problem
stringlengths 18
4.46k
| answer
stringlengths 1
942
| pass_at_n
float64 0.08
0.92
|
|---|---|---|
In triangle \(ABC\), \(AB = 32\), \(AC = 35\), and \(BC = x\). What is the smallest positive integer \(x\) such that \(1 + \cos^2 A\), \(\cos^2 B\), and \(\cos^2 C\) form the sides of a non-degenerate triangle? | 48 | 0.25 |
Given a cube $ABCD-A_{1}B_{1}C_{1}D_{1}$ with edge length 1, find the length of the path traced by a moving point $P$ on the surface of the cube such that the volume of the tetrahedron $P-BDD_{1}B_{1}$ is $\frac{1}{3}$. | 2 | 0.25 |
Given that the sequences $\left\{a_{n}\right\}$, $\left\{b_{n}\right\}$, and $\left\{c_{n}\right\}$ are all arithmetic sequences, and
$$
a_{1}+b_{1}+c_{1}=0, \quad a_{2}+b_{2}+c_{2}=1,
$$
find $a_{2015}+b_{2015}+c_{2015}$. | 2014 | 0.875 |
Find the maximum value of \(3 \sin \left(x+\frac{\pi}{9}\right)+5 \sin \left(x+\frac{4 \pi}{9}\right)\), where \(x\) ranges over all real numbers. | 7 | 0.875 |
Given a positive integer $n \geqslant 2$, find the maximum value of the constant $C(n)$ such that for all real numbers $x_{1}, x_{2}, \cdots, x_{n}$ satisfying $x_{i} \in (0, 1)$ for $i = 1, 2, \cdots, n$ and
$$
\left(1 - x_{i}\right)\left(1 - x_{j}\right) \geqslant \frac{1}{4} \quad \text{for} \quad 1 \leqslant i < j \leqslant n,
$$
the following inequality holds
$$
\sum_{i=1}^{n} x_{i} \geqslant C(n) \sum_{1 \leqslant i < j \leqslant n}\left(2x_{i}x_{j} + \sqrt{x_{i}x_{j}}\right).
$$ | \frac{1}{n-1} | 0.75 |
For what values of the parameter \( a \) will the minimum value of the function
$$
f(x)=|7x - 3a + 8| + |5x + 4a - 6| + |x - a - 8| - 24
$$
be the smallest? | a = \frac{82}{43} | 0.375 |
Let \( p \) and \( q \) represent two consecutive prime numbers. For some fixed integer \( n \), the set \( \{ n - 1, 3n - 19, 38 - 5n, 7n - 45 \} \) represents \( \{ p, 2p, q, 2q \} \), but not necessarily in that order. Find the value of \( n \). | 7 | 0.75 |
Given the functions \( f(x) = x^2 + 4x + 3 \) and \( g(x) = x^2 + 2x - 1 \), find all integer solutions to the equation \( f(g(f(x))) = g(f(g(x))) \). | x = -2 | 0.625 |
The first term of the geometric sequence \( \{a_n\} \) is \( a_1 = 1536 \), and the common ratio is \( q = -\frac{1}{2} \). Let \( f(n) \) denote the product of the first \( n \) terms. For which \( n \) is \( f(n) \) maximized? | n = 12 | 0.875 |
On a board, the numbers $1, 2, 3, \ldots, 235$ were written. Petya erased several of them. It turned out that among the remaining numbers, no number is divisible by the difference of any two others. What is the maximum number of numbers that could remain on the board? | 118 | 0.375 |
In the number system with base $A+1$,
$$
(A A A A)^{2}=A A A C B B B C,
$$
where different letters represent different digits. Determine the values of $A$, $B$, and $C$. | A=2, B=0, C=1 | 0.875 |
Solve the equation \(2021 \cdot \sqrt[202]{x^{2020}} - 1 = 2020 x\) for \(x \geq 0\). | x = 1 | 0.5 |
A pack of \( n \) cards, including three aces, is well shuffled. Cards are turned over in turn. Show that the expected number of cards that must be turned over to reach the second ace is \( \frac{n+1}{2} \). | \frac{n+1}{2} | 0.25 |
We know that the number of factors of 2013, 2014, and 2015 are the same. What is the smallest value of \( n \) among three consecutive natural numbers \( n, n+1, \) and \( n+2 \) that have the same number of factors? | 33 | 0.25 |
Denote by \(\langle x\rangle\) the fractional part of the real number \(x\) (for instance, \(\langle 3.2\rangle = 0.2\)). A positive integer \(N\) is selected randomly from the set \(\{1, 2, 3, \ldots, M\}\), with each integer having the same probability of being picked, and \(\left\langle\frac{87}{303} N\right\rangle\) is calculated. This procedure is repeated \(M\) times and the average value \(A(M)\) is obtained. What is \(\lim_{M \rightarrow \infty} A(M)\)? | \frac{50}{101} | 0.25 |
Four small animals change seats. Initially, the mouse sits in seat 1, the monkey in seat 2, the rabbit in seat 3, and the cat in seat 4. They continuously exchange seats. The first time, the top and bottom rows swap. The second time, the left and right columns swap after the first exchange. The third time, the top and bottom rows swap again. The fourth time, the left and right columns swap again, and so on. After the tenth exchange, in which seat is the rabbit sitting? | 2 | 0.625 |
Let \(\alpha, \beta \in \left(0, \frac{\pi}{2}\right)\), and satisfy the equation \(\sin^2 \alpha + \sin^2 \beta - \frac{\sqrt{6}}{2} \sin \alpha - \frac{\sqrt{10}}{2} \sin \beta + 1 = 0\). Find \(\alpha + \beta\). | \frac{\pi}{2} | 0.875 |
In an aquarium, there are three types of fish: gold, silver, and red fish. If a cat eats all the gold fish, the number of fish becomes 1 less than $\frac{2}{3}$ of the original number. If a cat eats all the red fish, the number of fish becomes 4 more than $\frac{2}{3}$ of the original number. Which type of fish—gold or silver—are there more of, and by how many? | 2 | 0.5 |
Adva van egy egyenlő oldalú háromszög egyik oldala \(a\). Szerkesszünk e háromszög fölé egy merőleges hasábot és egy merőleges gúlát. Mekkora e testek köbtartalma és felülete, ha mindegyiknek a magassága \(m\)? Határozzuk meg \(m\)-et úgy, hogy a hasáb oldalfelülete egyenlő legyen a gúla oldalfelületével. | m = \frac{a}{6} | 0.875 |
Find the value of \(\tan \left(\tan^{-1} \frac{1}{2} + \tan^{-1} \frac{1}{2 \times 2^2} + \tan^{-1} \frac{1}{2 \times 3^2} + \cdots + \tan^{-1} \frac{1}{2 \times 2009^2}\right)\). | \frac{2009}{2010} | 0.875 |
How many values of the parameter \( a \) exist such that the equation
$$
4a^{2} + 3x \log x + 3 \log^{2} x = 13a \log x + ax
$$
has a unique solution? | 2 | 0.75 |
If the acute angle $\theta$ satisfies $\sin (\pi \cos \theta) = \cos (\pi \sin \theta)$, then what is $\sin 2\theta$? | \frac{3}{4} | 0.875 |
Vasya cut a triangle out of cardboard and numbered its vertices with the numbers 1, 2, and 3. It turns out that if Vasya rotates his triangle 15 times clockwise around the vertex numbered 1 by an angle equal to the angle at this vertex, the triangle returns to its original position. If Vasya rotates his triangle 6 times clockwise around the vertex numbered 2 by an angle equal to the angle at this vertex, the triangle returns to its original position. Vasya claims that if he rotates his triangle \( n \) times around the vertex numbered 3 by an angle equal to the angle at this vertex, the triangle will return to its original position. What is the minimal \( n \) that Vasya could name so that his statement is true for at least some cardboard triangle?
| 5 | 0.875 |
Given that point \( P \) lies on the hyperbola \( \frac{x^2}{16} - \frac{y^2}{9} = 1 \), and the distance from \( P \) to the right directrix of this hyperbola is the arithmetic mean of the distances from \( P \) to the two foci of the hyperbola, find the x-coordinate of \( P \). | -\frac{64}{5} | 0.25 |
Consider a rectangle \( ABCD \) where the side lengths are \( \overline{AB}=4 \) and \( \overline{BC}=8 \). Points \( M \) and \( N \) are fixed on sides \( BC \) and \( AD \), respectively, such that the quadrilateral \( BMDN \) is a rhombus. Calculate the area of this rhombus. | 20 | 0.625 |
In the diagram below, \(ABCD\) is a square. The points \(A, B\), and \(G\) are collinear. The line segments \(AC\) and \(DG\) intersect at \(E\), and the line segments \(DG\) and \(BC\) intersect at \(F\). Suppose that \(DE = 15 \text{ cm}\), \(EF = 9 \text{ cm}\), and \(FG = x \text{ cm}\). Find the value of \(x\). | 16 | 0.5 |
For which natural numbers \( n \) is the integer \( n^2 + n + 1 \) a perfect square? | 0 | 0.5 |
Given the following numbers: 20172017, 20172018, 20172019, 20172020, and 20172021. Is there a number among them that is relatively prime to all the others? If so, which one? | 20172019 | 0.75 |
A rectangle can be divided into \( n \) equal squares. The same rectangle can also be divided into \( n + 76 \) equal squares. Find all possible values of \( n \). | 324 | 0.875 |
Find all positive integers \( n \) and \( m \) that satisfy the equation
\[
n^{5} + n^{4} = 7^{m} - 1.
\] | (2, 2) | 0.375 |
Find the sum of all four-digit numbers, in which only the digits $1, 2, 3, 4, 5$ are used, and each digit occurs no more than once. | 399960 | 0.5 |
Vasya has \( n \) candies of several types, where \( n \geq 145 \). It is known that if any group of at least 145 candies is chosen from these \( n \) candies (in particular, one can choose the group of all \( n \) candies), there exists a type of candy such that the chosen group contains exactly 10 candies of that type. Find the largest possible value of \( n \). | 160 | 0.125 |
The teacher wrote the following numbers on the board:
$$
1, 4, 7, 10, 13, 16, 19, 22, 25, 28, 31, 34, 37, 40, 43
$$
Two neighboring numbers always differ by the same value, in this case by 3. Then, she erased all the numbers except for 1, 19, and 43. Next, she added several integers between these three numbers such that each pair of neighboring numbers again differ by the same value and no number is written more than once.
In how many ways could the teacher have added the numbers?
(Note: Can the number 5 be included among the added numbers?) | 4 | 0.625 |
There is one three-digit number and two two-digit numbers written on the board. The sum of the numbers containing the digit seven is 208. The sum of the numbers containing the digit three is 76. Find the sum of all three numbers. | 247 | 0.5 |
In triangle \(ABC\), angle \(A\) is \(40^\circ\). The triangle is randomly thrown onto a table. Find the probability that vertex \(A\) ends up east of the other two vertices. | \frac{7}{18} | 0.375 |
In how many ways can an \( n \times n \) grid be filled with zeros and ones such that each row and each column contains an even number of ones? Each cell of the grid must contain either a zero or a one. | 2^{(n-1)^2} | 0.875 |
Arrange the elements of the set \(\left\{2^{x}+2^{y} \mid x, y \in \mathbf{N}, x < y\right\}\) in ascending order. What is the 60th number? (Answer with a number). | 2064 | 0.125 |
Seryozha and Misha, while walking in the park, stumbled upon a meadow surrounded by linden trees. Seryozha walked around the meadow, counting the trees. Misha did the same, but started at a different tree (although he walked in the same direction). The tree that was the 20th for Seryozha was the 7th for Misha, and the tree that was the 7th for Seryozha was the 94th for Misha. How many trees were growing around the meadow? | 100 | 0.625 |
Find all pairs of integers $(m, n)$ such that
$$
m^{3}-n^{3}=2 m n+8
$$ | (0, -2), (2, 0) | 0.5 |
Let $a=\sqrt{4+\sqrt{5-a}}, b=\sqrt{4+\sqrt{5+b}}, c=\sqrt{4-\sqrt{5-c}},$ and $d=\sqrt{4-\sqrt{5+d}}.$
Calculate $a b c d$. | 11 | 0.5 |
Calculate the contour integrals:
1) \(\oint_{-i} 2 x \, dx - (x + 2 y) \, dy\)
2) \(\oint_{+i} y \cos x \, dx + \sin x \, dy\)
along the perimeter of the triangle with vertices \(A(-1, 0)\), \(B(0, 2)\), and \(C(2, 0)\). | 0 | 0.625 |
A non-empty finite set is called a trivial set if the sum of the squares of all its elements is an odd number. Given the set \( A = \{1, 2, 3, \ldots, 2016, 2017\} \), find the number of trivial subsets of \( A \) that are proper subsets (i.e., not equal to \( A \) itself). (You may express the answer using exponents.) | 2^{2016} - 1 | 0.75 |
Maria travels to school by a combination of walking and skateboarding. She can get there in 38 minutes if she walks for 25 minutes and skateboards for 13 minutes, or in 31 minutes if she walks for 11 minutes and skateboards for 20 minutes. How long (in minutes) would it take her to walk to school? | 51 | 0.875 |
The function \( y = f(t) \) is such that the sum of the roots of the equation \( f(\sin x) = 0 \) in the interval \([3 \pi / 2, 2 \pi]\) is \( 33 \pi \), and the sum of the roots of the equation \( f(\cos x) = 0 \) in the interval \([\pi, 3 \pi / 2]\) is \( 23 \pi \). What is the sum of the roots of the second equation in the interval \([\pi / 2, \pi]?\) | 17 \pi | 0.375 |
Let $[x]$ denote the greatest integer not exceeding the real number $x$. If
\[ A = \left[\frac{7}{8}\right] + \left[\frac{7^2}{8}\right] + \cdots + \left[\frac{7^{2019}}{8}\right] + \left[\frac{7^{2020}}{8}\right], \]
what is the remainder when $A$ is divided by 50? | 40 | 0.375 |
How many roots does \( \arctan x = x^{2} - 1.6 \) have, where the arctan function is defined in the range \( -\frac{\pi}{2} < \arctan x < \frac{\pi}{2} \)? | 2 | 0.625 |
Six pirates - a captain and five crew members - are sitting around a campfire facing the center. They need to divide a treasure of 180 gold coins. The captain proposes a way to divide the coins (i.e., how many coins each pirate should receive: each pirate must receive a whole non-negative number of coins; different pirates may receive different amounts of coins). After this, the other five pirates vote on the captain's proposal. A pirate will vote "yes" only if he receives more coins than each of his two neighbors. The proposal is accepted if at least three out of the five crew members vote "yes".
What is the maximum number of coins the captain can receive under these rules? | 59 | 0.375 |
A set of five volumes of an encyclopedia is arranged in ascending order on a shelf, i.e., from left to right, volumes 1 through 5 are lined up. We want to rearrange them in descending order, i.e., from left to right, volumes 5 through 1, but each time we are only allowed to swap the positions of two adjacent volumes. What is the minimum number of such swaps required to achieve the desired arrangement? | 10 | 0.625 |
In the Cartesian coordinate system $xOy$, the graph of the parabola $y=ax^2 - 3x + 3 \ (a \neq 0)$ is symmetric with the graph of the parabola $y^2 = 2px \ (p > 0)$ with respect to the line $y = x + m$. Find the product of the real numbers $a$, $p$, and $m$. | -3 | 0.875 |
a) \((x+y)\left(x^{2}-xy+y^{2}\right)\)
b) \((x+3)\left(x^{2}-3x+9\right)\)
c) \((x-1)\left(x^{2}+x+1\right)\)
d) \((2x-3)\left(4x^{2}+6x+9\right)\) | 8x^3 - 27 | 0.25 |
Given that $a$ and $b$ are positive integers, and $a - b \sqrt{3} = (2 - \sqrt{3})^{100}$, find the unit digit of $a \cdot b$. | 2 | 0.625 |
Find all functions \( f: \mathbf{Z}^{*} \rightarrow \mathbf{R} \) (where \(\mathbf{Z}^{*}\) is the set of non-negative integers) that satisfy \( f(n+m) + f(n-m) = f(3n) \) for all \( m, n \in \mathbf{Z}^{*} \) with \( n \geqslant m \). | f(n) = 0 | 0.75 |
Given $\triangle ABC$ with $AB = 2$, $AC = 1$, and $\angle BAC = 120^\circ$, $O$ is the circumcenter of $\triangle ABC$. If $\overrightarrow{AO} = \lambda \overrightarrow{AB} + \mu \overrightarrow{AC}$, find $\lambda + \mu$. | \frac{13}{6} | 0.875 |
Solve the following system of equations:
$$
\begin{aligned}
& x + y = a \\
& \frac{1}{x} + \frac{1}{y} = \frac{1}{b}
\end{aligned}
$$
Determine the values of \(a\) and \(b\) for which the roots are real. What relationship must exist between \(a\) and \(b\) for one root to be three times the other? | b = \frac{3a}{16} | 0.5 |
Numbers from 1 to 6 are placed on the faces of a cube. The cube is rolled twice. The first time, the sum of the numbers on the four lateral faces was 12, and the second time it was 15. What number is written on the face opposite the one where the number 3 is written? | 6 | 0.625 |
One-and-a-half-liter mineral water bottles are narrowed in the shape of an "hourglass" to make them easier to grip. The normal circumference of the bottle is $27.5 \mathrm{~cm}$, while at the waist - which is a $1 \mathrm{~cm}$ tall cylindrical section - it is only $21.6 \mathrm{~cm}$. The cylindrical sections with different circumferences are connected above and below the waist by truncated cone sections each $2 \mathrm{~cm}$ high. How much taller are such bottles compared to their regular counterparts with the same volume and normal circumference, but without the narrowing? | 1.18 \text{ cm} | 0.125 |
Find all functions \( f: \mathbb{R} \rightarrow \mathbb{R} \) such that for all real numbers \( x \) and \( y \):
\[
f\left(x^{2} + f(y)\right) = y + f(x)^{2}
\] | f(x) = x | 0.875 |
On a plate, there are different candies of three types: 2 lollipops, 3 chocolate candies, and 5 jelly candies. Sveta ate all of them one by one, choosing each next candy at random. Find the probability that the first and last candies she ate were of the same type. | \frac{14}{45} | 0.75 |
In $\triangle ABC$, $\angle ABC = \angle ACB = 40^\circ$, and $P$ is a point inside the triangle such that $\angle PAC = 20^\circ$ and $\angle PCB = 30^\circ$. Find the measure of $\angle PBC$. | 20^\circ | 0.5 |
Five identical balls are moving in one direction in a straight line at some distance from each other, while five other identical balls are moving towards them. The speeds of all the balls are the same. Upon collision, any two balls bounce off in opposite directions with the same speed they had before the collision. How many total collisions will occur between the balls? | 25 | 0.625 |
A positive unknown number less than 2022 was written on the board next to the number 2022. Then, one of the numbers on the board was replaced by their arithmetic mean. This replacement was done 9 more times, and the arithmetic mean was always an integer. Find the smaller of the numbers that were initially written on the board. | 998 | 0.875 |
Points \( P \) and \( Q \) are located on side \( BC \) of triangle \( ABC \), with \( BP: PQ: QC = 1: 2: 3 \). Point \( R \) divides side \( AC \) of this triangle such that \( AR: RC = 1: 2 \). What is the ratio of the area of quadrilateral \( PQST \) to the area of triangle \( ABC \), if \( S \) and \( T \) are the intersection points of line \( BR \) with lines \( AQ \) and \( AP \), respectively? | \frac{5}{24} | 0.375 |
Solve the following inequalities
1. \(3x + 1 \geq -2\)
2. \(y \geq 1\) and \(-2y \geq -2\)
3. \(y^2(x^2 + 1) - 1 \leq x^2\) | -1 \leq y \leq 1 | 0.375 |
Let the side lengths of a triangle be integers \( l \), \( m \), and \( n \), with \( l > m > n \). It is given that \( \left\{\frac{3^{l}}{10^{4}}\right\} = \left\{\frac{3^{m}}{10^{4}}\right\} = \left\{\frac{3^{n}}{10^{4}}\right\} \), where \(\{x\} = x - [x]\) and \([x]\) denotes the greatest integer less than or equal to \( x \). Find the minimum perimeter of such a triangle. | 3003 | 0.625 |
Let triangle \(ABC\) have \(AB=5\), \(BC=6\), and \(AC=7\), with circumcenter \(O\). Extend ray \(AB\) to point \(D\) such that \(BD=5\), and extend ray \(BC\) to point \(E\) such that \(OD=OE\). Find \(CE\). | \sqrt{59} - 3 | 0.5 |
From the 11 positive integers $1, 2, 3, \cdots, 11$, if you randomly choose 3 different positive integers $a, b, c$, what is the probability that their product $abc$ is divisible by 4? | \frac{20}{33} | 0.75 |
Several people were seated around a round table such that the distances between neighboring people were equal. One of them was given a card with the number 1, and the rest were given cards with numbers 2, 3, and so on, in a clockwise direction.
The person with the card numbered 31 noticed that the distance from him to the person with the card numbered 7 is the same as the distance to the person with the card numbered 14. How many people are seated at the table in total? | 41 | 0.5 |
The base of an isosceles triangle is $4 \sqrt{2} \text{ cm}$, and the median of one of the equal sides is $5 \text{ cm}$. Find the lengths of the equal sides. | 6 \text{ cm} | 0.875 |
Given $M$ as the set of all rational numbers within the open interval $(0,1)$. Does there exist a subset $A$ of $M$ such that every element in $M$ can be uniquely represented as the sum of a finite number of distinct elements from $A$? | \text{No} | 0.625 |
Calculate the limit of the function:
\[
\lim _{x \rightarrow \frac{\pi}{3}} \frac{1-2 \cos x}{\pi-3 x}
\] | -\frac{\sqrt{3}}{3} | 0.875 |
Choose 3 different numbers from 1 to 300 such that their sum is divisible by 3. How many such combinations are there? | 1485100 | 0.375 |
\( P(x) \) is a fourth-degree polynomial with integer coefficients, with a leading coefficient that is positive. It is known that \( P(\sqrt{3}) = P(\sqrt{5}) \). Find the value(s) of \( x \) at which \( P(x) \) takes its minimum value. | \pm 2 | 0.625 |
Randomly select a permutation \(\sigma = \left(x_{1}, x_{2}, \ldots, x_{n}\right)\) of \(1, 2, \ldots, n\). If \(x_{i} = i\), then \(i\) is called a fixed point of \(\sigma\). Let the number of fixed points in the permutation be denoted as the random variable \(X_{n}\). Find the expectation \(E\left(X_{n}\right)\) of \(X_{n}\). | 1 | 0.875 |
In the diagram below, \( \triangle ABC \) is a triangle with \( AB = 39 \text{ cm}, BC = 45 \text{ cm}, \) and \( CA = 42 \text{ cm} \). The tangents at \( A \) and \( B \) to the circumcircle of \( \triangle ABC \) meet at the point \( P \). The point \( D \) lies on \( BC \) such that \( PD \) is parallel to \( AC \). It is given that the area of \( \triangle ABD \) is \( x \text{ cm}^2 \). Find the value of \( x \). | 168 | 0.75 |
An equilateral triangle is subdivided into 4 smaller equilateral triangles. Using red and yellow to paint the vertices of the triangles, each vertex must be colored and only one color can be used per vertex. If two colorings are considered the same when they can be made identical by rotation, how many different colorings are possible? | 24 | 0.75 |
A torpedo boat is anchored 9 km from the nearest point on the shore. A messenger needs to be sent from the boat to a camp located 15 km along the shore from the nearest point of the boat. If the messenger travels on foot at a speed of 5 km/h and rows at a speed of 4 km/h, at which point on the shore should he land to reach the camp in the shortest possible time? | 12 | 0.75 |
Two cars, Car A and Car B, start from point A and point B, respectively, and move towards each other simultaneously. They meet after 3 hours, after which Car A turns back towards point A and Car B continues moving forward. Once Car A reaches point A, it turns towards point B again and meets Car B half an hour later. How many minutes does Car B take to travel from point A to point B? | 432 \text{ minutes} | 0.75 |
Given \(\sin x + \sin y = 0.6\) and \(\cos x + \cos y = 0.8\), find \(\cos x \cdot \cos y\). | -\frac{11}{100} | 0.5 |
Given \( a, b \in \mathbb{Z} \), and \( a + b \) is a root of the equation \( x^{2} + ax + b = 0 \), what is the maximum possible value of \( b \)? | 9 | 0.625 |
Buying an album. Masha was short of 2 kopecks, Kolya was short of 34 kopecks, and Fedya was short of 35 kopecks. When they pooled their money, it was still not enough to buy the album. How much does the album cost? | 35 | 0.875 |
There are two coal mines, Mine A and Mine B. Each gram of coal from Mine A releases 4 calories of heat when burned, and each gram of coal from Mine B releases 6 calories of heat when burned. The price per ton of coal at the production site is 20 yuan for Mine A and 24 yuan for Mine B. It is known that the transportation cost for a ton of coal from Mine A to city $N$ is 8 yuan. What should the transportation cost per ton of coal from Mine B to city $N$ be for it to be more economical than transporting coal from Mine A? | 18 \text{ Yuan} | 0.75 |
Let \([x]\) denote the greatest integer less than or equal to \(x\), e.g. \([\pi] = 3\), \([5.31] = 5\), and \([2009] = 2009\). Evaluate \(\left[\sqrt{2009^2 + 1} + \sqrt{2009^2 + 2} + \cdots + \sqrt{2009^2 + 4018}\right]\). | 8074171 | 0.5 |
Let the function \( f(x) \) be defined on \([0,1]\), satisfying \( f(0) = f(1) \) and for any \( x, y \in [0,1] \) having \( |f(x) - f(y)| < |x - y| \). Find the smallest real number \( m \) such that for any function \( f(x) \) meeting the above conditions and for any \( x, y \in [0,1] \), we have \( |f(x) - f(y)| < m \). | \frac{1}{2} | 0.875 |
What is "Romney"? If
$$
\frac{N}{O}=. \text{Romney Romney Romney} \ldots
$$
is the decimal representation of a certain proper fraction, where each letter represents some decimal digit, find the value of the word Romney (the letters $N$ and $n$ represent the same digit; the same applies to $O$ and $o$). | 571428 | 0.375 |
A semicircle with a radius of 1 is drawn inside a semicircle with a radius of 2. A circle is drawn such that it touches both semicircles and their common diameter. What is the radius of this circle? | \frac{8}{9} | 0.5 |
Given the sequence elements \( a_{n} \) such that \( a_{1}=1337 \) and \( a_{2n+1}=a_{2n}=n-a_{n} \) for all positive integers \( n \). Determine the value of \( a_{2004} \). | 2004 | 0.25 |
Calculate the volumes of the solids formed by the rotation of the figures bounded by the graphs of the functions. Axis of rotation is \(O x\).
$$
y=1-x^{2}, x=0, x=\sqrt{y-2}, x=1
$$ | 5 \pi | 0.75 |
Chester traveled from Hualien to Lukang in Changhua to participate in the Hua Luogeng Gold Cup Math Competition. Before leaving, his father checked the car’s odometer, which displayed a palindromic number of 69,696 kilometers (a palindromic number reads the same forward and backward). After driving for 5 hours, they arrived at the destination with the odometer showing another palindromic number. During the journey, the father's driving speed never exceeded 85 kilometers per hour. What is the maximum possible average speed (in kilometers per hour) that Chester's father could have driven? | 82.2 | 0.25 |
The edges of a rectangular parallelepiped are in the ratio $3: 4: 12$. A diagonal cross-section is made through the largest edge. Find the sine of the angle between the plane of this cross-section and the diagonal of the parallelepiped that does not lie in that plane. | \frac{24}{65} | 0.375 |
In triangle \(ABC\), \(\angle A = 2 \angle C\). Suppose that \(AC = 6\), \(BC = 8\), and \(AB = \sqrt{a} - b\), where \(a\) and \(b\) are positive integers. Compute \(100a + b\). | 7303 | 0.875 |
Compute the area of the figure bounded by one arch of the cycloid given by the parametric equations $x = 2(t - \sin t)$ and $y = 2(1 - \cos t)$. | 12\pi | 0.875 |
Defined on $\mathbf{R}$, the function $f$ satisfies
$$
f(1+x)=f(9-x)=f(9+x).
$$
Given $f(0)=0$, and $f(x)=0$ has $n$ roots in the interval $[-4020, 4020]$, find the minimum value of $n$. | 2010 | 0.25 |
Two spheres with equal mass and charge are initially at a distance of \( l \) from each other when they are released without an initial velocity. After \( t \) seconds, the distance between them doubles. How much time will it take for the distance between the spheres to double if they are initially released from a distance of \( 3l \)? Air resistance can be neglected. | 3 \sqrt{3} t | 0.25 |
Subset \( S \subseteq \{1, 2, 3, \ldots, 1000\} \) is such that if \( m \) and \( n \) are distinct elements of \( S \), then \( m + n \) does not belong to \( S \). What is the largest possible number of elements in \( S \)? | 501 | 0.5 |
Given a regular 2017-sided polygon \(A_{1} A_{2} \cdots A_{2017}\) inscribed in a unit circle \(\odot O\), choose any two different vertices \(A_{i}, A_{j}\). Find the probability that \(\overrightarrow{O A_{i}} \cdot \overrightarrow{O A_{j}} > \frac{1}{2}\). | \frac{1}{3} | 0.625 |
Given the parabola \( y^2 = 4p(x + p) \) (where \( p > 0 \)), two mutually perpendicular chords \( AB \) and \( CD \) pass through the origin \( O \). Find the minimum value of \( |AB| + |CD| \). | 16p | 0.875 |
Let \(\mathbb{Q}^{+}\) denote the set of positive rational numbers. Find, with proof, all functions \(f: \mathbb{Q}^{+} \rightarrow \mathbb{Q}^{+}\) such that, for all positive rational numbers \(x\) and \(y\), we have
\[
f(x) = f(x + y) + f\left(x + x^2 f(y)\right).
\] | f(x) = \frac{1}{x} | 0.875 |
There are $N$ prime numbers written on a board (not necessarily distinct). It turns out that the sum of any three numbers on the board is also a prime number. What is the largest possible value of $N$? | 4 | 0.375 |
The positive integers \( x \) and \( y \) satisfy the equation \( y x^{2} + x y^{2} = 70 \). What is the value of \( x^{4} + y^{4} \)? | 641 | 0.875 |
Let \( n \) be a natural number. For any real numbers \( x, y, z \), the inequality \(\left(x^{2}+y^{2}+z^{2}\right) \leq n\left(x^{4}+y^{4}+z^{4}\right)\) always holds. Find the minimum value of \( n \). | 3 | 0.25 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.