problem stringlengths 10 5.15k | answer dict |
|---|---|
A garden fence, similar to the one shown in the picture, had in each section (between two vertical posts) the same number of columns, and each vertical post (except for the two end posts) divided one of the columns in half. When we absentmindedly counted all the columns from end to end, counting two halves as one whole column, we found that there were a total of 1223 columns. We also noticed that the number of sections was 5 more than twice the number of whole columns in each section.
How many columns were there in each section? | {
"answer": "23",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a diagram, $\triangle ABC$ and $\triangle BDC$ are right-angled, with $\angle ABC = \angle BDC = 45^\circ$, and $AB = 16$. Determine the length of $BC$. | {
"answer": "8\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The school plans to set up two computer labs, each equipped with one teacher's computer and several student computers. In a standard lab, the teacher's computer costs 8000 yuan, and each student computer costs 3500 yuan; in an advanced lab, the teacher's computer costs 11500 yuan, and each student computer costs 7000 yuan. It is known that the total investment for purchasing computers in both labs is equal and is between 200,000 yuan and 210,000 yuan. How many student computers should be prepared for each lab? | {
"answer": "27",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine the value of
\[3003 + \frac{1}{3} \left( 3002 + \frac{1}{3} \left( 3001 + \dots + \frac{1}{3} \left( 4 + \frac{1}{3} \cdot 3 \right) \right) \dotsb \right).\] | {
"answer": "9006",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A bag contains four balls, each labeled with one of the characters "美", "丽", "惠", "州". Balls are drawn with replacement until both "惠" and "州" are drawn, at which point the drawing stops. Use a random simulation method to estimate the probability that the drawing stops exactly on the third draw. Use a computer to randomly generate integer values between 0 and 3, with 0, 1, 2, and 3 representing "惠", "州", "美", and "丽" respectively. Each group of three random numbers represents the result of three draws. The following 16 groups of random numbers were generated:
232 321 230 023 123 021 132 220
231 130 133 231 331 320 122 233
Estimate the probability that the drawing stops exactly on the third draw. | {
"answer": "\\frac{1}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given an ellipse $E:\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1(a>b>0)$ with a major axis length of $4$, and the point $P(1,\frac{3}{2})$ lies on the ellipse $E$. <br/>$(1)$ Find the equation of the ellipse $E$; <br/>$(2)$ A line $l$ passing through the right focus $F$ of the ellipse $E$ is drawn such that it does not coincide with the two coordinate axes. The line intersects $E$ at two distinct points $M$ and $N$. The perpendicular bisector of segment $MN$ intersects the $y$-axis at point $T$. Find the minimum value of $\frac{|MN|}{|OT|}$ (where $O$ is the origin) and determine the equation of line $l$ at this point. | {
"answer": "24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A fly is being chased by three spiders on the edges of a regular octahedron. The fly has a speed of $50$ meters per second, while each of the spiders has a speed of $r$ meters per second. The spiders choose their starting positions, and choose the fly's starting position, with the requirement that the fly must begin at a vertex. Each bug knows the position of each other bug at all times, and the goal of the spiders is for at least one of them to catch the fly. What is the maximum $c$ so that for any $r<c,$ the fly can always avoid being caught?
*Author: Anderson Wang* | {
"answer": "25",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( X = \{1, 2, \ldots, 98\} \). Call a subset of \( X \) good if it satisfies the following conditions:
1. It has 10 elements.
2. If it is partitioned in any way into two subsets of 5 elements each, then one subset has an element coprime to each of the other 4, and the other subset has an element which is not coprime to any of the other 4.
Find the smallest \( n \) such that any subset of \( X \) of \( n \) elements has a good subset. | {
"answer": "50",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A block of wood has the shape of a right circular cylinder with a radius of $8$ and a height of $10$. The entire surface of the block is painted red. Points $P$ and $Q$ are chosen on the edge of one of the circular faces such that the arc $\overarc{PQ}$ measures $180^\text{o}$. The block is then sliced in half along the plane passing through points $P$, $Q$, and the center of the cylinder, revealing a flat, unpainted face on each half. Determine the area of one of these unpainted faces expressed in terms of $\pi$ and $\sqrt{d}$. Find $a + b + d$ where the area is expressed as $a \pi + b\sqrt{d}$, with $a$, $b$, $d$ as integers, and $d$ not being divisible by the square of any prime. | {
"answer": "193",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the regular octagon $ABCDEFGH$ with its center at $J$, and each of the vertices and the center associated with the digits 1 through 9, with each digit used once, such that the sums of the numbers on the lines $AJE$, $BJF$, $CJG$, and $DJH$ are equal, determine the number of ways in which this can be done. | {
"answer": "1152",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the sequence of even counting numbers starting from $2$, find the sum of the first $3000$ terms and the sequence of odd counting numbers starting from $3$, find the sum of the first $3000$ terms, and then calculate their difference. | {
"answer": "-3000",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $\sin \alpha + \cos \alpha = \frac{1}{5}$, and $- \frac{\pi}{2} \leqslant \alpha \leqslant \frac{\pi}{2}$, find the value of $\tan \alpha$. | {
"answer": "- \\frac{3}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A polyhedron has faces that all either triangles or squares. No two square faces share an edge, and no two triangular faces share an edge. What is the ratio of the number of triangular faces to the number of square faces? | {
"answer": "4:3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A clothing retailer offered a discount of $\frac{1}{4}$ on all jackets tagged at a specific price. If the cost of the jackets was $\frac{2}{3}$ of the price they were actually sold for and considering this price included a sales tax of $\frac{1}{10}$, what would be the ratio of the cost to the tagged price?
**A)** $\frac{1}{3}$
**B)** $\frac{2}{5}$
**C)** $\frac{11}{30}$
**D)** $\frac{3}{10}$
**E)** $\frac{1}{2}$ | {
"answer": "\\frac{11}{30}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given an ellipse $C:\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b>0)$ with left and right foci $F_{1}$, $F_{2}$. There exists a point $M$ in the first quadrant of ellipse $C$ such that $|MF_{1}|=|F_{1}F_{2}|$. Line $F_{1}M$ intersects the $y$-axis at point $A$, and $F_{2}A$ bisects $\angle MF_{2}F_{1}$. Find the eccentricity of ellipse $C$. | {
"answer": "\\frac{\\sqrt{5} - 1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that point $P$ is a moving point on the parabola $y^2=2x$, find the minimum value of the sum of the distance from point $P$ to point $A(0,2)$ and the distance from $P$ to the directrix of the parabola. | {
"answer": "\\frac{\\sqrt{17}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine the value of $a^3 + b^3$ given that $a+b=12$ and $ab=20$, and also return the result for $(a+b-c)(a^3+b^3)$, where $c=a-b$. | {
"answer": "4032",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Square $ABCD$ has an area of $256$ square units. Point $E$ lies on side $\overline{BC}$ and divides it in the ratio $3:1$. Points $F$ and $G$ are the midpoints of $\overline{AE}$ and $\overline{DE}$, respectively. Given that quadrilateral $BEGF$ has an area of $48$ square units, what is the area of triangle $GCD$? | {
"answer": "48",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A solid right prism $PQRSTU$ has a height of 20. Its bases are equilateral triangles with side length 10. Points $M$, $N$, and $O$ are the midpoints of edges $PQ$, $QR$, and $RT$, respectively. Calculate the perimeter of triangle $MNO$. | {
"answer": "5 + 10\\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given an ellipse with the equation $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1 (a > b > 0)$ and an eccentricity of $\frac{1}{2}$. $F\_1$ and $F\_2$ are the left and right foci of the ellipse, respectively. A line $l$ passing through $F\_2$ intersects the ellipse at points $A$ and $B$. The perimeter of $\triangle F\_1AB$ is $8$.
(I) Find the equation of the ellipse.
(II) If the slope of line $l$ is $0$, and its perpendicular bisector intersects the $y$-axis at $Q$, find the range of the $y$-coordinate of $Q$.
(III) Determine if there exists a point $M(m, 0)$ on the $x$-axis such that the $x$-axis bisects $\angle AMB$. If it exists, find the value of $m$; otherwise, explain the reason. | {
"answer": "m = 4",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Triangle $ABC$ is equilateral with side length $12$ . Point $D$ is the midpoint of side $\overline{BC}$ . Circles $A$ and $D$ intersect at the midpoints of side $AB$ and $AC$ . Point $E$ lies on segment $\overline{AD}$ and circle $E$ is tangent to circles $A$ and $D$ . Compute the radius of circle $E$ .
*2015 CCA Math Bonanza Individual Round #5* | {
"answer": "3\\sqrt{3} - 6",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A hemisphere is placed on a sphere of radius \(100 \text{ cm}\). The second hemisphere is placed on the first one, and the third hemisphere is placed on the second one (as shown below). Find the maximum height of the tower (in cm). | {
"answer": "300",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the equation $x^{2}+4ax+3a+1=0 (a > 1)$, whose two roots are $\tan \alpha$ and $\tan \beta$, with $\alpha, \beta \in (-\frac{\pi}{2}, \frac{\pi}{2})$, find $\tan \frac{\alpha + \beta}{2}$. | {
"answer": "-2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the value of $2 \cdot \sqrt[4]{2^7 + 2^7 + 2^8}$? | {
"answer": "8 \\cdot \\sqrt[4]{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a point $P$ on the curve $y= \frac{1}{2}e^{x}$ and a point $Q$ on the curve $y=\ln (2x)$, determine the minimum value of $|PQ|$. | {
"answer": "\\sqrt{2}(1-\\ln 2)",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A square is completely covered by a large circle and each corner of the square touches a smaller circle of radius \( r \). The side length of the square is 6 units. What is the radius \( R \) of the large circle? | {
"answer": "3\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $a,$ $b,$ and $c$ be three positive real numbers whose sum is 1. If no one of these numbers is more than three times any other, find the minimum value of the product $abc.$ | {
"answer": "\\frac{9}{343}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Point $F$ is taken on the extension of side $AD$ of parallelogram $ABCD$. $BF$ intersects diagonal $AC$ at $E$ and side $DC$ at $G$. If $EF = 40$ and $GF = 30$, find the length of $BE$. | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the largest solution to \[
\lfloor x \rfloor = 7 + 150 \{ x \},
\] where $\{x\} = x - \lfloor x \rfloor$. | {
"answer": "156.9933",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $\triangle ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $2a\sin B= \sqrt{3}b$ and $\cos C = \frac{5}{13}$:
(1) Find the value of $\sin A$;
(2) Find the value of $\cos B$. | {
"answer": "\\frac{12\\sqrt{3} - 5}{26}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In rectangle $LMNO$, points $P$ and $Q$ quadruple $\overline{LN}$, and points $R$ and $S$ quadruple $\overline{MO}$. Point $P$ is at $\frac{1}{4}$ the length of $\overline{LN}$ from $L$, and point $Q$ is at $\frac{1}{4}$ length from $P$. Similarly, $R$ is $\frac{1}{4}$ the length of $\overline{MO}$ from $M$, and $S$ is $\frac{1}{4}$ length from $R$. Given $LM = 4$, and $LO = MO = 3$. Find the area of quadrilateral $PRSQ$. | {
"answer": "0.75",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x) = \frac{\ln x}{x}$.
(1) Find the monotonic intervals of the function $f(x)$;
(2) Let $m > 0$, find the maximum value of $f(x)$ on $[m, 2m]$. | {
"answer": "\\frac{1}{e}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $ABC$, $AB = 12$, $AC = 10$, and $BC = 16$. The centroid $G$ of triangle $ABC$ divides each median in the ratio $2:1$. Calculate the length $GP$, where $P$ is the foot of the perpendicular from point $G$ to side $BC$. | {
"answer": "\\frac{\\sqrt{3591}}{24}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many positive integers, not exceeding 200, are multiples of 5 or 7 but not 10? | {
"answer": "43",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given vectors $\overrightarrow {a}, \overrightarrow {b}$ that satisfy $\overrightarrow {a}\cdot ( \overrightarrow {a}+ \overrightarrow {b})=5$, and $|\overrightarrow {a}|=2$, $|\overrightarrow {b}|=1$, find the angle between vectors $\overrightarrow {a}$ and $\overrightarrow {b}$. | {
"answer": "\\frac{\\pi}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given plane vectors $\vec{a}, \vec{b}, \vec{c}$ that satisfy the following conditions: $|\vec{a}| = |\vec{b}| \neq 0$, $\vec{a} \perp \vec{b}$, $|\vec{c}| = 2 \sqrt{2}$, and $|\vec{c} - \vec{a}| = 1$, determine the maximum possible value of $|\vec{a} + \vec{b} - \vec{c}|$. | {
"answer": "3\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are 4 willow trees and 4 poplar trees planted in a row. How many ways can they be planted alternately? | {
"answer": "1152",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a pyramid-like structure with a rectangular base consisting of $4$ apples by $7$ apples, each apple above the first level resting in a pocket formed by four apples below, and the stack topped off with a single row of apples, determine the total number of apples in the stack. | {
"answer": "60",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Last year, Australian Suzy Walsham won the annual women's race up the 1576 steps of the Empire State Building in New York for a record fifth time. Her winning time was 11 minutes 57 seconds. Approximately how many steps did she climb per minute? | {
"answer": "130",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the domains of functions f(x) and g(x) are both R, and f(x) + g(2-x) = 5, g(x) - f(x-4) = 7. If the graph of y=g(x) is symmetric about the line x=2, g(2) = 4, find the sum of the values of f(k) for k from 1 to 22. | {
"answer": "-24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In our daily life, we often use passwords, such as when making payments through Alipay. There is a type of password generated using the "factorization" method, which is easy to remember. The principle is to factorize a polynomial. For example, the polynomial $x^{3}+2x^{2}-x-2$ can be factorized as $\left(x-1\right)\left(x+1\right)\left(x+2\right)$. When $x=29$, $x-1=28$, $x+1=30$, $x+2=31$, and the numerical password obtained is $283031$.<br/>$(1)$ According to the above method, when $x=15$ and $y=5$, for the polynomial $x^{3}-xy^{2}$, after factorization, what numerical passwords can be formed?<br/>$(2)$ Given a right-angled triangle with a perimeter of $24$, a hypotenuse of $11$, and the two legs being $x$ and $y$, find a numerical password obtained by factorizing the polynomial $x^{3}y+xy^{3}$ (only one is needed). | {
"answer": "24121",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Last month, Xiao Ming's household expenses were 500 yuan for food, 200 yuan for education, and 300 yuan for other expenses. This month, the costs of these three categories increased by 6%, 20%, and 10%, respectively. What is the percentage increase in Xiao Ming's household expenses for this month compared to last month? | {
"answer": "10\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
An $10 \times 25$ rectangle is divided into two congruent polygons, and these polygons are rearranged to form a rectangle again. Determine the length of the smaller side of the resulting rectangle. | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $P$ be the portion of the graph of $$ y=\frac{6x+1}{32x+8} - \frac{2x-1}{32x-8} $$ located in the first quadrant (not including the $x$ and $y$ axes). Let the shortest possible distance between the origin and a point on $P$ be $d$ . Find $\lfloor 1000d \rfloor$ .
*Proposed by **Th3Numb3rThr33*** | {
"answer": "433",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many positive integer multiples of $77$ can be expressed in the form $10^{j} - 10^{i}$, where $i$ and $j$ are integers and $0 \leq i < j \leq 49$? | {
"answer": "182",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let the set $I = \{1,2,3,4,5\}$. Choose two non-empty subsets $A$ and $B$ from $I$. How many different ways are there to choose $A$ and $B$ such that the smallest number in $B$ is greater than the largest number in $A$? | {
"answer": "49",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a box contains a total of 180 marbles, 25% are silver, 20% are gold, 15% are bronze, 10% are sapphire, and 10% are ruby, and the remainder are diamond marbles. If 10% of the gold marbles are removed, calculate the number of marbles left in the box. | {
"answer": "176",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the digits 1, 2, 3, 7, 8, 9, find the smallest sum of two 3-digit numbers that can be obtained by placing each of these digits in one of the six boxes in the given addition problem, with the condition that each number must contain one digit from 1, 2, 3 and one digit from 7, 8, 9. | {
"answer": "417",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that \( r, s, t \) are integers, and the set \( \{a \mid a = 2^r + 2^s + 2^t, 0 \leq t < s < r\} \) forms a sequence \(\{a_n\} \) from smallest to largest as \(7, 11, 13, 14, \cdots\), find \( a_{36} \). | {
"answer": "131",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two different natural numbers are selected from the set $\{1, 2, 3, \ldots, 8\}$. What is the probability that the greatest common factor of these two numbers is one? Express your answer as a common fraction. | {
"answer": "\\frac{5}{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The 12 numbers from 1 to 12 on a clock face divide the circumference into 12 equal parts. Using any 4 of these division points as vertices to form a quadrilateral, find the total number of rectangles that can be formed. | {
"answer": "15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If a four-digit natural number $\overline{abcd}$ has digits that are all different and not equal to $0$, and satisfies $\overline{ab}-\overline{bc}=\overline{cd}$, then this four-digit number is called a "decreasing number". For example, the four-digit number $4129$, since $41-12=29$, is a "decreasing number"; another example is the four-digit number $5324$, since $53-32=21\neq 24$, is not a "decreasing number". If a "decreasing number" is $\overline{a312}$, then this number is ______; if the sum of the three-digit number $\overline{abc}$ formed by the first three digits and the three-digit number $\overline{bcd}$ formed by the last three digits of a "decreasing number" is divisible by $9$, then the maximum value of the number that satisfies the condition is ______. | {
"answer": "8165",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the Cartesian coordinate system, the center of circle $C$ is at $(2,0)$, and its radius is $\sqrt{2}$. Establish a polar coordinate system with the origin as the pole and the positive half-axis of $x$ as the polar axis. The parametric equation of line $l$ is:
$$
\begin{cases}
x=-t \\
y=1+t
\end{cases} \quad (t \text{ is a parameter}).
$$
$(1)$ Find the polar coordinate equations of circle $C$ and line $l$;
$(2)$ The polar coordinates of point $P$ are $(1,\frac{\pi}{2})$, line $l$ intersects circle $C$ at points $A$ and $B$, find the value of $|PA|+|PB|$. | {
"answer": "3\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, point $E$ is on $AB$, point $F$ is on $AC$, and $BF$ intersects $CE$ at point $P$. If the areas of quadrilateral $AEPF$ and triangles $BEP$ and $CFP$ are all equal to 4, what is the area of $\triangle BPC$? | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the parabola $y^2=4x$ and the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1 (a > 0, b > 0)$ have the same focus $F$, $O$ is the coordinate origin, points $A$ and $B$ are the intersection points of the two curves. If $(\overrightarrow{OA} + \overrightarrow{OB}) \cdot \overrightarrow{AF} = 0$, find the length of the real axis of the hyperbola. | {
"answer": "2\\sqrt{2}-2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A company buys an assortment of 150 pens from a catalog for \$15.00. Shipping costs an additional \$5.50. Furthermore, they receive a 10% discount on the total price due to a special promotion. What is the average cost, in cents, for each pen? | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A woman buys a property for $150,000 with a goal to achieve a $7\%$ annual return on her investment. She sets aside $15\%$ of each month's rent for maintenance costs, and pays property taxes at $0.75\%$ of the property's value each year. Calculate the monthly rent she needs to charge to meet her financial goals. | {
"answer": "1139.71",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A farmer has a right-angled triangular farm with legs of lengths 3 and 4. At the right-angle corner, the farmer leaves an unplanted square area $S$. The shortest distance from area $S$ to the hypotenuse of the triangle is 2. What is the ratio of the area planted with crops to the total area of the farm? | {
"answer": "$\\frac{145}{147}$",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the diagram, \( PQR \) is a straight line segment and \( QS = QT \). Also, \( \angle PQS = x^\circ \) and \( \angle TQR = 3x^\circ \). If \( \angle QTS = 76^\circ \), the value of \( x \) is: | {
"answer": "38",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the arithmetic mean of the reciprocals of the first four prime numbers, including the number 7 instead of 5. | {
"answer": "\\frac{493}{1848}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Four-digit "progressive numbers" are arranged in ascending order, determine the 30th number. | {
"answer": "1359",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A cooperative farm can purchase two types of feed mixtures from a neighboring farm to feed its animals. The Type I feed costs $30 per sack and contains 10 kg of component A and 10 kg of component B. The Type II feed costs $50 per sack and contains 10 kg of component A, 20 kg of component B, and 5 kg of component C. It has been determined that for healthy development of the animals, the farm needs at least 45 kg of component A, 60 kg of component B, and 5 kg of component C daily. How much of each feed mixture should they purchase to minimize the cost while meeting the nutritional requirements? | {
"answer": "165",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two adjacent faces of a tetrahedron, representing equilateral triangles with side length 3, form a dihedral angle of 30 degrees. The tetrahedron rotates around the common edge of these faces. Find the maximum area of the projection of the rotating tetrahedron onto a plane containing this edge. | {
"answer": "\\frac{9\\sqrt{3}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the diagram, $\triangle ABC$ is isosceles with $AB = AC$ and $BC = 30 \mathrm{~cm}$. Square $EFGH$, which has a side length of $12 \mathrm{~cm}$, is inscribed in $\triangle ABC$, as shown. The area of $\triangle AEF$, in $\mathrm{cm}^{2}$, is | {
"answer": "48",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the parametric equation of curve \\(C_{1}\\) as \\(\begin{cases}x=3\cos \alpha \\ y=\sin \alpha\end{cases} (\alpha\\) is the parameter\\()\\), and taking the origin \\(O\\) of the Cartesian coordinate system \\(xOy\\) as the pole and the positive half-axis of \\(x\\) as the polar axis to establish a polar coordinate system, the polar equation of curve \\(C_{2}\\) is \\(\rho\cos \left(\theta+ \dfrac{\pi}{4}\right)= \sqrt{2} \\).
\\((\\)Ⅰ\\()\\) Find the Cartesian equation of curve \\(C_{2}\\) and the maximum value of the distance \\(|OP|\\) from the moving point \\(P\\) on curve \\(C_{1}\\) to the origin \\(O\\);
\\((\\)Ⅱ\\()\\) If curve \\(C_{2}\\) intersects curve \\(C_{1}\\) at points \\(A\\) and \\(B\\), and intersects the \\(x\\)-axis at point \\(E\\), find the value of \\(|EA|+|EB|\\). | {
"answer": "\\dfrac{6 \\sqrt{3}}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $\binom{18}{8}=31824$, $\binom{18}{9}=48620$, and $\binom{18}{10}=43758$, calculate $\binom{20}{10}$. | {
"answer": "172822",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a sequence $\{a_n\}$ that satisfies $a_n-(-1)^n a_{n-1}=n$ ($n\geqslant 2, n\in \mathbb{N}^*$), and $S_n$ is the sum of the first $n$ terms of $\{a_n\}$, then $S_{40}=$_______. | {
"answer": "440",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a Go championship participated by three players: A, B, and C, the matches are conducted according to the following rules: the first match is between A and B; the second match is between the winner of the first match and C; the third match is between the winner of the second match and the loser of the first match; the fourth match is between the winner of the third match and the loser of the second match. Based on past records, the probability of A winning over B is 0.4, B winning over C is 0.5, and C winning over A is 0.6.
(1) Calculate the probability of B winning four consecutive matches to end the competition;
(2) Calculate the probability of C winning three consecutive matches to end the competition. | {
"answer": "0.162",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A club consists of five leaders and some regular members. Each year, all leaders leave the club and each regular member recruits three new people to join as regular members. Subsequently, five new leaders are elected from outside the club to join. Initially, there are eighteen people in total in the club. How many people will be in the club after five years? | {
"answer": "3164",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Gauss is a famous German mathematician, known as the "Prince of Mathematics". There are 110 achievements named after his name "Gauss". For $x\in R$, let $[x]$ represent the largest integer not greater than $x$, and let $\{x\}=x-[x]$ represent the non-negative fractional part of $x$. Then, $y=[x]$ is called the Gauss function. It is known that the sequence $\{a_{n}\}$ satisfies: $a_{1}=\sqrt{3}$, $a_{n+1}=[a_{n}]+\frac{1}{\{a_{n}\}}$, $(n∈N^{*})$, then $a_{2019}=$\_\_\_\_\_\_\_\_. | {
"answer": "3027+ \\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The altitudes of an acute isosceles triangle, where \(AB = BC\), intersect at point \(H\). Find the area of triangle \(ABC\), given \(AH = 5\) and the altitude \(AD\) is 8. | {
"answer": "40",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If the equation with respect to \( x \), \(\frac{x \lg^2 a - 1}{x + \lg a} = x\), has a solution set that contains only one element, then \( a \) equals \(\quad\) . | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For any sequence of real numbers $A=\{a_1, a_2, a_3, \ldots\}$, define $\triangle A$ as the sequence $\{a_2 - a_1, a_3 - a_2, a_4 - a_3, \ldots\}$, where the $n$-th term is $a_{n+1} - a_n$. Assume that all terms of the sequence $\triangle (\triangle A)$ are $1$ and $a_{18} = a_{2017} = 0$, find the value of $a_{2018}$. | {
"answer": "1000",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The line $l_1$: $x+my+6=0$ is parallel to the line $l_2$: $(m-2)x+3y+2m=0$. Find the value of $m$. | {
"answer": "-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Estimate the population of Island X in the year 2045, given that the population doubles every 15 years and the population in 2020 was 500. | {
"answer": "1587",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two semicircles, each with radius \(\sqrt{2}\), are tangent to each other. If \( AB \parallel CD \), determine the length of segment \( AD \). | {
"answer": "4\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Five years from now, Billy's age will be twice Joe's current age. Currently, the sum of their ages is 60. How old is Billy right now? | {
"answer": "38\\frac{1}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the triangular pyramid $P-ABC$ where $PA\bot $ plane $ABC$, $PA=AB=2$, and $\angle ACB=30^{\circ}$, find the surface area of the circumscribed sphere of the triangular pyramid $P-ABC$. | {
"answer": "20\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $ABC$, $a=3$, $\angle C = \frac{2\pi}{3}$, and the area of $ABC$ is $\frac{3\sqrt{3}}{4}$. Find the lengths of sides $b$ and $c$. | {
"answer": "\\sqrt{13}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Four rectangular strips each measuring $4$ by $16$ inches are laid out with two vertical strips crossing two horizontal strips forming a single polygon which looks like a tic-tack-toe pattern. What is the perimeter of this polygon?
[asy]
size(100);
draw((1,0)--(2,0)--(2,1)--(3,1)--(3,0)--(4,0)--(4,1)--(5,1)--(5,2)--(4,2)--(4,3)--(5,3)--(5,4)--(4,4)--(4,5)--(3,5)--(3,4)--(2,4)--(2,5)--(1,5)--(1,4)--(0,4)--(0,3)--(1,3)--(1,2)--(0,2)--(0,1)--(1,1)--(1,0));
draw((2,2)--(2,3)--(3,3)--(3,2)--cycle);
[/asy] | {
"answer": "80",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A cowboy is 6 miles south of a stream which flows due east. He is also 12 miles west and 10 miles north of his cabin. Before returning to his cabin, he wishes to fill his water barrel from the stream and also collect firewood 5 miles downstream from the point directly opposite his starting point. Find the shortest distance (in miles) he can travel to accomplish all these tasks.
A) $11 + \sqrt{276}$
B) $11 + \sqrt{301}$
C) $11 + \sqrt{305}$
D) $12 + \sqrt{280}$ | {
"answer": "11 + \\sqrt{305}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Solve the equations:
1. $2x^{2}+4x+1=0$ (using the method of completing the square)
2. $x^{2}+6x=5$ (using the formula method) | {
"answer": "-3-\\sqrt{14}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given point $F(0,1)$, moving point $M$ lies on the line $l:y=-1$. The line passing through point $M$ and perpendicular to the $x$-axis intersects the perpendicular bisector of segment $MF$ at point $P$. Let the locus of point $P$ be curve $C$.
$(1)$ Find the equation of curve $C$;
$(2)$ Given that the circle $x^{2}+(y+2)^{2}=4$ has a diameter $AB$, extending $AO$ and $BO$ intersect curve $C$ at points $S$ and $T$ respectively, find the minimum area of quadrilateral $ABST$. | {
"answer": "36",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $2$ red and $2$ white balls, a total of $4$ balls are randomly arranged in a row. The probability that balls of the same color are adjacent to each other is $\_\_\_\_\_\_$. | {
"answer": "\\frac{1}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=\sin \left( \frac {5\pi}{6}-2x\right)-2\sin \left(x- \frac {\pi}{4}\right)\cos \left(x+ \frac {3\pi}{4}\right).$
$(1)$ Find the minimum positive period and the intervals of monotonic increase for the function $f(x)$;
$(2)$ If $x_{0}\in\left[ \frac {\pi}{3}, \frac {7\pi}{12}\right]$ and $f(x_{0})= \frac {1}{3}$, find the value of $\cos 2x_{0}.$ | {
"answer": "- \\frac {2 \\sqrt {6}+1}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On the edges \(AB\), \(BC\), and \(AD\) of the tetrahedron \(ABCD\), points \(K\), \(N\), and \(M\) are chosen, respectively, such that \(AK:KB = BN:NC = 2:1\) and \(AM:MD = 3:1\). Construct the section of the tetrahedron by the plane passing through points \(K\), \(M\), and \(N\). In what ratio does this plane divide edge \(CD\)? | {
"answer": "4:3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The arithmetic sequence \( a, a+d, a+2d, a+3d, \ldots, a+(n-1)d \) has the following properties:
- When the first, third, fifth, and so on terms are added, up to and including the last term, the sum is 320.
- When the first, fourth, seventh, and so on, terms are added, up to and including the last term, the sum is 224.
What is the sum of the whole sequence? | {
"answer": "608",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
When Person A has traveled 100 meters, Person B has traveled 50 meters. When Person A reaches point $B$, Person B is still 100 meters away from $B$. Person A immediately turns around and heads back towards $A$, and they meet 60 meters from point $B$. What is the distance between points $A$ and $B$ in meters? | {
"answer": "300",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, if $bc=3$, $a=2$, then the minimum value of the area of the circumcircle of $\triangle ABC$ is $\_\_\_\_\_\_$. | {
"answer": "\\frac{9\\pi}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Compute the number of ordered pairs of integers $(x,y)$ with $1\le x<y\le 50$ such that $i^x+i^y$ is a real number, and additionally, $x+y$ is divisible by $4$. | {
"answer": "288",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the line $2mx+ny-4=0$ passes through the point of intersection of the function $y=\log _{a}(x-1)+2$ where $a>0$ and $a\neq 1$, find the minimum value of $\frac{1}{m}+\frac{4}{n}$. | {
"answer": "3+2\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Five cards with different numbers are given: $-5$, $-4$, $0$, $+4$, $+6$. Two cards are drawn from them. The smallest quotient obtained by dividing the numbers on these two cards is ____. | {
"answer": "-\\dfrac{3}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the lines $l_{1}$: $x+ay-a+2=0$ and $l_{2}$: $2ax+(a+3)y+a-5=0$.
$(1)$ When $a=1$, find the coordinates of the intersection point of lines $l_{1}$ and $l_{2}$.
$(2)$ If $l_{1}$ is parallel to $l_{2}$, find the value of $a$. | {
"answer": "a = \\frac{3}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the game of *Winners Make Zeros*, a pair of positive integers $(m,n)$ is written on a sheet of paper. Then the game begins, as the players make the following legal moves:
- If $m\geq n$ , the player choose a positive integer $c$ such that $m-cn\geq 0$ , and replaces $(m,n)$ with $(m-cn,n)$ .
- If $m<n$ , the player choose a positive integer $c$ such that $n-cm\geq 0$ , and replaces $(m,n)$ with $(m,n-cm)$ .
When $m$ or $n$ becomes $0$ , the game ends, and the last player to have moved is declared the winner. If $m$ and $n$ are originally $2007777$ and $2007$ , find the largest choice the first player can make for $c$ (on his first move) such that the first player has a winning strategy after that first move. | {
"answer": "999",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the numbers $12534, 25341, 53412, 34125$, calculate their sum. | {
"answer": "125412",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $a < b < c < d < e$ be real numbers. We calculate all possible sums in pairs of these 5 numbers. Of these 10 sums, the three smaller ones are 32, 36, 37, while the two larger ones are 48 and 51. Determine all possible values that $e$ can take. | {
"answer": "27.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a mathematics competition conducted at a school, the scores $X$ of all participating students approximately follow the normal distribution $N(70, 100)$. It is known that there are 16 students with scores of 90 and above (inclusive of 90).
(1) What is the approximate total number of students who participated in the competition?
(2) If the school plans to reward students who scored 80 and above (inclusive of 80), how many students are expected to receive a reward in this competition?
Note: $P(|X-\mu| < \sigma)=0.683$, $P(|X-\mu| < 2\sigma)=0.954$, $P(|X-\mu| < 3\sigma)=0.997$. | {
"answer": "110",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Alli rolls a standard $8$-sided die twice. What is the probability of rolling integers that differ by $3$ on her first two rolls? Express your answer as a common fraction. | {
"answer": "\\frac{1}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the diagram below, $AB = 30$ and $\angle ADB = 90^\circ$. If $\sin A = \frac{3}{5}$ and $\sin C = \frac{1}{4}$, what is the length of $DC$? | {
"answer": "18\\sqrt{15}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A point $P(\frac{\pi}{12}, m)$ on the graph of the function $y = \sin 2x$ can be obtained by shifting a point $Q$ on the graph of the function $y = \cos (2x - \frac{\pi}{4})$ to the left by $n (n > 0)$ units. Determine the minimum value of $n \cdot m$. | {
"answer": "\\frac{5\\pi}{48}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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