problem stringlengths 10 5.15k | answer dict |
|---|---|
In the Cartesian coordinate system, given points A(1, -3), B(4, -1), P(a, 0), and N(a+1, 0), if the perimeter of the quadrilateral PABN is minimal, then find the value of a. | {
"answer": "a = \\frac{5}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the value of the expression \( \sqrt{\frac{16^{12} + 8^{15}}{16^5 + 8^{16}}} \). | {
"answer": "\\frac{3\\sqrt{2}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle \(ABC\), the angle at vertex \(B\) is \(\frac{\pi}{3}\), and the line segments connecting the incenter to vertices \(A\) and \(C\) are 4 and 6, respectively. Find the radius of the circle inscribed in triangle \(ABC\). | {
"answer": "\\frac{6 \\sqrt{3}}{\\sqrt{19}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The sequence $\{2n+1\}$ ($n\in\mathbb{N}^*$) is arranged sequentially in brackets such that the first bracket contains one number, the second bracket contains two numbers, the third bracket contains three numbers, the fourth bracket contains four numbers, the fifth bracket contains one number, the sixth bracket contains two numbers, and so on in a cycle. What is the sum of the numbers in the 104th bracket? | {
"answer": "2072",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that vehicles are not allowed to turn back at a crossroads, calculate the total number of driving routes. | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The traditional Chinese mathematical masterpiece "Nine Chapters on the Mathematical Art" records: "There are 5 cows and 2 sheep, worth 19 taels of silver; 2 cows and 5 sheep, worth 16 taels of silver. How much is each cow and each sheep worth in silver?" According to the translation above, answer the following two questions:
$(1)$ Find out how much each cow and each sheep are worth in silver.
$(2)$ If the cost of raising a cow is 2 taels of silver and the cost of raising a sheep is 1.5 taels of silver, villager Li wants to raise a total of 10 cows and sheep (the number of cows does not exceed the number of sheep). When he sells them all, how many cows and sheep should Li raise to earn the most silver? | {
"answer": "7.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a sequence where each term is either 1 or 2, starting with 1, and where between the \(k\)-th 1 and the \((k+1)\)-th 1 there are \(2^{k-1}\) 2's (i.e., the sequence is 1, 2, 1, 2, 2, 1, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 1, ...), determine the sum of the first 1998 terms of this sequence. | {
"answer": "3986",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For a four-digit natural number $M$, if the digit in the thousands place is $6$ more than the digit in the units place, and the digit in the hundreds place is $2$ more than the digit in the tens place, then $M$ is called a "naive number." For example, the four-digit number $7311$ is a "naive number" because $7-1=6$ and $3-1=2$. On the other hand, the four-digit number $8421$ is not a "naive number" because $8-1\neq 6$. Find the smallest "naive number" which is ______.
Let the digit in the thousands place of a "naive number" $M$ be $a$, the digit in the hundreds place be $b$, the digit in the tens place be $c$, and the digit in the units place be $d$. Define $P(M)=3(a+b)+c+d$ and $Q(M)=a-5$. If $\frac{{P(M)}}{{Q(M)}}$ is divisible by $10$, then the maximum value of $M$ that satisfies this condition is ______. | {
"answer": "9313",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given an ellipse $\frac {x^{2}}{a^{2}} + \frac {y^{2}}{b^{2}} = 1$ ($a > b > 0$) and a line $l: y = -\frac { \sqrt {3}}{3}x + b$ intersect at two distinct points P and Q. The distance from the origin to line $l$ is $\frac { \sqrt {3}}{2}$, and the eccentricity of the ellipse is $\frac { \sqrt {6}}{3}$.
(Ⅰ) Find the equation of the ellipse;
(Ⅱ) Determine whether there exists a real number $k$ such that the line $y = kx + 2$ intersects the ellipse at points P and Q, and the circle with diameter PQ passes through point D(1, 0). If it exists, find the value of $k$; if not, explain why. | {
"answer": "-\\frac {7}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Lesha's summer cottage has the shape of a nonagon with three pairs of equal and parallel sides. Lesha knows that the area of the triangle with vertices at the midpoints of the remaining sides of the nonagon is 12 sotkas. Help him find the area of the entire summer cottage. | {
"answer": "48",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that a person can click four times in sequence and receive one of three types of red packets each time, with the order of appearance corresponding to different prize rankings, calculate the number of different prize rankings that can be obtained if all three types of red packets are collected in any order before the fourth click. | {
"answer": "18",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a right triangle where one leg has length $4x + 2$ feet and the other leg has length $(x-3)^2$ feet, find the value of $x$ if the hypotenuse is $5x + 1$ feet. | {
"answer": "\\sqrt{\\frac{3}{2}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Thirty teams play a tournament where every team plays every other team exactly once. Each game results either in a win or a loss with a $50\%$ chance for either outcome. Calculate the probability that all teams win a unique number of games. Express your answer as $\frac{m}{n}$ where $m$ and $n$ are coprime integers and find $\log_2 n$. | {
"answer": "409",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In 1980, the per capita income in our country was $255; by 2000, the standard of living had reached a moderately prosperous level, meaning the per capita income had reached $817. What was the annual average growth rate? | {
"answer": "6\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine the real value of $t$ that minimizes the expression
\[
\sqrt{t^2 + (t^2 - 1)^2} + \sqrt{(t-14)^2 + (t^2 - 46)^2}.
\] | {
"answer": "7/2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many distinct trees with exactly 7 vertices are there? Here, a tree in graph theory refers to a connected graph without cycles, which can be simply understood as connecting \(n\) vertices with \(n-1\) edges. | {
"answer": "11",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a theater performance of King Lear, the locations of Acts II-V are drawn by lot before each act. The auditorium is divided into four sections, and the audience moves to another section with their chairs if their current section is chosen as the next location. Assume that all four sections are large enough to accommodate all chairs if selected, and each section is chosen with equal probability. What is the probability that the audience will have to move twice compared to the probability that they will have to move only once? | {
"answer": "1/2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the following product: $12 \times 0.5 \times 3 \times 0.2 =$
A) $\frac{20}{5}$
B) $\frac{22}{5}$
C) $\frac{16}{5}$
D) $\frac{18}{5}$
E) $\frac{14}{5}$ | {
"answer": "\\frac{18}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a press conference before a championship game, ten players from four teams will be taking questions. The teams are as follows: three Celtics, three Lakers, two Warriors, and two Nuggets. If teammates insist on sitting together and one specific Warrior must sit at the end of the row on the left, how many ways can the ten players be seated in a row? | {
"answer": "432",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A Gareth sequence is a sequence of numbers where each number after the second is the non-negative difference between the two previous numbers. For example, if a Gareth sequence begins 15, 12, then:
- The third number in the sequence is \(15 - 12 = 3\),
- The fourth number is \(12 - 3 = 9\),
- The fifth number is \(9 - 3 = 6\),
resulting in the sequence \(15, 12, 3, 9, 6, \ldots\).
If a Gareth sequence begins 10, 8, what is the sum of the first 30 numbers in the sequence? | {
"answer": "64",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a regular 15-gon, three distinct segments are chosen at random among the segments whose end-points are the vertices. What is the probability that the lengths of these three segments are the three side lengths of a triangle with positive area?
A) $\frac{345}{455}$
B) $\frac{100}{455}$
C) $\frac{310}{455}$
D) $\frac{305}{455}$
E) $\frac{450}{455}$ | {
"answer": "\\frac{345}{455}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that a recipe calls for \( 4 \frac{1}{2} \) cups of flour, calculate the amount of flour needed if only half of the recipe is made. | {
"answer": "2 \\frac{1}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $α$ and $β$ are both acute angles, and $\cos (α+β)=\dfrac{\sin α}{\sin β}$, the maximum value of $\tan α$ is \_\_\_\_. | {
"answer": "\\dfrac{ \\sqrt{2}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The equations
\[60x^4 + ax^3 + bx^2 + cx + 20 = 0\]and
\[20x^5 + dx^4 + ex^3 + fx^2 + gx + 60 = 0\]have a common rational root $r$ which is not an integer, and which is positive. What is $r?$ | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the fifth-largest divisor of 3,640,350,000? | {
"answer": "227,521,875",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The mean of one set of seven numbers is 15, and the mean of a separate set of eight numbers is 20. What is the mean of the set of all fifteen numbers? | {
"answer": "17.67",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ that satisfy $|\overrightarrow{a}| = 1$, $|\overrightarrow{b}| = \sqrt{3}$, and $(3\overrightarrow{a} - 2\overrightarrow{b}) \perp \overrightarrow{a}$, find the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$. | {
"answer": "\\frac{\\pi}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given an ellipse $C: \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1 \left( a > b > 0 \right)$ with eccentricity $\dfrac{\sqrt{3}}{2}$, and the length of its minor axis is $2$.
(I) Find the standard equation of the ellipse $C$.
(II) Suppose a tangent line $l$ of the circle $O: x^2 + y^2 = 1$ intersects curve $C$ at points $A$ and $B$. The midpoint of segment $AB$ is $M$. Find the maximum value of $|OM|$. | {
"answer": "\\dfrac{5}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Among the positive integers that can be expressed as the sum of 2005 consecutive integers, which occupies the 2005th position when arranged in order?
*Roland Hablutzel, Venezuela*
<details><summary>Remark</summary>The original question was: Among the positive integers that can be expressed as the sum of 2004 consecutive integers, and also as the sum of 2005 consecutive integers, which occupies the 2005th position when arranged in order?</details> | {
"answer": "2005 * 2004 * 2005",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find all values of \( a \) such that the roots \( x_1, x_2, x_3 \) of the polynomial
\[ x^3 - 6x^2 + ax + a \]
satisfy
\[ \left(x_1 - 3\right)^3 + \left(x_2 - 3\right)^3 + \left(x_3 - 3\right)^3 = 0. \] | {
"answer": "-9",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the diagram below, $ABCD$ is a trapezoid such that $\overline{AB}\parallel \overline{CD}$ and $\overline{AC}\perp\overline{CD}$. If $CD = 15$, $\tan D = 2$, and $\tan B = 2.5$, then what is $BC$? | {
"answer": "2\\sqrt{261}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A 6x6x6 cube is formed by assembling 216 unit cubes. Two 1x6 stripes are painted on each of the six faces of the cube parallel to the edges, with one stripe along the top edge and one along the bottom edge of each face. How many of the 216 unit cubes have no paint on them? | {
"answer": "144",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a cultural performance, there are already 10 programs arranged in the program list. Now, 3 more programs are to be added, with the requirement that the relative order of the originally scheduled 10 programs remains unchanged. How many different arrangements are there for the program list? (Answer with a number). | {
"answer": "1716",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A line segment starts at $(2, 4)$ and ends at the point $(7, y)$ with $y > 0$. The segment is 6 units long. Find the value of $y$. | {
"answer": "4 + \\sqrt{11}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let planes \( \alpha \) and \( \beta \) be parallel to each other. Four points are selected on plane \( \alpha \) and five points are selected on plane \( \beta \). What is the maximum number of planes that can be determined by these points? | {
"answer": "72",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Suppose we want to divide 12 puppies into three groups where one group has 4 puppies, one has 6 puppies, and one has 2 puppies. Determine how many ways we can form the groups such that Coco is in the 4-puppy group and Rocky is in the 6-puppy group. | {
"answer": "2520",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
At a school cafeteria, Jenny wants to buy a meal consisting of one main dish, one drink, one dessert, and one side dish. The list below contains Jenny's preferred choices available:
\begin{tabular}{ |c|c|c|c| }
\hline
\textbf{Main Dishes} & \textbf{Drinks} & \textbf{Desserts} & \textbf{Side Dishes} \\
\hline
Spaghetti & Water & Cookie & Salad \\
\hline
Turkey Sandwich & Juice & Cake & Fruit Cup \\
\hline
Veggie Burger & & & Chips \\
\hline
Mac and Cheese & & & \\
\hline
\end{tabular}
How many distinct possible meals can Jenny arrange from these options? | {
"answer": "48",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( M(x, y, z) \) represent the minimum of the three numbers \( x, y, z \). If the quadratic function \( f(x) = ax^2 + bx + c \) (where \( a, b, c > 0 \)) has a zero, determine the maximum value of \( M \left( \frac{b+c}{a}, \frac{c+a}{b}, \frac{a+b}{c} \right) \). | {
"answer": "\\frac{5}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $y=\cos(2x+ \frac {\pi}{3})$, determine the horizontal shift required to obtain this function from the graph of $y=\sin 2x$. | {
"answer": "\\frac{5\\pi}{12}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The ratio of the area of the rectangle to the area of the decagon can be calculated given that a regular decagon $ABCDEFGHIJ$ contains a rectangle $AEFJ$. | {
"answer": "\\frac{2}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The numbers from 1 to 9 are placed in the cells of a 3x3 table so that the sum of the numbers on one diagonal is 7 and the sum on the other diagonal is 21. What is the sum of the numbers in the five shaded cells? | {
"answer": "25",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the maximum real number \( k \) such that for any positive numbers \( a \) and \( b \), the following inequality holds:
$$
(a+b)(ab+1)(b+1) \geqslant k \, ab^2.
$$ | {
"answer": "27/4",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Segments $BD$ and $AE$ intersect at $C$, with $AB = BC$ and $CD = DE = EC$. Additionally, $\angle A = 4 \angle B$. Determine the degree measure of $\angle D$.
A) 45
B) 50
C) 52.5
D) 55
E) 60 | {
"answer": "52.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $f(x)$ have a domain of $R$, $f(x+1)$ be an odd function, and $f(x+2)$ be an even function. When $x\in [1,2]$, $f(x)=ax^{2}+b$. If $f(0)+f(3)=6$, then calculate the value of $f\left(\frac{9}{2}\right)$. | {
"answer": "\\frac{5}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Three boys and three girls are lined up for a photo. Boy A is next to boy B, and exactly two girls are next to each other. Calculate the total number of different ways they can be arranged. | {
"answer": "144",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The length of edge PQ of a tetrahedron PQRS measures 51 units, and the lengths of the other edges are 12, 19, 24, 33, and 42 units. Determine the length of edge RS. | {
"answer": "24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Increase Grisha's yield by 40% and Vasya's yield by 20%.
Grisha, the most astute among them, calculated that in the first case their total yield would increase by 1 kg; in the second case, it would decrease by 0.5 kg; in the third case, it would increase by 4 kg. What was the total yield of the friends (in kilograms) before their encounter with Hottabych? | {
"answer": "15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the smallest positive integer $n$ that is divisible by $100$ and has exactly $100$ divisors. | {
"answer": "162000",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the diagram below, $WXYZ$ is a trapezoid such that $\overline{WX}\parallel \overline{ZY}$ and $\overline{WY}\perp\overline{ZY}$. If $YZ = 15$, $\tan Z = \frac{4}{3}$, and $\tan X = \frac{3}{2}$, what is the length of $XY$? | {
"answer": "\\frac{20\\sqrt{13}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Solve the following equations:
(1) $x^{2}-3x=4$;
(2) $x(x-2)+x-2=0$. | {
"answer": "-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given 6 digits: \(0, 1, 2, 3, 4, 5\). Find the sum of all four-digit even numbers that can be written using these digits (the same digit can be repeated in a number). | {
"answer": "1769580",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the center of an ellipse is at the origin, the focus is on the $x$-axis, and the eccentricity $e= \frac { \sqrt {2}}{2}$, the area of the quadrilateral formed by connecting the four vertices of the ellipse in order is $2 \sqrt {2}$.
(1) Find the standard equation of the ellipse;
(2) Given that line $l$ intersects the ellipse at points $M$ and $N$, and $O$ is the origin. If point $O$ is on the circle with $MN$ as the diameter, find the distance from point $O$ to line $l$. | {
"answer": "\\frac{\\sqrt{6}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Three squares, $ABCD$, $EFGH$, and $GHIJ$, each have side length $s$. Point $C$ is located at the midpoint of side $HG$, and point $D$ is located at the midpoint of side $EF$. The line segment $AJ$ intersects the line segment $GH$ at point $X$. Determine the ratio of the area of the shaded region formed by triangle $AXD$ and trapezoid $JXCB$ to the total area of the three squares. | {
"answer": "\\frac{1}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Define $m(n)$ to be the greatest proper natural divisor of $n\in \mathbb{N}$ . Find all $n \in \mathbb{N} $ such that $n+m(n) $ is a power of $10$ .
*N. Agakhanov* | {
"answer": "75",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If the function $f(x) = \frac{1}{2}(m-2)x^2 + (n-8)x + 1$ with $m \geq 0$ and $n \geq 0$ is monotonically decreasing in the interval $\left[\frac{1}{2}, 2\right]$, then the maximum value of $mn$ is __________. | {
"answer": "18",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the sum of the binomial coefficients in the expansion of {(5x-1/√x)^n} is 64, determine the constant term in its expansion. | {
"answer": "375",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
An integer is called a "good number" if it has 8 positive divisors and the sum of these 8 positive divisors is 3240. For example, 2006 is a good number because the sum of its divisors 1, 2, 17, 34, 59, 118, 1003, and 2006 is 3240. Find the smallest good number. | {
"answer": "1614",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $2^{3x} = 128$, calculate the value of $2^{-x}$. | {
"answer": "\\frac{1}{2^{\\frac{7}{3}}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle XYZ$, a triangle $\triangle MNO$ is inscribed such that vertices $M, N, O$ lie on sides $YZ, XZ, XY$, respectively. The circumcircles of $\triangle XMO$, $\triangle YNM$, and $\triangle ZNO$ have centers $P_1, P_2, P_3$, respectively. Given that $XY = 26, YZ = 28, XZ = 27$, and $\stackrel{\frown}{MO} = \stackrel{\frown}{YN}, \stackrel{\frown}{NO} = \stackrel{\frown}{XM}, \stackrel{\frown}{NM} = \stackrel{\frown}{ZO}$. The length of $ZO$ can be written as $\frac{p}{q}$, where $p$ and $q$ are relatively prime integers. Find $p+q$. | {
"answer": "15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider the sequence created by intermixing the following sets of numbers: the first $1000$ odd numbers, and the squares of the first $100$ integers. What is the median of the new list of $1100$ numbers?
- $1, 3, 5, \ldots, 1999$
- $1^2, 2^2, \ldots, 100^2$
A) $1089$
B) $1095$
C) $1100$
D) $1102$
E) $1105$ | {
"answer": "1100",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the slope angle of the tangent line to the curve $f(x)=\frac{1}{3}{x}^{3}-{x}^{2}+5$ at $x=1$. | {
"answer": "\\frac{3\\pi}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find $d$, given that $\lfloor d\rfloor$ is a solution to \[3x^2 + 19x - 70 = 0\] and $\{d\} = d - \lfloor d\rfloor$ is a solution to \[4x^2 - 12x + 5 = 0.\] | {
"answer": "-8.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Define the derivative of the $(n-1)$th derivative as the $n$th derivative $(n \in N^{*}, n \geqslant 2)$, that is, $f^{(n)}(x)=[f^{(n-1)}(x)]'$. They are denoted as $f''(x)$, $f'''(x)$, $f^{(4)}(x)$, ..., $f^{(n)}(x)$. If $f(x) = xe^{x}$, then the $2023$rd derivative of the function $f(x)$ at the point $(0, f^{(2023)}(0))$ has a $y$-intercept on the $x$-axis of ______. | {
"answer": "-\\frac{2023}{2024}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $E(n)$ denote the sum of the even digits of $n$. Modify $E(n)$ such that if $n$ is prime, $E(n)$ is counted as zero, and if $n$ is not prime, $E(n)$ is counted twice. Calculate $E'(1)+E'(2)+E'(3)+\cdots+E'(200)$.
A) 1200
B) 1320
C) 1360
D) 1400
E) 1500 | {
"answer": "1360",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many different positive values of \( x \) will make this statement true: there are exactly 3 three-digit multiples of \( x \)? | {
"answer": "84",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The 2-digit integers from 31 to 75 are written consecutively to form the integer $M = 313233\cdots7475$. Suppose that $3^m$ is the highest power of 3 that is a factor of $M$. What is $m$?
A) 0
B) 1
C) 2
D) 3
E) more than 3 | {
"answer": "\\text{(A) } 0",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A regular octagon $ABCDEFGH$ has its sides' midpoints connected to form a smaller octagon inside it. Determine the fraction of the area of the larger octagon $ABCDEFGH$ that is enclosed by this smaller octagon. | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a circle with 800 points labeled in sequence clockwise as \(1, 2, \ldots, 800\), dividing the circle into 800 arcs. Initially, one point is painted red, and subsequently, additional points are painted red according to the following rule: if the \(k\)-th point is already red, the next point to be painted red is found by moving clockwise \(k\) arcs from \(k\). What is the maximum number of red points that can be obtained on the circle? Explain the reasoning. | {
"answer": "25",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $S$ be the set of positive real numbers. Let $g : S \to \mathbb{R}$ be a function such that
\[g(x) g(y) = g(xy) + 3003 \left( \frac{1}{x} + \frac{1}{y} + 3002 \right)\]for all $x,$ $y > 0.$
Let $m$ be the number of possible values of $g(2),$ and let $t$ be the sum of all possible values of $g(2).$ Find $m \times t.$ | {
"answer": "\\frac{6007}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and $4b\sin A= \sqrt {7}a$.
(I) Find the value of $\sin B$;
(II) If $a$, $b$, and $c$ form an arithmetic sequence with a common difference greater than $0$, find the value of $\cos A-\cos C$. | {
"answer": "\\frac { \\sqrt {7}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $\overrightarrow{a}$ and $\overrightarrow{b}$ are unit vectors, and $(2\overrightarrow{a}+ \overrightarrow{b})\cdot (\overrightarrow{a}-2\overrightarrow{b})=- \frac {3 \sqrt {3}}{2}$, calculate the angle between vectors $\overrightarrow{a}$ and $\overrightarrow{b}$. | {
"answer": "\\frac{\\pi}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and satisfy $\frac{c}{\cos C}=\frac{a+b}{\cos A+\cos B}$. Point $D$ is the midpoint of side $BC$.
$(1)$ Find the measure of angle $C$.
$(2)$ If $AC=2$ and $AD=\sqrt{7}$, find the length of side $AB$. | {
"answer": "2\\sqrt{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
ABC is a triangle. D is the midpoint of AB, E is a point on the side BC such that BE = 2 EC and ∠ADC = ∠BAE. Find ∠BAC. | {
"answer": "30",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)= \sqrt{3}\sin \omega x+\cos (\omega x+ \frac{\pi}{3})+\cos (\omega x- \frac{\pi}{3})-1$ ($\omega > 0$, $x\in\mathbb{R}$),
and the smallest positive period of the function $f(x)$ is $\pi$.
$(1)$ Find the analytical expression of the function $f(x)$;
$(2)$ In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $\alpha$ are $l$, $\alpha$, and $l$ respectively. If $\alpha$, $(\vec{BA}\cdot \vec{BC}= \frac{3}{2})$, and $a+c=4$, find the value of $b$. | {
"answer": "\\sqrt{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the sum of all integers between 50 and 450 that end in 1 or 7. | {
"answer": "19920",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $P$ is a moving point on the parabola $y^{2}=4x$, and $Q$ is a moving point on the circle $x^{2}+(y-4)^{2}=1$, the minimum value of the sum of the distance from point $P$ to point $Q$ and the distance from point $P$ to the directrix of the parabola is ______. | {
"answer": "\\sqrt{17}-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the number of permutations \((b_1, b_2, b_3, b_4, b_5, b_6)\) of \((1,2,3,4,5,6)\) such that
\[
\frac{b_1 + 6}{2} \cdot \frac{b_2 + 5}{2} \cdot \frac{b_3 + 4}{2} \cdot \frac{b_4 + 3}{2} \cdot \frac{b_5 + 2}{2} \cdot \frac{b_6 + 1}{2} > 6!.
\] | {
"answer": "719",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A parallelogram $ABCD$ is inscribed in the ellipse $\frac{x^{2}}{4}+y^{2}=1$, where the slope of the line $AB$ is $k_{1}=1$. Determine the slope of the line $AD$. | {
"answer": "-\\frac{1}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
One TV was sold for a 12% profit and the other for a 12% loss at a selling price of 3080 yuan each. Determine the net profit or loss from these transactions. | {
"answer": "-90",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For rational numbers $x$, $y$, $a$, $t$, if $|x-a|+|y-a|=t$, then $x$ and $y$ are said to have a "beautiful association number" of $t$ with respect to $a$. For example, $|2-1|+|3-1|=3$, then the "beautiful association number" of $2$ and $3$ with respect to $1$ is $3$. <br/> $(1)$ The "beautiful association number" of $-1$ and $5$ with respect to $2$ is ______; <br/> $(2)$ If the "beautiful association number" of $x$ and $5$ with respect to $3$ is $4$, find the value of $x$; <br/> $(3)$ If the "beautiful association number" of $x_{0}$ and $x_{1}$ with respect to $1$ is $1$, the "beautiful association number" of $x_{1}$ and $x_{2}$ with respect to $2$ is $1$, the "beautiful association number" of $x_{2}$ and $x_{3}$ with respect to $3$ is $1$, ..., the "beautiful association number" of $x_{1999}$ and $x_{2000}$ with respect to $2000$ is $1$, ... <br/> ① The minimum value of $x_{0}+x_{1}$ is ______; <br/> ② What is the minimum value of $x_{1}+x_{2}+x_{3}+x_{4}+...+x_{2000}$? | {
"answer": "2001000",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are three flavors of chocolates in a jar: hazelnut, liquor, and milk. There are 12 chocolates that are not hazelnut, 18 chocolates that are not liquor, and 20 chocolates that are not milk. How many chocolates are there in total in the jar? | {
"answer": "50",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
When 1524 shi of rice is mixed with an unknown amount of wheat, and in a sample of 254 grains, 28 are wheat grains, calculate the estimated amount of wheat mixed with this batch of rice. | {
"answer": "168",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
(1) Among the following 4 propositions:
① The converse of "If $a$, $G$, $b$ form a geometric sequence, then $G^2=ab$";
② The negation of "If $x^2+x-6\geqslant 0$, then $x > 2$";
③ In $\triangle ABC$, the contrapositive of "If $A > B$, then $\sin A > \sin B$";
④ When $0\leqslant \alpha \leqslant \pi$, if $8x^2-(8\sin \alpha)x+\cos 2\alpha\geqslant 0$ holds for $\forall x\in \mathbb{R}$, then the range of $\alpha$ is $0\leqslant \alpha \leqslant \frac{\pi}{6}$.
The numbers of the true propositions are ______.
(2) Given an odd function $f(x)$ whose graph is symmetric about the line $x=3$, and when $x\in [0,3]$, $f(x)=-x$, then $f(-16)=$ ______.
(3) The graph of the function $f(x)=a^{x-1}+4$ ($a > 0$ and $a\neq 1$) passes through a fixed point, then the coordinates of this point are ______.
(4) Given a point $P$ on the parabola $y^2=2x$, the minimum value of the sum of the distance from point $P$ to the point $(0,2)$ and the distance from $P$ to the directrix of the parabola is ______. | {
"answer": "\\frac{\\sqrt{17}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The length of a rectangular yard exceeds twice its width by 30 feet, and the perimeter of the yard is 700 feet. What is the area of the yard in square feet? | {
"answer": "25955.56",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given: $\cos\left(\alpha+ \frac{\pi}{4}\right) = \frac{3}{5}$, $\frac{\pi}{2} < \alpha < \frac{3\pi}{2}$, find $\cos\left(2\alpha+ \frac{\pi}{4}\right)$. | {
"answer": "-\\frac{31\\sqrt{2}}{50}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine the greatest number m such that the
system $x^2$ + $y^2$ = 1; | $x^3$ - $y^3$ |+|x-y|= $m^3$ has a solution. | {
"answer": "\\sqrt[3]{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $F$ is the right focus of the hyperbola $C$: $x^{2}- \frac {y^{2}}{8}=1$, and $P$ is a point on the left branch of $C$, $A(0,6 \sqrt {6})$. When the perimeter of $\triangle APF$ is minimized, the area of this triangle is \_\_\_\_\_\_. | {
"answer": "12 \\sqrt {6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider the sum $$ S =\sum^{2021}_{j=1} \left|\sin \frac{2\pi j}{2021}\right|. $$ The value of $S$ can be written as $\tan \left( \frac{c\pi}{d} \right)$ for some relatively prime positive integers $c, d$ , satisfying $2c < d$ . Find the value of $c + d$ . | {
"answer": "3031",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $a+b=1$, find the supremum of $$- \frac {1}{2a}- \frac {2}{b}.$$ | {
"answer": "- \\frac {9}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Four positive integers $p$, $q$, $r$, $s$ satisfy $p \cdot q \cdot r \cdot s = 9!$ and $p < q < r < s$. What is the smallest possible value of $s-p$? | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider a trapezoid field with base lengths of 120 meters and 180 meters and non-parallel sides each measuring 130 meters. The angles adjacent to the longer base are $60^\circ$. At harvest, the crops at any point in the field are brought to the nearest point on the field's perimeter. Determine the fraction of the crop that is brought to the longest base. | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The sum of the digits of the integer equal to \( 777777777777777^2 - 222222222222223^2 \) is | {
"answer": "74",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Medians $\overline{DP}$ and $\overline{EQ}$ of $\triangle DEF$ are perpendicular. If $DP= 15$ and $EQ = 20$, then what is ${DF}$? | {
"answer": "\\frac{20\\sqrt{13}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find \( g(2021) \) if for any real numbers \( x \) and \( y \) the following equality holds:
\[ g(x-y) = 2021(g(x) + g(y)) - 2022xy \] | {
"answer": "2043231",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On January 15 in the stormy town of Stormville, there is a $50\%$ chance of rain. Every day, the probability of it raining has a $50\%$ chance of being $\frac{2017}{2016}$ times that of the previous day (or $100\%$ if this new quantity is over $100\%$ ) and a $50\%$ chance of being $\frac{1007}{2016}$ times that of the previous day. What is the probability that it rains on January 20?
*2018 CCA Math Bonanza Lightning Round #3.3* | {
"answer": "243/2048",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the number of positive integers $n$ that satisfy
\[(n - 2)(n - 4)(n - 6) \dotsm (n - 98) < 0.\] | {
"answer": "23",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ and the parabola $y^{2} = 4cx$, where $c = \sqrt{a^{2} + b^{2}}$, find the eccentricity of the hyperbola given that $|AB| = 4c$. | {
"answer": "\\sqrt{2} + 1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A and B are playing a series of Go games, with the first to win 3 games declared the winner. Assuming in a single game, the probability of A winning is 0.6 and the probability of B winning is 0.4, with the results of each game being independent. It is known that in the first two games, A and B each won one game.
(1) Calculate the probability of A winning the match;
(2) Let $\xi$ represent the number of games played from the third game until the end of the match. Calculate the distribution and the mathematical expectation of $\xi$. | {
"answer": "2.48",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $f(x) = \sin \left( \frac{\pi}{3}x \right)$, and the set $A = \{1, 2, 3, 4, 5, 6, 7, 8\}$. Now, choose any two distinct elements $s$ and $t$ from set $A$. Find out the number of possible pairs $(s, t)$ such that $f(s)\cdot f(t) = 0$. | {
"answer": "13",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $P=\{1,2,\ldots,6\}$, and let $A$ and $B$ be two non-empty subsets of $P$. Find the number of pairs of sets $(A,B)$ such that the maximum number in $A$ is less than the minimum number in $B$. | {
"answer": "129",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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