problem stringlengths 10 5.15k | answer dict |
|---|---|
Inside the triangle \(ABC\), a point \(M\) is taken such that \(\angle MBA = 30^\circ\) and \(\angle MAB = 10^\circ\). Find \(\angle AMC\) if \(\angle ACB = 80^\circ\) and \(AC = BC\). | {
"answer": "70",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
(1) Let the function $f(x)=|x+2|+|x-a|$. If the inequality $f(x) \geqslant 3$ always holds for $x$ in $\mathbb{R}$, find the range of the real number $a$.
(2) Given positive numbers $x$, $y$, $z$ satisfying $x+2y+3z=1$, find the minimum value of $\frac{3}{x}+\frac{2}{y}+\frac{1}{z}$. | {
"answer": "16+8\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The sixth graders were discussing how old their principal is. Anya said, "He is older than 38 years." Borya said, "He is younger than 35 years." Vova: "He is younger than 40 years." Galya: "He is older than 40 years." Dima: "Borya and Vova are right." Sasha: "You are all wrong." It turned out that the boys and girls were wrong the same number of times. Can we determine how old the principal is? | {
"answer": "39",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the greatest integer less than or equal to $\frac{5^{98} + 2^{104}}{5^{95} + 2^{101}}$. | {
"answer": "125",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Simplify first, then evaluate: $({1-\frac{2}{{x+1}}})÷\frac{{{x^2}-1}}{{3x+3}}$, where $x=\sqrt{3}-1$. | {
"answer": "\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the definite integral:
$$
\int_{0}^{2 \pi} \sin ^{2}\left(\frac{x}{4}\right) \cos ^{6}\left(\frac{x}{4}\right) d x
$$ | {
"answer": "\\frac{5\\pi}{64}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
5 people are standing in a row for a photo, among them one person must stand in the middle. There are ways to arrange them. | {
"answer": "24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine the distance that the origin $O(0,0)$ moves under the dilation transformation that sends the circle of radius $4$ centered at $B(3,1)$ to the circle of radius $6$ centered at $B'(7,9)$. | {
"answer": "0.5\\sqrt{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Gari is seated in a jeep, and at the moment, has one 10-peso coin, two 5-peso coins, and six 1-peso coins in his pocket. If he picks four coins at random from his pocket, what is the probability that these will be enough to pay for his jeepney fare of 8 pesos? | {
"answer": "37/42",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The distance from home to work is $s = 6$ km. At the moment Ivan left work, his favorite dog dashed out of the house and ran to meet him. They met at a distance of one-third of the total route from work. The dog immediately turned back and ran home. Upon reaching home, the dog turned around instantly and ran back towards Ivan, and so on. Assuming Ivan and his dog move at constant speeds, determine the distance the dog will run by the time Ivan arrives home. | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If organisms do not die but only divide, then the population will certainly never die out.
The conditions are satisfied by the function whose graph is highlighted in the image.
$$
x(p)=\left\{\begin{array}{l}
1, \text { if } 0 \leq p \leq \frac{1}{2} \\
\frac{q}{p}, \text { if } \frac{1}{2}<p \leq 1
\end{array}\right.
$$
In our case $p=0.6>\frac{1}{2}$, therefore, $x=\frac{q}{p}=\frac{2}{3}$. | {
"answer": "\\frac{2}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the largest integer $x$ such that the number
$$
4^{27} + 4^{1000} + 4^{x}
$$
is a perfect square. | {
"answer": "1972",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Twelve candidates for mayor are participating in a televised debate. At some point, one of them says, "So far, we've lied once." A second then says, "Now it's twice." A third exclaims, "Three times now," and so on, up to the twelfth who claims that before him, they lied twelve times. The presenter then stops the discussion. Given that at least one of the candidates correctly stated how many times lies had been told before their turn to speak, determine how many candidates lied in total. | {
"answer": "11",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
An infinite geometric series has a sum of 2020. If the first term, the third term, and the fourth term form an arithmetic sequence, find the first term. | {
"answer": "1010(1+\\sqrt{5})",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Ten identical books cost no more than 11 rubles, whereas 11 of the same books cost more than 12 rubles. How much does one book cost? | {
"answer": "110",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The novel takes 630 minutes to read aloud. The disc can hold 80 minutes of reading with at most 4 minutes of unused space. Calculate the number of minutes of reading each disc will contain. | {
"answer": "70",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Simplify \[\frac{1}{\dfrac{2}{\sqrt{5}+2} + \dfrac{3}{\sqrt{7}-2}}.\] | {
"answer": "\\frac{2\\sqrt{5} + \\sqrt{7} + 2}{23 + 4\\sqrt{35}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If a sequence of numbers \(a_{1}, a_{2}, \cdots\) satisfies, for any positive integer \(n\),
$$
a_{n}=\frac{n^{2}+n-2-\sqrt{2}}{n^{2}-2},
$$
then what is the value of \(a_{1} a_{2} \cdots a_{2016}\)? | {
"answer": "2016\\sqrt{2} - 2015",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $a > 0$, $b > 0$, and it satisfies the equation $3a + b = a^2 + ab$. Find the minimum value of $2a + b$. | {
"answer": "2\\sqrt{2} + 3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given \\(\alpha \in (0^{\circ}, 90^{\circ})\\) and \\(\sin (75^{\circ} + 2\alpha) = -\frac{3}{5}\\), calculate \\(\sin (15^{\circ} + \alpha) \cdot \sin (75^{\circ} - \alpha)\\). | {
"answer": "\\frac{\\sqrt{2}}{20}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a triangle \( \triangle ABC \) with sides \( a, b, c \) opposite to angles \( A, B, C \) respectively, and \( a^{2} + b^{2} = c^{2} + \frac{2}{3}ab \). If the circumradius of \( \triangle ABC \) is \( \frac{3\sqrt{2}}{2} \), what is the maximum possible area of \( \triangle ABC \)? | {
"answer": "4\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Each of the ten cards has a real number written on it. For every non-empty subset of these cards, the sum of all the numbers written on the cards in that subset is calculated. It is known that not all of the obtained sums turned out to be integers. What is the largest possible number of integer sums that could have resulted? | {
"answer": "511",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine the total surface area of a cube if the distance between the non-intersecting diagonals of two adjacent faces of this cube is 8. If the answer is not an integer, round it to the nearest whole number. | {
"answer": "1152",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the rectangular prism \(ABCD-A_1B_1C_1D_1\), \(AB=2\), \(AA_1=AD=1\). Points \(E\), \(F\), and \(G\) are the midpoints of edges \(AA_1\), \(C_1D_1\), and \(BC\) respectively. What is the volume of the tetrahedron \(B_1-EFG\)? | {
"answer": "\\frac{3}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a triangular prism \( S-ABC \) with a base that is an isosceles right triangle with \( AB \) as the hypotenuse, and \( SA = SB = SC = AB = 2 \). If the points \( S, A, B, C \) all lie on the surface of a sphere centered at \( O \), what is the surface area of this sphere? | {
"answer": "\\frac{16 \\pi}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The function \[f(x) = \left\{ \begin{aligned} 2x + 1 & \quad \text{ if } x < 3 \\ x^2 & \quad \text{ if } x \ge 3 \end{aligned} \right.\] has an inverse $f^{-1}.$ Compute the value of $f^{-1}(-3) + f^{-1}(0) + \dots + f^{-1}(4) + f^{-1}(9).$ | {
"answer": "3.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $a$, $b$, and $c$ are the three sides of $\triangle ABC$, and $3a^2+3b^2-3c^2+2ab=0$, then $\tan C= \_\_\_\_\_\_$. | {
"answer": "-2 \\sqrt {2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Crystal decides to alter her running routine slightly. She heads due north for 2 miles, then goes northwest for 2 miles, followed by a southwest direction for 2 miles, and she finishes with a final segment directly back to her starting point. Calculate the distance of this final segment of her run. | {
"answer": "2\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine the total surface area of a cone with a diameter of 8 cm and a height of 12 cm. Express your answer in terms of \(\pi\). | {
"answer": "16\\pi (\\sqrt{10} + 1)",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How can 13 rectangles of sizes $1 \times 1, 2 \times 1, 3 \times 1, \ldots, 13 \times 1$ be combined to form a rectangle, where all sides are greater than 1? | {
"answer": "13 \\times 7",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( M \) and \( m \) be the maximum and minimum elements, respectively, of the set \( \left\{\left.\frac{3}{a}+b \right\rvert\, 1 \leq a \leq b \leq 2\right\} \). Find the value of \( M - m \). | {
"answer": "5 - 2\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the polar coordinate system, curve $C$: $\rho =2a\cos \theta (a > 0)$, line $l$: $\rho \cos \left( \theta -\frac{\pi }{3} \right)=\frac{3}{2}$, $C$ and $l$ have exactly one common point. $O$ is the pole, $A$ and $B$ are two points on $C$, and $\angle AOB=\frac{\pi }{3}$, then the maximum value of $|OA|+|OB|$ is __________. | {
"answer": "2 \\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The volume of a cube in cubic meters and its surface area in square meters is numerically equal to four-thirds of the sum of the lengths of its edges in meters. What is the total volume in cubic meters of twenty-seven such cubes? | {
"answer": "216",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the random variable $\xi$ follows the normal distribution $N(1, 4)$, if $P(\xi > 4) = 0.1$, then $P(-2 \leq \xi \leq 4)$ equals _______. | {
"answer": "0.8",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Evaluate the expression $\sqrt{5+4\sqrt{3}} - \sqrt{5-4\sqrt{3}} + \sqrt{7 + 2\sqrt{10}} - \sqrt{7 - 2\sqrt{10}}$.
A) $4\sqrt{3}$
B) $2\sqrt{2}$
C) $6$
D) $4\sqrt{2}$
E) $2\sqrt{5}$ | {
"answer": "2\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the value of $\sqrt{5! \cdot (5!)^2}$ expressed as a positive integer? | {
"answer": "240\\sqrt{30}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given positive real numbers $a$ and $b$ satisfying $a+b=2$, the minimum value of $\dfrac{1}{a}+\dfrac{2}{b}$ is ______. | {
"answer": "\\dfrac{3+2 \\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Carl decided to fence his rectangular flowerbed using 24 fence posts, including one on each corner. He placed the remaining posts spaced exactly 3 yards apart along the perimeter of the bed. The bed’s longer side has three times as many posts compared to the shorter side, including the corner posts. Calculate the area of Carl’s flowerbed, in square yards. | {
"answer": "144",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A bag contains 70 balls that differ only in color: 20 red, 20 blue, 20 yellow, and the rest are black and white.
What is the minimum number of balls that must be drawn from the bag, without looking, to ensure that among them there are at least 10 balls of a single color? | {
"answer": "38",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle \(ABC\), points \(P\) and \(Q\) are taken on the base \(AC\) such that \(AP < AQ\). The lines \(BP\) and \(BQ\) divide the median \(AM\) into three equal parts. It is known that \(PQ = 3\).
Find \(AC\). | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The sequence starts with 800,000; each subsequent term is obtained by dividing the previous term by 3. What is the last integer in this sequence? | {
"answer": "800000",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There is a sphere with a radius of $\frac{\sqrt{3}}{2}$, on which 4 points $A, B, C, D$ form a regular tetrahedron. What is the maximum distance from the center of the sphere to the faces of the regular tetrahedron $ABCD$? | {
"answer": "\\frac{\\sqrt{3}}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively. It is known that $c=a\cos B+b\sin A$.
(I) Find angle $A$.
(II) If $a=2$, find the maximum area of $\triangle ABC$. | {
"answer": "\\sqrt {2}+1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A team of loggers was scheduled to harvest $216 \mathrm{~m}^{3}$ of wood over several days. For the first three days, the team met the daily target set by the plan. Then, they harvested an additional $8 \mathrm{~m}^{3}$ above the daily target each day. As a result, they harvested $232 \mathrm{~m}^{3}$ of wood one day ahead of schedule. How many cubic meters of wood per day was the team supposed to harvest according to the plan? | {
"answer": "24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given points $A(-1,1)$, $B(1,2)$, $C(-2,-1)$, $D(2,2)$, the projection of vector $\overrightarrow{AB}$ in the direction of $\overrightarrow{CD}$ is ______. | {
"answer": "\\dfrac{11}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The numbers \(1, 2, 3, \ldots, 7\) are randomly divided into two non-empty subsets. What is the probability that the sum of the numbers in the two subsets is equal? If the probability is expressed as \(\frac{p}{q}\) in its lowest terms, find \(p + q\). | {
"answer": "67",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $y=\sin(\omega x)$ ($\omega>0$) is increasing in the interval $[0, \frac{\pi}{3}]$ and its graph is symmetric about the point $(3\pi, 0)$, the maximum value of $\omega$ is \_\_\_\_\_\_. | {
"answer": "\\frac{4}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a certain area, there are 100,000 households, among which there are 99,000 ordinary households and 1,000 high-income households. A simple random sampling method is used to select 990 households from the ordinary households and 100 households from the high-income households for a survey. It was found that a total of 120 households own 3 or more sets of housing, among which there are 40 ordinary households and 80 high-income households. Based on these data and combining your statistical knowledge, what do you think is a reasonable estimate of the proportion of households in the area that own 3 or more sets of housing? | {
"answer": "4.8\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a rectangular prism $A^{\prime}C$, with $AB=5$, $BC=4$, and $B^{\prime}B=6$, $E$ is the midpoint of $AA^{\prime}$. Find the distance between the skew lines $BE$ and $A^{\prime}C^{\prime}$. | {
"answer": "\\frac{60}{\\sqrt{769}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A city adopts a lottery system for "price-limited housing," where winning families can randomly draw a house number from the available housing in a designated community. It is known that two friendly families, Family A and Family B, have both won the lottery and decided to go together to a certain community to draw their house numbers. Currently, there are $5$ houses left in this community, spread across the $4^{th}$, $5^{th}$, and $6^{th}$ floors of a building, with $1$ house on the $4^{th}$ floor and $2$ houses each on the $5^{th}$ and $6^{th}$ floors.
(Ⅰ) Calculate the probability that Families A and B will live on the same floor.
(Ⅱ) Calculate the probability that Families A and B will live on adjacent floors. | {
"answer": "\\dfrac{3}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( XYZ \) be a triangle with \( \angle X = 60^\circ \) and \( \angle Y = 45^\circ \). A circle with center \( P \) passes through points \( A \) and \( B \) on side \( XY \), \( C \) and \( D \) on side \( YZ \), and \( E \) and \( F \) on side \( ZX \). Suppose \( AB = CD = EF \). Find \( \angle XPY \) in degrees. | {
"answer": "120",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In how many different ways can four couples sit around a circular table such that no couple sits next to each other? | {
"answer": "1488",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a positive real number \(\alpha\), determine the greatest real number \(C\) such that the inequality
$$
\left(1+\frac{\alpha}{x^{2}}\right)\left(1+\frac{\alpha}{y^{2}}\right)\left(1+\frac{\alpha}{z^{2}}\right) \geq C \cdot\left(\frac{x}{z}+\frac{z}{x}+2\right)
$$
holds for all positive real numbers \(x, y\), and \(z\) that satisfy \(xy + yz + zx = \alpha\). When does equality hold? | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
At a physical education lesson, 29 seventh graders attended, some of whom brought one ball each. During the lesson, sometimes one seventh grader would give their ball to another seventh grader who did not have a ball.
At the end of the lesson, $N$ seventh graders said, "I received balls less often than I gave them away!" Find the largest possible value of $N$, given that no one lied. | {
"answer": "14",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the lowest prime number that is thirteen more than a cube? | {
"answer": "229",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the regression equation $y = 0.849x - 85.712$, where $x$ represents the height in cm and $y$ represents the weight in kg, determine the predicted weight of a female student who is 172 cm tall. | {
"answer": "60.316",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the random variables \( X \sim N(1,2) \) and \( Y \sim N(3,4) \), if \( P(X < 0) = P(Y > a) \), find the value of \( a \). | {
"answer": "3 + \\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If \( a + b + c = 1 \), what is the maximum value of \( \sqrt{3a+1} + \sqrt{3b+1} + \sqrt{3c+1} \)? | {
"answer": "3\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the production of a certain item, its weight \( X \) is subject to random fluctuations. The standard weight of the item is 30 g, its standard deviation is 0.7, and the random variable \( X \) follows a normal distribution. Find the probability that the weight of a randomly selected item is within the range from 28 to 31 g. | {
"answer": "0.9215",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $S-ABC$ be a triangular prism with circumscribed sphere centered at $O$. The midpoints of $SB$ and $AC$ are $N$ and $M$, respectively. The midpoint of line segment $MN$ is $P$, and it is given that $SA^{2} + SB^{2} + SC^{2} = AB^{2} + BC^{2} + AC^{2}$. If $SP = 3\sqrt{7}$ and $OP = \sqrt{21}$, find the radius of sphere $O$. | {
"answer": "2 \\sqrt{21}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the perimeter of an equilateral triangle inscribed in a circle, given that a chord of this circle, equal to 2, is at a distance of 3 from its center. | {
"answer": "3 \\sqrt{30}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
It is given that \( k \) is a positive integer not exceeding 99. There are no natural numbers \( x \) and \( y \) such that \( x^{2} - k y^{2} = 8 \). Find the difference between the maximum and minimum possible values of \( k \). | {
"answer": "96",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( S-ABC \) be a triangular prism with the base being an isosceles right triangle \( ABC \) with \( AB \) as the hypotenuse, and \( SA = SB = SC = 2 \) and \( AB = 2 \). If \( S \), \( A \), \( B \), and \( C \) are points on a sphere centered at \( O \), find the distance from point \( O \) to the plane \( ABC \). | {
"answer": "\\frac{\\sqrt{3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a circle circumscribed around triangle \(FDC\), a tangent \(FK\) is drawn such that \(\angle KFC = 58^\circ\). Points \(K\) and \(D\) lie on opposite sides of line \(FC\) as shown in the diagram. Find the acute angle between the angle bisectors of \(\angle CFD\) and \(\angle FCD\). Provide your answer in degrees. | {
"answer": "61",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For what is the smallest $n$ such that there exist $n$ numbers within the interval $(-1, 1)$ whose sum is 0 and the sum of their squares is 42? | {
"answer": "44",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that player A needs to win 2 more games and player B needs to win 3 more games, and the probability of winning each game for both players is $\dfrac{1}{2}$, calculate the probability of player A ultimately winning. | {
"answer": "\\dfrac{11}{16}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( A B C \) be an isosceles triangle with \( B \) as the vertex of the equal angles. Let \( F \) be a point on the bisector of \( \angle A B C \) such that \( (A F) \) is parallel to \( (B C) \). Let \( E \) be the midpoint of \([B C]\), and let \( D \) be the symmetric point of \( A \) with respect to \( F \). Calculate the ratio of the distances \( E F / B D \). | {
"answer": "1/2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
(1) Solve the inequality $\log_{\frac{1}{2}}(x+2) > -3$
(2) Calculate: $(\frac{1}{8})^{\frac{1}{3}} \times (-\frac{7}{6})^{0} + 8^{0.25} \times \sqrt[4]{2} + (\sqrt[3]{2} \times \sqrt{3})^{6}$. | {
"answer": "\\frac{221}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
During an underwater archaeology activity, a diver needs to dive 50 meters to the seabed to carry out archaeological work. The oxygen consumption includes the following three aspects:
1. The average descent speed is $x$ meters per minute, and the oxygen consumption per minute is $\frac{1}{100}x^2$ liters;
2. The diver's working time on the seabed ranges from at least 10 minutes to a maximum of 20 minutes, with an oxygen consumption of 0.3 liters per minute;
3. When returning to the surface, the average speed is $\frac{1}{2}x$ meters per minute, with an oxygen consumption of 0.32 liters per minute. The total oxygen consumption of the diver in this archaeological activity is $y$ liters.
(1) If the working time on the seabed is 10 minutes, express $y$ as a function of $x$;
(2) If $x \in [6,10]$ and the working time on the seabed is 20 minutes, find the range of the total oxygen consumption $y$;
(3) If the diver carries 13.5 liters of oxygen, what is the maximum number of minutes the diver can stay underwater? (Round the result to the nearest integer). | {
"answer": "18",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $u$ and $v$ be real numbers satisfying the inequalities $2u + 3v \le 10$ and $4u + v \le 9.$ Find the largest possible value of $u + 2v$. | {
"answer": "6.1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. It is given that $a = b\cos C + c\sin B$.
(1) Find angle $B$.
(2) If $b = 4$, find the maximum area of triangle $ABC$. | {
"answer": "4\\sqrt{2} + 4",
"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 circle with center at point (b, 0) and radius a is drawn. The circle intersects with one of the asymptotes of the hyperbola at points M and N, and ∠MPN = 90°. Calculate the eccentricity of the hyperbola. | {
"answer": "\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine the sum of all positive integers \( N < 1000 \) for which \( N + 2^{2015} \) is divisible by 257. | {
"answer": "2058",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the circumcenter of triangle $ABC$ is $O$, and $2 \overrightarrow{O A} + 3 \overrightarrow{O B} + 4 \overrightarrow{O C} = 0$, determine the value of $\cos \angle BAC$. | {
"answer": "\\frac{1}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many multiples of 4 are between 70 and 300? | {
"answer": "57",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On the game show $\text{\emph{Wheel of Fortune Redux}}$, you see the following spinner. Given that each region is of the same area, what is the probability that you will earn exactly $\$3200$ in your first three spins? The spinner includes the following sections: $"\$2000"$, $"\$300"$, $"\$700"$, $"\$1500"$, $"\$500"$, and "$Bankrupt"$. Express your answer as a common fraction. | {
"answer": "\\frac{1}{36}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A point \(A_{1}\) is taken on the side \(AC\) of triangle \(ABC\), and a point \(C_{1}\) is taken on the extension of side \(BC\) beyond point \(C\). The length of segment \(A_{1}C\) is 85% of the length of side \(AC\), and the length of segment \(BC_{1}\) is 120% of the length of side \(BC\). What percentage of the area of triangle \(ABC\) is the area of triangle \(A_{1}BC_{1}\)? | {
"answer": "102",
"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. If the angles \( A \), \( B \), and \( C \) form a geometric progression, and \( b^{2} - a^{2} = ac \), then the radian measure of angle \( B \) is equal to ________. | {
"answer": "\\frac{2\\pi}{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function \( f(x) = 4 \pi \arcsin x - (\arccos(-x))^2 \), find the difference between its maximum value \( M \) and its minimum value \( m \). Specifically, calculate \( M - m \). | {
"answer": "3\\pi^2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find all real numbers \( x \) that satisfy the equation
$$
\frac{x-2020}{1} + \frac{x-2019}{2} + \cdots + \frac{x-2000}{21} = \frac{x-1}{2020} + \frac{x-2}{2019} + \cdots + \frac{x-21}{2000},
$$
and simplify your answer(s) as much as possible. Justify your solution. | {
"answer": "2021",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The sequence $\{a_n\}$ satisfies $a_n=13-3n$, $b_n=a_n⋅a_{n+1}⋅a_{n+2}$, $S_n$ is the sum of the first $n$ terms of $\{b_n\}$. Find the maximum value of $S_n$. | {
"answer": "310",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The base of a rectangular parallelepiped is a square with a side length of \(2 \sqrt{3}\). The diagonal of a lateral face forms an angle of \(30^\circ\) with the plane of an adjacent lateral face. Find the volume of the parallelepiped. | {
"answer": "72",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate: $\sqrt[3]{27}+|-\sqrt{2}|+2\sqrt{2}-(-\sqrt{4})$. | {
"answer": "5 + 3\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The domain of the function $y=\sin x$ is $[a,b]$, and its range is $\left[-1, \frac{1}{2}\right]$. Calculate the maximum value of $b-a$. | {
"answer": "\\frac{4\\pi}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
With all angles measured in degrees, the product $\prod_{k=1}^{22} \sec^2(4k)^\circ=p^q$, where $p$ and $q$ are integers greater than 1. Find the value of $p+q$. | {
"answer": "46",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate $\lim _{n \rightarrow \infty}\left(\sqrt[3^{2}]{3} \cdot \sqrt[3^{3}]{3^{2}} \cdot \sqrt[3^{4}]{3^{3}} \ldots \sqrt[3^{n}]{3^{n-1}}\right)$. | {
"answer": "\\sqrt[4]{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In an equilateral triangle \(ABC\), a point \(P\) is chosen such that \(AP = 10\), \(BP = 8\), and \(CP = 6\). Find the area of this triangle. | {
"answer": "36 + 25\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
An archipelago consists of \( N \geq 7 \) islands. Any two islands are connected by no more than one bridge. It is known that no more than 5 bridges lead from each island, and among any 7 islands, there are necessarily two that are connected by a bridge. What is the maximum value that \( N \) can take? | {
"answer": "36",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that five volunteers are randomly assigned to conduct promotional activities in three communities, A, B, and C, at least 2 volunteers are assigned to community A, and at least 1 volunteer is assigned to each of communities B and C, calculate the number of different arrangements. | {
"answer": "80",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
At the CleverCat Academy, there are three skills that the cats can learn: jump, climb, and hunt. Out of the cats enrolled in the school:
- 40 cats can jump.
- 25 cats can climb.
- 30 cats can hunt.
- 10 cats can jump and climb.
- 15 cats can climb and hunt.
- 12 cats can jump and hunt.
- 5 cats can do all three skills.
- 6 cats cannot perform any of the skills.
How many cats are in the academy? | {
"answer": "69",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The sequence $(a_{n})$ is defined by the following relations: $a_{1}=1$, $a_{2}=3$, $a_{n}=a_{n-1}-a_{n-2}+n$ (for $n \geq 3$). Find $a_{1000}$. | {
"answer": "1002",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $α+β= \frac {π}{3}$ and $tanα+tanβ=2$, find the value of $cos(α-β)$. | {
"answer": "\\frac { \\sqrt {3}-1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the greatest negative value of the expression \( x - y \) for all pairs of numbers \((x, y)\) satisfying the equation
$$
(\sin x + \sin y)(\cos x - \cos y) = \frac{1}{2} + \sin(x - y) \cos(x + y)
$$ | {
"answer": "-\\frac{\\pi}{6}",
"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 $c \sin A = \sqrt{3}a \cos C$ and $(a-c)(a+c)=b(b-c)$, find the period and the monotonically increasing interval of the function $f(x) = 2 \sin x \cos (\frac{\pi}{2} - x) - \sqrt{3} \sin (\pi + x) \cos x + \sin (\frac{\pi}{2} + x) \cos x$. Also, find the value of $f(B)$. | {
"answer": "\\frac{5}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the complex number \( z \) satisfying
$$
\left|\frac{z^{2}+1}{z+\mathrm{i}}\right|+\left|\frac{z^{2}+4 \mathrm{i}-3}{z-\mathrm{i}+2}\right|=4,
$$
find the minimum value of \( |z - 1| \). | {
"answer": "\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the function $f(x+1)$ is an odd function, and the function $f(x-1)$ is an even function, and $f(0) = 2$, determine the value of $f(4)$. | {
"answer": "-2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many ways are there to choose 6 different numbers from $1, 2, \cdots, 49$, where at least two numbers are consecutive? | {
"answer": "\\binom{49}{6} - \\binom{44}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a dart board is a regular hexagon divided into regions, the center of the board is another regular hexagon formed by joining the midpoints of the sides of the larger hexagon, and the dart is equally likely to land anywhere on the board, find the probability that the dart lands within the center hexagon. | {
"answer": "\\frac{1}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a round-robin chess tournament, only grandmasters and masters participated. The number of masters was three times the number of grandmasters, and the total points scored by the masters was 1.2 times the total points scored by the grandmasters.
How many people participated in the tournament? What can be said about the tournament's outcome? | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The sum of all real roots of the equation $|x^2-3x+2|+|x^2+2x-3|=11$ is . | {
"answer": "\\frac{5\\sqrt{97}-19}{20}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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