problem stringlengths 10 5.15k | answer dict |
|---|---|
Persons A, B, and C set out from location $A$ to location $B$ at the same time. Their speed ratio is 4: 5: 12, respectively, where A and B travel by foot, and C travels by bicycle. C can carry one person with him on the bicycle (without changing speed). In order for all three to reach $B$ at the same time in the shortest time possible, what is the ratio of the walking distances covered by A and B? | {
"answer": "7/10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the acute angle \( x \) that satisfies the equation \( \sin^3 x + \cos^3 x = \frac{\sqrt{2}}{2} \), find \( x \). | {
"answer": "\\frac{\\pi}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Suppose the function \( y= \left| \log_{2} \frac{x}{2} \right| \) has a domain of \([m, n]\) and a range of \([0,2]\). What is the minimum length of the interval \([m, n]\)? | {
"answer": "3/2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that \(\tan \frac{\alpha+\beta}{2}=\frac{\sqrt{6}}{2}\) and \(\cot \alpha \cdot \cot \beta=\frac{7}{13}\), find the value of \(\cos (\alpha-\beta)\). | {
"answer": "\\frac{2}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( f(n) = 3n^2 - 3n + 1 \). Find the last four digits of \( f(1) + f(2) + \cdots + f(2010) \). | {
"answer": "1000",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Triangles \(ABC\) and \(ABD\) are inscribed in a semicircle with diameter \(AB = 5\). A perpendicular from \(D\) to \(AB\) intersects segment \(AC\) at point \(Q\), ray \(BC\) at point \(R\), and segment \(AB\) at point \(P\). It is known that \(PR = \frac{27}{10}\), and \(PQ = \frac{5}{6}\). Find the length of segment \(DP\). If necessary, round the answer to hundredths. | {
"answer": "1.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the product of two approximate numbers: $0.3862 \times 0.85$. | {
"answer": "0.33",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( F_{1} \) and \( F_{2} \) be the left and right foci of the hyperbola \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1 \) (where \( a > 0 \) and \( b > 0 \)). There exists a point \( P \) on the right branch of the hyperbola such that \( \left( \overrightarrow{OP} + \overrightarrow{OF_{2}} \right) \cdot \overrightarrow{PF_{2}} = 0 \), where \( O \) is the origin. Additionally, \( \left| \overrightarrow{PF_{1}} \right| = \sqrt{3} \left| \overrightarrow{PF_{2}} \right| \). Determine the eccentricity of the hyperbola. | {
"answer": "\\sqrt{3} + 1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the coordinate plane \(xOy\), given points \(A(1,3)\), \(B\left(8 \frac{1}{3}, 1 \frac{2}{3}\right)\), and \(C\left(7 \frac{1}{3}, 4 \frac{2}{3}\right)\), the extended lines \(OA\) and \(BC\) intersect at point \(D\). Points \(M\) and \(N\) are on segments \(OD\) and \(BD\) respectively, with \(OM = MN = BN\). Find the length of line segment \(MN\). | {
"answer": "\\frac{5 \\sqrt{10}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There is a cube of size \(10 \times 10 \times 10\) made up of small unit cubes. A grasshopper is sitting at the center \(O\) of one of the corner cubes. It can jump to the center of a cube that shares a face with the one in which the grasshopper is currently located, provided that the distance to point \(O\) increases. How many ways can the grasshopper jump to the cube opposite to the original one? | {
"answer": "\\frac{27!}{(9!)^3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle \( \triangle ABC \), \( M \) is the midpoint of side \( BC \), and \( N \) is the midpoint of segment \( BM \). Given \( \angle A = \frac{\pi}{3} \) and the area of \( \triangle ABC \) is \( \sqrt{3} \), find the minimum value of \( \overrightarrow{AM} \cdot \overrightarrow{AN} \). | {
"answer": "\\sqrt{3} + 1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The number $\overline{x y z t}$ is a perfect square such that the number $\overline{t z y x}$ is also a perfect square, and the quotient of the numbers $\overline{x y z t}$ and $\overline{t z y x}$ is also a perfect square. Determine the number $\overline{x y z t}$. (The overline indicates that the number is written in the decimal system.) | {
"answer": "9801",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
30 students from 5 grades participated in answering 40 questions. Each student answered at least 1 question. Every two students from the same grade answered the same number of questions, and students from different grades answered a different number of questions. How many students answered only 1 question? | {
"answer": "26",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( G \) be the centroid of \(\triangle ABC\). Given \( BG \perp CG \) and \( BC = \sqrt{2} \), find the maximum value of \( AB + AC \). | {
"answer": "2\\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a park, there is a row of flags arranged in the sequence of 3 yellow flags, 2 red flags, and 4 pink flags. Xiaohong sees that the row ends with a pink flag. Given that the total number of flags does not exceed 200, what is the maximum number of flags in this row? | {
"answer": "198",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Severus Snape, the potions professor, prepared three potions, each in an equal volume of 400 ml. The first potion makes the drinker smarter, the second makes them more beautiful, and the third makes them stronger. To ensure the effect of any potion, it is sufficient to drink at least 30 ml of that potion. Snape intended to drink the potions himself, but he was called to see the headmaster and had to leave, leaving the labeled potions on his desk in large jugs. Harry, Hermione, and Ron took advantage of his absence and began to taste the potions.
Hermione was the first to try the potions: she approached the first jug with the intelligence potion and drank half of it, then poured the remaining potion into the second jug with the beauty potion, stirred the contents of the jug thoroughly, and drank half of it. Next, it was Harry's turn: he drank half of the third jug with the strength potion, poured the remaining potion into the second jug, stirred everything in this jug thoroughly, and drank half of it. Now all the contents are in the second jug, which went to Ron. What percentage of the contents of this jug does Ron need to drink to ensure that each of the three potions will have an effect on him? | {
"answer": "60",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The base of the quadrangular pyramid \( M A B C D \) is a parallelogram \( A B C D \). Given that \( \overline{D K} = \overline{K M} \) and \(\overline{B P} = 0.25 \overline{B M}\), the point \( X \) is the intersection of the line \( M C \) and the plane \( A K P \). Find the ratio \( M X: X C \). | {
"answer": "3 : 4",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are 5 integers written on the board. By summing them in pairs, the following set of 10 numbers is obtained: 2, 6, 10, 10, 12, 14, 16, 18, 20, 24. Determine which numbers are written on the board and write their product as the answer. | {
"answer": "-3003",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let the function
$$
f(x) = A \sin(\omega x + \varphi) \quad (A>0, \omega>0).
$$
If \( f(x) \) is monotonic on the interval \(\left[\frac{\pi}{6}, \frac{\pi}{2}\right]\) and
$$
f\left(\frac{\pi}{2}\right) = f\left(\frac{2\pi}{3}\right) = -f\left(\frac{\pi}{6}\right),
$$
then the smallest positive period of \( f(x) \) is ______. | {
"answer": "\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a company, some pairs of people are friends (if $A$ is friends with $B$, then $B$ is friends with $A$). It turns out that among every set of 100 people in the company, the number of pairs of friends is odd. Find the largest possible number of people in such a company. | {
"answer": "101",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A subscriber forgot the last digit of a phone number and therefore dials it randomly. What is the probability that they will have to dial the number no more than three times? | {
"answer": "0.3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In how many ways can you form 5 quartets from 5 violinists, 5 violists, 5 cellists, and 5 pianists? | {
"answer": "(5!)^3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the smallest natural number that cannot be written in the form \(\frac{2^{a} - 2^{b}}{2^{c} - 2^{d}}\), where \(a\), \(b\), \(c\), and \(d\) are natural numbers. | {
"answer": "11",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two decks, each containing 36 cards, were placed on a table. The first deck was shuffled and placed on top of the second deck. For each card in the first deck, the number of cards between it and the same card in the second deck was counted. What is the sum of these 36 numbers? | {
"answer": "1260",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
An aluminum part and a copper part have the same volume. The density of aluminum is $\rho_{A} = 2700 \, \text{kg/m}^3$, and the density of copper is $\rho_{M} = 8900 \, \text{kg/m}^3$. Find the mass of the aluminum part, given that the masses of the parts differ by $\Delta m = 60 \, \text{g}$. | {
"answer": "26.13",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Choose six out of the ten digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 to fill in the blanks below, so that the equation is true. Each blank is filled with a single digit, and no two digits are the same.
$\square + \square \square = \square \square \square$.
What is the largest possible three-digit number in the equation? | {
"answer": "105",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Anya, Vanya, Dania, Sanya, and Tanya were collecting apples. It turned out that each of them collected an integer percentage of the total number of apples, and all these percentages are different and greater than zero. What is the minimum number of apples that could have been collected? | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The population of a city increases annually by $1 / 50$ of the current number of inhabitants. In how many years will the population triple? | {
"answer": "55",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \(a, b, c, d, e\) be positive integers. Their sum is 2345. Let \(M = \max (a+b, b+c, c+d, d+e)\). Find the smallest possible value of \(M\). | {
"answer": "782",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two people, A and B, take turns to draw candies from a bag. A starts by taking 1 candy, then B takes 2 candies, A takes 4 candies next, then B takes 8 candies, and so on. This continues. When the number of candies remaining in the bag is less than the number they are supposed to take, they take all the remaining candies. If A has taken a total of 90 candies, how many candies were there initially in the bag? | {
"answer": "260",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are 8 different positive integers. Among them, at least 6 are multiples of 2, at least 5 are multiples of 3, at least 3 are multiples of 5, and at least 1 is a multiple of 7. To minimize the largest number among these 8 integers, what is this largest number? | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Kolya started playing WoW when the hour and minute hands were opposite each other. He finished playing after a whole number of minutes, at which point the minute hand coincided with the hour hand. How long did he play (assuming he played for less than 12 hours)? | {
"answer": "360",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given \(\alpha, \beta \in [0, \pi]\), find the maximum value of \((\sin \alpha + \sin (\alpha + \beta)) \cdot \sin \beta\). | {
"answer": "\\frac{8\\sqrt{3}}{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Compute
$$
\int_{1}^{2} \frac{9x+4}{x^{5}+3x^{2}+x} \, dx. | {
"answer": "\\ln \\frac{80}{23}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $[x]$ denote the greatest integer less than or equal to the real number $x$,
$$
\begin{array}{c}
S=\left[\frac{1}{1}\right]+\left[\frac{2}{1}\right]+\left[\frac{1}{2}\right]+\left[\frac{2}{2}\right]+\left[\frac{3}{2}\right]+ \\
{\left[\frac{4}{2}\right]+\left[\frac{1}{3}\right]+\left[\frac{2}{3}\right]+\left[\frac{3}{3}\right]+\left[\frac{4}{3}\right]+} \\
{\left[\frac{5}{3}\right]+\left[\frac{6}{3}\right]+\cdots}
\end{array}
$$
up to 2016 terms, where each segment for a denominator $k$ contains $2k$ terms $\left[\frac{1}{k}\right],\left[\frac{2}{k}\right], \cdots,\left[\frac{2k}{k}\right]$, and only the last segment might have less than $2k$ terms. Find the value of $S$. | {
"answer": "1078",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On an infinite tape, numbers are written in a row. The first number is one, and each subsequent number is obtained from the previous one by adding the smallest nonzero digit of its decimal representation. How many digits are in the decimal representation of the number that is in the $9 \cdot 1000^{1000}$-th position in this row? | {
"answer": "3001",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The school plans to arrange 6 leaders to be on duty from May 1st to May 3rd. Each leader must be on duty for 1 day, with 2 leaders assigned each day. If leader A cannot be on duty on the 2nd, and leader B cannot be on duty on the 3rd, how many different methods are there to arrange the duty schedule? | {
"answer": "42",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
It is known that
$$
\sqrt{9-8 \sin 50^{\circ}}=a+b \sin c^{\circ}
$$
for exactly one set of positive integers \((a, b, c)\), where \(0 < c < 90\). Find the value of \(\frac{b+c}{a}\). | {
"answer": "14",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In an arithmetic sequence \(\{a_{n}\}\), if \(\frac{a_{11}}{a_{10}} < -1\) and the sum of the first \(n\) terms \(S_{n}\) has a maximum value, then the value of \(n\) when \(S_{n}\) attains its smallest positive value is \(\qquad\). | {
"answer": "19",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a three-dimensional Cartesian coordinate system, there is a sphere with its center at the origin and a radius of 3 units. How many lattice points lie on the surface of the sphere? | {
"answer": "30",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a physical education class, 25 students from class 5"B" stood in a line. Each student is either an honor student who always tells the truth or a troublemaker who always lies.
Honor student Vlad took the 13th position. All the students, except Vlad, said: "There are exactly 6 troublemakers between me and Vlad." How many troublemakers are in the line in total? | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the integral
$$
\int_{0}^{0.1} \cos \left(100 x^{2}\right) d x
$$
with an accuracy of $\alpha=0.001$. | {
"answer": "0.090",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the minimum value of the function
$$
f(x)=x^{2}+(x-2)^{2}+(x-4)^{2}+\ldots+(x-104)^{2}
$$
If the result is a non-integer, round it to the nearest integer. | {
"answer": "49608",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given 6000 cards, each with a unique natural number from 1 to 6000 written on it. It is required to choose two cards such that the sum of the numbers on them is divisible by 100. In how many ways can this be done? | {
"answer": "179940",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The numbers $1978^{n}$ and $1978^{m}$ have the same last three digits. Find the positive integers $n$ and $m$ such that $m+n$ is minimized, given that $n > m \geq 1$. | {
"answer": "106",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For non-negative integers \( x \), the function \( f(x) \) is defined as follows:
$$
f(0) = 0, \quad f(x) = f\left(\left\lfloor \frac{x}{10} \right\rfloor\right) + \left\lfloor \log_{10} \left( \frac{10}{x - 10\left\lfloor \frac{x-1}{10} \right\rfloor} \right) \right\rfloor .
$$
For \( 0 \leqslant x \leqslant 2006 \), what is the value of \( x \) that maximizes \( f(x) \)? | {
"answer": "1111",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Petya places "+" and "-" signs in all possible ways into the expression $1 * 2 * 3 * 4 * 5 * 6$ at the positions of the asterisks. For each arrangement of signs, he calculates the resulting value and writes it on the board. Some numbers may appear on the board multiple times. Petya then sums all the numbers on the board. What is the sum that Petya obtains? | {
"answer": "32",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Point \( O \) is located on side \( AC \) of triangle \( ABC \) such that \( CO : CA = 2 : 3 \). When this triangle is rotated by a certain angle around point \( O \), vertex \( B \) moves to vertex \( C \), and vertex \( A \) moves to point \( D \), which lies on side \( AB \). Find the ratio of the areas of triangles \( BOD \) and \( ABC \). | {
"answer": "1/6",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the coordinate plane, a rectangle is drawn with vertices at $(34,0),(41,0),(34,9),(41,9)$. Find the smallest value of the parameter $a$ for which the line $y=ax$ divides this rectangle into two parts such that the area of one part is twice the area of the other. If the answer is not an integer, write it as a decimal fraction. | {
"answer": "0.08",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Masha came up with the number \( A \), and Pasha came up with the number \( B \). It turned out that \( A + B = 2020 \), and the fraction \( \frac{A}{B} \) is less than \( \frac{1}{4} \). What is the maximum value that the fraction \( \frac{A}{B} \) can take? | {
"answer": "403/1617",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \(a, b, c\) be the side lengths of a right triangle, with \(a \leqslant b < c\). Determine the maximum constant \(k\) such that the inequality \(a^{2}(b+c) + b^{2}(c+a) + c^{2}(a+b) \geqslant k a b c\) holds for all right triangles, and specify when equality occurs. | {
"answer": "2 + 3\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are 1000 lamps and 1000 switches, each switch simultaneously controls all lamps whose number is a multiple of the switch's number. Initially, all lamps are on. Now, if the switches numbered 2, 3, and 5 are flipped, how many lamps remain on? | {
"answer": "499",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the total length of the intervals on the number line where the inequalities $x < 1$ and $\sin (\log_{2} x) < 0$ hold. | {
"answer": "\\frac{2^{\\pi}}{1+2^{\\pi}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let's call a natural number "remarkable" if it is the smallest among natural numbers with the same sum of digits. What is the sum of the digits of the two-thousand-and-first remarkable number? | {
"answer": "2001",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a math lesson, each gnome needs to find a three-digit number without zero digits, divisible by 3, such that when 297 is added to it, the resulting number consists of the same digits but in reverse order. What is the minimum number of gnomes that must be in the lesson so that among the numbers they find, there are always at least two identical ones? | {
"answer": "19",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a regular hexagon \( ABCDEF \), the diagonals \( AC \) and \( CE \) are divided by interior points \( M \) and \( N \) in the following ratio: \( AM : AC = CN : CE = r \). If the points \( B, M, N \) are collinear, find the ratio \( r \). | {
"answer": "\\frac{\\sqrt{3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the value of the sum $\left[\log _{2} 1\right]+\left[\log _{2} 2\right]+\left[\log _{2} 3\right]+\cdots+\left[\log _{2} 2002\right]$? | {
"answer": "17984",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two painters are painting a fence that surrounds garden plots. They come every other day and paint one plot (there are 100 plots) in either red or green. The first painter is colorblind and mixes up the colors; he remembers which plots he painted, but cannot distinguish the color painted by the second painter. The first painter aims to maximize the number of places where a green plot borders a red plot. What is the maximum number of such transitions he can achieve (regardless of how the second painter acts)?
Note: The garden plots are arranged in a single line. | {
"answer": "49",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The vertices and midpoints of the sides of a regular decagon (thus a total of 20 points marked) are noted.
How many triangles can be formed with vertices at the marked points? | {
"answer": "1130",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a certain year, a specific date was never a Sunday in any month. Determine this date. | {
"answer": "31",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On an island, there are 1000 villages, each with 99 inhabitants. Each inhabitant is either a knight, who always tells the truth, or a liar, who always lies. It is known that the island has exactly 54,054 knights. One day, each inhabitant was asked the question: "Are there more knights or liars in your village?" It turned out that in each village, 66 people answered that there are more knights in the village, and 33 people answered that there are more liars. How many villages on the island have more knights than liars? | {
"answer": "638",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The set
$$
A=\{\sqrt[n]{n} \mid n \in \mathbf{N} \text{ and } 1 \leq n \leq 2020\}
$$
has the largest element as $\qquad$ . | {
"answer": "\\sqrt[3]{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A target is a triangle divided by three sets of parallel lines into 100 equal equilateral triangles with unit sides. A sniper shoots at the target. He aims at a triangle and hits either it or one of the adjacent triangles sharing a side. He can see the results of his shots and can choose when to stop shooting. What is the maximum number of triangles he can hit exactly five times with certainty? | {
"answer": "25",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many of the integers from \(2^{10}\) to \(2^{18}\) inclusive are divisible by \(2^{9}\)? | {
"answer": "511",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The line \( l: (2m+1)x + (m+1)y - 7m - 4 = 0 \) intersects the circle \( C: (x-1)^{2} + (y-2)^{2} = 25 \) to form the shortest chord length of \(\qquad \). | {
"answer": "4 \\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many integers between 100 and 10,000 contain exactly 3 identical digits in their representation? | {
"answer": "324",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Among 150 schoolchildren, only boys collect stamps. 67 people collect USSR stamps, 48 people collect African stamps, and 32 people collect American stamps. 11 people collect only USSR stamps, 7 people collect only African stamps, 4 people collect only American stamps, and only Ivanov collects stamps from the USSR, Africa, and America. Find the maximum number of girls. | {
"answer": "66",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( [x] \) denote the greatest integer not exceeding \( x \), e.g., \( [\pi]=3 \), \( [5.31]=5 \), and \( [2010]=2010 \). Given \( f(0)=0 \) and \( f(n)=f\left(\left[\frac{n}{2}\right]\right)+n-2\left[\frac{n}{2}\right] \) for any positive integer \( n \). If \( m \) is a positive integer not exceeding 2010, find the greatest possible value of \( f(m) \). | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given \( x_{i} \in \mathbf{R}, x_{i} \geq 0 \) for \( i=1,2,3,4,5 \), and \( \sum_{i=1}^{5} x_{i} = 1 \), find the minimum value of \(\max \left\{ x_{1} + x_{2}, x_{2} + x_{3}, x_{3} + x_{4}, x_{4} + x_{5} \right\} \). | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine the value of the following expression:
$$
\left\lfloor\frac{11}{2010}\right\rfloor+\left\lfloor\frac{11 \times 2}{2010}\right\rfloor+\left\lfloor\frac{11 \times 3}{2010}\right\rfloor+\\left\lfloor\frac{11 \times 4}{2010}\right\rfloor+\cdots+\left\lfloor\frac{11 \times 2009}{2010}\right\rfloor,
$$
where \(\lfloor y\rfloor\) denotes the greatest integer less than or equal to \(y\). | {
"answer": "10045",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Square \(ABCD\) has side length 2, and \(X\) is a point outside the square such that \(AX = XB = \sqrt{2}\). What is the length of the longest diagonal of pentagon \(AXB\)?
| {
"answer": "\\sqrt{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The brakes of a car allow it to stay stationary on an inclined asphalt surface with a base angle not exceeding $30^{\circ}$. Determine the minimum braking distance of this car when traveling at a speed of $30 \, \text{m/s}$ on a flat horizontal road with the same surface. The acceleration due to gravity is $g=10 \, \text{m/s}^2$, $\cos 30^{\circ} \approx 0.866$, and $\sin 30^{\circ} = 0.5$. (15 points) | {
"answer": "78",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
27 identical dice were glued together to form a $3 \times 3 \times 3$ cube in such a way that any two adjacent small dice have the same number of dots on the touching faces. How many dots are there on the surface of the large cube? | {
"answer": "189",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the smallest positive integer $n$ that satisfies the following conditions: For $n$, there exists a positive integer $k$ such that $\frac{8}{15} < \frac{n}{n+k} < \frac{7}{13}$. | {
"answer": "15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two circles are externally tangent to each other at point \( A \), and both are tangent to a third circle at points \( B \) and \( C \). The extension of chord \( AB \) of the first circle intersects the second circle at point \( D \), and the extension of chord \( AC \) intersects the first circle at point \( E \). The extensions of chords \( BE \) and \( CD \) intersect the third circle at points \( F \) and \( G \) respectively. Find \( BG \) if \( BC = 5 \) and \( BF = 12 \). | {
"answer": "13",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A number of trucks with the same capacity were requested to transport cargo from one place to another. Due to road issues, each truck had to carry 0.5 tons less than planned, which required 4 additional trucks. The mass of the transported cargo was at least 55 tons but did not exceed 64 tons. How many tons of cargo were transported on each truck? | {
"answer": "2.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Into each row of a \( 9 \times 9 \) grid, Nigel writes the digits \( 1, 2, 3, 4, 5, 6, 7, 8, 9 \) in order, starting at one of the digits and returning to 1 after 9: for example, one row might contain \( 7, 8, 9, 1, 2, 3, 4, 5, 6 \). The grid is gorgeous if each nine-digit number read along a row or column or along the diagonal from the top-left corner to the bottom-right corner or the diagonal from the bottom-left corner to the top-right corner is divisible by 9. How many of the \( 9^{9} \) possible grids are gorgeous? | {
"answer": "9^8",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On the hypotenuse \( AB \) of a right triangle \( ABC \), square \( ABDE \) is constructed externally with \( AC=2 \) and \( BC=5 \). In what ratio does the angle bisector of angle \( C \) divide side \( DE \)? | {
"answer": "2 : 5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Three friends are driving cars on a road in the same direction. At a certain moment, they are positioned relative to each other as follows: Andrews is at a certain distance behind Brooks, and Carter is at a distance twice the distance from Andrews to Brooks, ahead of Brooks. Each driver is traveling at a constant speed, and Andrews catches up with Brooks in 7 minutes, and then after 5 more minutes catches up with Carter.
How many minutes after Andrews will Brooks catch up with Carter? | {
"answer": "6.666666666666667",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What fraction of the volume of a parallelepiped is the volume of a tetrahedron whose vertices are the centroids of the tetrahedra cut off by the planes of a tetrahedron inscribed in the parallelepiped? | {
"answer": "1/24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that points $\mathbf{A}$ and $\mathbf{B}$ lie on the curves $C_{1}: x^{2} - y + 1 = 0$ and $C_{2}: y^{2} - x + 1 = 0$ respectively, determine the minimum value of $|AB|$. | {
"answer": "\\frac{3 \\sqrt{2}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Mice built an underground house consisting of chambers and tunnels:
- Each tunnel leads from one chamber to another (i.e., none are dead ends).
- From each chamber, exactly three tunnels lead to three different chambers.
- From each chamber, it is possible to reach any other chamber through tunnels.
- There is exactly one tunnel such that, if it is filled in, the house will be divided into two separate parts.
What is the minimum number of chambers the mice's house could have? Draw a possible configuration of how the chambers could be connected. | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Every second, the computer displays a number equal to the sum of the digits of the previous number multiplied by 31. On the first second, the number 2020 was displayed. What number will be displayed on the screen on the 2020th second? | {
"answer": "310",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the center of a circular field stands a geologists' house. From it, 8 straight roads extend, dividing the field into 8 equal sectors. Two geologists embark on a journey from their house at a speed of 5 km/h, each choosing a road at random. Determine the probability that the distance between them will be more than 8 km after one hour. | {
"answer": "0.375",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a triangular pyramid \( S-ABC \) with vertex \( S \). The projection of \( S \) onto the base \( \triangle ABC \) is the orthocenter \( H \) of \( \triangle ABC \). Additionally, \( BC = 2 \), \( SB = SC \), and the dihedral angle between the face \( SBC \) and the base is \( 60^\circ \). Determine the volume of the pyramid. | {
"answer": "\\frac{\\sqrt{3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The regular hexagon \(ABCDEF\) has diagonals \(AC\) and \(CE\). The internal points \(M\) and \(N\) divide these diagonals such that \(AM: AC = CN: CE = r\). Determine \(r\) if it is known that points \(B\), \(M\), and \(N\) are collinear. | {
"answer": "\\frac{1}{\\sqrt{3}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given four points \(O, A, B, C\) on a plane, such that \(OA = 4\), \(OB = 3\), \(OC = 2\), and \(\overrightarrow{OB} \cdot \overrightarrow{OC} = 3\), find the maximum value of the area \(S_{\triangle ABC}\). | {
"answer": "2\\sqrt{7} + \\frac{3\\sqrt{3}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the number of ways to pave a $1 \times 10$ block with tiles of sizes $1 \times 1, 1 \times 2$ and $1 \times 4$, assuming tiles of the same size are indistinguishable. It is not necessary to use all the three kinds of tiles. | {
"answer": "169",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In quadrilateral \(ABCD\), \(\angle DAC = 98^\circ\), \(\angle DBC = 82^\circ\), \(\angle BCD = 70^\circ\), and \(BC = AD\). Find \(\angle ACD\). | {
"answer": "28",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Inside the cube \(ABCD A_{1}B_{1}C_{1}D_{1}\) there is a sphere with center \(O\) and radius 10. The sphere intersects the face \(AA_{1}D_{1}D\) in a circle of radius 1, the face \(A_{1}B_{1}C_{1}D_{1}\) in a circle of radius 1, and the face \(CD D_{1}C_{1}\) in a circle of radius 3. Find the length of the segment \(OD_{1}\). | {
"answer": "17",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the definite integral:
$$
\int_{0}^{\frac{\pi}{2}} \frac{\sin x \, dx}{(1+\sin x)^{2}}
$$ | {
"answer": "\\frac{1}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle \( ABC \), let \( E \) be the point where the side \( AC \) is divided into quarters closest to \( C \), and let \( F \) be the midpoint of side \( BC \). The line passing through points \( E \) and \( F \) intersects line \( AB \) at point \( D \). What percentage of the area of triangle \( ABC \) is the area of triangle \( ADE \)? | {
"answer": "112.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the sets \( M=\{x, xy, \lg(xy)\} \) and \( N=\{0, |x|, y\} \), and that \( M=N \), find the value of \( \left(x+\frac{1}{y}\right)+\left(x^{2}+\frac{1}{y^{2}}\right)+\left(x^{3}+\frac{1}{y^{3}}\right)+\cdots+\left(x^{2001}+\frac{1}{y^{2001}}\right) \). | {
"answer": "-2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $L$ be the intersection point of the diagonals $CE$ and $DF$ of a regular hexagon $ABCDEF$ with side length 2. Point $K$ is defined such that $\overrightarrow{LK} = \overrightarrow{AC} - 3 \overrightarrow{BC}$. Determine whether point $K$ lies inside, on the boundary, or outside of $ABCDEF$, and find the length of the segment $KB$. | {
"answer": "\\frac{2\\sqrt{3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the minimum value for \(a, b > 0\) of the expression
$$
\frac{|2a - b + 2a(b - a)| + |b + 2a - a(b + 4a)|}{\sqrt{4a^2 + b^2}}
$$ | {
"answer": "\\frac{\\sqrt{5}}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For a natural number \( N \), if at least eight out of the nine natural numbers from 1 to 9 can divide \( N \), then \( N \) is called a "Ba Xian number". What is the smallest "Ba Xian number" greater than 2000? | {
"answer": "2016",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are exactly 120 ways to color five cells in a \( 5 \times 5 \) grid such that each row and each column contains exactly one colored cell.
There are exactly 96 ways to color five cells in a \( 5 \times 5 \) grid without the corner cell such that each row and each column contains exactly one colored cell.
How many ways are there to color five cells in a \( 5 \times 5 \) grid without two corner cells such that each row and each column contains exactly one colored cell? | {
"answer": "78",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the volumes of the solids formed by rotating the regions bounded by the graphs of the functions around the y-axis.
$$
y = \arcsin x, \quad y = \arccos x, \quad y = 0
$$ | {
"answer": "\\frac{\\pi}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
An archipelago consists of $N \geqslant 7$ islands. Any two islands are connected by at most one bridge. It is known that no more than 5 bridges lead from each island, and among any 7 islands, there are always at least two connected by a bridge. What is the maximum possible value of $N$? | {
"answer": "36",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two right triangles \( \triangle AXY \) and \( \triangle BXY \) have a common hypotenuse \( XY \) and side lengths (in units) \( AX=5 \), \( AY=10 \), and \( BY=2 \). Sides \( AY \) and \( BX \) intersect at \( P \). Determine the area (in square units) of \( \triangle PXY \). | {
"answer": "25/3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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