problem stringlengths 10 5.15k | answer dict |
|---|---|
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given vectors $\vec{m}=(a,c)$ and $\vec{n}=(\cos C,\cos A)$.
1. If $\vec{m}\parallel \vec{n}$ and $a= \sqrt {3}c$, find angle $A$;
2. If $\vec{m}\cdot \vec{n}=3b\sin B$ and $\cos A= \frac {3}{5}$, find the value of $\cos C$. | {
"answer": "\\frac {4-6 \\sqrt {2}}{15}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
From post office $A$, a car leaves heading towards post office $B$. After 20 minutes, a motorcyclist departs in pursuit of the car, traveling at a speed of 60 km/h. Upon catching up with the car, the motorcyclist delivers a package to the driver's cab and immediately turns back. The car reaches $B$ at the moment when the motorcyclist is halfway back from the rendezvous point to $A$. Determine the speed of the car, given that the distance from $A$ to $B$ is 82.5 km. | {
"answer": "45",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 (a > 0, b > 0)$, a line passing through its right focus $F$ and parallel to the asymptote $y = -\frac{b}{a}x$ intersects the right branch of the hyperbola and the other asymptote at points $A$ and $B$ respectively, with $\overrightarrow{FA} = \overrightarrow{AB}$. Find the eccentricity of the hyperbola.
A) $\frac{3}{2}$
B) $\sqrt{2}$
C) $\sqrt{3}$
D) $2$ | {
"answer": "\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the limit of the function:
\[ \lim _{x \rightarrow \frac{\pi}{3}} \frac{1-2 \cos x}{\sin (\pi-3 x)} \] | {
"answer": "-\\frac{\\sqrt{3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider that for integers from 1 to 1500, $x_1+2=x_2+4=x_3+6=\cdots=x_{1500}+3000=\sum_{n=1}^{1500}x_n + 3001$. Find the value of $\left\lfloor|S|\right\rfloor$, where $S=\sum_{n=1}^{1500}x_n$. | {
"answer": "1500",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The cross-section of a sphere passing through points $A$, $B$, and $C$, whose distance from the center of the sphere is equal to half the radius, and $AB \perp BC$, $AB=1$, $BC=\sqrt{2}$. Calculate the surface area of the sphere. | {
"answer": "4\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a parabola $C$ that passes through the point $(4,4)$ and its focus lies on the $x$-axis.
$(1)$ Find the standard equation of parabola $C$.
$(2)$ Let $P$ be any point on parabola $C$. Find the minimum distance between point $P$ and the line $x - y + 4 = 0$. | {
"answer": "\\frac{3\\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A teacher received letters on Monday through Friday with counts of $10$, $6$, $8$, $5$, $6$ respectively. Calculate the variance (${s^{2}} =$) of this data set. | {
"answer": "3.2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the value of: \(\frac{\left(\sqrt{3} \cdot \tan 12^{\circ} - 3\right) \cdot \csc 12^{\circ}}{4 \cos ^{2} 12^{\circ} - 2}\). | {
"answer": "-4 \\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The roots of the equation $x^2 + kx + 8 = 0$ differ by 10. Find the greatest possible value of $k$. | {
"answer": "2\\sqrt{33}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the number \(2016 * * * * 02 *\), each of the 5 asterisks needs to be replaced with any of the digits \(0, 2, 4, 7, 8, 9\) (digits can be repeated) so that the resulting 11-digit number is divisible by 6. In how many ways can this be done? | {
"answer": "1728",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Points \( M, N, P, Q \) are taken on the diagonals \( D_1A, A_1B, B_1C, C_1D \) of the faces of cube \( ABCD A_1B_1C_1D_1 \) respectively, such that:
\[ D_1M: D_1A = BA_1: BN = B_1P: B_1C = DQ: DC_1 = \mu, \]
and the lines \( MN \) and \( PQ \) are mutually perpendicular. Find \( \mu \). | {
"answer": "\\frac{1}{\\sqrt{2}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Elisa creates a sequence in a manner similar to Jacob but starts with the first term as 10. Each succeeding term depends on the outcome of flipping a fair coin: If it lands heads, the next term is obtained by doubling the previous term and subtracting 1; if it lands tails, the next term is half of the previous term, subtracting 1. What is the probability that the fourth term in Elisa's sequence is an integer?
A) $\frac{1}{4}$
B) $\frac{1}{3}$
C) $\frac{1}{2}$
D) $\frac{3}{4}$
E) $\frac{5}{8}$ | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the Cartesian coordinate system $xOy$, it is known that the line $l_1$ is defined by the parametric equations $\begin{cases}x=t\cos \alpha\\y=t\sin \alpha\end{cases}$ (where $t$ is the parameter), and the line $l_2$ by $\begin{cases}x=t\cos(\alpha + \frac{\pi}{4})\\y=t\sin(\alpha + \frac{\pi}{4})\end{cases}$ (where $t$ is the parameter), with $\alpha\in(0, \frac{3\pi}{4})$. Taking the point $O$ as the pole and the non-negative $x$-axis as the polar axis, a polar coordinate system is established with the same length unit. The polar equation of curve $C$ is $\rho-4\cos \theta=0$.
$(1)$ Write the polar equations of $l_1$, $l_2$ and the rectangular coordinate equation of curve $C$.
$(2)$ Suppose $l_1$ and $l_2$ intersect curve $C$ at points $A$ and $B$ (excluding the coordinate origin), calculate the value of $|AB|$. | {
"answer": "2\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $P(x) = b_0 + b_1x + \dots + b_nx^n$ be a polynomial with integer coefficients, and $0 \le b_i < 5$ for all $0 \le i \le n$.
Given that $P(\sqrt{5}) = 40 + 31\sqrt{5}$, compute $P(3)$. | {
"answer": "381",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The integer parts of two finite decimals are 7 and 10, respectively. How many possible values are there for the integer part of the product of these two finite decimals? | {
"answer": "18",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A column of infantry stretched out over 1 km. Sergeant Kim, riding on a hoverboard from the end of the column, reached its front and returned to the end. During this time, the infantrymen walked 2 km 400 meters. What distance did the sergeant cover during this time? | {
"answer": "3.6",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( A = \{1, 2, 3, 4, 5, 6\} \). Find the number of distinct functions \( f: A \rightarrow A \) such that \( f(f(f(n))) = n \) for all \( n \in A \). | {
"answer": "81",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The Rotokas alphabet has twelve letters: A, E, G, I, K, O, P, R, S, T, U, and V. Design license plates of five letters using only these letters where the license plate ends with either G or K, starts with S, cannot contain T, and where no letters repeat. How many such license plates are possible? | {
"answer": "1008",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( a, b, c \) be the side lengths of a right triangle, with \( a \leqslant b < c \). Determine the maximum constant \( k \) such that \( a^{2}(b+c) + b^{2}(c+a) + c^{2}(a+b) \geqslant k a b c \) holds for all right triangles, and identify when equality occurs. | {
"answer": "2 + 3\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that a regular tetrahedron has all edge lengths of $\sqrt {2}$, and all four vertices are on the same spherical surface, find the surface area of this sphere. | {
"answer": "3\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Real numbers \( x, y, z \) satisfy \( x \geq y \geq z \geq 0 \) and \( 6x + 5y + 4z = 120 \). Find the sum of the maximum and minimum values of \( x + y + z \). | {
"answer": "44",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the ellipse $$\frac {x^{2}}{a^{2}}+ \frac {y^{2}}{b^{2}}=1(a＞b＞0)$$ with eccentricity $$e= \frac { \sqrt {3}}{2}$$, A and B are the left and right vertices of the ellipse, respectively, and P is a point on the ellipse different from A and B. The angles of inclination of lines PA and PB are $\alpha$ and $\beta$, respectively. Then, $$\frac {cos(\alpha-\beta)}{cos(\alpha +\beta )}$$ equals \_\_\_\_\_\_. | {
"answer": "\\frac {3}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The height $BL$ of the rhombus $ABCD$, dropped perpendicular to the side $AD$, intersects the diagonal $AC$ at point $E$. Find $AE$ if $BL = 8$ and $AL:LD = 3:2$. | {
"answer": "3\\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Defined on $\mathbf{R}$, the function $f$ satisfies
$$
f(1+x)=f(9-x)=f(9+x).
$$
Given $f(0)=0$, and $f(x)=0$ has $n$ roots in the interval $[-4020, 4020]$, find the minimum value of $n$. | {
"answer": "2010",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\Delta ABC$, $c=2a$, $B={120}^{\circ}$, and the area of $\Delta ABC$ is $\frac{\sqrt{3}}{2}$.
(I) Find the value of $b$;
(II) Find the value of $\tan A$. | {
"answer": "\\frac{\\sqrt{3}}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The area of a trapezoid is 1. What is the minimum possible length of the longest diagonal of this trapezoid? | {
"answer": "\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Natasha and Inna each bought the same box of tea bags. It is known that one tea bag is enough for either two or three cups of tea. This box lasted Natasha for 41 cups of tea, and Inna for 58 cups of tea. How many tea bags were in the box? | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The reciprocal of the opposite number of \(-(-3)\) is \(\frac{1}{3}\). | {
"answer": "-\\frac{1}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the largest $n$ for which it is possible to construct two bi-infinite sequences $A$ and $B$ such that any subsequence of $B$ of length $n$ is contained in $A$, $A$ has a period of 1995, and $B$ does not have this property (is either non-periodic or has a period of a different length)? | {
"answer": "1995",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the quadratic equation $ax^{2}+bx+c=0$ with $a > 0$ and $b, c \in \mathbb{R}$, and the roots of the equation lying in the interval $(0, 2)$, determine the minimum value of the real number $a$ given that $25a+10b+4c \geqslant 4$ for $c \geqslant 1$. | {
"answer": "\\frac{16}{25}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let the set \(M = \{1,2,\cdots, 1000\}\). For any non-empty subset \(X\) of \(M\), let \(\alpha_X\) denote the sum of the largest and smallest numbers in \(X\). Find the arithmetic mean of all such \(\alpha_X\). | {
"answer": "1001",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If the polynomial $x^{3}+x^{10}=a_{0}+a_{1}(x+1)+...+a_{9}(x+1)^{9}+a_{10}(x+1)^{10}$, then $a_{9}=$ \_\_\_\_\_\_. | {
"answer": "-10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The problem is related to coordinate systems and parametric equations. In the rectangular coordinate system $xOy$, a pole is established at the coordinate origin, and the positive semi-axis of the $x$-axis is used as the polar axis to build a polar coordinate system. The polar coordinate equation of the curve $C1$ is $ρ\cos θ=4$.
(1) Point $M$ is a moving point on curve $C1$, and point $P$ is on line segment $OM$ such that $|OM|\cdot|OP|=16$. Determine the rectangular coordinate equation of the trajectory of point $P$ ($C2$).
(2) Let point $A$ have polar coordinates $\left(2, \dfrac{π}{3}\right)$, and point $B$ is on curve $C2$. Determine the maximum area of $\triangle OAB$. | {
"answer": "2+ \\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Evaluate $$\lceil\sqrt{10}\rceil + \lceil\sqrt{11}\rceil + \lceil\sqrt{12}\rceil + \cdots + \lceil\sqrt{34}\rceil$$
Note: For a real number $x,$ $\lceil x \rceil$ denotes the smallest integer that is greater than or equal to $x.$ | {
"answer": "127",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that F is the right focus of the ellipse $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1(a>b>0)$, and A is one endpoint of the ellipse's minor axis. If F is the trisection point of the chord of the ellipse that passes through AF, calculate the eccentricity of the ellipse. | {
"answer": "\\frac{\\sqrt{3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Four football teams participate in a round-robin tournament, where each team plays a match against every other team. In each match, the winning team earns 3 points, the losing team earns 0 points, and in the case of a draw, both teams earn 1 point each. After all matches are completed, it is known that the total points of the four teams are exactly four consecutive positive integers. Find the product of these four numbers. | {
"answer": "120",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a country with 15 cities, some of which are connected by airlines belonging to three different companies, it is known that even if any one of the airlines ceases operations, it will still be possible to travel between any two cities (possibly with transfers) using the remaining two companies' flights. What is the minimum number of airline routes in the country? | {
"answer": "21",
"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, satisfying $2a\sin A = (2\sin B - \sqrt{3}\sin C)b + (2\sin C - \sqrt{3}\sin B)c$.
(1) Find the measure of angle $A$.
(2) If $a=2$ and $b=2\sqrt{3}$, find the area of $\triangle ABC$. | {
"answer": "\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A hotelier wanted to equip the dining room with new chairs. In the catalog, he chose a type of chair. When placing the order, he learned from the manufacturer that as part of a discount event, every fourth chair is offered at half price and that, therefore, he could save the equivalent cost of seven and a half chairs from the original plan. The hotelier calculated that for the originally planned amount, he could purchase nine more chairs than he intended.
How many chairs did the hotelier originally want to buy?
(L. Simünek)
Hint: First solve the problem without the information that, for the originally planned amount, nine more chairs could be bought. | {
"answer": "63",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)$ and its derivative $f''(x)$ on $\mathbb{R}$, and for any real number $x$, it satisfies $f(x)+f(-x)=2x^{2}$, and for $x < 0$, $f''(x)+1 < 2x$, find the minimum value of the real number $a$ such that $f(a+1) \leqslant f(-a)+2a+1$. | {
"answer": "-\\dfrac {1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate Mr. $X$'s net gain or loss from the transactions, given that he sells his home valued at $12,000$ to Mr. $Y$ for a $20\%$ profit and then buys it back from Mr. $Y$ at a $15\%$ loss. | {
"answer": "2160",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a triangle $ABC$ with sides opposite to angles $A$, $B$, and $C$ being $a$, $b$, and $c$, respectively, and it is given that $(3b-c)\cos A = a\cos C$.
(1) Find the value of $\cos A$;
(2) If the area of $\triangle ABC$ is $S=2\sqrt{2}$, find the minimum value of the perimeter of $\triangle ABC$. | {
"answer": "2\\sqrt{6}+2\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, $a$, $b$, $c$ are the sides opposite to angles $A$, $B$, $C$ respectively, and it is given that $a\sin A - b\sin B = (\sqrt{3}a - c)\sin C$, with $a:b = 2:3$.
1. Find the value of $\sin C$.
2. If $b = 6$, find the area of $\triangle ABC$. | {
"answer": "2\\sqrt{3} + 4\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Initially, there are some red balls and some black balls in a box. After adding some black balls, the red balls account for one-fourth of the total number of balls. Then, after adding some red balls, the number of red balls becomes two-thirds the number of black balls. If the number of added black balls and red balls is the same, find the original ratio of the number of red balls to black balls in the box. | {
"answer": "1:2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If Xiao Zhang's daily sleep time is uniformly distributed between 6 to 9 hours, what is the probability that his average sleep time over two consecutive days is at least 7 hours? | {
"answer": "7/9",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many numbers between 10 and 13000, when read from left to right, are formed by consecutive digits in ascending order? For example, 456 is one of these numbers, but 7890 is not.
(a) 10
(b) 13
(c) 18
(d) 22
(e) 25 | {
"answer": "22",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let the set \( A = \{1, 2, 3, 4, 5, 6\} \) and the mapping \( f: A \rightarrow A \). If the triple composition \( f \cdot f \cdot f \) is an identity mapping, how many such functions \( f \) are there? | {
"answer": "81",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
V is the pyramidal region defined by the inequalities \( x, y, z \geq 0 \) and \( x + y + z \leq 1 \). Evaluate the integral:
\[ \int_V x y^9 z^8 (1 - x - y - z)^4 \, dx \, dy \, dz. \ | {
"answer": "\\frac{9! 8! 4!}{25!}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the complex numbers \( z_{1} = -\sqrt{3} - i \), \( z_{2} = 3 + \sqrt{3} i \), and \( z = (2 + \cos \theta) + i \sin \theta \), find the minimum value of \( \left|z - z_{1}\right| + \left|z - z_{2}\right| \). | {
"answer": "2 + 2\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On a particular street in Waterloo, there are exactly 14 houses, each numbered with an integer between 500 and 599, inclusive. The 14 house numbers form an arithmetic sequence in which 7 terms are even and 7 terms are odd. One of the houses is numbered 555 and none of the remaining 13 numbers has two equal digits. What is the smallest of the 14 house numbers?
(An arithmetic sequence is a sequence in which each term after the first is obtained from the previous term by adding a constant. For example, 3, 5, 7, 9 is an arithmetic sequence with four terms.) | {
"answer": "506",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given acute angles \\(\alpha\\) and \\(\beta\\) satisfy \\((\tan \alpha-1)(\tan \beta-1)=2\\), then the value of \\(\alpha+\beta\\) is \_\_\_\_\_\_. | {
"answer": "\\dfrac {3\\pi}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the number of permutations $(a_1, a_2, a_3, a_4, a_5, a_6, a_7)$ of $(1,2,3,4,5,6,7)$ that satisfy
\[\frac{a_1 + 1}{2} \cdot \frac{a_2 + 2}{2} \cdot \frac{a_3 + 3}{2} \cdot \frac{a_4 + 4}{2} \cdot \frac{a_5 + 5}{2} \cdot \frac{a_6 + 6}{2} \cdot \frac{a_7 + 7}{2} > 7!.\] | {
"answer": "5039",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the expansion of the binomial ${(\sqrt{x}-\frac{1}{{2x}}})^n$, only the coefficient of the 4th term is the largest. The constant term in the expansion is ______. | {
"answer": "\\frac{15}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( f(x) = a^x - 1 \). Find the largest value of \( a > 1 \) such that if \( 0 \leq x \leq 3 \), then \( 0 \leq f(x) \leq 3 \). | {
"answer": "\\sqrt[3]{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $e > 0$ be a given real number. Find the least value of $f(e)$ (in terms of $e$ only) such that the inequality $a^{3}+ b^{3}+ c^{3}+ d^{3} \leq e^{2}(a^{2}+b^{2}+c^{2}+d^{2}) + f(e)(a^{4}+b^{4}+c^{4}+d^{4})$ holds for all real numbers $a, b, c, d$ . | {
"answer": "\\frac{1}{4e^2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
$a$ , $b$ , $c$ are real. What is the highest value of $a+b+c$ if $a^2+4b^2+9c^2-2a-12b+6c+2=0$ | {
"answer": "\\frac{17}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $\triangle ABC$, $\cos C=\frac{2}{3}$, $AC=4$, $BC=3$, calculate the value of $\tan B$. | {
"answer": "4\\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A bus at a certain station departs punctually at 7:00 and 7:30 in the morning. Student Xiao Ming arrives at the station to catch the bus between 6:50 and 7:30, and his arrival time is random. The probability that he waits for less than 10 minutes for the bus is ______. | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the approximate change in the function \( y = 3x^{2} + 2 \) at \( x = 2 \) with \( \Delta x = 0.001 \). Determine the absolute and relative errors of the calculation. | {
"answer": "0.012",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the sum of the positive solutions to \( 2x^2 - x \lfloor x \rfloor = 5 \), where \( \lfloor x \rfloor \) is the largest integer less than or equal to \( x \)? | {
"answer": "\\frac{3 + \\sqrt{41} + 2\\sqrt{11}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A certain company implements an annual salary system, where an employee's annual salary consists of three components: basic salary, housing allowance, and medical expenses, as specified below:
| Item | Salary in the First Year (in ten thousand yuan) | Calculation Method After One Year |
|---------------|--------------------------------------------------|-----------------------------------|
| Basic Salary | $1$ | Increases at the same rate each year |
| Housing Allowance | $0.04$ | Increases by $0.04$ each year |
| Medical Expenses | $0.1384$ | Fixed and unchanged |
(1) Let the annual growth rate of the basic salary be $x$. Express the basic salary in the third year in terms of $x$.
(2) A person has been working in the company for $3$ years. He calculated that the sum of the housing allowance and medical expenses he received in these $3$ years is exactly $18\%$ of the total basic salary for these $3$ years. What is the annual growth rate of the basic salary? | {
"answer": "20\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The integer points $(x, y)$ in the first quadrant satisfy $x + y > 8$ and $x \leq y \leq 8$. How many such integer points $(x, y)$ are there? | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
An isosceles triangle has its vertex at $(0,5)$ and a base between points $(3,5)$ and $(13,5)$. The two equal sides are each 10 units long. If the third vertex (top vertex) is in the first quadrant, what is the y-coordinate? | {
"answer": "5 + 5\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The height of a right truncated quadrilateral pyramid is 3 cm, its volume is 38 cm³, and the areas of its bases are in the ratio 4:9. Determine the lateral surface area of the truncated pyramid. | {
"answer": "10 \\sqrt{19}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle \(A B C\), side \(B C\) equals 4, and the median drawn to this side equals 3. Find the length of the common chord of two circles, each of which passes through point \(A\) and is tangent to \(B C\), with one tangent at point \(B\) and the other at point \(C\). | {
"answer": "\\frac{5}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the minimum number of planes that divide a cube into at least 300 parts. | {
"answer": "13",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the rhombus \(ABCD\), the angle \(BCD\) is \(135^{\circ}\), and the sides are 8. A circle touches the line \(CD\) and intersects side \(AB\) at two points located 1 unit away from \(A\) and \(B\). Find the radius of this circle. | {
"answer": "\\frac{41 \\sqrt{2}}{16}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Suppose \(ABCD\) is a rectangle whose diagonals meet at \(E\). The perimeter of triangle \(ABE\) is \(10\pi\) and the perimeter of triangle \(ADE\) is \(n\). Compute the number of possible integer values of \(n\). | {
"answer": "47",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the sequence: \(\frac{2}{3}, \frac{2}{9}, \frac{4}{9}, \frac{6}{9}, \frac{8}{9}, \frac{2}{27}, \frac{4}{27}, \cdots, \frac{26}{27}, \cdots, \frac{2}{3^{n}}, \frac{4}{3^{n}}, \cdots, \frac{3^{n}-1}{3^{n}}, \cdots\). Then, \(\frac{2020}{2187}\) is the \(n\)-th term of this sequence. | {
"answer": "1553",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $\overrightarrow{a}=(2,3)$, $\overrightarrow{b}=(-4,7)$, and $\overrightarrow{a}+\overrightarrow{c}=\overrightarrow{0}$, find the projection of $\overrightarrow{c}$ on the direction of $\overrightarrow{b}$. | {
"answer": "-\\frac{\\sqrt{65}}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In an isosceles right triangle \( \triangle ABC \), \( CA = CB = 1 \). Let point \( P \) be any point on the boundary of \( \triangle ABC \). Find the maximum value of \( PA \cdot PB \cdot PC \). | {
"answer": "\\frac{\\sqrt{2}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the limit of the function:
$$\lim _{x \rightarrow \pi}\left(\operatorname{ctg}\left(\frac{x}{4}\right)\right)^{1 / \cos \left(\frac{x}{2}\right)}$$ | {
"answer": "e",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The function \( f(x) \) has a domain of \( \mathbf{R} \). For any \( x \in \mathbf{R} \) and \( y \neq 0 \), \( f(x+y)=f\left(x y-\frac{x}{y}\right) \), and \( f(x) \) is a periodic function. Find one of its positive periods. | {
"answer": "\\frac{1 + \\sqrt{5}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the value of $b$ such that the following equation in base $b$ is true:
$$\begin{array}{c@{}c@{}c@{}c@{}c@{}c@{}c}
&&8&7&3&6&4_b\\
&+&9&2&4&1&7_b\\
\cline{2-7}
&1&8&5&8&7&1_b.
\end{array}$$ | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a cube $ABCDEFGH$ with an edge length of $6$, let $M$ and $N$ be points on $BB_1$ and $B_1C_1$ respectively, such that $B_1M = B_1N = 2$. Let $S$ and $P$ be the midpoints of segments $AD$ and $MN$ respectively. Find the distance between the skew lines $SP$ and $AC_1$. | {
"answer": "\\frac{\\sqrt{114}}{38}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a small office, each worker has a probability of being late once every 40 days due to traffic. Calculate the probability that among three randomly chosen workers on a given day, exactly two are late while the third one is on time. Express your answer as a percent to the nearest tenth. | {
"answer": "0.2\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The cost of purchasing a car is 150,000 yuan, and the annual expenses for insurance, tolls, and gasoline are about 15,000 yuan. The maintenance cost for the first year is 3,000 yuan, which increases by 3,000 yuan each year thereafter. Determine the best scrap year limit for this car. | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the area of a trapezoid with diagonals of 7 cm and 8 cm, and bases of 3 cm and 6 cm. | {
"answer": "12 \\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On the diagonals $AC$ and $CE$ of a regular hexagon $ABCDEF$, points $M$ and $N$ are taken respectively, such that $\frac{AM}{AC} = \frac{CN}{CE} = \lambda$. It is known that points $B, M$, and $N$ lie on one line. Find $\lambda$. | {
"answer": "\\frac{\\sqrt{3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Complex numbers $p$, $q$, and $r$ form an equilateral triangle with side length 24 in the complex plane. If $|p + q + r| = 48$, find $|pq + pr + qr|$. | {
"answer": "768",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A circle can be circumscribed around the quadrilateral $ABCD$. Additionally, $AB = 3$, $BC = 4$, $CD = 5$, and $AD = 2$.
Find $AC$. | {
"answer": "\\sqrt{\\frac{299}{11}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The quadrilateral \(ABCD\) is circumscribed around a circle with a radius of \(1\). Find the greatest possible value of \(\left| \frac{1}{AC^2} + \frac{1}{BD^2} \right|\). | {
"answer": "1/4",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The whole numbers from 1 to \( 2k \) are split into two equal-sized groups in such a way that any two numbers from the same group share no more than two distinct prime factors. What is the largest possible value of \( k \)? | {
"answer": "44",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
It is known that $\sin y = 2 \cos x + \frac{5}{2} \sin x$ and $\cos y = 2 \sin x + \frac{5}{2} \cos x$. Find $\sin 2x$. | {
"answer": "-\\frac{37}{20}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Suppose you have a Viennese pretzel lying on the table. What is the maximum number of parts you can cut it into with one straight swing of the knife? In which direction should this cut be made? | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Last year, the East Sea Crystal City World's business hall made a profit of 3 million yuan. At the beginning of this year, it relocated to the new Crystal City business hall, expanding its scope of operations. To achieve higher profits, it is necessary to increase advertising efforts. It is expected that starting from this year, the profit will grow at an annual rate of 26%, while on December 30th of each year, an advertising fee of x million yuan will be paid. To achieve the goal of doubling the profit after 10 years, find the maximum value of the annual advertising fee x in million yuan. (Note: $1.26^{10} \approx 10.$) | {
"answer": "52",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $ABC$ be an acute triangle with circumcircle $\omega$ . Let $D$ and $E$ be the feet of the altitudes from $B$ and $C$ onto sides $AC$ and $AB$ , respectively. Lines $BD$ and $CE$ intersect $\omega$ again at points $P \neq B$ and $Q \neq C$ . Suppose that $PD=3$ , $QE=2$ , and $AP \parallel BC$ . Compute $DE$ .
*Proposed by Kyle Lee* | {
"answer": "\\sqrt{23}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A triangle has sides with lengths of $18$, $24$, and $30$. Calculate the length of the shortest altitude. | {
"answer": "18",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Divide the product of the first six positive composite integers by the product of the next six composite integers. Express your answer as a common fraction. | {
"answer": "\\frac{1}{49}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The diagonal of a square is 10 inches, and the diameter of a circle is also 10 inches. Additionally, an equilateral triangle is inscribed within the square. Find the difference in area between the circle and the combined area of the square and the equilateral triangle. Express your answer as a decimal to the nearest tenth. | {
"answer": "-14.8",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $n$ be the largest real solution to the equation
\[\dfrac{4}{x-2} + \dfrac{6}{x-6} + \dfrac{13}{x-13} + \dfrac{15}{x-15} = x^2 - 7x - 6\]
There are positive integers $p, q,$ and $r$ such that $n = p + \sqrt{q + \sqrt{r}}$. Find $p+q+r$. | {
"answer": "103",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many four-digit whole numbers are there such that the leftmost digit is a prime number, the last digit is a perfect square, and all four digits are different? | {
"answer": "288",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a regular quadrilateral pyramid \(P-ABCD\) with a volume of 1, points \(E\), \(F\), \(G\), and \(H\) are the midpoints of segments \(AB\), \(CD\), \(PB\), and \(PC\), respectively. Find the volume of the polyhedron \(BEG-CFH\). | {
"answer": "5/16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $\overrightarrow{a}=(1,-1)$ and $\overrightarrow{b}=(1,2)$, calculate the projection of $\overrightarrow{b}$ onto $\overrightarrow{a}$. | {
"answer": "-\\frac{\\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine the exact value of
\[
\sqrt{\left( 2 - \sin^2 \frac{\pi}{9} \right) \left( 2 - \sin^2 \frac{2 \pi}{9} \right) \left( 2 - \sin^2 \frac{4 \pi}{9} \right)}.
\] | {
"answer": "\\frac{\\sqrt{619}}{16}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The probability of getting rain on any given day in August in Beach Town is \(\frac{1}{5}\). What is the probability that it rains on at most 3 days in the first week of August? | {
"answer": "0.813",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many integers $n$ satisfy the inequality $-\frac{9\pi}{2} \leq n \leq 12\pi$? | {
"answer": "53",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the smallest positive four-digit number divisible by 8 which has three odd and one even digit? | {
"answer": "1032",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the number $52674.1892$, calculate the ratio of the value of the place occupied by the digit 6 to the value of the place occupied by the digit 8. | {
"answer": "10,000",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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