What is the area of this set? Note that the spherical distance between two points on a sphere is the length of the shortest arc of a great circle connecting these points.",
"role": "user"
}
] | 18\pi |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.\nFind all real numbers \\( x, y, z \\) satisfying the system of equations:\n\n$$\n\\left\\{\n\\begin{array}{l}\n(x+1) y z = 12 \\\\\n(y+1) z x = 4 \\\\\n(z+1) x y = 4\n\\end{array}\n\\right.\n$$",
"role": "user"
}
] | (2, -2, -2) \quad \text{and} \quad \left(\frac{1}{3}, 3, 3\right) |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.\na, b, c, and d can complete a piece of work in 24, 6, 12, and 10 days respectively. Working together, how long will they take to complete the same work?",
"role": "user"
}
] | 2.55 |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.\nLet $a_{1}, a_{2}, \\cdots, a_{n}, b_{1}, b_{2}, \\cdots, b_{n}$ be complex numbers. Does there always exist a $k \\in \\{1, 2, \\cdots, n\\}$ such that\n$$\n\\sum_{i=1}^{n}\\left|a_{i} - a_{k}\\right|^{2} \\leqslant \\sum_{i=1}^{n}\\left|b_{i} - a_{k}\\right|^{2} ?\n$$",
"role": "user"
}
] | k \text{ does not necessarily exist. |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.\nGiven proposition $p$: The equation $\\frac{x^2}{2m}-\\frac{y^2}{m-1}=1$ represents an ellipse with foci on the $y$-axis; proposition $q$: The eccentricity $e$ of the hyperbola $\\frac{y^2}{5}-\\frac{x^2}{m}=1$ is in the range $(1,2)$. If $p \\vee q$ is true, find the range of the real number $m$.",
"role": "user"
}
] | (0,15) |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.\nA certain clothing store purchases a batch of clothing at a price of $a$ yuan each, and sells them at a markup of $50\\%$. The final two pieces of clothing are sold at a discount of $20\\%$ off the selling price. The selling price of the final two pieces of clothing is ______ yuan. (Express your answer in terms of $a$ using algebraic expressions)",
"role": "user"
}
] | 2.4a |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.\nGiven that $S_{n}$ is the sum of the first $n$ terms of the sequence ${a_{n}}$, and $S_{3}=1$, $S_{4}=11$, $a_{n+3}=2a_{n}(n∈N^{∗})$, find $S_{3n+1}=$\\_\\_\\_\\_\\_\\_.",
"role": "user"
}
] | 3×2^{n+1}-1 |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.\nBrenda is a vet who needs to spay some cats and dogs. She needs to spay 21 animals total today, and she needs to spay 7 cats. What is the ratio of the number of dogs she needs to spay to the number of cats she needs to spay?",
"role": "user"
}
] | 2:1 |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.\nA spring, when a 100 N weight is hung on it, stretches by 20 cm. When a 150 N weight is hung on it, the spring stretches by ______ cm.",
"role": "user"
}
] | 30 \text{ cm} |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.\nA shopkeeper sold some articles at the cost price of 12 articles and made a 20% profit. How many articles did the shopkeeper sell?",
"role": "user"
}
] | 15 |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.\nIn $\\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $\\dfrac {2b- \\sqrt {3}c}{ \\sqrt {3}a}= \\dfrac {\\cos C}{\\cos A}$, find the measure of angle $A$.",
"role": "user"
}
] | \dfrac{\pi}{6} |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.\nGiven the function $f(x)=|\\log(x+1)|$, and real numbers $a$, $b$ satisfying $a < b$ and $f(a)=f\\left(-\\frac{b+1}{b+2}\\right)$, determine the value of $a+b$ when $f(8a+2b+11)$ takes its minimum value.",
"role": "user"
}
] | a+b=-\frac{1}{2} |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.\nIf x and y are sets of integers, x # y denotes the set of integers that belong to set x or set y, but not both. If x consists of 8 integers, y consists of 18 integers, and z consists of 12 integers, with 6 of the integers being in both x and y, 4 of the integers being in both y and z, and 3 of the integers being in both x and z, then how many integers are in (x # y) # z? Assume that there is no integer present in all three sets x, y, and z.",
"role": "user"
}
] | 19 |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.\n$QL$ is the angle bisector of triangle $PQR$, and $M$ is the circumcenter of triangle $PQL$. It turns out that points $M$ and $L$ are symmetric with respect to line $PQ$. Find the angles of triangle $PQL$.",
"role": "user"
}
] | \angle PLQ = 120^{\circ}, \angle QPL = 30^{\circ}, \angle PQL = 30^{\circ} |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.\nGiven a hyperbola $\\frac{x^{2}}{a^{2}} - \\frac{y^{2}}{b^{2}} = 1$ whose one asymptote intersects the parabola $y=x^{2}+1$ at only one point, find the eccentricity of the hyperbola.",
"role": "user"
}
] | e = \sqrt{5} |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.\nMr. Salazar had seven dozen oranges. He reserved 1/4 of it for his first friend and another 1/6 for his second friend. Then, he was able to sell 3/7 of the remaining oranges yesterday. Today, he found that birds had eaten 1/10 of the remaining oranges and that four of the oranges had gone rotten. How many oranges are left to be sold today?",
"role": "user"
}
] | 22 |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.\nthe side of a rhombus is 24 m and length of one of its diagonals is 18 m . the area of the rhombus is ?",
"role": "user"
}
] | 400.97 |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.\nIn a factory, there are 500 employees. 300 of them earn $15 per hour for the first 8 hours, and time and a half for any hours beyond that. Of the remaining employees, 100 of them earn $18 per hour for a 10-hour shift, with double time for hours beyond that. The remaining 100 employees earn $20 per hour, work for 8 hours, but also receive an additional flat rate of $50 per day for health benefits, irrespective of the number of hours they work. If all the employees worked for 12 hours, what is the cost to employ all these people for one day?",
"role": "user"
}
] | $109,200 |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.\nIn Anchuria, the presidential election is being prepared, and President Miraflores wants to win. Exactly half of the numerous voters support Miraflores, and the other half support Dick Maloney. Miraflores himself is also a voter. By law, he has the right to divide all voters into two voting districts at his discretion. In each district, the voting procedure is as follows: each voter marks the name of their candidate on the ballot; all ballots are placed in an urn. Then a random ballot is drawn from the urn, and the candidate whose name is on it wins in that district. A candidate wins the election only if they win in both districts. If no winner is determined, the next round of voting is scheduled under the same rules. How should Miraflores divide the voters to maximize his chance of winning in the first round?",
"role": "user"
}
] | Place himself alone in one district and the other electors in the second district. |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.\nIn a grid with an $m \\times n$ rectangular section, what is the largest perimeter of a non-self-intersecting lattice polygon if every side of the polygon is either vertical or horizontal?",
"role": "user"
}
] | k = (m + 1)(n + 1) + \frac{(-1)^{(m+1)(n+1)} - 1}{2} |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.\nIn a village in Sri Lanka, a certain percentage of people died by bombardment, 15% of the remainder left the village on account of fear. The population is now reduced to 3553. If the population in the beginning was 4543, what percentage of people died by bombardment?",
"role": "user"
}
] | 7.99\% |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.\nA solution with a certain percentage of salt by weight had one fourth of it replaced by a second solution resulting in a solution that was 16 percent sugar by weight. The second solution was 25 percent salt by weight. What was the percentage of salt by weight in the first solution?",
"role": "user"
}
] | 13\% |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.\nIn triangle $ABC$, the angle bisectors $AL$, $BM$, and $CN$ are drawn such that $\\angle ANM = \\angle ALC$. Find the radius of the circumcircle of triangle $LMN$, two sides of which are 3 and 4.",
"role": "user"
}
] | 2 |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.\nIn a game, Wang Liang drew the cards $7, 7, 7, 3$ and found that $7 + 7 + 7 + 3 = 24$. If a combination of 4 cards that adds up to 24 is called a \"friendly hand,\" how many different friendly hands have a largest number of 7?",
"role": "user"
}
] | 4 \text{ groups} |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.\nA tank with a capacity of 8000 gallons is 3/4 full. Daxton empty's the tank by 40% of the total volume of water in the tank to water his vegetable farm. He then fills the tank with a certain percentage of the volume of water remaining in the tank. The final volume of water in the tank is 4680 gallons. What percentage of the remaining water did Daxton use to refill the tank?",
"role": "user"
}
] | 30\% |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.\nIf a / b = 4 / 3, and the ratio of (3a + 2b) to a certain expression involving a and b is 3, what is that expression?",
"role": "user"
}
] | 2b |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.\nThe value of the definite integral $\\int_{0}^{-1} (\\sqrt{1-x^{2}}+x)dx$ is ______.",
"role": "user"
}
] | \frac{\pi}{4}+ \frac{1}{2} |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.\nGiven \\\\(a > 0\\\\), \\\\(b > 0\\\\), and the point \\\\((a,b)\\\\) is on the line \\\\(6x+3y-1=0\\\\), then the minimum value of \\\\( \\dfrac{1}{5a+2b}+ \\dfrac{2}{a+b}\\\\) is \\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_.",
"role": "user"
}
] | 3+2 \sqrt{2} |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.\nThe range of the function $y= \\frac {2e^{x}-1}{e^{x}+2}$ is $\\_\\_\\_\\_\\_\\_$",
"role": "user"
}
] | (- \frac {1}{2}, 2) |
[
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.\nA wire cube. Each edge of a wire cube has a resistance of 1 ohm. What is the resistance between two opposite vertices of this cube?",
"role": "user"
}
] | \frac{5}{6} \text{ Ohms} |
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