problem stringlengths 10 5.15k | answer dict |
|---|---|
Given an ellipse $M$ with its axes of symmetry being the coordinate axes, and its eccentricity is $\frac{\sqrt{2}}{2}$, and one of its foci is at $(\sqrt{2}, 0)$.
$(1)$ Find the equation of the ellipse $M$;
$(2)$ Suppose a line $l$ intersects the ellipse $M$ at points $A$ and $B$, and a parallelogram $OAPB$ is formed with $OA$ and $OB$ as adjacent sides, where point $P$ is on the ellipse $M$ and $O$ is the origin. Find the minimum distance from point $O$ to line $l$. | {
"answer": "\\frac{\\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( a, b, c, x, y, z \) be nonzero complex numbers such that
\[ a = \frac{b+c}{x-3}, \quad b = \frac{a+c}{y-3}, \quad c = \frac{a+b}{z-3}, \]
and \( xy + xz + yz = 10 \) and \( x + y + z = 6 \), find \( xyz \). | {
"answer": "15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the numbers - 2, 5, 8, 11, and 14 are arranged in a specific cross-like structure, find the maximum possible sum for the numbers in either the row or the column. | {
"answer": "36",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
$Q$ is the point of intersection of the diagonals of one face of a cube whose edges have length 2 units. Calculate the length of $QR$. | {
"answer": "\\sqrt{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The pony wants to cross a bridge where there are two monsters, A and B. Monster A is awake for 2 hours and rests for 1 hour. Monster B is awake for 3 hours and rests for 2 hours. The pony can only cross the bridge when both monsters are resting; otherwise, it will be eaten by the awake monster. When the pony arrives at the bridge, both monsters have just finished their rest periods. How long does the pony need to wait, in hours, to cross the bridge with the least amount of waiting time? | {
"answer": "15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider a five-digit integer in the form $AB,BCA$, where $A$, $B$, and $C$ are distinct digits. What is the largest possible value of $AB,BCA$ that is divisible by 7? | {
"answer": "98,879",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let the sum of the first $n$ terms of an arithmetic sequence $\{a_n\}$ be $S_n$, and it satisfies $S_{2016} > 0, S_{2017} < 0$. For any positive integer $n$, it holds that $|a_n| \geqslant |a_k|$, find the value of $k$. | {
"answer": "1009",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find all real numbers \( p \) such that the cubic equation \( 5x^3 - 5(p+1)x^2 + (71p-1)x + 1 = 66p \) has two roots that are natural numbers. | {
"answer": "76",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that signals are composed of the digits $0$ and $1$ with equal likelihood of transmission, the probabilities of error in transmission are $0.9$ and $0.1$ for signal $0$ being received as $1$ and $0$ respectively, and $0.95$ and $0.05$ for signal $1$ being received as $1$ and $0$ respectively. | {
"answer": "0.525",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that Kira needs to store 25 files onto disks, each with 2.0 MB of space, where 5 files take up 0.6 MB each, 10 files take up 1.0 MB each, and the rest take up 0.3 MB each, determine the minimum number of disks needed to store all 25 files. | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A certain item has a cost price of $4$ yuan and is sold at a price of $5$ yuan. The merchant is preparing to offer a discount on the selling price, but the profit margin must not be less than $10\%$. Find the maximum discount rate that can be offered. | {
"answer": "8.8\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a school library with four types of books: A, B, C, and D, and a student limit of borrowing at most 3 books, determine the minimum number of students $m$ such that there must be at least two students who have borrowed the same type and number of books. | {
"answer": "15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the Fibonacci sequence $\{a_n\}$, where each number from the third one is equal to the sum of the two preceding numbers, find the term of the Fibonacci sequence that corresponds to $\frac{{a_1}^2 + {a_2}^2 + {a_3}^2 + … + {a_{2017}}^2}{a_{2017}}$. | {
"answer": "2018",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a function $f(x)$ that satisfies: For any $x \in (0, +\infty)$, it always holds that $f(2x) = 2f(x)$; (2) When $x \in (1, 2]$, $f(x) = 2 - x$. If $f(a) = f(2020)$, find the smallest positive real number $a$. | {
"answer": "36",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $m = 2^{20}5^{15}.$ How many positive integer divisors of $m^2$ are less than $m$ but do not divide $m$? | {
"answer": "299",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Each side of a cube has a stripe drawn diagonally from one vertex to the opposite vertex. The stripes can either go from the top-right to bottom-left or from top-left to bottom-right, chosen at random for each face. What is the probability that there is at least one continuous path following the stripes that goes from one vertex of the cube to the diagonally opposite vertex, passing through consecutively adjacent faces?
- A) $\frac{1}{128}$
- B) $\frac{1}{32}$
- C) $\frac{3}{64}$
- D) $\frac{1}{16}$
- E) $\frac{1}{8}$ | {
"answer": "\\frac{3}{64}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the least positive integer $n$ such that $7350$ is a factor of $n!$? | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Xiao Ming has multiple 1 yuan, 2 yuan, and 5 yuan banknotes. He wants to buy a kite priced at 18 yuan using no more than 10 of these banknotes and must use at least two different denominations. How many different ways can he pay? | {
"answer": "11",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A truck delivered 4 bags of cement. They are stacked in the truck. A worker can carry one bag at a time either from the truck to the gate or from the gate to the shed. The worker can carry the bags in any order, each time taking the top bag, carrying it to the respective destination, and placing it on top of the existing stack (if there are already bags there). If given a choice to carry a bag from the truck or from the gate, the worker randomly chooses each option with a probability of 0.5. Eventually, all the bags end up in the shed.
a) (7th grade level, 1 point). What is the probability that the bags end up in the shed in the reverse order compared to how they were placed in the truck?
b) (7th grade level, 1 point). What is the probability that the bag that was second from the bottom in the truck ends up as the bottom bag in the shed? | {
"answer": "\\frac{1}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $T$ be a subset of $\{1,2,3,...,100\}$ such that no pair of distinct elements in $T$ has a sum divisible by $5$. What is the maximum number of elements in $T$? | {
"answer": "60",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( g : \mathbb{R} \to \mathbb{R} \) be a function such that
\[ g(g(x) - y) = 2g(x) + g(g(y) - g(-x)) + y \] for all real numbers \( x \) and \( y \).
Let \( n \) be the number of possible values of \( g(2) \), and let \( s \) be the sum of all possible values of \( g(2) \). Find \( n \times s \). | {
"answer": "-2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Xiao Hua needs to attend an event at the Youth Palace at 2 PM, but his watch gains 4 minutes every hour. He reset his watch at 10 AM. When Xiao Hua arrives at the Youth Palace according to his watch at 2 PM, how many minutes early is he actually? | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the smallest number $n \geq 5$ for which there can exist a set of $n$ people, such that any two people who are acquainted have no common acquaintances, and any two people who are not acquainted have exactly two common acquaintances.
*Bulgaria* | {
"answer": "11",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For how many integers $a$ with $|a| \leq 2005$ , does the system
$x^2=y+a$
$y^2=x+a$
have integer solutions? | {
"answer": "90",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A sphere with a radius of $1$ is placed inside a cone and touches the base of the cone. The minimum volume of the cone is \_\_\_\_\_\_. | {
"answer": "\\dfrac{8\\pi}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that there is a point P (x, -1) on the terminal side of ∠Q (x ≠ 0), and $\tan\angle Q = -x$, find the value of $\sin\angle Q + \cos\angle Q$. | {
"answer": "-\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A and B began riding bicycles from point A to point C, passing through point B on the way. After a while, A asked B, "How many kilometers have we ridden?" B responded, "We have ridden a distance equivalent to one-third of the distance from here to point B." After riding another 10 kilometers, A asked again, "How many kilometers do we have left to ride to reach point C?" B answered, "We have a distance left to ride equivalent to one-third of the distance from here to point B." What is the distance between point A and point C? (Answer should be in fraction form.) | {
"answer": "\\frac{40}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine the maximal size of a set of positive integers with the following properties:
(1) The integers consist of digits from the set \(\{1,2,3,4,5,6\}\).
(2) No digit occurs more than once in the same integer.
(3) The digits in each integer are in increasing order.
(4) Any two integers have at least one digit in common (possibly at different positions).
(5) There is no digit which appears in all the integers. | {
"answer": "32",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The orthocenter of triangle $DEF$ divides altitude $\overline{DM}$ into segments with lengths $HM = 10$ and $HD = 24.$ Calculate $\tan E \tan F.$ | {
"answer": "3.4",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle ABC, medians AD and BE intersect at centroid G. The midpoint of segment AB is F. Given that the area of triangle GFC is l times the area of triangle ABC, find the value of l. | {
"answer": "\\frac{1}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given vectors $\vec{a}$ and $\vec{b}$ satisfy $|\vec{a}|=|\vec{b}|=2$, and $\vec{b}$ is perpendicular to $(2\vec{a}+\vec{b})$, find the angle between vector $\vec{a}$ and $\vec{b}$. | {
"answer": "\\frac{2\\pi}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $x$ and $y$ be distinct real numbers such that
\[
\begin{vmatrix} 2 & 3 & 7 \\ 4 & x & y \\ 4 & y & x+1 \end{vmatrix}
= 0.\]Find $x + y.$ | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the smallest positive integer \(n\) such that \(\frac{n}{n+51}\) is equal to a terminating decimal? | {
"answer": "74",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On a circle, there are 2018 points. Each of these points is labeled with an integer. Each number is greater than the sum of the two numbers that immediately precede it in a clockwise direction.
Determine the maximum possible number of positive numbers that can be among the 2018 numbers. | {
"answer": "1008",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the quadratic equation $2x^{2}-1=6x$, the coefficient of the quadratic term is ______, the coefficient of the linear term is ______, and the constant term is ______. | {
"answer": "-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $\triangle ABC$ have sides $a$, $b$, and $c$ opposite to angles $A$, $B$, and $C$, respectively, and satisfy the equation $a\sin B = \sqrt{3}b\cos A$.
$(1)$ Find the measure of angle $A$.
$(2)$ Choose one set of conditions from the following three sets to ensure the existence and uniqueness of $\triangle ABC$, and find the area of $\triangle ABC$.
Set 1: $a = \sqrt{19}$, $c = 5$;
Set 2: The altitude $h$ on side $AB$ is $\sqrt{3}$, $a = 3$;
Set 3: $\cos C = \frac{1}{3}$, $c = 4\sqrt{2}$. | {
"answer": "4\\sqrt{3} + 3\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine how many "super prime dates" occurred in 2007, where a "super prime date" is defined as a date where both the month and day are prime numbers, and additionally, the day is less than or equal to the typical maximum number of days in the respective prime month. | {
"answer": "50",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=\sin 2x+2\cos ^{2}x-1$.
$(1)$ Find the smallest positive period of $f(x)$;
$(2)$ When $x∈[0,\frac{π}{2}]$, find the minimum value of $f(x)$ and the corresponding value of the independent variable $x$. | {
"answer": "\\frac{\\pi}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A cinema is showing four animated movies: Toy Story, Ice Age, Shrek, and Monkey King. The ticket prices are 50 yuan, 55 yuan, 60 yuan, and 65 yuan, respectively. Each viewer watches at least one movie and at most two movies. However, due to time constraints, viewers cannot watch both Ice Age and Shrek. Given that there are exactly 200 people who spent the exact same amount of money on movie tickets today, what is the minimum total number of viewers the cinema received today? | {
"answer": "1792",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Write the number in the form of a fraction (if possible):
$$
x=0.5123412341234123412341234123412341234 \ldots
$$
Can you generalize this method to all real numbers with a periodic decimal expansion? And conversely? | {
"answer": "\\frac{51229}{99990}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Suppose that $\sec y - \tan y = \frac{15}{8}$ and that $\csc y - \cot y = \frac{p}{q},$ where $\frac{p}{q}$ is in lowest terms. Find $p+q.$ | {
"answer": "30",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the absolute value of the difference of single-digit integers \( C \) and \( D \) such that in base \( 5 \):
$$ \begin{array}{c@{}c@{\;}c@{}c@{}c@{}c}
& & & D & D & C_5 \\
& & & \mathbf{3} & \mathbf{2} & D_5 \\
& & + & C & \mathbf{2} & \mathbf{4_5} \\
\cline{2-6}
& & C & \mathbf{2} & \mathbf{3} & \mathbf{1_5} \\
\end{array} $$ | {
"answer": "1_5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
An urn initially contains two red balls and one blue ball. George undertakes the operation of randomly drawing a ball and then adding two more balls of the same color from a box into the urn. This operation is done three times. After these operations, the urn has a total of nine balls. What is the probability that there are exactly five red balls and four blue balls in the urn?
A) $\frac{1}{10}$
B) $\frac{2}{10}$
C) $\frac{3}{10}$
D) $\frac{4}{10}$
E) $\frac{5}{10}$ | {
"answer": "\\frac{3}{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider $7$ points on a circle. Compute the number of ways there are to draw chords between pairs of points such that two chords never intersect and one point can only belong to one chord. It is acceptable to draw no chords. | {
"answer": "127",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The equation \(2008=1111+444+222+99+77+55\) is an example of decomposing the number 2008 as a sum of distinct numbers with more than one digit, where each number's representation (in the decimal system) uses only one digit.
i) Find a similar decomposition for the number 2009.
ii) Determine all possible such decompositions of the number 2009 that use the minimum number of terms (the order of terms does not matter). | {
"answer": "1111 + 777 + 66 + 55",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the largest prime factor of $18^4 + 12^5 - 6^6$. | {
"answer": "11",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For a positive real number $a$ , let $C$ be the cube with vertices at $(\pm a, \pm a, \pm a)$ and let $T$ be the tetrahedron with vertices at $(2a,2a,2a),(2a, -2a, -2a),(-2a, 2a, -2a),(-2a, -2a, -2a)$ . If the intersection of $T$ and $C$ has volume $ka^3$ for some $k$ , find $k$ . | {
"answer": "4/3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Alice is jogging north at a speed of 6 miles per hour, and Tom is starting 3 miles directly south of Alice, jogging north at a speed of 9 miles per hour. Moreover, assume Tom changes his path to head north directly after 10 minutes of eastward travel. How many minutes after this directional change will it take for Tom to catch up to Alice? | {
"answer": "60",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Point $D$ lies on side $AC$ of equilateral triangle $ABC$ such that the measure of angle $DBC$ is 30 degrees. What is the ratio of the area of triangle $ADB$ to the area of triangle $CDB$? | {
"answer": "\\frac{1}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Concentric circles $\Omega_1$ and $\Omega_2$ with radii $1$ and $100$ , respectively, are drawn with center $O$ . Points $A$ and $B$ are chosen independently at random on the circumferences of $\Omega_1$ and $\Omega_2$ , respectively. Denote by $\ell$ the tangent line to $\Omega_1$ passing through $A$ , and denote by $P$ the reflection of $B$ across $\ell$ . Compute the expected value of $OP^2$ .
*Proposed by Lewis Chen* | {
"answer": "10004",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the diagram, pentagon \( PQRST \) has \( PQ = 13 \), \( QR = 18 \), \( ST = 30 \), and a perimeter of 82. Also, \( \angle QRS = \angle RST = \angle STP = 90^\circ \). The area of the pentagon \( PQRST \) is: | {
"answer": "270",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given an arithmetic sequence $\{a_n\}$, the sum of the first $n$ terms is denoted as $S_n$. It is known that $a_1=9$, $a_2$ is an integer, and $S_n \leqslant S_5$. The sum of the first $9$ terms of the sequence $\left\{ \frac{1}{a_na_{n+1}} \right\}$ is ______. | {
"answer": "- \\frac{1}{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Starting with the display "1," calculate the fewest number of keystrokes needed to reach "400". | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
To enhance and beautify the city, all seven streetlights on a road are to be changed to colored lights. If there are three colors available for the colored lights - red, yellow, and blue - and the installation requires that no two adjacent streetlights are of the same color, with at least two lights of each color, there are ____ different installation methods. | {
"answer": "114",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Xiao Ming must stand in the very center, and Xiao Li and Xiao Zhang must stand together in a graduation photo with seven students. Find the number of different arrangements. | {
"answer": "192",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A school selects 4 teachers from 8 to teach in 4 remote areas at the same time (one person per area), where teacher A and teacher B cannot go together, and teacher A and teacher C can only go together or not go at all. The total number of different dispatch plans is ___. | {
"answer": "600",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The results of asking 50 students if they participate in music or sports are shown in the Venn diagram. Calculate the percentage of the 50 students who do not participate in music and do not participate in sports. | {
"answer": "20\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the smallest natural number \( n \) such that both \( n^2 \) and \( (n+1)^2 \) contain the digit 7. | {
"answer": "27",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Right triangle $ABC$ has one leg of length 9 cm, another leg of length 12 cm, and a right angle at $A$. A square has one side on the hypotenuse of triangle $ABC$ and a vertex on each of the two legs of triangle $ABC$. What is the length of one side of the square, in cm? Express your answer as a common fraction. | {
"answer": "\\frac{180}{37}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two circles, circle $A$ with radius 2 and circle $B$ with radius 1.5, are to be constructed with the following process: The center of circle $A$ is chosen uniformly and at random from the line segment joining $(0,0)$ to $(3,0)$. The center of circle $B$ is chosen uniformly and at random, and independently from the first choice, from the line segment joining $(1,2)$ to $(4,2)$. What is the probability that circles $A$ and $B$ intersect?
A) 0.90
B) 0.95
C) 0.96
D) 1.00 | {
"answer": "0.96",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Identical matches of length 1 are used to arrange the following pattern. If \( c \) denotes the total length of matches used, find \( c \). | {
"answer": "700",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given non-zero vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfy $|\overrightarrow{a}|=2|\overrightarrow{b}|$, and $(\overrightarrow{a}-\overrightarrow{b})\bot \overrightarrow{b}$, calculate the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$. | {
"answer": "\\frac{\\pi}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If set $A=\{x\in N\left|\right.-1 \lt x\leqslant 2\}$, $B=\{x\left|\right.x=ab,a,b\in A\}$, then the number of non-empty proper subsets of set $B$ is ______. | {
"answer": "14",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
An entrepreneur invested $\$20,\!000$ in a nine-month term deposit that paid a simple annual interest rate of $8\%$. After the term ended, she reinvested all the proceeds into another nine-month term deposit. At the end of the second term, her total investment had grown to $\$22,\!446.40$. If the annual interest rate of the second term deposit is $s\%$, what is $s?$ | {
"answer": "7.840",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the sum of three numbers, all equally likely to be $1$, $2$, $3$, or $4$, drawn from an urn with replacement, is $9$, calculate the probability that the number $3$ was drawn each time. | {
"answer": "\\frac{1}{13}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Monsieur Dupont remembered that today is their wedding anniversary and invited his wife to dine at a fine restaurant. Upon leaving the restaurant, he noticed that he had only one fifth of the money he initially took with him. He found that the centimes he had left were equal to the francs he initially had (1 franc = 100 centimes), while the francs he had left were five times less than the initial centimes he had.
How much did Monsieur Dupont spend at the restaurant? | {
"answer": "7996",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Define a function $A(m, n)$ in line with the Ackermann function and compute $A(3, 2)$. | {
"answer": "11",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A four-digit number satisfies the following conditions:
(1) If you simultaneously swap its unit digit with the hundred digit and the ten digit with the thousand digit, the value increases by 5940;
(2) When divided by 9, the remainder is 8.
Find the smallest odd four-digit number that satisfies these conditions.
(Shandong Province Mathematics Competition, 1979) | {
"answer": "1979",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider the function $y=a\sqrt{1-x^2} + \sqrt{1+x} + \sqrt{1-x}$ ($a\in\mathbb{R}$), and let $t= \sqrt{1+x} + \sqrt{1-x}$ ($\sqrt{2} \leq t \leq 2$).
(1) Express $y$ as a function of $t$, denoted as $m(t)$.
(2) Let the maximum value of the function $m(t)$ be $g(a)$. Find $g(a)$.
(3) For $a \geq -\sqrt{2}$, find all real values of $a$ that satisfy $g(a) = g\left(\frac{1}{a}\right)$. | {
"answer": "a = 1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are 94 safes and 94 keys. Each key can open only one safe, and each safe can be opened by only one key. We place randomly one key into each safe. 92 safes are then randomly chosen, and then locked. What is the probability that we can open all the safes with the two keys in the two remaining safes?
(Once a safe is opened, the key inside the safe can be used to open another safe.) | {
"answer": "1/47",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $A$, $B$, $R$, $M$, and $L$ be positive real numbers such that
\begin{align*}
\log_{10} (AB) + \log_{10} (AM) &= 2, \\
\log_{10} (ML) + \log_{10} (MR) &= 3, \\
\log_{10} (RA) + \log_{10} (RB) &= 5.
\end{align*}
Compute the value of the product $ABRML$. | {
"answer": "100",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine the smallest integer $B$ such that there exist several consecutive integers, including $B$, that add up to 2024. | {
"answer": "-2023",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For how many integer values of $n$ between 1 and 1000 inclusive does the decimal representation of $\frac{n}{2520}$ terminate? | {
"answer": "47",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the diagram, $\triangle QRS$ is an isosceles right-angled triangle with $QR=SR$ and $\angle QRS=90^{\circ}$. Line segment $PT$ intersects $SQ$ at $U$ and $SR$ at $V$. If $\angle PUQ=\angle RVT=y^{\circ}$, the value of $y$ is | {
"answer": "67.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $ABCD$ be a cyclic quadrilateral, and $E$ be the intersection of its diagonals. If $m(\widehat{ADB}) = 22.5^\circ$ , $|BD|=6$ , and $|AD|\cdot|CE|=|DC|\cdot|AE|$ , find the area of the quadrilateral $ABCD$ . | {
"answer": "9\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
John earned scores of 92, 85, and 91 on his first three physics examinations. If John receives a score of 95 on his fourth exam, then by how much will his average increase? | {
"answer": "1.42",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If the direction vector of line $l$ is $\overrightarrow{e}=(-1,\sqrt{3})$, calculate the inclination angle of line $l$. | {
"answer": "\\frac{2\\pi}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For any positive integer $a$ , define $M(a)$ to be the number of positive integers $b$ for which $a+b$ divides $ab$ . Find all integer(s) $a$ with $1\le a\le 2013$ such that $M(a)$ attains the largest possible value in the range of $a$ . | {
"answer": "1680",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The equations $x^3 + Cx - 20 = 0$ and $x^3 + Dx^2 - 40 = 0$ have two roots in common. Find the product of these common roots, which can be expressed in the form $p \sqrt[q]{r}$, where $p$, $q$, and $r$ are positive integers. What is $p + q + r$? | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the sequence \(\left\{a_{n}\right\}\) that satisfies
\[ a_{n}=\left[(2+\sqrt{5})^{n}+\frac{1}{2^{n}}\right] \quad \text{for} \quad n \in \mathbf{Z}_{+}, \]
where \([x]\) denotes the greatest integer less than or equal to the real number \(x\). Let \(C\) be a real number such that for any positive integer \(n\),
\[ \sum_{k=1}^{n} \frac{1}{a_{k} a_{k+2}} \leqslant C. \]
Find the minimum value of \(C\). | {
"answer": "\\frac{\\sqrt{5} - 2}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that in the rectangular coordinate system $(xOy)$, the origin is the pole and the positive semi-axis of $x$ is the polar axis to establish a polar coordinate system, the polar coordinate equation of the conic section $(C)$ is $p^{2}= \frac {12}{3+\sin ^{2}\theta }$, the fixed point $A(0,- \sqrt {3})$, $F\_{1}$, $F\_{2}$ are the left and right foci of the conic section $(C)$, and the line $l$ passes through point $F\_{1}$ and is parallel to the line $AF\_{2}$.
(I) Find the rectangular coordinate equation of conic section $(C)$ and the parametric equation of line $l$;
(II) If line $l$ intersects conic section $(C)$ at points $M$ and $N$, find $|F\_{1}M|⋅|F\_{1}N|$. | {
"answer": "\\frac {12}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the set $M$ consisting of all functions $f(x)$ that satisfy the property: there exist real numbers $a$ and $k$ ($k \neq 0$) such that for all $x$ in the domain of $f$, $f(a+x) = kf(a-x)$. The pair $(a,k)$ is referred to as the "companion pair" of the function $f(x)$.
1. Determine whether the function $f(x) = x^2$ belongs to set $M$ and explain your reasoning.
2. If $f(x) = \sin x \in M$, find all companion pairs $(a,k)$ for the function $f(x)$.
3. If $(1,1)$ and $(2,-1)$ are both companion pairs of the function $f(x)$, where $f(x) = \cos(\frac{\pi}{2}x)$ for $1 \leq x < 2$ and $f(x) = 0$ for $x=2$. Find all zeros of the function $y=f(x)$ when $2014 \leq x \leq 2016$. | {
"answer": "2016",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f\left( x \right)=2\sin (\omega x+\varphi )\left( \omega \gt 0,\left| \varphi \right|\lt \frac{\pi }{2} \right)$, the graph passes through point $A(0,-1)$, and is monotonically increasing on $\left( \frac{\pi }{18},\frac{\pi }{3} \right)$. The graph of $f\left( x \right)$ is shifted to the left by $\pi$ units and coincides with the original graph. When ${x}_{1}$, ${x}_{2} \in \left( -\frac{17\pi }{12},-\frac{2\pi }{3} \right)$ and ${x}_{1} \ne {x}_{2}$, if $f\left( {x}_{1} \right)=f\left( {x}_{2} \right)$, find $f({x}_{1}+{x}_{2})$. | {
"answer": "-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A circle with radius 100 is drawn on squared paper with unit squares. It does not touch any of the grid lines or pass through any of the lattice points. What is the maximum number of squares it can pass through? | {
"answer": "800",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a circle with a radius of 5 units, \( CD \) and \( AB \) are mutually perpendicular diameters. A chord \( CH \) intersects \( AB \) at \( K \) and has a length of 8 units, calculate the lengths of the two segments into which \( AB \) is divided. | {
"answer": "8.75",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Real numbers \(a, b, c\) and a positive number \(\lambda\) satisfy \(f(x) = x^3 + a x^2 + b x + c\), which has 3 real roots \(x_1, x_2, x_3\), such that:
(1) \(x_2 - x_1 = \lambda\);
(2) \(x_3 > \frac{1}{2}(x_1 + x_2)\).
Find the maximum value of \(\frac{2a^3 + 27c - 9ab}{\lambda^3}\). | {
"answer": "\\frac{3 \\sqrt{3}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
You are given the numbers $0$, $2$, $3$, $4$, $6$. Use these numbers to form different combinations and calculate the following:
$(1)$ How many unique three-digit numbers can be formed?
$(2)$ How many unique three-digit numbers that can be divided by $3$ can be formed? (Note: Write the result of each part in data form) | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
I live on a very short street with 14 small family houses. The odd-numbered houses from 1 are on one side of the street, and the even-numbered houses from 2 are on the opposite side (e.g., 1 and 2 are opposite each other).
On one side of the street, all families have surnames that are colors, and on the other side, the surnames indicate professions.
Szabó and Fazekas live opposite to Zöld and Fehér, respectively, who are both neighbors of Fekete.
Kovács is the father-in-law of Lakatos.
Lakatos lives in a higher-numbered house than Barna. The sum of the house numbers of Lakatos and Barna is equal to the sum of the house numbers of Fehér and Fazekas. Kádárné's house number is twice the house number of her sister, Kalaposné.
Sárga lives opposite Pék.
If Bordóné's house number is two-digit and she lives opposite her sister, Kádárné, what is the house number of Mr. Szürke? | {
"answer": "13",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many of the numbers from the set $\{1, 2, 3, \ldots, 100\}$ have a perfect square factor other than one? | {
"answer": "41",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the textbook, students were once asked to explore the coordinates of the midpoint of a line segment: In a plane Cartesian coordinate system, given two points $A(x_{1}, y_{1})$ and $B(x_{2}, y_{2})$, the midpoint of the line segment $AB$ is $M$, then the coordinates of $M$ are ($\frac{{x}_{1}+{x}_{2}}{2}$, $\frac{{y}_{1}+{y}_{2}}{2}$). For example, if point $A(1,2)$ and point $B(3,6)$, then the coordinates of the midpoint $M$ of line segment $AB$ are ($\frac{1+3}{2}$, $\frac{2+6}{2}$), which is $M(2,4)$. Using the above conclusion to solve the problem: In a plane Cartesian coordinate system, if $E(a-1,a)$, $F(b,a-b)$, the midpoint $G$ of the line segment $EF$ is exactly on the $y$-axis, and the distance to the $x$-axis is $1$, then the value of $4a+b$ is ____. | {
"answer": "4 \\text{ or } 0",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the number of functions of the form $f(x) = ax^3 + bx^2 + cx + d$ such that
\[f(x) f(-x) = f(x^3).\] | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Define a positive integer $n$ to be a factorial tail if there is some positive integer $m$ such that the decimal representation of $m!$ ends with exactly $n$ zeroes. How many positive integers less than $2500$ are not factorial tails? | {
"answer": "500",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=\begin{cases} x+2 & (x\leqslant -1) \\ x^{2} & (-1< x < 2) \\ 2x & (x\geqslant 2) \end{cases}$
$(1)$ Find $f(2)$, $f\left(\dfrac{1}{2}\right)$, $f[f(-1)]$;
$(2)$ If $f(a)=3$, find the value of $a$. | {
"answer": "\\sqrt {3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the greatest prime factor of $15! + 18!$? | {
"answer": "17",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, the sides opposite to $\angle A$, $\angle B$, and $\angle C$ are $a$, $b$, and $c$ respectively. $M$ is the midpoint of $BC$ with $BM=MC=2$, and $AM=b-c$. Find the maximum area of $\triangle ABC$. | {
"answer": "2\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The solution to the inequality
$$
(x-1)^{[\sqrt{1}]}(x-2)^{[\sqrt{2}]} \ldots(x-k)^{[\sqrt{k}]} \ldots(x-150)^{[\sqrt{150}]}<0
$$
is a union of several non-overlapping intervals. Find the sum of their lengths. If necessary, round the answer to the nearest 0.01.
Recall that $[x]$ denotes the greatest integer less than or equal to $x$. | {
"answer": "78.00",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What percent of square $EFGH$ is shaded? All angles in the diagram are right angles. [asy]
import graph;
defaultpen(linewidth(0.7));
xaxis(0,8,Ticks(1.0,NoZero));
yaxis(0,8,Ticks(1.0,NoZero));
fill((0,0)--(2,0)--(2,2)--(0,2)--cycle);
fill((3,0)--(5,0)--(5,5)--(0,5)--(0,3)--(3,3)--cycle);
fill((6,0)--(7,0)--(7,7)--(0,7)--(0,6)--(6,6)--cycle);
label("$E$",(0,0),SW);
label("$F$",(0,7),N);
label("$G$",(7,7),NE);
label("$H$",(7,0),E);
[/asy] | {
"answer": "67\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the smallest five-digit positive integer congruent to $2 \pmod{17}$? | {
"answer": "10013",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A bag contains 4 red, 3 blue, and 6 yellow marbles. One marble is drawn and removed from the bag but is only considered in the new count if it is yellow. What is the probability, expressed as a fraction, of then drawing one marble which is either red or blue from the updated contents of the bag? | {
"answer": "\\frac{91}{169}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the largest natural number in which all the digits are different and each pair of adjacent digits differs by 6 or 7. | {
"answer": "60718293",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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