problem stringlengths 10 5.15k | answer dict |
|---|---|
Given the function $f(x)=\begin{cases} 2^{x}, & x < 0 \\ f(x-1)+1, & x\geqslant 0 \end{cases}$, calculate the value of $f(2)$. | {
"answer": "\\dfrac{5}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A ferry boat shuttles tourists to an island every half-hour from 10 AM to 3 PM, with 100 tourists on the first trip and 2 fewer tourists on each successive trip. Calculate the total number of tourists taken to the island that day. | {
"answer": "990",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Under normal circumstances, for people aged between 18 and 38 years old, the regression equation for weight $y$ (kg) based on height $x$ (cm) is $y=0.72x-58.5$. Zhang Honghong, who is neither fat nor thin, has a height of 1.78 meters. His weight should be around \_\_\_\_\_ kg. | {
"answer": "70",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider a revised dataset given in the following stem-and-leaf plot, where $7|1$ represents 71:
\begin{tabular}{|c|c|}\hline
\textbf{Tens} & \textbf{Units} \\ \hline
2 & $0 \hspace{2mm} 0 \hspace{2mm} 1 \hspace{2mm} 1 \hspace{2mm} 2$ \\ \hline
3 & $3 \hspace{2mm} 6 \hspace{2mm} 6 \hspace{2mm} 7$ \\ \hline
4 & $3 \hspace{2mm} 5 \hspace{2mm} 7 \hspace{2mm} 9$ \\ \hline
6 & $2 \hspace{2mm} 4 \hspace{2mm} 5 \hspace{2mm} 6 \hspace{2mm} 8$ \\ \hline
7 & $1 \hspace{2mm} 3 \hspace{2mm} 5 \hspace{2mm} 9$ \\ \hline
\end{tabular}
What is the positive difference between the median and the mode of the new dataset? | {
"answer": "23",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $|\vec{a}|=1$, $|\vec{b}|= \sqrt{2}$, and $\vec{a} \perp (\vec{a} - \vec{b})$, find the angle between the vectors $\vec{a}$ and $\vec{b}$. | {
"answer": "\\frac{\\pi}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that a tetrahedron $ABCD$ is inscribed in a sphere $O$, and $AD$ is the diameter of the sphere $O$. If triangles $\triangle ABC$ and $\triangle BCD$ are equilateral triangles with side length 1, calculate the volume of tetrahedron $ABCD$. | {
"answer": "\\frac{\\sqrt{3}}{12}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the tower function $T(n)$ defined by $T(1) = 3$ and $T(n + 1) = 3^{T(n)}$ for $n \geq 1$, calculate the largest integer $k$ for which $\underbrace{\log_3\log_3\log_3\ldots\log_3B}_{k\text{ times}}$ is defined, where $B = (T(2005))^A$ and $A = (T(2005))^{T(2005)}$. | {
"answer": "2005",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A marathon of 42 km started at 11:30 AM and the winner finished at 1:45 PM on the same day. What was the average speed of the winner, in km/h? | {
"answer": "18.6",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the least positive integer with exactly $12$ positive factors? | {
"answer": "96",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $ABCD$ be a square with side length $16$ and center $O$ . Let $\mathcal S$ be the semicircle with diameter $AB$ that lies outside of $ABCD$ , and let $P$ be a point on $\mathcal S$ so that $OP = 12$ . Compute the area of triangle $CDP$ .
*Proposed by Brandon Wang* | {
"answer": "120",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A parking lot in Flower Town is a square with $7 \times 7$ cells, each of which can accommodate a car. The parking lot is enclosed by a fence, and one of the corner cells has an open side (this is the gate). Cars move along paths that are one cell wide. Neznaika was asked to park as many cars as possible in such a way that any car can exit while the others remain parked. Neznaika parked 24 cars as shown in the diagram. Try to arrange the cars differently so that more can fit. | {
"answer": "28",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A 10 by 10 checkerboard has alternating black and white squares. How many distinct squares, with sides on the grid lines of the checkerboard (horizontal and vertical) and containing at least 6 black squares, can be drawn on the checkerboard? | {
"answer": "140",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the random variable $X$ follows a normal distribution $N(2,\sigma^{2})$, and its normal distribution density curve is the graph of the function $f(x)$, and $\int_{0}^{2} f(x)dx=\dfrac{1}{3}$, calculate $P(x > 4)$. | {
"answer": "\\dfrac{1}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Monsieur and Madame Dubois are traveling from Paris to Deauville, where their children live. Each is driving their own car. They depart together and arrive in Deauville at the same time. However, Monsieur Dubois spent on stops one-third of the time during which his wife continued driving, while Madame Dubois spent on stops one-quarter of the time during which her husband was driving.
What is the ratio of the average speeds of each of their cars? | {
"answer": "8/9",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are respectively $a$, $b$, and $c$. It is known that $c\sin A= \sqrt {3}a\cos C$.
(I) Find $C$;
(II) If $c= \sqrt {7}$ and $\sin C+\sin (B-A)=3\sin 2A$, find the area of $\triangle ABC$. | {
"answer": "\\frac {3 \\sqrt {3}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If the seven digits 1, 1, 3, 5, 5, 5, and 9 are arranged to form a seven-digit positive integer, what is the probability that the integer is divisible by 25? | {
"answer": "\\frac{1}{14}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many of the numbers from the set $\{1,\ 2,\ 3,\ldots,\ 100\}$ have a perfect square factor other than one? | {
"answer": "41",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A dice is thrown twice. Let $a$ be the number of dots that appear on the first throw and $b$ be the number of dots that appear on the second throw. Given the system of equations $\begin{cases} ax+by=2 \\\\ 2x+y=3\\end{cases}$.
(I) Find the probability that the system of equations has only one solution.
(II) If each solution of the system of equations corresponds to a point $P(x,y)$ in the Cartesian plane, find the probability that point $P$ lies in the fourth quadrant. | {
"answer": "\\frac{7}{12}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a certain DNA fragment consists of 500 base pairs, with A+T making up 34% of the total number of bases, calculate the total number of free cytosine deoxyribonucleotide molecules required when this DNA fragment is replicated twice. | {
"answer": "1320",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In an isosceles trapezoid, the larger base is equal to the sum of the smaller base and the length of the altitude, and every diagonal is equal to the length of the smaller base plus half the altitude. Find the ratio of the smaller base to the larger base.
A) $\frac{1}{2}$
B) $\frac{\sqrt{5}}{2}$
C) $\frac{2+\sqrt{5}}{4}$
D) $\frac{2-\sqrt{5}}{2}$
E) $\frac{3-\sqrt{5}}{2}$ | {
"answer": "\\frac{2-\\sqrt{5}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many positive odd integers greater than 1 and less than $200$ are square-free? | {
"answer": "79",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If point \( P \) is the circumcenter of \(\triangle ABC\) and \(\overrightarrow{PA} + \overrightarrow{PB} + \lambda \overrightarrow{PC} = \mathbf{0}\), where \(\angle C = 120^\circ\), then find the value of the real number \(\lambda\). | {
"answer": "-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $\triangle ABC$ be a triangle in the plane, and let $D$ be a point outside the plane of $\triangle ABC$, so that $DABC$ is a pyramid whose faces are all triangles.
Suppose that every edge of $DABC$ has length $20$ or $45$, but no face of $DABC$ is equilateral. Then what is the surface area of $DABC$? | {
"answer": "40 \\sqrt{1925}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A row consists of 10 chairs, but chair #5 is broken and cannot be used. Mary and James each sit in one of the available chairs, choosing their seats at random from the remaining chairs. What is the probability that they don't sit next to each other? | {
"answer": "\\frac{5}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle \(ABC\), angle bisectors \(AA_{1}\), \(BB_{1}\), and \(CC_{1}\) are drawn. \(L\) is the intersection point of segments \(B_{1}C_{1}\) and \(AA_{1}\), \(K\) is the intersection point of segments \(B_{1}A_{1}\) and \(CC_{1}\). Find the ratio \(LM: MK\) if \(M\) is the intersection point of angle bisector \(BB_{1}\) with segment \(LK\), and \(AB: BC: AC = 2: 3: 4\). (16 points) | {
"answer": "11/12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $N \ge 5$ be given. Consider all sequences $(e_1,e_2,...,e_N)$ with each $e_i$ equal to $1$ or $-1$ . Per move one can choose any five consecutive terms and change their signs. Two sequences are said to be similar if one of them can be transformed into the other in finitely many moves. Find the maximum number of pairwise non-similar sequences of length $N$ .
| {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For some real number $c,$ the graphs of the equation $y=|x-20|+|x+18|$ and the line $y=x+c$ intersect at exactly one point. What is $c$ ? | {
"answer": "18",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine the time in hours it will take to fill a 32,000 gallon swimming pool using three hoses that deliver 3 gallons of water per minute. | {
"answer": "59",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate $x$ such that the sum \[1 \cdot 1979 + 2 \cdot 1978 + 3 \cdot 1977 + \dots + 1978 \cdot 2 + 1979 \cdot 1 = 1979 \cdot 990 \cdot x.\] | {
"answer": "661",
"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 $b\sin^{2}\frac{A}{2}+a\sin^{2}\frac{B}{2}=\frac{C}{2}$.
1. If $c=2$, find the perimeter of $\triangle ABC$.
2. If $C=\frac{\pi}{3}$ and the area of $\triangle ABC$ is $2\sqrt{3}$, find $c$. | {
"answer": "2\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $21^{-1} \equiv 15 \pmod{61}$, find $40^{-1} \pmod{61}$, as a residue modulo 61. (Provide a number between 0 and 60, inclusive.) | {
"answer": "46",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $PQR$, angle $R$ is a right angle and the altitude from $R$ meets $\overline{PQ}$ at $S$. The lengths of the sides of $\triangle PQR$ are integers, $PS=17^3$, and $\cos Q = a/b$, where $a$ and $b$ are relatively prime positive integers. Find $a+b$. | {
"answer": "18",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A spider is on the edge of a ceiling of a circular room with a radius of 65 feet. The spider moves straight across the ceiling to the opposite edge, passing through the circle's center. It then moves directly to another point on the edge of the circle, not passing through the center. The final segment of the journey is straight back to the starting point and is 90 feet long. How many total feet did the spider travel during the entire journey? | {
"answer": "220 + 20\\sqrt{22}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $S$ be the set of integers between $1$ and $2^{40}$ whose binary expansions have exactly two $1$'s. If a number is chosen at random from $S,$ the probability that it is divisible by $15$ is $p/q,$ where $p$ and $q$ are relatively prime positive integers. Find $p+q.$ | {
"answer": "49",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $T$ be a triangle with side lengths $1, 1, \sqrt{2}$. Two points are chosen independently at random on the sides of $T$. The probability that the straight-line distance between the points is at least $\dfrac{\sqrt{2}}{2}$ is $\dfrac{d-e\pi}{f}$, where $d$, $e$, and $f$ are positive integers with $\gcd(d,e,f)=1$. What is $d+e+f$? | {
"answer": "17",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the ellipse $c_{1}$: $\frac{x^{2}}{8} + \frac{y^{2}}{4} = 1$ with left and right focal points $F_{1}$ and $F_{2}$, a line $l_{1}$ is drawn through point $F_{1}$ perpendicular to the x-axis. A line $l_{2}$ intersects $l_{1}$ perpendicularly at point $P$. The perpendicular bisector of the line segment $PF_{2}$ intersects $l_{2}$ at point $M$.
1. Find the equation of the locus $C_{2}$ of point $M$.
2. Draw two mutually perpendicular lines $AC$ and $BD$ through point $F_{2}$, intersecting the ellipse at points $A$, $B$, $C$, and $D$. Find the minimum value of the area of the quadrilateral $ABCD$. | {
"answer": "\\frac{64}{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
I live on the ground floor of a ten-story building. Each friend of mine lives on a different floor. One day, I put the numbers $1, 2, \ldots, 9$ into a hat and drew them randomly, one by one. I visited my friends in the order in which I drew their floor numbers. On average, how many meters did I travel by elevator, if the distance between each floor is 4 meters, and I took the elevator from each floor to the next one drawn? | {
"answer": "440/3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many multiples of 5 are between 100 and 400? | {
"answer": "60",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A six-digit number begins with digit 1 and ends with digit 7. If the digit in the units place is decreased by 1 and moved to the first place, the resulting number is five times the original number. Find this number. | {
"answer": "142857",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a triangle \( \triangle ABC \) with sides \( a, b, c \) and corresponding medians \( m_a, m_b, m_c \), and angle bisectors \( w_a, w_b, w_c \). Let \( w_a \cap m_b = P \), \( w_b \cap m_c = Q \), and \( w_c \cap m_a = R \). Denote the area of \( \triangle PQR \) by \( \delta \) and the area of \( \triangle ABC \) by \( F \). Determine the smallest positive constant \( \lambda \) such that the inequality \( \frac{\delta}{F} < \lambda \) holds. | {
"answer": "\\frac{1}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The average score of 60 students is 72. After disqualifying two students whose scores are 85 and 90, calculate the new average score for the remaining class. | {
"answer": "71.47",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Five brothers divided their father's inheritance equally. The inheritance included three houses. Since it was not possible to split the houses, the three older brothers took the houses, and the younger brothers were given money: each of the three older brothers paid $2,000. How much did one house cost in dollars? | {
"answer": "3000",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The sides of a non-degenerate isosceles triangle are \(x\), \(x\), and \(24\) units. How many integer values of \(x\) are possible? | {
"answer": "11",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If 3913 were to be expressed as a sum of distinct powers of 2, what would be the least possible sum of the exponents of these powers? | {
"answer": "47",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given non-zero vectors $a=(-x, x)$ and $b=(2x+3, 1)$, where $x \in \mathbb{R}$.
$(1)$ If $a \perp b$, find the value of $x$;
$(2)$ If $a \nparallel b$, find $|a - b|$. | {
"answer": "3\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $a$, $b$, $c$, $d$, and $e$ be positive integers with $a+b+c+d+e=2020$ and let $M$ be the largest of the sum $a+b$, $b+c$, $c+d$, and $d+e$. What is the smallest possible value of $M$? | {
"answer": "675",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the number of diagonals in a polygon with 120 sides, and calculate the length of one diagonal of this polygon assuming it is regular and each side length is 5 cm. | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Paint both sides of a small wooden board. It takes 1 minute to paint one side, but you must wait 5 minutes for the paint to dry before painting the other side. How many minutes will it take to paint 6 wooden boards in total? | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC, AB = 10, BC = 8, CA = 7$ and side $BC$ is extended to a point $P$ such that $\triangle PAB$ is similar to $\triangle PCA$. The length of $PC$ is | {
"answer": "\\frac{56}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
From a certain airport, 100 planes (1 command plane and 99 supply planes) take off at the same time. When their fuel tanks are full, each plane can fly 1000 kilometers. During the flight, planes can refuel each other, and after completely transferring their fuel, planes can land as planned. How should the flight be arranged to enable the command plane to fly as far as possible? | {
"answer": "100000",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the smallest number $a$ such that a square of side $a$ can contain five disks of radius $1$ , so that no two of the disks have a common interior point. | {
"answer": "2 + 2\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that \(\alpha\) is an acute angle and \(\beta\) is an obtuse angle, and \(\sec (\alpha - 2\beta)\), \(\sec \alpha\), and \(\sec (\alpha + 2\beta)\) form an arithmetic sequence, find the value of \(\frac{\cos \alpha}{\cos \beta}\). | {
"answer": "\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Every card in a deck displays one of three symbols - star, circle, or square, each filled with one of three colors - red, yellow, or blue, and each color is shaded in one of three intensities - light, normal, or dark. The deck contains 27 unique cards, each representing a different symbol-color-intensity combination. A set of three cards is considered harmonious if:
i. Each card has a different symbol or all three have the same symbol.
ii. Each card has a different color or all three have the same color.
iii. Each card has a different intensity or all three have the same intensity.
Determine how many different harmonious three-card sets are possible. | {
"answer": "702",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given an ellipse $$\frac {x^{2}}{a^{2}}$$ + $$\frac {y^{2}}{b^{2}}$$ = 1 with its right focus F, a line passing through the origin O intersects the ellipse C at points A and B. If |AF| = 2, |BF| = 4, and the eccentricity of the ellipse C is $$\frac {\sqrt {7}}{3}$$, calculate the area of △AFB. | {
"answer": "2\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What percent of square $PQRS$ is shaded? All angles in the diagram are right angles. [asy]
import graph;
defaultpen(linewidth(0.8));
xaxis(0,7,Ticks(1.0,NoZero));
yaxis(0,7,Ticks(1.0,NoZero));
fill((0,0)--(2,0)--(2,2)--(0,2)--cycle);
fill((3,0)--(5,0)--(5,5)--(0,5)--(0,3)--(3,3)--cycle);
fill((6,0)--(7,0)--(7,7)--(0,7)--(0,6)--(6,6)--cycle);
label("$P$",(0,0),SW);
label("$Q$",(0,7),N);
label("$R$",(7,7),NE);
label("$S$",(7,0),E);[/asy] | {
"answer": "67.35\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are 8 students arranged in two rows, with 4 people in each row. If students A and B must be arranged in the front row, and student C must be arranged in the back row, then the total number of different arrangements is ___ (answer in digits). | {
"answer": "5760",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a grade, Class 1, Class 2, and Class 3 each select two students (one male and one female) to form a group of high school students. Two students are randomly selected from this group to serve as the chairperson and vice-chairperson. Calculate the probability of the following events:
- The two selected students are not from the same class;
- The two selected students are from the same class;
- The two selected students are of different genders and not from the same class. | {
"answer": "\\dfrac{2}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( n = \overline{abc} \) be a three-digit number, where \( a, b, \) and \( c \) are the digits of the number. If \( a, b, \) and \( c \) can form an isosceles triangle (including equilateral triangles), how many such three-digit numbers \( n \) are there? | {
"answer": "165",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the smallest positive integer $k$ such that $1^2+2^2+3^2+\ldots+k^2$ is a multiple of $360$. | {
"answer": "360",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\Delta XYZ$, $\overline{MN} \parallel \overline{XY}$, $XM = 5$ cm, $MY = 8$ cm, and $NZ = 9$ cm. What is the length of $\overline{YZ}$? | {
"answer": "23.4",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Shirley has a magical machine. If she inputs a positive even integer $n$ , the machine will output $n/2$ , but if she inputs a positive odd integer $m$ , the machine will output $m+3$ . The machine keeps going by automatically using its output as a new input, stopping immediately before it obtains a number already processed. Shirley wants to create the longest possible output sequence possible with initial input at most $100$ . What number should she input? | {
"answer": "67",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If two lines $l' $ and $m'$ have equations $y = -3x + 9$, and $y = -5x + 9$, what is the probability that a point randomly selected in the 1st quadrant and below $l'$ will fall between $l'$ and $m'$? Express your answer as a decimal to the nearest hundredth. | {
"answer": "0.4",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine $\sqrt[5]{102030201}$ without a calculator. | {
"answer": "101",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find all odd natural numbers greater than 500 but less than 1000, each of which has the sum of the last digits of all its divisors (including 1 and the number itself) equal to 33. | {
"answer": "729",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider a sequence $F_0=2$ , $F_1=3$ that has the property $F_{n+1}F_{n-1}-F_n^2=(-1)^n\cdot2$ . If each term of the sequence can be written in the form $a\cdot r_1^n+b\cdot r_2^n$ , what is the positive difference between $r_1$ and $r_2$ ?
| {
"answer": "\\frac{\\sqrt{17}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $S$ be the set of positive real numbers. Let $f : S \to \mathbb{R}$ be a function such that
\[f(x) f(y) = f(xy) + 2023 \left( \frac{1}{x} + \frac{1}{y} + 2022 \right)\]
for all $x, y > 0.$
Let $n$ be the number of possible values of $f(2)$, and let $s$ be the sum of all possible values of $f(2)$. Find $n \times s.$ | {
"answer": "\\frac{4047}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The symphony orchestra has more than 200 members but fewer than 300 members. When they line up in rows of 6, there are two extra members; when they line up in rows of 8, there are three extra members; and when they line up in rows of 9, there are four extra members. How many members are in the symphony orchestra? | {
"answer": "260",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The table shows the vertical drops of six roller coasters in Fibonacci Fun Park:
\begin{tabular}{|l|c|}
\hline
Speed Demon & 150 feet \\
\hline
Looper & 230 feet \\
\hline
Dare Devil & 160 feet \\
\hline
Giant Drop & 190 feet \\
\hline
Sky Scream & 210 feet \\
\hline
Hell Spiral & 180 feet \\
\hline
\end{tabular}
What is the positive difference between the mean and the median of these values? | {
"answer": "1.67",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The number 119 has the following property:
- Division by 2 leaves a remainder of 1;
- Division by 3 leaves a remainder of 2;
- Division by 4 leaves a remainder of 3;
- Division by 5 leaves a remainder of 4;
- Division by 6 leaves a remainder of 5.
How many positive integers less than 2007 satisfy this property? | {
"answer": "32",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the area bounded by the graph of \( y = \arcsin(\cos x) \) and the \( x \)-axis on the interval \( \left[0, 2\pi\right] \). | {
"answer": "\\frac{\\pi^2}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Call a lattice point visible if the line segment connecting the point and the origin does not pass through another lattice point. Given a positive integer \(k\), denote by \(S_{k}\) the set of all visible lattice points \((x, y)\) such that \(x^{2}+y^{2}=k^{2}\). Let \(D\) denote the set of all positive divisors of \(2021 \cdot 2025\). Compute the sum
$$
\sum_{d \in D}\left|S_{d}\right|
$$
Here, a lattice point is a point \((x, y)\) on the plane where both \(x\) and \(y\) are integers, and \(|A|\) denotes the number of elements of the set \(A\). | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
At some time between 9:30 and 10 o'clock, the triangle determined by the minute hand and the hour hand is an isosceles triangle. If the two equal angles in this triangle are each twice as large as the third angle, what is the time? | {
"answer": "9:36",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider a fictional language with ten letters in its alphabet: A, B, C, D, F, G, H, J, L, M. Suppose license plates of six letters utilize only letters from this alphabet. How many license plates of six letters are possible that begin with either B or D, end with J, cannot contain any vowels (A), and have no letters that repeat? | {
"answer": "1680",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The slant height of a cone forms an angle $\alpha$ with the plane of its base, where $\cos \alpha = \frac{1}{4}$. A sphere is inscribed in the cone, and a plane is drawn through the circle of tangency of the sphere and the lateral surface of the cone. The volume of the part of the cone enclosed between this plane and the base plane of the cone is 37. Find the volume of the remaining part of the cone. | {
"answer": "37",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On November 1, 2019, at exactly noon, two students, Gavrila and Glafira, set their watches (both have standard watches where the hands make a full rotation in 12 hours) accurately. It is known that Glafira's watch gains 12 seconds per day, and Gavrila's watch loses 18 seconds per day. After how many days will their watches simultaneously show the correct time again? Give the answer as a whole number, rounding if necessary. | {
"answer": "1440",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
**The first term of a sequence is $2089$. Each succeeding term is the sum of the squares of the digits of the previous term. What is the $2089^{\text{th}}$ term of the sequence?** | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A plane flies from city A to city B against a wind in 120 minutes. On the return trip with the wind, it takes 10 minutes less than it would in still air. Determine the time in minutes for the return trip. | {
"answer": "110",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the least positive integer $n$ such that $7350$ is a factor of $n!$? | {
"answer": "15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A grocery store manager decides to design a more compact pyramid-like stack of apples with a rectangular base of 4 apples by 6 apples. Each apple above the first level still rests in a pocket formed by four apples below, and the stack is completed with a double row of apples on top. Determine the total number of apples in the stack. | {
"answer": "53",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a square $ABCD$ with side length $4$, find the probability that $\angle AMB$ is an acute angle. | {
"answer": "1-\\dfrac{\\pi}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Three male students and three female students, a total of six students, stand in a row. If male student A does not stand at either end, and exactly two of the three female students stand next to each other, then the number of different arrangements is ______. | {
"answer": "288",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The letter T is formed by placing a $2\:\text{inch}\!\times\!6\:\text{inch}$ rectangle vertically and a $3\:\text{inch}\!\times\!2\:\text{inch}$ rectangle horizontally on top of the vertical rectangle at its middle, as shown. What is the perimeter of this T, in inches?
```
[No graphic input required]
``` | {
"answer": "22",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The lateral face of a regular triangular pyramid \( SABC \) is inclined to the base \( ABC \) at an angle \(\alpha = \arctan \frac{3}{4}\). Points \( M, N, K \) are the midpoints of the sides of the base \( ABC \). The triangle \( MNK \) forms the lower base of a right prism. The edges of the upper base of the prism intersect the lateral edges of the pyramid \( SABC \) at points \( F, P \), and \( R \) respectively. The total surface area of the polyhedron with vertices at points \( M, N, K, F, P, R \) is \( 53 \sqrt{3} \). Find the side length of the triangle \( ABC \). (16 points) | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
John and Mary select a natural number each and tell that to Bill. Bill wrote their sum and product in two papers hid one paper and showed the other to John and Mary.
John looked at the number (which was $2002$ ) and declared he couldn't determine Mary's number. Knowing this Mary also said she couldn't determine John's number as well.
What was Mary's Number? | {
"answer": "1001",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the smallest natural number \( n \) such that both \( n^2 \) and \( (n+1)^2 \) contain the digit 7. | {
"answer": "27",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $\mathcal{A}$ be the set of finite sequences of positive integers $a_1,a_2,\dots,a_k$ such that $|a_n-a_{n-1}|=a_{n-2}$ for all $3\leqslant n\leqslant k$ . If $a_1=a_2=1$ , and $k=18$ , determine the number of elements of $\mathcal{A}$ . | {
"answer": "1597",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A boy named Vasya wrote down the nonzero coefficients of a tenth-degree polynomial \( P(x) \) in his notebook. He then calculated the derivative of the resulting polynomial and wrote down its nonzero coefficients, and continued this process until he arrived at a constant, which he also wrote down.
What is the minimum number of different numbers he could have ended up with?
Coefficients are written down with their signs, and constant terms are also recorded. If there is a term of the form \(\pm x^n\), \(\pm 1\) is written down. | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A mouse has a wheel of cheese which is cut into $2018$ slices. The mouse also has a $2019$ -sided die, with faces labeled $0,1,2,\ldots, 2018$ , and with each face equally likely to come up. Every second, the mouse rolls the dice. If the dice lands on $k$ , and the mouse has at least $k$ slices of cheese remaining, then the mouse eats $k$ slices of cheese; otherwise, the mouse does nothing. What is the expected number of seconds until all the cheese is gone?
*Proposed by Brandon Wang* | {
"answer": "2019",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Three cards are dealt successively without replacement from a standard deck of 52 cards. What is the probability that the first card is a $\heartsuit$, the second card is a King, and the third card is a $\spadesuit$? | {
"answer": "\\frac{13}{2550}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $|\vec{a}|=1$, $|\vec{b}|=2$, and $(\vec{a}+\vec{b})\cdot \vec{b}=3$, find the angle between $\vec{b}$ and $\vec{a}$. | {
"answer": "\\frac{2\\pi}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A right rectangular prism has integer side lengths $a$ , $b$ , and $c$ . If $\text{lcm}(a,b)=72$ , $\text{lcm}(a,c)=24$ , and $\text{lcm}(b,c)=18$ , what is the sum of the minimum and maximum possible volumes of the prism?
*Proposed by Deyuan Li and Andrew Milas* | {
"answer": "3024",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given an ellipse $C:\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a > b > 0)$ with its right focus $F$ lying on the line $2x-y-2=0$, where $A$ and $B$ are the left and right vertices of $C$, and $|AF|=3|BF|$.<br/>$(1)$ Find the standard equation of $C$;<br/>$(2)$ A line $l$ passing through point $D(4,0)$ intersects $C$ at points $P$ and $Q$, with the midpoint of segment $PQ$ denoted as $N$. If the slope of line $AN$ is $\frac{2}{5}$, find the slope of line $l$. | {
"answer": "-\\frac{1}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A point is marked one quarter of the way along each side of a triangle. What fraction of the area of the triangle is shaded? | {
"answer": "$\\frac{5}{8}$",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. It is known that $4a = \sqrt{5}c$ and $\cos C = \frac{3}{5}$.
$(Ⅰ)$ Find the value of $\sin A$.
$(Ⅱ)$ If $b = 11$, find the area of $\triangle ABC$. | {
"answer": "22",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that in the expansion of \\((1+2x)^{n}\\), only the coefficient of the fourth term is the maximum, then the constant term in the expansion of \\((1+ \dfrac {1}{x^{2}})(1+2x)^{n}\\) is \_\_\_\_\_\_. | {
"answer": "61",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two adjacent faces of a tetrahedron, which are equilateral triangles with a side length of 1, form a dihedral angle of 45 degrees. The tetrahedron is rotated around the common edge of these faces. Find the maximum area of the projection of the rotating tetrahedron onto the plane containing this edge. | {
"answer": "\\frac{\\sqrt{3}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A subset \( S \) of the set of integers \{ 0, 1, 2, ..., 99 \} is said to have property \( A \) if it is impossible to fill a 2x2 crossword puzzle with the numbers in \( S \) such that each number appears only once. Determine the maximal number of elements in sets \( S \) with property \( A \). | {
"answer": "25",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For how many integers $n$ with $1 \le n \le 2023$ is the product
\[
\prod_{k=0}^{n-1} \left( \left( 1 + e^{2 \pi i k / n} \right)^n + 1 \right)^2
\]equal to zero? | {
"answer": "337",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A student's score on a 150-point test is directly proportional to the hours she studies. If she scores 90 points after studying for 2 hours, what would her score be if she studied for 5 hours? | {
"answer": "225",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the maximum number of different integers that can be selected from the set $ \{1,2,...,2013\}$ so that no two exist that their difference equals to $17$ . | {
"answer": "1010",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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