problem stringlengths 10 5.15k | answer dict |
|---|---|
Given that the magnitude of the star Altair is $0.75$ and the magnitude of the star Vega is $0$, determine the ratio of the luminosity of Altair to Vega. | {
"answer": "10^{-\\frac{3}{10}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that in triangle $\triangle ABC$, $a=2$, $\angle A=\frac{π}{6}$, $b=2\sqrt{3}$, find the measure of angle $\angle C$. | {
"answer": "\\frac{\\pi}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the parabola $C: x^{2}=2py\left(p \gt 0\right)$ with focus $F$, and the minimum distance between $F$ and a point on the circle $M: x^{2}+\left(y+4\right)^{2}=1$ is $4$.<br/>$(1)$ Find $p$;<br/>$(2)$ If point $P$ lies on $M$, $PA$ and $PB$ are two tangents to $C$ with points $A$ and $B$ as the points of tangency, find the maximum area of $\triangle PAB$. | {
"answer": "20\\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Arrange 5 people to be on duty from Monday to Friday, with each person on duty for one day and one person arranged for each day. The conditions are: A and B are not on duty on adjacent days, while B and C are on duty on adjacent days. The number of different arrangements is $\boxed{\text{answer}}$. | {
"answer": "36",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the ellipse $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1 (a > b > 0)$ with left and right foci $F\_1$ and $F\_2$, respectively. Point $A(4,2\sqrt{2})$ lies on the ellipse, and $AF\_2$ is perpendicular to the $x$-axis.
1. Find the equation of the ellipse.
2. A line passing through point $F\_2$ intersects the ellipse at points $B$ and $C$. Find the maximum area of triangle $COB$. | {
"answer": "8\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function \( f(x)=\frac{\sin (\pi x)-\cos (\pi x)+2}{\sqrt{x}} \) for \( \frac{1}{4} \leqslant x \leqslant \frac{5}{4} \), find the minimum value of \( f(x) \). | {
"answer": "\\frac{4\\sqrt{5}}{5} - \\frac{2\\sqrt{10}}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a tetrahedron V-ABC with edge length 10, point O is the center of the base ABC. Segment MN has a length of 2, with one endpoint M on segment VO and the other endpoint N inside face ABC. If point T is the midpoint of segment MN, then the area of the trajectory formed by point T is __________. | {
"answer": "2\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Three distinct diameters are drawn on a unit circle such that chords are drawn as shown. If the length of one chord is \(\sqrt{2}\) units and the other two chords are of equal lengths, what is the common length of these chords? | {
"answer": "\\sqrt{2-\\sqrt{2}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The Lions are competing against the Eagles in a seven-game championship series. The Lions have a probability of $\dfrac{2}{3}$ of winning a game whenever it rains and a probability of $\dfrac{1}{2}$ of winning when it does not rain. Assume it's forecasted to rain for the first three games and the remaining will have no rain. What is the probability that the Lions will win the championship series? Express your answer as a percent, rounded to the nearest whole percent. | {
"answer": "76\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Assuming that the clock hands move without jumps, determine how many minutes after the clock shows 8:00 will the minute hand catch up with the hour hand. | {
"answer": "43 \\frac{7}{11}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine the number of relatively prime dates in the month with the second fewest relatively prime dates. | {
"answer": "11",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $x \sim N(-1,36)$ and $P(-3 \leqslant \xi \leqslant -1) = 0.4$, calculate $P(\xi \geqslant 1)$. | {
"answer": "0.1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A man and a dog are walking. The dog waits for the man to start walking along a path and then runs to the end of the path and back to the man a total of four times, always moving at a constant speed. The last time the dog runs back to the man, it covers the remaining distance of 81 meters. The distance from the door to the end of the path is 625 meters. Find the speed of the dog if the man is walking at a speed of 4 km/h. | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let the set \( S = \{1, 2, 3, \cdots, 50\} \). Find the smallest positive integer \( n \) such that every subset of \( S \) with \( n \) elements contains three numbers that can be the side lengths of a right triangle. | {
"answer": "42",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find four consecutive odd numbers, none of which are divisible by 3, such that their sum is divisible by 5. What is the smallest possible value of this sum? | {
"answer": "40",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)= \begin{cases} \sqrt {x}+3,x\geqslant 0 \\ ax+b,x < 0 \end{cases}$ that satisfies the condition: for all $x_{1}∈R$ and $x_{1}≠ 0$, there exists a unique $x_{2}∈R$ and $x_{1}≠ x_{2}$ such that $f(x_{1})=f(x_{2})$, determine the value of the real number $a+b$ when $f(2a)=f(3b)$ holds true. | {
"answer": "-\\dfrac{\\sqrt{6}}{2}+3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the ratio of women to men is $7$ to $5$, and the average age of women is $30$ years and the average age of men is $35$ years, determine the average age of the community. | {
"answer": "32\\frac{1}{12}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that in a mathematics test, $20\%$ of the students scored $60$ points, $25\%$ scored $75$ points, $20\%$ scored $85$ points, $25\%$ scored $95$ points, and the rest scored $100$ points, calculate the difference between the mean and the median score of the students' scores on this test. | {
"answer": "6.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a triangle configuration, each row consists of increasing multiples of 3 unit rods. The number of connectors in a triangle always forms an additional row than the rods, with connectors enclosing each by doubling the requirements of connecting joints from the previous triangle. How many total pieces are required to build a four-row triangle? | {
"answer": "60",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A circle passes through the vertices $K$ and $P$ of triangle $KPM$ and intersects its sides $KM$ and $PM$ at points $F$ and $B$, respectively. Given that $K F : F M = 3 : 1$ and $P B : B M = 6 : 5$, find $K P$ given that $B F = \sqrt{15}$. | {
"answer": "2 \\sqrt{33}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If an irrational number $a$ multiplied by $\sqrt{8}$ is a rational number, write down one possible value of $a$ as ____. | {
"answer": "\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A mischievous child mounted the hour hand on the minute hand's axle and the minute hand on the hour hand's axle of a correctly functioning clock. The question is, how many times within a day does this clock display the correct time? | {
"answer": "22",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a dark room, a drawer contains 120 red socks, 100 green socks, 70 blue socks, and 50 black socks. A person selects socks one by one from the drawer without being able to see their color. What is the minimum number of socks that must be selected to ensure that at least 15 pairs of socks are selected, with no sock being counted in more than one pair? | {
"answer": "33",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine how many integer values of $n$ between 1 and 180 inclusive ensure that the decimal representation of $\frac{n}{180}$ terminates. | {
"answer": "60",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Tetrahedron $PQRS$ is such that $PQ=6$, $PR=5$, $PS=4\sqrt{2}$, $QR=3\sqrt{2}$, $QS=5$, and $RS=4$. Calculate the volume of tetrahedron $PQRS$.
**A)** $\frac{130}{9}$
**B)** $\frac{135}{9}$
**C)** $\frac{140}{9}$
**D)** $\frac{145}{9}$ | {
"answer": "\\frac{140}{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many points does one have to place on a unit square to guarantee that two of them are strictly less than 1/2 unit apart? | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $p,$ $q,$ $r,$ $s$ be distinct real numbers such that the roots of $x^2 - 12px - 13q = 0$ are $r$ and $s,$ and the roots of $x^2 - 12rx - 13s = 0$ are $p$ and $q.$ Find the value of $p + q + r + s.$ | {
"answer": "2028",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given any point $P$ on the ellipse $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1\; \; (a > b > 0)$ with foci $F\_{1}$ and $F\_{2}$, if $\angle PF\_1F\_2=\alpha$, $\angle PF\_2F\_1=\beta$, $\cos \alpha= \frac{ \sqrt{5}}{5}$, and $\sin (\alpha+\beta)= \frac{3}{5}$, find the eccentricity of this ellipse. | {
"answer": "\\frac{\\sqrt{5}}{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the expansion of $(1+\frac{a}{x}){{(2x-\frac{1}{x})}^{5}}$, find the constant term. | {
"answer": "80",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Xiao Ming arranges chess pieces in a two-layer hollow square array (the diagram shows the top left part of the array). After arranging the inner layer, 60 chess pieces are left. After arranging the outer layer, 32 chess pieces are left. How many chess pieces does Xiao Ming have in total? | {
"answer": "80",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $\{a_{n}\}$ be an integer sequence such that for any $n \in \mathbf{N}^{*}$, the condition \((n-1) a_{n+1} = (n+1) a_{n} - 2 (n-1)\) holds. Additionally, \(2008 \mid a_{2007}\). Find the smallest positive integer \(n \geqslant 2\) such that \(2008 \mid a_{n}\). | {
"answer": "501",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For which $x$ and $y$ is the number $x x y y$ a square of a natural number? | {
"answer": "7744",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
You have a whole cake in your pantry. On your first trip to the pantry, you eat one-third of the cake. On each successive trip, you eat one-third of the remaining cake. After four trips to the pantry, what fractional part of the cake have you eaten? | {
"answer": "\\frac{40}{81}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The line joining $(2,3)$ and $(5,1)$ divides the square shown into two parts. What fraction of the area of the square is above this line? The square has vertices at $(2,1)$, $(5,1)$, $(5,4)$, and $(2,4)$. | {
"answer": "\\frac{2}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
$n$ balls are placed independently uniformly at random into $n$ boxes. One box is selected at random, and is found to contain $b$ balls. Let $e_n$ be the expected value of $b^4$ . Find $$ \lim_{n \to
\infty}e_n. $$ | {
"answer": "15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the least possible sum of two positive integers $a$ and $b$ where $a \cdot b = 10! ?$ | {
"answer": "3960",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given an increasing sequence $\{a_{n}\}$ where all terms are positive integers, the sum of the first $n$ terms is $S_{n}$. If $a_{1}=3$ and $S_{n}=2023$, calculate the value of $a_{n}$ when $n$ takes its maximum value. | {
"answer": "73",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine the volume of the released gas:
\[ \omega\left(\mathrm{SO}_{2}\right) = n\left(\mathrm{SO}_{2}\right) \cdot V_{m} = 0.1122 \cdot 22.4 = 2.52 \text{ liters} \] | {
"answer": "2.52",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A pentagon is formed by placing an equilateral triangle on top of a rectangle. The side length of the equilateral triangle is equal to the width of the rectangle, and the height of the rectangle is twice the side length of the triangle. What percent of the area of the pentagon is the area of the equilateral triangle? | {
"answer": "\\frac{\\sqrt{3}}{\\sqrt{3} + 8} \\times 100\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a class with 21 students, such that at least two of any three students are friends, determine the largest possible value of k. | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $ABC$, $\angle C = 4\angle A$, $a = 36$, and $c = 60$. Determine the length of side $b$. | {
"answer": "45",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Evaluate $\sqrt[3]{1+27} \cdot \sqrt[3]{1+\sqrt[3]{27}}$. | {
"answer": "\\sqrt[3]{112}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are 7 cards, each with a number written on it: 1, 2, 2, 3, 4, 5, 6. Three cards are randomly drawn from these 7 cards. Let the minimum value of the numbers on the drawn cards be $\xi$. Find $P\left(\xi =2\right)=$____ and $E\left(\xi \right)=$____. | {
"answer": "\\frac{12}{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A rectangular piece of paper with dimensions 8 cm by 6 cm is folded in half horizontally. After folding, the paper is cut vertically at 3 cm and 5 cm from one edge, forming three distinct rectangles. Calculate the ratio of the perimeter of the smallest rectangle to the perimeter of the largest rectangle. | {
"answer": "\\frac{5}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Arrange the letters a, a, b, b, c, c into three rows and two columns, such that in each row and each column, the letters are different. How many different arrangements are there? | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the parallelepiped $ABCD-A_{1}B_{1}C_{1}D_{1}$, where $AB=4$, $AD=3$, $AA_{1}=3$, $\angle BAD=90^{\circ}$, $\angle BAA_{1}=60^{\circ}$, $\angle DAA_{1}=60^{\circ}$, find the length of $AC_{1}$. | {
"answer": "\\sqrt{55}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $n$ be an odd integer with exactly 12 positive divisors. Find the number of positive divisors of $27n^3$. | {
"answer": "256",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let's say you have three numbers A, B, and C, which satisfy the equations $2002C + 4004A = 8008$, and $3003B - 5005A = 7007$. What is the average of A, B, and C?
**A)** $\frac{7}{3}$
**B)** $7$
**C)** $\frac{26}{9}$
**D)** $\frac{22}{9}$
**E)** $0$ | {
"answer": "\\frac{22}{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On the board, the number 0 is written. Two players take turns appending to the expression on the board: the first player appends a + or - sign, and the second player appends one of the natural numbers from 1 to 1993. The players make 1993 moves each, and the second player uses each of the numbers from 1 to 1993 exactly once. At the end of the game, the second player receives a reward equal to the absolute value of the algebraic sum written on the board. What is the maximum reward the second player can guarantee for themselves? | {
"answer": "1993",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A collection of seven positive integers has a mean of 6, a unique mode of 4, and a median of 6. If a 12 is added to this collection, what is the new median? | {
"answer": "6.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Jo and Blair take turns counting numbers, starting with Jo who says "1". Blair follows by saying the next number in the sequence, which is normally the last number said by Jo plus one. However, every third turn, the speaker will say the next number plus two instead of one. Express the $53^{\text{rd}}$ number said in this new sequence. | {
"answer": "71",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the Cartesian coordinate plane $(xOy)$, the curve $y=x^{2}-6x+1$ intersects the coordinate axes at points that lie on circle $C$.
(1) Find the equation of circle $C$;
(2) Given point $A(3,0)$, and point $B$ is a moving point on circle $C$, find the maximum value of $\overrightarrow{OA} \cdot \overrightarrow{OB}$, and find the length of the chord cut by line $OB$ on circle $C$ at this time. | {
"answer": "\\frac{36\\sqrt{37}}{37}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( n \) be a fixed integer, \( n \geqslant 2 \).
(a) Determine the minimal constant \( c \) such that the inequality
$$
\sum_{1 \leqslant i < j \leqslant n} x_i x_j \left(x_i^2 + x_j^2\right) \leqslant c \left( \sum_{1 \leqslant i \leqslant n} x_i \right)^4
$$
holds for all non-negative real numbers \( x_1, x_2, \cdots, x_n \geqslant 0 \).
(b) For this constant \( c \), determine the necessary and sufficient conditions for equality to hold. | {
"answer": "\\frac{1}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In trapezoid \(A B C D\), the bases \(A D\) and \(B C\) are 8 and 18, respectively. It is known that the circumscribed circle of triangle \(A B D\) is tangent to lines \(B C\) and \(C D\). Find the perimeter of the trapezoid. | {
"answer": "56",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $g_0(x) = x + |x - 150| - |x + 150|$, and for $n \geq 1$, let $g_n(x) = |g_{n-1}(x)| - 2$. For how many values of $x$ is $g_{100}(x) = 0$? | {
"answer": "299",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
$\triangle PQR$ is inscribed inside $\triangle XYZ$ such that $P, Q, R$ lie on $YZ, XZ, XY$, respectively. The circumcircles of $\triangle PYZ, \triangle QXR, \triangle RQP$ have centers $O_4, O_5, O_6$, respectively. Also, $XY = 29, YZ = 35, XZ=28$, and $\stackrel{\frown}{YR} = \stackrel{\frown}{QZ},\ \stackrel{\frown}{XR} = \stackrel{\frown}{PY},\ \stackrel{\frown}{XP} = \stackrel{\frown}{QY}$. The length of $QY$ can be written in the form $\frac{p}{q}$, where $p$ and $q$ are relatively prime integers. Find $p+q$. | {
"answer": "31",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A certain school club has 10 members, and two of them are put on duty each day from Monday to Friday. Given that members A and B must be scheduled on the same day, and members C and D cannot be scheduled together, the total number of different possible schedules is (▲). Choices:
A) 21600
B) 10800
C) 7200
D) 5400 | {
"answer": "5400",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In parallelogram ABCD, $\angle BAD=60^\circ$, $AB=1$, $AD=\sqrt{2}$, and P is a point inside the parallelogram such that $AP=\frac{\sqrt{2}}{2}$. If $\overrightarrow{AP}=\lambda\overrightarrow{AB}+\mu\overrightarrow{AD}$ ($\lambda,\mu\in\mathbb{R}$), then the maximum value of $\lambda+\sqrt{2}\mu$ is \_\_\_\_\_\_. | {
"answer": "\\frac{\\sqrt{6}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the polygon shown, each side is perpendicular to its adjacent sides, and all 24 of the sides are congruent. The perimeter of the polygon is 48. Find the area of the polygon. | {
"answer": "128",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the graph of the function $y=\cos (x+\frac{4\pi }{3})$ is translated $\theta (\theta > 0)$ units to the right, and the resulting graph is symmetrical about the $y$-axis, determine the smallest possible value of $\theta$. | {
"answer": "\\frac{\\pi }{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $O$ is the circumcenter of $\triangle ABC$, $AC \perp BC$, $AC = 3$, and $\angle ABC = \frac{\pi}{6}$, find the dot product of $\overrightarrow{OC}$ and $\overrightarrow{AB}$. | {
"answer": "-9",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many (possibly empty) sets of lattice points $\{P_1, P_2, ... , P_M\}$ , where each point $P_i =(x_i, y_i)$ for $x_i
, y_i \in \{0, 1, 2, 3, 4, 5, 6\}$ , satisfy that the slope of the line $P_iP_j$ is positive for each $1 \le i < j \le M$ ? An infinite slope, e.g. $P_i$ is vertically above $P_j$ , does not count as positive. | {
"answer": "3432",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the diagram, \( S \) lies on \( R T \), \( \angle Q T S = 40^{\circ} \), \( Q S = Q T \), and \( \triangle P R S \) is equilateral. Find the value of \( x \). | {
"answer": "80",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Jenny and Jack run on a circular track. Jenny runs counterclockwise and completes a lap every 75 seconds, while Jack runs clockwise and completes a lap every 70 seconds. They start at the same place and at the same time. Between 15 minutes and 16 minutes from the start, a photographer standing outside the track takes a picture that shows one-third of the track, centered on the starting line. What is the probability that both Jenny and Jack are in the picture?
A) $\frac{23}{60}$
B) $\frac{12}{60}$
C) $\frac{13}{60}$
D) $\frac{46}{60}$
E) $\frac{120}{60}$ | {
"answer": "\\frac{23}{60}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the product of the base nine numbers $35_9$ and $47_9$, express it in base nine, and find the base nine sum of the digits of this product. Additionally, subtract $2_9$ from the sum of the digits. | {
"answer": "22_9",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that Jo climbs a flight of 8 stairs, and Jo can take the stairs 1, 2, 3, or 4 at a time, or a combination of these steps that does not exceed 4 with a friend, determine the number of ways Jo and the friend can climb the stairs together, including instances where either Jo or the friend climbs alone or together. | {
"answer": "108",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Suppose we flip five coins simultaneously: a penny, a nickel, a dime, a quarter, and a half-dollar. What is the probability that at least 30 cents worth of coins come up heads? | {
"answer": "\\dfrac{3}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The condition for three line segments to form a triangle is: the sum of the lengths of any two line segments is greater than the length of the third line segment. Now, there is a wire 144cm long, and it needs to be cut into $n$ small segments ($n>2$), each segment being no less than 1cm in length. If any three of these segments cannot form a triangle, then the maximum value of $n$ is ____. | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A circle with a radius of 2 units rolls around the inside of a triangle with sides 9, 12, and 15 units. The circle is always tangent to at least one side of the triangle. Calculate the total distance traveled by the center of the circle when it returns to its starting position. | {
"answer": "24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If the coefficient of the $x^2$ term in the expansion of $(1-ax)(1+2x)^4$ is $4$, then $\int_{\frac{e}{2}}^{a}{\frac{1}{x}}dx =$ . | {
"answer": "\\ln(5) - 1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $\lg 2=0.3010$ and $\lg 3=0.4771$, at which decimal place does the first non-zero digit of $\left(\frac{6}{25}\right)^{100}$ occur?
(Shanghai Middle School Mathematics Competition, 1984) | {
"answer": "62",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There is a garden with 3 rows and 2 columns of rectangular flower beds, each measuring 6 feet long and 2 feet wide. Between the flower beds, as well as around the garden, there is a 1-foot wide path. What is the total area \( S \) of the path in square feet? | {
"answer": "78",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On three faces of a cube, diagonals are drawn such that a triangle is formed. Find the angles of this triangle. | {
"answer": "60",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $S$ be the set of triples $(a,b,c)$ of non-negative integers with $a+b+c$ even. The value of the sum
\[\sum_{(a,b,c)\in S}\frac{1}{2^a3^b5^c}\]
can be expressed as $\frac{m}{n}$ for relative prime positive integers $m$ and $n$ . Compute $m+n$ .
*Proposed by Nathan Xiong* | {
"answer": "37",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $C$ be a unit cube and let $p$ denote the orthogonal projection onto the plane. Find the maximum area of $p(C)$ . | {
"answer": "\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the height of a cylinder is $2$, and the circumferences of its two bases lie on the surface of the same sphere with a diameter of $2\sqrt{6}$, calculate the surface area of the cylinder. | {
"answer": "(10+4\\sqrt{5})\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find one third of 7.2, expressed as a simplified fraction or a mixed number. | {
"answer": "2 \\frac{2}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given vectors $\overrightarrow{a}=(\cos 25^{\circ},\sin 25^{\circ})$ and $\overrightarrow{b}=(\sin 20^{\circ},\cos 20^{\circ})$, let $t$ be a real number and $\overrightarrow{u}=\overrightarrow{a}+t\overrightarrow{b}$. Determine the minimum value of $|\overrightarrow{u}|$. | {
"answer": "\\frac{\\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Except for the first two terms, each term of the sequence $2000, y, 2000 - y,\ldots$ is obtained by subtracting the preceding term from the one before that. The last term of the sequence is the first negative term encountered. What positive integer $y$ produces a sequence of maximum length? | {
"answer": "1236",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Team X and team Y play a series where the first team to win four games wins the series. Each team is equally likely to win each game, there are no ties, and the outcomes of the individual games are independent. If team Y wins the third game and team X wins the series, what is the probability that team Y wins the first game?
**A) $\frac{5}{12}$**
**B) $\frac{1}{2}$**
**C) $\frac{1}{3}$**
**D) $\frac{1}{4}$**
**E) $\frac{2}{3}$** | {
"answer": "\\frac{5}{12}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Suppose $a$, $b$, and $c$ are positive integers such that $a \geq b \geq c$ and $a+b+c=2010$. Furthermore, $a!b!c! = m \cdot 10^n$, where $m$ and $n$ are integers, and $m$ is not divisible by $10$. What is the smallest possible value of $n$?
A) 499
B) 500
C) 502
D) 504
E) 506 | {
"answer": "500",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $x^{3}=4$, solve for $x$. | {
"answer": "\\sqrt[3]{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the line $y=kx+1$, the maximum length of the chord intercepted by the ellipse $\frac{x^{2}}{4}+y^{2}=1$ as $k$ varies, determine the maximum length of the chord. | {
"answer": "\\frac{4\\sqrt{3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $f(x)=\frac{1}{2}\cos^{2}x-\frac{1}{2}\sin^{2}x+1-\sqrt{3}\sin x \cos x$.
$(1)$ Find the period and the interval where $f(x)$ is monotonically decreasing.
$(2)$ Find the minimum value of $f(x)$ on $[0,\frac{\pi}{2}]$ and the corresponding set of independent variables. | {
"answer": "\\left\\{\\frac{\\pi}{3}\\right\\}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The Chinese government actively responds to changes in gas emissions and has set a goal to reduce carbon emission intensity by 40% by 2020 compared to 2005. It is known that in 2005, China's carbon emission intensity was about 3 tons per 10,000 yuan, and each year thereafter, the carbon emission intensity decreases by 0.08 tons per 10,000 yuan.
(1) Can the emission reduction target be achieved by 2020? Explain your reasoning;
(2) If the GDP of China in 2005 was a million yuan, and thereafter increases by 8% annually, from which year will the carbon dioxide emissions start to decrease?
(Note: "Carbon emission intensity" refers to the amount of carbon dioxide emissions per 10,000 yuan of GDP) | {
"answer": "2030",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $n$ be a positive integer, and let $b_0, b_1, \dots, b_n$ be a sequence of real numbers such that $b_0 = 54$, $b_1 = 81$, $b_n = 0$, and $$ b_{k+1} = b_{k-1} - \frac{4.5}{b_k} $$ for $k = 1, 2, \dots, n-1$. Find $n$. | {
"answer": "972",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
We are given 5771 weights weighing 1,2,3,...,5770,5771. We partition the weights into $n$ sets of equal weight. What is the maximal $n$ for which this is possible? | {
"answer": "2886",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On the complex plane, the parallelogram formed by the points 0, $z,$ $\frac{1}{z},$ and $z + \frac{1}{z}$ has an area of $\frac{12}{13}.$ If the real part of $z$ is positive, compute the smallest possible value of $\left| z + \frac{1}{z} \right|^2.$ | {
"answer": "\\frac{36}{13}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A middle school cafeteria regularly purchases rice from a grain store at a price of 1500 yuan per ton. Each time rice is purchased, a transportation fee of 100 yuan is required. The cafeteria needs 1 ton of rice per day, and the storage cost for rice is 2 yuan per ton per day (less than one day is counted as one day). Assuming the cafeteria purchases rice on the day it runs out.
(1) How often should the cafeteria purchase rice to minimize the total daily cost?
(2) The grain store offers a discount: if the purchase quantity is not less than 20 tons at a time, the price of rice can enjoy a 5% discount (i.e., 95% of the original price). Can the cafeteria accept this discount condition? Please explain your reason. | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
22. In the polar coordinate system, the polar equation of circle $C$ is $\rho =4\cos \theta$. Taking the pole as the origin and the direction of the polar axis as the positive direction of the $x$-axis, a Cartesian coordinate system is established using the same unit length. The parametric equation of line $l$ is $\begin{cases} & x=\dfrac{1}{2}+\dfrac{\sqrt{2}}{2}t \\ & y=\dfrac{\sqrt{2}}{2}t \end{cases}$ (where $t$ is the parameter).
(Ⅰ) Write the Cartesian coordinate equation of circle $C$ and the general equation of line $l$;
(Ⅱ) Given point $M\left( \dfrac{1}{2},0 \right)$, line $l$ intersects circle $C$ at points $A$ and $B$. Find the value of $\left| \left| MA \right|-\left| MB \right| \right|$. | {
"answer": "\\dfrac{\\sqrt{46}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On each side of an equilateral triangle with side length $n$ units, where $n$ is an integer, $1 \leq n \leq 100$ , consider $n-1$ points that divide the side into $n$ equal segments. Through these points, draw lines parallel to the sides of the triangle, obtaining a net of equilateral triangles of side length one unit. On each of the vertices of these small triangles, place a coin head up. Two coins are said to be adjacent if the distance between them is 1 unit. A move consists of flipping over any three mutually adjacent coins. Find the number of values of $n$ for which it is possible to turn all coins tail up after a finite number of moves. | {
"answer": "67",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many ways can a schedule of 4 mathematics courses - algebra, geometry, number theory, and calculus - be created in an 8-period day if exactly one pair of these courses can be taken in consecutive periods, and the other courses must not be consecutive? | {
"answer": "1680",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In rectangle $WXYZ$, $P$ is a point on $WY$ such that $\angle WPZ=90^{\circ}$. $UV$ is perpendicular to $WY$ with $WU=UP$, as shown. $PZ$ intersects $UV$ at $Q$. Point $R$ is on $YZ$ such that $WR$ passes through $Q$. In $\triangle PQW$, $PW=15$, $WQ=20$ and $QP=25$. Find $VZ$. (Express your answer as a common fraction.) | {
"answer": "\\dfrac{20}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A cuckoo clock strikes the number of times corresponding to the current hour (for example, at 19:00, it strikes 7 times). One morning, Max approached the clock when it showed 9:05. He started turning the minute hand until it moved forward by 7 hours. How many times did the cuckoo strike during this period? | {
"answer": "43",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Suppose three hoses, X, Y, and Z, are used to fill a pool. Hoses X and Y together take 3 hours, while hose Y working alone takes 9 hours. Hoses X and Z together take 4 hours to fill the same pool. All three hoses working together can fill the pool in 2.5 hours. How long does it take for hose Z working alone to fill the pool? | {
"answer": "15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many of the 2401 smallest positive integers written in base 7 use 3 or 6 (or both) as a digit? | {
"answer": "1776",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider a circle of radius $4$ with center $O_1$ , a circle of radius $2$ with center $O_2$ that lies on the circumference of circle $O_1$ , and a circle of radius $1$ with center $O_3$ that lies on the circumference of circle $O_2$ . The centers of the circle are collinear in the order $O_1$ , $O_2$ , $O_3$ . Let $A$ be a point of intersection of circles $O_1$ and $O_2$ and $B$ be a point of intersection of circles $O_2$ and $O_3$ such that $A$ and $B$ lie on the same semicircle of $O_2$ . Compute the length of $AB$ . | {
"answer": "\\sqrt{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a regular hexagon $ABCDEF$, points $P$, $Q$, $R$, and $S$ are chosen on sides $\overline{AB}$, $\overline{CD}$, $\overline{DE}$, and $\overline{FA}$ respectively, so that lines $PC$ and $RA$, as well as $QS$ and $EB$, are parallel. Moreover, the distances between these parallel lines are equal and constitute half of the altitude of the triangles formed by drawing diagonals from the vertices $B$ and $D$ to the opposite side. Calculate the ratio of the area of hexagon $APQRSC$ to the area of hexagon $ABCDEF$. | {
"answer": "\\frac{3}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the system of equations
\begin{align*}
4x+2y &= c, \\
6y - 12x &= d,
\end{align*}
where \(d \neq 0\), find the value of \(\frac{c}{d}\). | {
"answer": "-\\frac{1}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The angle of inclination of the line $$\begin{cases} \left.\begin{matrix}x=3- \frac { \sqrt {2}}{2}t \\ y= \sqrt {5}- \frac { \sqrt {2}}{2}t\end{matrix}\right.\end{cases}$$ is ______. | {
"answer": "\\frac{\\pi}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.