problem stringlengths 10 5.15k | answer dict |
|---|---|
How many triangles with positive area can be formed with vertices at points $(i,j)$ in the coordinate plane, where $i$ and $j$ are integers between $1$ and $6$, inclusive? | {
"answer": "6788",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A \(3 \times 3 \times 3\) cube composed of 27 unit cubes rests on a horizontal plane. Determine the number of ways of selecting two distinct unit cubes (order is irrelevant) from a \(3 \times 3 \times 1\) block with the property that the line joining the centers of the two cubes makes a \(45^{\circ}\) angle with the horizontal plane. | {
"answer": "18",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the least positive integer such that when its leftmost digit is deleted, the resulting integer is 1/19 of the original integer. | {
"answer": "950",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $\triangle ABC$, it is known that $\overrightarrow{CD}=2\overrightarrow{DB}$, $P$ is a point on segment $AD$, and satisfies $\overrightarrow{CP}=\frac{1}{2}\overrightarrow{CA}+m\overrightarrow{CB}$. If the area of $\triangle ABC$ is $\sqrt{3}$ and $∠ACB=\frac{π}{3}$, then the minimum value of the length of segment $CP$ is ______. | {
"answer": "\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the total surface area of two hemispheres of radius 8 cm each, joined at their bases to form a complete sphere. Assume that one hemisphere is made of a reflective material that doubles the effective surface area for purposes of calculation. Express your answer in terms of $\pi$. | {
"answer": "384\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the weights (in kilograms) of 4 athletes are all integers, and they weighed themselves in pairs for a total of 5 times, obtaining weights of 99, 113, 125, 130, 144 kilograms respectively, and there are two athletes who did not weigh together, determine the weight of the heavier one among these two athletes. | {
"answer": "66",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $[x]$ represent the greatest integer less than or equal to $x$. Given that the natural number $n$ satisfies $\left[\frac{1}{15}\right] + \left[\frac{2}{15}\right] + \left[\frac{3}{15}\right] + \cdots + \left[\frac{n-1}{15}\right] + \left[\frac{n}{15}\right] > 2011$, what is the smallest value of $n$? | {
"answer": "253",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Four fair dice are tossed at random. What is the probability that the four numbers on the dice can be arranged to form an arithmetic progression with a common difference of one? | {
"answer": "\\frac{1}{432}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=\ln x-mx (m\in R)$.
(I) Discuss the monotonic intervals of the function $f(x)$;
(II) When $m\geqslant \frac{ 3 \sqrt {2}}{2}$, let $g(x)=2f(x)+x^{2}$ and its two extreme points $x\_1$, $x\_2 (x\_1 < x\_2)$ are exactly the zeros of $h(x)=\ln x-cx^{2}-bx$. Find the minimum value of $y=(x\_1-x\_2)h′( \frac {x\_1+x\_2}{2})$. | {
"answer": "-\\frac{2}{3} + \\ln 2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
We call any eight squares in a diagonal of a chessboard as a fence. The rook is moved on the chessboard in such way that he stands neither on each square over one time nor on the squares of the fences (the squares which the rook passes is not considered ones it has stood on). Then what is the maximum number of times which the rook jumped over the fence? | {
"answer": "47",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Alvin rides on a flat road at $25$ kilometers per hour, downhill at $35$ kph, and uphill at $10$ kph. Benny rides on a flat road at $35$ kph, downhill at $45$ kph, and uphill at $15$ kph. Alvin goes from town $X$ to town $Y$, a distance of $15$ km all uphill, then from town $Y$ to town $Z$, a distance of $25$ km all downhill, and then back to town $X$, a distance of $30$ km on the flat. Benny goes the other way around using the same route. What is the time in minutes that it takes Alvin to complete the $70$-km ride, minus the time in minutes it takes Benny to complete the same ride? | {
"answer": "33",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two circles touch in $M$ , and lie inside a rectangle $ABCD$ . One of them touches the sides $AB$ and $AD$ , and the other one touches $AD,BC,CD$ . The radius of the second circle is four times that of the first circle. Find the ratio in which the common tangent of the circles in $M$ divides $AB$ and $CD$ . | {
"answer": "1:1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On Tony's map, the distance from Saint John, NB to St. John's, NL is $21 \mathrm{~cm}$. The actual distance between these two cities is $1050 \mathrm{~km}$. What is the scale of Tony's map? | {
"answer": "1:5 000 000",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the smallest positive integer that has exactly eight distinct positive divisors, where all divisors are powers of prime numbers? | {
"answer": "24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many ways are there to arrange the letters of the word $\text{ZOO}_1\text{M}_1\text{O}_2\text{M}_2\text{O}_3$, in which the three O's and the two M's are considered distinct? | {
"answer": "5040",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Fix a sequence $a_1,a_2,a_3\ldots$ of integers satisfying the following condition:for all prime numbers $p$ and all positive integers $k$ ,we have $a_{pk+1}=pa_k-3a_p+13$ .Determine all possible values of $a_{2013}$ . | {
"answer": "13",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Convert the binary number $110110100_2$ to base 4. | {
"answer": "31220_4",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the Cartesian coordinate system $xOy$, let $D$ be the region represented by the inequality $|x| + |y| \leq 1$, and $E$ be the region consisting of points whose distance from the origin is no greater than 1. If a point is randomly thrown into $E$, the probability that this point falls into $D$ is. | {
"answer": "\\frac{4}{\\pi}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a circle $C$: $(x+1)^{2}+y^{2}=r^{2}$ and a parabola $D$: $y^{2}=16x$ intersect at points $A$ and $B$, and $|AB|=8$, calculate the area of circle $C$. | {
"answer": "25\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A rectangular sheet of paper is 20 cm long and 12 cm wide. It is folded along its diagonal. What is the perimeter of the shaded region? | {
"answer": "64",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Inside the circle \(\omega\), there is a circle \(\omega_{1}\) that touches it at point \(K\). Circle \(\omega_{2}\) touches circle \(\omega_{1}\) at point \(L\) and intersects circle \(\omega\) at points \(M\) and \(N\). It turns out that points \(K\), \(L\), and \(M\) are collinear. Find the radius of circle \(\omega\) if the radii of circles \(\omega_{1}\) and \(\omega_{2}\) are 4 and 7, respectively. | {
"answer": "11",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine the number of 6-digit even numbers that can be formed using the digits 1, 2, 3, 4, 5, 6 without repetition and ensuring that 1, 3, 5 are not adjacent to one another. | {
"answer": "36",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Compute \(\arccos(\cos 8.5)\). All functions are in radians. | {
"answer": "2.217",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Compute the number of ordered pairs of integers $(x,y)$ with $1\le x<y\le 200$ such that $i^x+i^y$ is a real number. | {
"answer": "4950",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many sequences of $0$s and $1$s of length $20$ are there that begin with a $0$, end with a $0$, contain no two consecutive $0$s, and contain no four consecutive $1$s?
A) 65
B) 75
C) 85
D) 86
E) 95 | {
"answer": "86",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Around a circle, an isosceles trapezoid \(ABCD\) is described. The side \(AB\) touches the circle at point \(M\), and the base \(AD\) touches the circle at point \(N\). The segments \(MN\) and \(AC\) intersect at point \(P\), such that \(NP: PM=2\). Find the ratio \(AD: BC\). | {
"answer": "3:1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Compute $e^{\pi}+\pi^e$ . If your answer is $A$ and the correct answer is $C$ , then your score on this problem will be $\frac{4}{\pi}\arctan\left(\frac{1}{\left|C-A\right|}\right)$ (note that the closer you are to the right answer, the higher your score is).
*2017 CCA Math Bonanza Lightning Round #5.2* | {
"answer": "45.5999",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are eight different symbols designed on $n\geq 2$ different T-shirts. Each shirt contains at least one symbol, and no two shirts contain all the same symbols. Suppose that for any $k$ symbols $(1\leq k\leq 7)$ the number of shirts containing at least one of the $k$ symbols is even. Determine the value of $n$ . | {
"answer": "255",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the product of $1101_2 \cdot 111_2$. Express your answer in base 2. | {
"answer": "1000111_2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that \(15^{-1} \equiv 31 \pmod{53}\), find \(38^{-1} \pmod{53}\), as a residue modulo 53. | {
"answer": "22",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The sequence $(a_n)$ is defined recursively by $a_0=1$, $a_1=\sqrt[23]{3}$, and $a_n=a_{n-1}a_{n-2}^3$ for $n\geq 2$. Determine the smallest positive integer $k$ such that the product $a_1a_2\cdots a_k$ is an integer. | {
"answer": "22",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=a\frac{x+1}{x}+\ln{x}$, the equation of the tangent line at the point (1, f(1)) is y=bx+5.
1. Find the real values of a and b.
2. Find the maximum and minimum values of the function $f(x)$ on the interval $[\frac{1}{e}, e]$, where $e$ is the base of the natural logarithm. | {
"answer": "3+\\ln{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the sum $E(1)+E(2)+E(3)+\cdots+E(200)$ where $E(n)$ denotes the sum of the even digits of $n$, and $5$ is added to the sum if $n$ is a multiple of $10$. | {
"answer": "902",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that one air conditioner sells for a 10% profit and the other for a 10% loss, and the two air conditioners have the same selling price, determine the percentage change in the shopping mall's overall revenue. | {
"answer": "1\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
So, Xiao Ming's elder brother was born in a year that is a multiple of 19. In 2013, determine the possible ages of Xiao Ming's elder brother. | {
"answer": "18",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the polar coordinate system, the equation of curve C is $\rho^2= \frac{3}{1+2\sin^2\theta}$. Point R is at $(2\sqrt{2}, \frac{\pi}{4})$.
P is a moving point on curve C, and side PQ of rectangle PQRS, with PR as its diagonal, is perpendicular to the polar axis. Find the maximum and minimum values of the perimeter of rectangle PQRS and the polar angle of point P when these values occur. | {
"answer": "\\frac{\\pi}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given 60 feet of fencing, what is the greatest possible number of square feet in the area of a pen, if the pen is designed as a rectangle subdivided evenly into two square areas? | {
"answer": "450",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For $a>0$ , let $f(a)=\lim_{t\to\+0} \int_{t}^1 |ax+x\ln x|\ dx.$ Let $a$ vary in the range $0 <a< +\infty$ , find the minimum value of $f(a)$ . | {
"answer": "\\frac{\\ln 2}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Where is Tony's cheese? The bottom right corner of a 3x3 grid :0 We’re doing weekly posts of problems until the start of the in-person tournament on April 13th (Sign up **[here](https://www.stanfordmathtournament.com/competitions/smt-2024)**!) Registrations for our synchronous Online tournament are also open **[here](https://www.stanfordmathtournament.com/competitions/smt-2024-online)**. Also if you’re coming in-person and want to get some pre-competition action, submit your solutions to PoTW [here](https://forms.gle/2FpDoHFeVHg2aqWS6), and the top 3 people with the most correct solutions will receive surprise prize at the tournament :)**Easy:** Consider the curves $x^2 + y^2 = 1$ and $2x^2 + 2xy + y^2 - 2x -2y = 0$ . These curves intersect at two points, one of which is $(1, 0)$ . Find the other one. **Hard:** Tony the mouse starts in the top left corner of a $3x3$ grid. After each second, he randomly moves to an adjacent square with equal probability. What is the probability he reaches the cheese in the bottom right corner before he reaches the mousetrap in the center.
Again feel free to reach out to us at [stanford.math.tournament@gmail.com](mailto:stanford.math.tournament@gmail.com) with any questions, or ask them in this thread!
Best,
[The SMT Team](https://www.stanfordmathtournament.com/our-team)
| {
"answer": "\\frac{1}{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The regular octagon $ABCDEFGH$ has its center at $J$. Each of the vertices and the center are to be associated with one of the digits $1$ through $9$, with each digit used once, in such a way that the sums of the numbers on the lines $AJE$, $BJF$, $CJG$, and $DJH$ are equal. In how many ways can this be done? [asy]
size(175);
defaultpen(linewidth(0.8));
path octagon;
string labels[]={"A","B","C","D","E","F","G","H","I"};
for(int i=0;i<=7;i=i+1)
{
pair vertex=dir(135-45/2-45*i);
octagon=octagon--vertex;
label(" $"+labels[i]+"$ ",vertex,dir(origin--vertex));
}
draw(octagon--cycle);
dot(origin);
label(" $J$ ",origin,dir(0));
[/asy] | {
"answer": "1152",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
\( S \) is the set of all ordered tuples \((a, b, c, d, e, f)\) where \(a, b, c, d, e, f\) are integers and \(a^2 + b^2 + c^2 + d^2 + e^2 = f^2\). Find the largest \( k \) such that \( k \) divides \( a b c d e f \) for all elements in \( S \). | {
"answer": "24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
From the 6 finalists, 1 first prize, 2 second prizes, and 3 third prizes are to be awarded. Calculate the total number of possible outcomes. | {
"answer": "60",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $ABC$ be the triangle with vertices located at the center of masses of Vincent Huang's house, Tristan Shin's house, and Edward Wan's house; here, assume the three are not collinear. Let $N = 2017$ , and define the $A$ -*ntipodes* to be the points $A_1,\dots, A_N$ to be the points on segment $BC$ such that $BA_1 = A_1A_2 = \cdots = A_{N-1}A_N = A_NC$ , and similarly define the $B$ , $C$ -ntipodes. A line $\ell_A$ through $A$ is called a *qevian* if it passes through an $A$ -ntipode, and similarly we define qevians through $B$ and $C$ . Compute the number of ordered triples $(\ell_A, \ell_B, \ell_C)$ of concurrent qevians through $A$ , $B$ , $C$ , respectively.
*Proposed by Brandon Wang* | {
"answer": "2017^3 - 2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the greatest root of the polynomial $f(x) = 16x^4 - 8x^3 + 9x^2 - 3x + 1$. | {
"answer": "0.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that Bill's age in two years will be three times his current age, and the digits of both Jack's and Bill's ages are reversed, find the current age difference between Jack and Bill. | {
"answer": "18",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Pentagon $ABCDE$ is inscribed in a circle such that $ACDE$ is a square with area $12$. Determine the largest possible area of pentagon $ABCDE$. | {
"answer": "9 + 3\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Tam created the mosaic shown using a regular hexagon, squares, and equilateral triangles. If the side length of the hexagon is \( 20 \text{ cm} \), what is the outside perimeter of the mosaic? | {
"answer": "240",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Inside a right triangle \(ABC\) with hypotenuse \(AC\), a point \(M\) is chosen such that the areas of triangles \(ABM\) and \(BCM\) are one-third and one-quarter of the area of triangle \(ABC\) respectively. Find \(BM\) if \(AM = 60\) and \(CM = 70\). If the answer is not an integer, round it to the nearest whole number. | {
"answer": "38",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $|x| + x + y = 14$ and $x + |y| - y = 16,$ find $x + y.$ | {
"answer": "-2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given \( 0 \leq m-n \leq 1 \) and \( 2 \leq m+n \leq 4 \), when \( m - 2n \) reaches its maximum value, what is the value of \( 2019m + 2020n \)? | {
"answer": "2019",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a modified game similar to Deal or No Deal, participants choose a box at random from a set of 30 boxes, each containing one of the following values:
\begin{tabular}{|c|c|}
\hline
\$0.50 & \$2,000 \\
\hline
\$2 & \$10,000 \\
\hline
\$10 & \$20,000 \\
\hline
\$20 & \$40,000 \\
\hline
\$50 & \$100,000 \\
\hline
\$100 & \$200,000 \\
\hline
\$500 & \$400,000 \\
\hline
\$1,000 & \$800,000 \\
\hline
\$1,500 & \$1,000,000 \\
\hline
\end{tabular}
After choosing a box, participants eliminate other boxes by opening them. What is the minimum number of boxes a participant needs to eliminate to have at least a 50% chance of holding a box containing at least \$200,000? | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the ancient Chinese mathematical text "The Mathematical Classic of Sunzi", there is a problem stated as follows: "Today, a hundred deer enter the city. Each family takes one deer, but not all are taken. Then, three families together take one deer, and all deer are taken. The question is: how many families are there in the city?" In this problem, the number of families in the city is ______. | {
"answer": "75",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Katie writes a different positive integer on the top face of each of the fourteen cubes in the pyramid shown. The sum of the nine integers written on the cubes in the bottom layer is 50. The integer written on each of the cubes in the middle and top layers of the pyramid is equal to the sum of the integers on the four cubes underneath it. What is the greatest possible integer that she can write on the top cube? | {
"answer": "118",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the equations:
\[
\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 4 \quad \text{and} \quad \frac{a}{x} + \frac{b}{y} + \frac{c}{z} = 3,
\]
find the value of \(\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2}\). | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine the value of
\[2023 + \frac{1}{2} \left( 2022 + \frac{1}{2} \left( 2021 + \dots + \frac{1}{2} \left( 4 + \frac{1}{2} \cdot (3 + 1) \right) \right) \dotsb \right).\] | {
"answer": "4044",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A bridge needs to be constructed over a river with a required elevation of 800 feet from one side to the other. Determine the additional bridge length required if the gradient is reduced from 2% to 1.5%. | {
"answer": "13333.33",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
It is known that the vertex of angle $\theta$ is at the origin of coordinates, its initial side coincides with the positive half-axis of the x-axis, and its terminal side falls on the ray $y= \frac {1}{2}x$ ($x\leq0$).
(Ⅰ) Find the value of $\cos\left( \frac {\pi}{2}+\theta \right)$;
(Ⅱ) If $\cos\left( \alpha+ \frac {\pi}{4} \right)=\sin\theta$, find the value of $\sin\left(2\alpha+ \frac {\pi}{4}\right)$. | {
"answer": "- \\frac { \\sqrt {2}}{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A number like 45132 is called a "wave number," which means the tens and thousands digits are both larger than their respective neighboring digits. What is the probability of forming a non-repeating five-digit "wave number" using the digits 1, 2, 3, 4, 5? | {
"answer": "\\frac{1}{15}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Medians $\overline{DP}$ and $\overline{EQ}$ of isosceles $\triangle DEF$, where $DE=EF$, are perpendicular. If $DP= 21$ and $EQ = 28$, then what is ${DE}$? | {
"answer": "\\frac{70}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are $5$ people arranged in a row. Among them, persons A and B must be adjacent, and neither of them can be adjacent to person D. How many different arrangements are there? | {
"answer": "36",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the center of a square, there is a police officer, and in one of the vertices, there is a gangster. The police officer can run throughout the whole square, while the gangster can only run along its sides. It is known that the ratio of the maximum speed of the police officer to the maximum speed of the gangster is: 0.5; 0.49; 0.34; 1/3. Can the police officer run in such a way that at some point he will be on the same side of the square as the gangster? | {
"answer": "1/3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the Cartesian coordinate system $(xOy)$, let the line $l: \begin{cases} x=2-t \\ y=2t \end{cases} (t \text{ is a parameter})$, and the curve $C_{1}: \begin{cases} x=2+2\cos \theta \\ y=2\sin \theta \end{cases} (\theta \text{ is a parameter})$. In the polar coordinate system with $O$ as the pole and the positive $x$-axis as the polar axis:
(1) Find the polar equations of $C_{1}$ and $l$:
(2) Let curve $C_{2}: \rho=4\sin\theta$. The curve $\theta=\alpha(\rho > 0, \frac{\pi}{4} < \alpha < \frac{\pi}{2})$ intersects with $C_{1}$ and $C_{2}$ at points $A$ and $B$, respectively. If the midpoint of segment $AB$ lies on line $l$, find $|AB|$. | {
"answer": "\\frac{4\\sqrt{10}}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On a circular track with a perimeter of 360 meters, three individuals A, B, and C start from the same point: A starts first, running counterclockwise. Before A completes one lap, B and C start simultaneously, running clockwise. When A and B meet for the first time, C is exactly halfway between them. After some time, when A and C meet for the first time, B is also exactly halfway between them. If B's speed is four times that of A's, how many meters has A run when B and C started? | {
"answer": "90",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many different 4-edge trips are there from $A$ to $B$ in a cube, where the trip can visit one vertex twice (excluding start and end vertices)? | {
"answer": "36",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the number of pairs of natural numbers \((x, y)\) such that \(1 \leq x, y \leq 1000\) and \(x^2 + y^2\) is divisible by 5. | {
"answer": "200000",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the ellipse $C$: $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1 (a > b > 0)$ passes through the point $(1, \dfrac{2\sqrt{3}}{3})$, with the left and right foci being $F_1$ and $F_2$ respectively. The circle $x^2 + y^2 = 2$ intersects with the line $x + y + b = 0$ at a chord of length $2$.
(Ⅰ) Find the standard equation of the ellipse $C$;
(Ⅱ) Let $Q$ be a moving point on the ellipse $C$ that is not on the $x$-axis, $O$ be the origin. A line parallel to $OQ$ passing through $F_2$ intersects the ellipse $C$ at two distinct points $M$ and $N$. Investigate whether the value of $\dfrac{|MN|}{|OQ|^2}$ is a constant. If it is, find this constant; if not, please explain why. | {
"answer": "\\dfrac{2\\sqrt{3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the sequence $\{a_k\}_{k=1}^{11}$ of real numbers defined by $a_1=0.5$, $a_2=(0.51)^{a_1}$, $a_3=(0.501)^{a_2}$, $a_4=(0.511)^{a_3}$, and in general,
$a_k=\begin{cases}
(0.\underbrace{501\cdots 01}_{k+1\text{ digits}})^{a_{k-1}} & \text{if } k \text{ is odd,} \\
(0.\underbrace{501\cdots 011}_{k+1\text{ digits}})^{a_{k-1}} & \text{if } k \text{ is even.}
\end{cases}$
Rearrange the numbers in the sequence $\{a_k\}_{k=1}^{11}$ in decreasing order to produce a new sequence $\{b_k\}_{k=1}^{11}$. Find the sum of all integers $k$, $1\le k \le 11$, such that $a_k = b_k$. | {
"answer": "30",
"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 $2c+b-2a\cos B=0$.
$(1)$ Find angle $A$;
$(2)$ If $a=2\sqrt{3}$, $\overrightarrow{BA}\cdot \overrightarrow{AC}=\frac{3}{2}$, and $AD$ is the median of $\triangle ABC$, find the length of $AD$. | {
"answer": "\\frac{\\sqrt{6}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the number of different possible rational roots of the polynomial:
\[6x^4 + b_3 x^3 + b_2 x^2 + b_1 x + 10 = 0.\] | {
"answer": "22",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The number of minutes in a week is closest to: | {
"answer": "10000",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the least positive integer value of $x$ such that $(3x)^2 + 3 \cdot 29 \cdot 3x + 29^2$ is a multiple of 43? | {
"answer": "19",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given:
$$ \frac{ \left( \frac{1}{3} \right)^2 + \left( \frac{1}{4} \right)^2 }{ \left( \frac{1}{5} \right)^2 + \left( \frac{1}{6} \right)^2} = \frac{37x}{73y} $$
Express $\sqrt{x} \div \sqrt{y}$ as a common fraction. | {
"answer": "\\frac{75 \\sqrt{73}}{6 \\sqrt{61} \\sqrt{37}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider real numbers $A$ , $B$ , \dots, $Z$ such that \[
EVIL = \frac{5}{31}, \;
LOVE = \frac{6}{29}, \text{ and }
IMO = \frac{7}{3}.
\] If $OMO = \tfrac mn$ for relatively prime positive integers $m$ and $n$ , find the value of $m+n$ .
*Proposed by Evan Chen* | {
"answer": "579",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Compute the sum of the squares of the roots of the equation \[x\sqrt{x} - 8x + 9\sqrt{x} - 2 = 0,\] given that all of the roots are real and nonnegative. | {
"answer": "46",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Juca is a scout exploring the vicinity of his camp. After collecting fruits and wood, he needs to fetch water from the river and return to his tent. Represent Juca by the letter $J$, the river by the letter $r$, and his tent by the letter $B$. The distance from the feet of the perpendiculars $C$ and $E$ on $r$ from points $J$ and $B$ is $180 m$. What is the shortest distance Juca can travel to return to his tent, passing through the river? | {
"answer": "180\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the volume of the region in three-dimensional space defined by the inequalities $|x|+|y|+|z|\le2$ and $|x|+|y|+|z-2|\le2$? | {
"answer": "\\frac{8}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the angle between the unit vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ is acute, and for any $(x,y)$ that satisfies $|x\overrightarrow{a}+y\overrightarrow{b}|=1$ and $xy\geqslant 0$, the inequality $|x+2y|\leqslant \frac{8}{\sqrt{15}}$ holds. Find the minimum value of $\overrightarrow{a}\cdot\overrightarrow{b}$. | {
"answer": "\\frac{1}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
We have $n$ positive integers greater than $1$ and less than $10000$ such that neither of them is prime but any two of them are relative prime. Find the maximum value of $n $ . | {
"answer": "25",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the terminal side of angle $\alpha$ ($0 < \alpha < \frac{\pi}{2}$) passes through the point $(\cos 2\beta, 1+\sin 3\beta \cos \beta - \cos 3\beta \sin \beta)$, where $\frac{\pi}{2} < \beta < \pi$ and $\beta \neq \frac{3\pi}{4}$, calculate $\alpha - \beta$. | {
"answer": "-\\frac{3\\pi}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Vasya and Petya are participating in a school sports-entertainment game. They only have one pair of roller skates between the two of them, and need to traverse a distance of 3 km as quickly as possible. They start simultaneously, with one running and the other roller-skating. At any moment, the one on roller skates can leave them for the other and continue running without them. This exchange can occur as many times as desired. Find the minimum time for both friends to complete the distance (which is determined by who finishes last), given that Vasya's running and roller skating speeds are 4 km/h and 8 km/h respectively, while Petya's speeds are 5 km/h and 10 km/h. Assume that no time is lost during the exchange of roller skates. | {
"answer": "0.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the parabola $C: x^{2}=2py\left(p \gt 0\right)$ with focus $F$, and the minimum distance between $F$ and a point on the circle $M: x^{2}+\left(y+4\right)^{2}=1$ is $4$.<br/>$(1)$ Find $p$;<br/>$(2)$ If point $P$ lies on $M$, $PA$ and $PB$ are two tangents to $C$ with points $A$ and $B$ as the points of tangency, find the maximum area of $\triangle PAB$. | {
"answer": "20\\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A six-digit palindrome is a positive integer with respective digits $abcdcba$, where $a$ is non-zero. Let $T$ be the sum of all six-digit palindromes. Calculate the sum of the digits of $T$. | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $p, q,$ and $r$ are three non-zero integers such that $p + q + r = 30$ and \[\frac{1}{p} + \frac{1}{q} + \frac{1}{r} + \frac{240}{pqr} = 1,\] compute $pqr$. | {
"answer": "1080",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a certain sequence, the first term is $a_1 = 101$ and the second term is $a_2 = 102$. Furthermore, the values of the remaining terms are chosen so that $a_n + a_{n+1} + a_{n+2} = n + 2$ for all $n \geq 1$. Determine $a_{50}$. | {
"answer": "117",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a polynomial with integer coefficients,
\[16x^5 + b_4x^4 + b_3x^3 + b_2x^2 + b_1x + 24 = 0,\]
find the number of different possible rational roots of this polynomial. | {
"answer": "32",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the values of the real number \( a \) such that all the roots of the polynomial in the variable \( x \),
\[ x^{3}-2x^{2}-25x+a \]
are integers. | {
"answer": "-50",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider a unit square $ABCD$ whose bottom left vertex is at the origin. A circle $\omega$ with radius $\frac{1}{3}$ is inscribed such that it touches the square's bottom side at point $M$. If $\overline{AM}$ intersects $\omega$ at a point $P$ different from $M$, where $A$ is at the top left corner of the square, find the length of $AP$. | {
"answer": "\\frac{1}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle \(ABC\), \(BK\) is the median, \(BE\) is the angle bisector, and \(AD\) is the altitude. Find the length of side \(AC\) if it is known that lines \(BK\) and \(BE\) divide segment \(AD\) into three equal parts and the length of \(AB\) is 4. | {
"answer": "2\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A power locomotive's hourly electricity consumption cost is directly proportional to the cube of its speed. It is known that when the speed is $20$ km/h, the hourly electricity consumption cost is $40$ yuan. Other costs amount to $400$ yuan per hour. The maximum speed of the locomotive is $100$ km/h. At what speed should the locomotive travel to minimize the total cost of traveling from city A to city B? | {
"answer": "20 \\sqrt[3]{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the three-dimensional Cartesian coordinate system, the equation of the plane passing through $P(x_{0}, y_{0}, z_{0})$ with normal vector $\overrightarrow{m}=(a, b, c)$ is $a(x-x_{0})+b(y-y_{0})+c(z-z_{0})=0$, and the equation of the line passing through $P(x_{0}, y_{0}, z_{0})$ with direction vector $\overrightarrow{n}=(A, B, C)$ is $\frac{x-x_{0}}{A}=\frac{y-y_{0}}{B}=\frac{z-z_{0}}{C}$. Read the above information and solve the following problem: Given the equation of the plane $\alpha$ as $3x+y-z-5=0$, and the equation of the line $l$ passing through the point $P(0,0,0)$ as $x=\frac{y}{2}=-z$, then one direction vector of the line $l$ is ______, and the cosine value of the angle between the line $l$ and the plane $\alpha$ is ______. | {
"answer": "\\frac{\\sqrt{55}}{11}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find maximum value of number $a$ such that for any arrangement of numbers $1,2,\ldots ,10$ on a circle, we can find three consecutive numbers such their sum bigger or equal than $a$ . | {
"answer": "18",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the radius of a sphere is 24cm, and the height of a cone is equal to the diameter of this sphere, and the surface area of the sphere is equal to the surface area of the cone, then the volume of this cone is \_\_\_\_\_\_ cm<sup>3</sup>. | {
"answer": "12288\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many whole numbers between 1 and 2000 do not contain the digits 1 or 2? | {
"answer": "1535",
"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{45}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the largest six-digit number in which all digits are distinct, and each digit, except for the extreme ones, is equal either to the sum or the difference of its neighboring digits. | {
"answer": "972538",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are 100 chips numbered from 1 to 100 placed in the vertices of a regular 100-gon in such a way that they follow a clockwise order. In each move, it is allowed to swap two chips located at neighboring vertices if their numbers differ by at most $k$. What is the smallest value of $k$ for which, by a series of such moves, it is possible to achieve a configuration where each chip is shifted one position clockwise relative to its initial position? | {
"answer": "50",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Define a sequence by \( a_0 = \frac{1}{3} \) and \( a_n = 1 + (a_{n-1} - 1)^3 \). Compute the infinite product \( a_0 a_1 a_2 \dotsm \). | {
"answer": "\\frac{3}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A biologist sequentially placed 150 beetles into ten jars. In each subsequent jar, he placed more beetles than in the previous one. The number of beetles in the first jar is at least half of the number of beetles in the tenth jar. How many beetles are in the sixth jar? | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $\\((2-x)^5 = a_0 + a_1x + a_2x^2 + \ldots + a_5x^5\\)$. Evaluate the value of $\dfrac{a_0 + a_2 + a_4}{a_1 + a_3}$. | {
"answer": "-\\dfrac{122}{121}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=(\sin x+\cos x)^{2}+2\cos ^{2}x$,
(1) Find the smallest positive period and the monotonically decreasing interval of the function $f(x)$;
(2) When $x\in[0, \frac{\pi}{2}]$, find the maximum and minimum values of $f(x)$. | {
"answer": "2+\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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