problem stringlengths 10 5.15k | answer dict |
|---|---|
Given that the first four terms of a geometric sequence $\{a\_n\}$ have a sum of $S\_4=5$, and $4a\_1,\;\; \frac {3}{2}a\_2\;,\;a\_2$ form an arithmetic sequence.
(I) Find the general term formula for $\{a\_n\}$;
(II) Let $\{b\_n\}$ be an arithmetic sequence with first term $2$ and common difference $-a\_1$. Its first $n$ terms' sum is $T\_n$. Find the maximum positive integer $n$ that satisfies $T_{n-1} > 0$. | {
"answer": "13",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine the product of the real parts of the solutions to the equation \(2x^2 + 4x = 1 + i\). | {
"answer": "\\frac{1 - 3}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $x, y, z \in \mathbb{R}$ are solutions to the system of equations $$ \begin{cases}
x - y + z - 1 = 0
xy + 2z^2 - 6z + 1 = 0
\end{cases} $$ what is the greatest value of $(x - 1)^2 + (y + 1)^2$ ? | {
"answer": "11",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A total area of \( 2500 \, \mathrm{m}^2 \) will be used to build identical houses. The construction cost for a house with an area \( a \, \mathrm{m}^2 \) is the sum of the material cost \( 100 p_{1} a^{\frac{3}{2}} \) yuan, labor cost \( 100 p_{2} a \) yuan, and other costs \( 100 p_{3} a^{\frac{1}{2}} \) yuan, where \( p_{1} \), \( p_{2} \), and \( p_{3} \) are consecutive terms of a geometric sequence. The sum of these terms is 21 and their product is 64. Given that building 63 of these houses would result in the material cost being less than the sum of the labor cost and the other costs, find the maximum number of houses that can be built to minimize the total construction cost. | {
"answer": "156",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let the real numbers \(a_1, a_2, \cdots, a_{100}\) satisfy the following conditions: (i) \(a_1 \geq a_2 \geq \cdots \geq a_{100} \geq 0\); (ii) \(a_1 + a_2 \leq 100\); (iii) \(a_3 + a_4 + \cdots + a_{100} \leq 100\). Find the maximum value of \(a_1^2 + a_2^2 + \cdots + a_{100}^2\) and the values of \(a_1, a_2, \cdots, a_{100}\) that achieve this maximum. | {
"answer": "10000",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given triangle \( \triangle ABC \), point \( P \) is an internal point such that \( \angle PBC = \angle PCB = 24^\circ \). If \( \angle ABP = 30^\circ \) and \( \angle ACP = 54^\circ \), find the measure of \( \angle BAP \). | {
"answer": "18",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the dihedral angle $\alpha - E F - \beta $, $AE \subset \alpha, BF \subset \beta$, and $AE \perp EF, BF \perp EF$. Given $EF = 1$, $AE = 2$, and $AB = \sqrt{2}$, find the maximum volume of the tetrahedron $ABEF$. | {
"answer": "\\frac{1}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the line integral
$$
\int_{L} \frac{y}{3} d x - 3 x d y + x d z
$$
along the curve \( L \), which is given parametrically by
$$
\begin{cases}
x = 2 \cos t \\
y = 2 \sin t \\
z = 1 - 2 \cos t - 2 \sin t
\end{cases}
\quad \text{for} \quad 0 \leq t \leq \frac{\pi}{2}
$$ | {
"answer": "2 - \\frac{13\\pi}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=\sin \omega x+\cos \left(\omega x+\dfrac{\pi }{6}\right)$, where $x\in R$, $\omega >0$.
(1) When $\omega =1$, find the value of $f\left(\dfrac{\pi }{3}\right)$;
(2) When the smallest positive period of $f(x)$ is $\pi $, find the value of $x$ when $f(x)$ reaches the maximum value in $\left[0,\dfrac{\pi }{4}\right]$. | {
"answer": "\\dfrac{\\pi }{12}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A triangle with vertices at \((1003,0), (1004,3),\) and \((1005,1)\) in the \(xy\)-plane is revolved all the way around the \(y\)-axis. Find the volume of the solid thus obtained. | {
"answer": "5020 \\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Each of the letters in "GEOMETRY" is written on its own square tile and placed in a bag. What is the probability that a tile randomly selected from the bag will have a letter on it that is in the word "RHYME"? Express your answer as a common fraction. | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If the graph of the function $f(x)=a^{x-2}-2a (a > 0, a \neq 1)$ always passes through the fixed point $\left(x\_0, \frac{1}{3}\right)$, then the minimum value of the function $f(x)$ on $[0,3]$ is equal to \_\_\_\_\_\_\_\_. | {
"answer": "-\\frac{1}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The circumference of the axial cross-section of a cylinder is $90 \text{ cm}$. What is the maximum possible volume of the cylinder? | {
"answer": "3375\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A convex polyhedron is bounded by 4 regular hexagonal faces and 4 regular triangular faces. At each vertex of the polyhedron, 2 hexagons and 1 triangle meet. What is the volume of the polyhedron if the length of its edges is one unit? | {
"answer": "\\frac{23\\sqrt{2}}{12}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $PQR,$ $PQ = 4,$ $PR = 9,$ $QR = 10,$ and a point $S$ lies on $\overline{QR}$ such that $\overline{PS}$ bisects $\angle QPR.$ Find $\cos \angle QPS.$ | {
"answer": "\\sqrt{\\frac{23}{48}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many 10-digit numbers exist in which at least two digits are the same? | {
"answer": "9 \\times 10^9 - 9 \\times 9!",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider the polynomial $49x^3 - 105x^2 + 63x - 10 = 0$ whose roots are in arithmetic progression. Determine the difference between the largest and smallest roots.
A) $\frac{2}{7}$
B) $\frac{1}{7}$
C) $\frac{3\sqrt{11}}{7}$
D) $\frac{2\sqrt{11}}{7}$
E) $\frac{4\sqrt{11}}{7}$ | {
"answer": "\\frac{2\\sqrt{11}}{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a sequence of positive integers $a_1, a_2, a_3, \ldots, a_{100}$, where the number of terms equal to $i$ is $k_i$ ($i=1, 2, 3, \ldots$), let $b_j = k_1 + k_2 + \ldots + k_j$ ($j=1, 2, 3, \ldots$),
define $g(m) = b_1 + b_2 + \ldots + b_m - 100m$ ($m=1, 2, 3, \ldots$).
(I) Given $k_1 = 40, k_2 = 30, k_3 = 20, k_4 = 10, k_5 = \ldots = k_{100} = 0$, calculate $g(1), g(2), g(3), g(4)$;
(II) If the maximum term in $a_1, a_2, a_3, \ldots, a_{100}$ is 50, compare the values of $g(m)$ and $g(m+1)$;
(III) If $a_1 + a_2 + \ldots + a_{100} = 200$, find the minimum value of the function $g(m)$. | {
"answer": "-100",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a pyramid \( S-ABCD \) with a square base where each side measures 2, and \( SD \perp \) plane \( ABCD \) and \( SD = AB \). Determine the surface area of the circumscribed sphere of the pyramid \( S-ABCD \). | {
"answer": "12\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What will be the length of the strip if a cubic kilometer is cut into cubic meters and laid out in a single line? | {
"answer": "1000000",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A parallelogram has its diagonals making an angle of \(60^{\circ}\) with each other. If two of its sides have lengths 6 and 8, find the area of the parallelogram. | {
"answer": "14\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In how many ways can four married couples sit around a circular table such that no man sits next to his wife? | {
"answer": "1488",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a regular 2017-sided polygon \(A_{1} A_{2} \cdots A_{2017}\) inscribed in a unit circle \(\odot O\), choose any two different vertices \(A_{i}, A_{j}\). Find the probability that \(\overrightarrow{O A_{i}} \cdot \overrightarrow{O A_{j}} > \frac{1}{2}\). | {
"answer": "1/3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the product
\[
24^{a} \cdot 25^{b} \cdot 26^{c} \cdot 27^{d} \cdot 28^{e} \cdot 29^{f} \cdot 30^{g}
\]
seven numbers \(1, 2, 3, 5, 8, 10, 11\) were assigned to the exponents \(a, b, c, d, e, f, g\) in some order. Find the maximum number of zeros that can appear at the end of the decimal representation of this product. | {
"answer": "32",
"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. Given that $\sin C + \sin(B - A) = \sqrt{2} \sin 2A$, and $A \neq \frac{\pi}{2}$.
(I) Find the range of values for angle $A$;
(II) If $a = 1$, the area of $\triangle ABC$ is $S = \frac{\sqrt{3} + 1}{4}$, and $C$ is an obtuse angle, find the measure of angle $A$. | {
"answer": "\\frac{\\pi}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given an arithmetic sequence $\left\{a_{n}\right\}$ with the sum of the first 12 terms being 60, find the minimum value of $\left|a_{1}\right| + \left|a_{2}\right| + \cdots + \left|a_{12}\right|$. | {
"answer": "60",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Convert the base 2 number \(1011111010_2\) to its base 4 representation. | {
"answer": "23322_4",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The solutions to the equation $(z-4)^6 = 64$ are connected in the complex plane to form a convex regular polygon, three of whose vertices are labelled $D, E,$ and $F$. What is the least possible area of triangle $DEF$? | {
"answer": "\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $$\alpha \in \left( \frac{5}{4}\pi, \frac{3}{2}\pi \right)$$ and it satisfies $$\tan\alpha + \frac{1}{\tan\alpha} = 8$$, then $\sin\alpha\cos\alpha = \_\_\_\_\_\_$; $\sin\alpha - \cos\alpha = \_\_\_\_\_\_$. | {
"answer": "-\\frac{\\sqrt{3}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Unconventional dice are to be designed such that the six faces are marked with numbers from $1$ to $6$ with $1$ and $2$ appearing on opposite faces. Further, each face is colored either red or yellow with opposite faces always of the same color. Two dice are considered to have the same design if one of them can be rotated to obtain a dice that has the same numbers and colors on the corresponding faces as the other one. Find the number of distinct dice that can be designed. | {
"answer": "48",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given real numbers $a$ and $b \gt 0$, if $a+2b=1$, then the minimum value of $\frac{3}{b}+\frac{1}{a}$ is ______. | {
"answer": "7 + 2\\sqrt{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Mary and James each sit in a row of 10 chairs. They choose their seats at random. Two of the chairs (chair number 4 and chair number 7) are broken and cannot be chosen. What is the probability that they do not sit next to each other? | {
"answer": "\\frac{3}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the prime factorization of $215^7$, $p^7 \cdot q^6 \cdot r^6$, where $p$, $q$, and $r$ are prime numbers, determine the number of positive integer divisors of $215^7$ that are perfect squares or perfect cubes (or both). | {
"answer": "21",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The monkey has 100 bananas and its home is 50 meters away. The monkey can carry at most 50 bananas at a time and eats one banana for every meter walked. Calculate the maximum number of bananas the monkey can bring home. | {
"answer": "25",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The base of a triangle is 20; the medians drawn to the lateral sides are 18 and 24. Find the area of the triangle. | {
"answer": "288",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a deck of three red cards labeled $A$, $B$, $C$, three green cards labeled $A$, $B$, $C$, and three blue cards labeled $A$, $B$, $C$, calculate the probability of drawing a winning pair. | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On the coordinate plane (\( x; y \)), a circle with radius 4 and center at the origin is drawn. A line given by the equation \( y = 4 - (2 - \sqrt{3}) x \) intersects the circle at points \( A \) and \( B \). Find the sum of the length of segment \( A B \) and the length of the shorter arc \( A B \). | {
"answer": "4\\sqrt{2 - \\sqrt{3}} + \\frac{2\\pi}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A regular $n$-gon has $n$ diagonals, its perimeter is $p$, and the sum of the lengths of all the diagonals is $q$. What is $\frac{p}{q} + \frac{q}{p}$? | {
"answer": "\\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $\cos(\frac{\pi}{6} - \alpha) = \frac{3}{5}$, find the value of $\cos(\frac{5\pi}{6} + \alpha)$:
A) $\frac{3}{5}$
B) $-\frac{3}{5}$
C) $\frac{4}{5}$
D) $-\frac{4}{5}$ | {
"answer": "-\\frac{3}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If for any $x \in D$, the inequality $f_1(x) \leq f(x) \leq f_2(x)$ holds, then the function $f(x)$ is called a "compromise function" of the functions $f_1(x)$ to $f_2(x)$ over the interval $D$. It is known that the function $f(x) = (k-1)x - 1$, $g(x) = 0$, $h(x) = (x+1)\ln x$, and $f(x)$ is a "compromise function" of $g(x)$ to $h(x)$ over the interval $[1, 2e]$, then the set of values of the real number $k$ is \_\_\_\_\_\_. | {
"answer": "\\{2\\}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
$AL$ and $BM$ are the angle bisectors of triangle $ABC$. The circumcircles of triangles $ALC$ and $BMC$ intersect again at point $K$, which lies on side $AB$. Find the measure of angle $ACB$. | {
"answer": "60",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On the base \(AC\) of an isosceles triangle \(ABC (AB = BC)\), point \(M\) is marked. It is known that \(AM = 7\), \(MB = 3\), \(\angle BMC = 60^\circ\). Find the length of segment \(AC\). | {
"answer": "17",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a convex 13-gon, all the diagonals are drawn. They divide it into polygons. Consider a polygon among them with the largest number of sides. What is the maximum number of sides it can have? | {
"answer": "13",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The six-digit number $M=\overline{abc321}$, where $a, b, c$ are three different numbers, and all are greater than 3. If $M$ is a multiple of 7, what is the smallest value of $M$? | {
"answer": "468321",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given an arithmetic sequence $\{a\_n\}$, where $a\_n \in \mathbb{N}^*$, and $S\_n = \frac{1}{8}(a\_n + 2)^2$. If $b\_n = \frac{1}{2}a\_n - 30$, find the minimum value of the sum of the first $\_\_\_\_\_\_$ terms of the sequence $\{b\_n\}$. | {
"answer": "15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On the sides $AB, AC, BC$ of an equilateral triangle $ABC$, with a side length of 2, points $C_{1}, B_{1}, A_{1}$ are chosen respectively.
What is the maximum possible value of the sum of the radii of the circles inscribed in the triangles $AB_{1}C_{1}$, $A_{1}BC_{1}$, and $A_{1}B_{1}C$? | {
"answer": "\\frac{\\sqrt{3}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
From the set \( \{1, 2, 3, \ldots, 999, 1000\} \), select \( k \) numbers. If among the selected numbers, there are always three numbers that can form the side lengths of a triangle, what is the smallest value of \( k \)? Explain why. | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $ABC$, let $a$, $b$, and $c$ be the lengths of the sides opposite to angles $A$, $B$, and $C$ respectively, with $b=4$ and $\frac{\cos B}{\cos C} = \frac{4}{2a - c}$.
(1) Find the measure of angle $B$;
(2) Find the maximum area of $\triangle ABC$. | {
"answer": "4\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In an isosceles triangle \(ABC \) (\(AB = BC\)), a point \(D\) is taken on the side \(BC\) such that \(BD : DC = 1 : 4\).
In what ratio does the line \(AD\) divide the height \(BE\) of the triangle \(ABC\), counted from the vertex \(B\)? | {
"answer": "1:2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the product of all possible real values for $k$ such that the system of equations $$ x^2+y^2= 80 $$ $$ x^2+y^2= k+2x-8y $$ has exactly one real solution $(x,y)$ .
*Proposed by Nathan Xiong* | {
"answer": "960",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a given triangle, for $\angle P$ to be the largest angle of the triangle, it must be that $a < y < b$. The side lengths are given by $y+6$, $2y+1$, and $5y-10$. What is the least possible value of $b-a$, expressed as a common fraction? | {
"answer": "4.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the hyperbola $\frac{x^{2}}{m} + \frac{y^{2}}{n} = 1 (m < 0 < n)$ with asymptote equations $y = \pm \sqrt{2}x$, calculate the hyperbola's eccentricity. | {
"answer": "\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a particular state, the design of vehicle license plates was changed from an old format to a new one. Under the old scheme, each license plate consisted of two letters followed by three digits (e.g., AB123). The new scheme is made up of four letters followed by two digits (e.g., ABCD12). Calculate by how many times has the number of possible license plates increased.
A) $\frac{26}{10}$
B) $\frac{26^2}{10^2}$
C) $\frac{26^2}{10}$
D) $\frac{26^3}{10^3}$
E) $\frac{26^3}{10^2}$ | {
"answer": "\\frac{26^2}{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are 7 parking spaces in a row in a parking lot, and now 4 cars need to be parked. If 3 empty spaces need to be together, calculate the number of different parking methods. | {
"answer": "120",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
(1) Point $P$ is any point on the curve $y=x^{2}-\ln x$. The minimum distance from point $P$ to the line $x-y-4=0$ is ______.
(2) If the tangent line to the curve $y=g(x)$ at the point $(1,g(1))$ is $y=2x+1$, then the equation of the tangent line to the curve $f(x)=g(x)+\ln x$ at the point $(1,f(1))$ is ______.
(3) Given that the distance from point $P(1,0)$ to one of the asymptotes of the hyperbola $C: \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1 (a > 0, b > 0)$ is $\frac{1}{2}$, the eccentricity of the hyperbola $C$ is ______.
(4) A line passing through point $M(1,1)$ with a slope of $-\frac{1}{2}$ intersects the ellipse $C: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 (a > b > 0)$ at points $A$ and $B$. If $M$ is the midpoint of segment $AB$, then the eccentricity of the ellipse $C$ is ______. | {
"answer": "\\frac{\\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the distance between the two (non-intersecting) face diagonals on adjacent faces of a unit cube? | {
"answer": "\\frac{\\sqrt{3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Petya and his three classmates started a 100-meter race simultaneously, and Petya finished first. Twelve seconds after the race began, no one had finished yet, and all four participants had collectively run a total of 288 meters. When Petya finished the race, the other three participants had a combined distance of 40 meters left to the finish line. How many meters did Petya run in the first 12 seconds? Justify your answer. It is assumed that each participant ran with a constant speed. | {
"answer": "80",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $A$ be the set of all real numbers $a$ that satisfy $\left(a-2\right)x^{2}+2\left(a-2\right)x-4 \lt 0$ for any $x\in R$. Let $B$ be the set of all real numbers $x$ that satisfy $\left(a-2\right)x^{2}+2\left(a-2\right)x-4 \lt 0$ for any $a\in \left[-2,2\right]$. Find $A\cap (\complement _{R}B)$. | {
"answer": "\\{-1\\}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A numerical sequence is defined by the conditions: \( a_{1}=1 \), \( a_{n+1}=a_{n}+\left \lfloor \sqrt{a_{n}} \right \rfloor \).
How many perfect squares are among the first terms of this sequence that do not exceed 1,000,000? | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A particle moves in a straight line inside a square of side 1. It is reflected from the sides, but absorbed by the four corners. It starts from an arbitrary point \( P \) inside the square. Let \( c(k) \) be the number of possible starting directions from which it reaches a corner after traveling a distance \( k \) or less. Find the smallest constant \( a_2 \), such that for some constants \( a_1 \) and \( a_0 \), \( c(k) \leq a_2 k^2 + a_1 k + a_0 \) for all \( P \) and all \( k \). | {
"answer": "\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Teacher Zhang led the students of class 6 (1) to plant trees. The students can be divided into 5 equal groups. It is known that each teacher and student plants the same number of trees, with a total of 527 trees planted. How many students are there in class 6 (1)? | {
"answer": "30",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For positive integers $n$, let $h(n)$ return the smallest positive integer $k$ such that $\frac{1}{k}$ has exactly $n$ digits after the decimal point, and $k$ is divisible by 3. How many positive integer divisors does $h(2010)$ have? | {
"answer": "4022",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $y = ax^{2} + x - b$ where $a \in \mathbb{R}$ and $b \in \mathbb{R}$.
$(1)$ If $b = 1$ and the set $\{x | y = 0\}$ has exactly one element, find the set of possible values for the real number $a$.
$(2)$ Solve the inequality with respect to $x$: $y < (a-1)x^{2} + (b+2)x - 2b$.
$(3)$ When $a > 0$ and $b > 1$, let $P$ be the solution set of the inequality $y > 0$, and $Q = \{x | -2-t < x < -2+t\}$. If for any positive number $t$, $P \cap Q \neq \varnothing$, find the maximum value of $\frac{1}{a} - \frac{1}{b}$. | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A positive integer \( m \) has the property that when multiplied by 12, the result is a four-digit number \( n \) of the form \( 20A2 \) for some digit \( A \). What is the four-digit number \( n \)? | {
"answer": "2052",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $f(x) = x^3 - 9x^2 + 27x - 25$ and let $g(f(x)) = 3x + 4$. What is the sum of all possible values of $g(7)$? | {
"answer": "39",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a round-robin tournament among $8$ chess players (each pair plays one match), the scoring rules are: the winner of a match earns $2$ points, a draw results in $1$ point for each player, and the loser scores $0$ points. The final scores of the players are all different, and the score of the player in second place equals the sum of the scores of the last four players. What is the score of the second-place player? | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Compute the smallest base-10 positive integer greater than 7 that is a palindrome when written in both base 3 and 5. | {
"answer": "26",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On the lateral side \( CD \) of trapezoid \( ABCD \) (\( AD \parallel BC \)), a point \( M \) is marked. From vertex \( A \), a perpendicular \( AH \) is drawn to segment \( BM \). It turns out that \( AD = HD \). Find the length of segment \( AD \), given that \( BC = 16 \), \( CM = 8 \), and \( MD = 9 \). | {
"answer": "18",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $0\leqslant x\_0 < 1$, for all integers $n > 0$, let $x\_n= \begin{cases} 2x_{n-1}, & 2x_{n-1} < 1 \\ 2x_{n-1}-1, & 2x_{n-1} \geqslant 1 \end{cases}$. Find the number of $x\_0$ that makes $x\_0=x\_6$ true. | {
"answer": "64",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Evaluate the sum $$\frac{3^1}{9^1 - 1} + \frac{3^2}{9^2 - 1} + \frac{3^4}{9^4 - 1} + \frac{3^8}{9^8 - 1} + \cdots.$$ | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Using five nines (9), arithmetic operations, and exponentiation, create the numbers from 1 to 13. | {
"answer": "13",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \(a, b, c\) be arbitrary real numbers such that \(a > b > c\) and \((a - b)(b - c)(c - a) = -16\). Find the minimum value of \(\frac{1}{a - b} + \frac{1}{b - c} - \frac{1}{c - a}\). | {
"answer": "\\frac{5}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Elena intends to buy 7 binders priced at $\textdollar 3$ each. Coincidentally, a store offers a 25% discount the next day and an additional $\textdollar 5$ rebate for purchases over $\textdollar 20$. Calculate the amount Elena could save by making her purchase on the day of the discount. | {
"answer": "10.25",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A shopkeeper purchases 2000 pens at a cost of $0.15 each. If the shopkeeper wants to sell them for $0.30 each, calculate the number of pens that need to be sold to make a profit of exactly $120.00. | {
"answer": "1400",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=|x-2|+|x-a^{2}|$.
$(1)$ If the inequality $f(x)\leqslant a$ has solutions for $x$, find the range of the real number $a$;
$(2)$ If the positive real numbers $m$, $n$ satisfy $m+2n=a$, when $a$ takes the maximum value from $(1)$, find the minimum value of $\left( \dfrac {1}{m}+ \dfrac {1}{n}\right)$. | {
"answer": "\\dfrac {3}{2}+ \\sqrt {2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
(1) Calculate: $\left( \frac{1}{8} \right)^{-\frac{1}{3}} - 3\log_{3}^{2}(\log_{3}4) \cdot (\log_{8}27) + 2\log_{\frac{1}{6}} \sqrt{3} - \log_{6}2$
(2) Calculate: $27^{\frac{2}{3}} - 2^{\log_{2}3} \times \log_{2}\frac{1}{8} + 2\lg \left( \sqrt{3+\sqrt{5}} + \sqrt{3-\sqrt{5}} \right)$ | {
"answer": "19",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Dasha added 158 numbers and obtained a sum of 1580. Then Sergey tripled the largest of these numbers and decreased another number by 20. The resulting sum remained unchanged. Find the smallest of the original numbers. | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The largest four-digit number whose digits add to 17 is 9800. The 5th largest four-digit number whose digits have a sum of 17 is: | {
"answer": "9611",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The shortest distance from a point on the parabola $x^2=y$ to the line $y=2x+m$ is $\sqrt{5}$. Find the value of $m$. | {
"answer": "-6",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A semicircle has diameter $XY$ . A square $PQRS$ with side length 12 is inscribed in the semicircle with $P$ and $S$ on the diameter. Square $STUV$ has $T$ on $RS$ , $U$ on the semicircle, and $V$ on $XY$ . What is the area of $STUV$ ? | {
"answer": "36",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Shift the graph of the function $f(x)=2\sin(2x+\frac{\pi}{6})$ to the left by $\frac{\pi}{12}$ units, and then shift it upwards by 1 unit to obtain the graph of $g(x)$. If $g(x_1)g(x_2)=9$, and $x_1, x_2 \in [-2\pi, 2\pi]$, then find the maximum value of $2x_1-x_2$. | {
"answer": "\\frac {49\\pi}{12}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $g(x)=ax^2+bx+c$, where $a$, $b$, and $c$ are integers. Suppose that $g(2)=0$, $90<g(9)<100$, $120<g(10)<130$, $7000k<g(150)<7000(k+1)$ for some integer $k$. What is $k$? | {
"answer": "k=6",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $\mathbf{a}, \mathbf{b},$ and $\mathbf{c}$ be nonzero vectors, no two of which are parallel, such that
\[(\mathbf{a} \times \mathbf{b}) \times \mathbf{c} = \frac{1}{2} \|\mathbf{b}\| \|\mathbf{c}\| \mathbf{a}.\]
Let $\theta$ be the angle between $\mathbf{b}$ and $\mathbf{c}.$ Determine $\cos \theta.$ | {
"answer": "-\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider the sequence
$$
a_{n}=\cos (\underbrace{100 \ldots 0^{\circ}}_{n-1})
$$
For example, $a_{1}=\cos 1^{\circ}, a_{6}=\cos 100000^{\circ}$.
How many of the numbers $a_{1}, a_{2}, \ldots, a_{100}$ are positive? | {
"answer": "99",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Compute the limit of the function:
$$
\lim _{x \rightarrow \pi} \frac{\ln (2+\cos x)}{\left(3^{\sin x}-1\right)^{2}}
$$ | {
"answer": "\\frac{1}{2 \\ln^2 3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a right triangle $ABC$ with equal legs $AC$ and $BC$, a circle is constructed with $AC$ as its diameter, intersecting side $AB$ at point $M$. Find the distance from vertex $B$ to the center of this circle if $BM = \sqrt{2}$. | {
"answer": "\\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Suppose \( A, B, C \) are three angles such that \( A \geq B \geq C \geq \frac{\pi}{8} \) and \( A + B + C = \frac{\pi}{2} \). Find the largest possible value of the product \( 720 \times (\sin A) \times (\cos B) \times (\sin C) \). | {
"answer": "180",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the greatest positive number $\lambda$ such that for any real numbers $a$ and $b$, the inequality $\lambda a^{2} b^{2}(a+b)^{2} \leqslant\left(a^{2}+ab+b^{2}\right)^{3}$ holds. | {
"answer": "\\frac{27}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many times during a day does the angle between the hour and minute hands measure exactly $17^{\circ}$? | {
"answer": "44",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The angle bisectors \( A L_{1} \) and \( B L_{2} \) of triangle \( A B C \) intersect at point \( I \). It is known that \( A I : I L_{1} = 3 \) and \( B I : I L_{2} = 2 \). Find the ratio of the sides of triangle \( A B C \). | {
"answer": "3:4:5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find all 4-digit numbers $\overline{abcd}$ that are multiples of $11$ , such that the 2-digit number $\overline{ac}$ is a multiple of $7$ and $a + b + c + d = d^2$ . | {
"answer": "3454",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
From a barrel, 4 liters of wine are drawn, and this is replaced with 4 liters of water. From the resulting mixture, 4 liters are drawn again and replaced with 4 liters of water. This operation is repeated a total of three times, and the final result is that there are 2.5 liters more water than wine. How many liters of wine were originally in the barrel? | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $a$, $b$, and $c$ represent the sides opposite to angles $A$, $B$, and $C$ of $\triangle ABC$, respectively, and $\overrightarrow{m}=(a,−\sqrt {3}b)$, $\overrightarrow{n}=(\sin B,\cos A)$, if $a= \sqrt {7}$, $b=2$, and $\overrightarrow{m} \perp \overrightarrow{n}$, then the area of $\triangle ABC$ is ______. | {
"answer": "\\frac{3\\sqrt{3}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A right octagonal pyramid has two cross sections obtained by slicing the pyramid with planes parallel to the octagonal base. The area of the smaller cross section is $256\sqrt{2}$ square feet and the area of the larger cross section is $576\sqrt{2}$ square feet. The distance between the two planes is $12$ feet. Determine the distance from the apex of the pyramid to the plane of the larger cross section. | {
"answer": "36",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the diagram, $AB$ is perpendicular to $BC$, and $CD$ is perpendicular to $AD$. Also, $AC = 625$ and $AD = 600$. If $\angle BAC = 2 \angle DAC$, what is the length of $BC$? | {
"answer": "336",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the definite integral:
$$
\int_{0}^{4} e^{\sqrt{(4-x) /(4+x)}} \cdot \frac{d x}{(4+x) \sqrt{16-x^{2}}}
$$ | {
"answer": "\\frac{1}{4}(e-1)",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Identify a six-digit number \( N \) composed of distinct digits such that the numbers \( 2N, 3N, 4N, 5N, \) and \( 6N \) are permutations of its digits. | {
"answer": "142857",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a finite increasing sequence \(a_{1}, a_{2}, \ldots, a_{n}\) of natural numbers (with \(n \geq 3\)), and the recurrence relation \(a_{k+2} = 3a_{k+1} - 2a_{k} - 2\) holds for all \(\kappa \leq n-2\). The sequence must contain \(a_{k} = 2022\). Determine the maximum number of three-digit numbers that are multiples of 4 that this sequence can contain. | {
"answer": "225",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the tetrahedron $ABCD$, $\triangle ABC$ is an equilateral triangle, $AD = BD = 2$, $AD \perp BD$, and $AD \perp CD$. Find the distance from point $D$ to the plane $ABC$. | {
"answer": "\\frac{2\\sqrt{3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the trapezoid \(ABCD\) (\(AD \parallel BC\)), a perpendicular \(EF\) is drawn from point \(E\) (the midpoint of \(CD\)) to line \(AB\). Find the area of the trapezoid if \(AB = 5\) and \(EF = 4\). | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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