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In how many different ways can a chess king move from square $e1$ to square $h5$, if it is only allowed to move one square to the right, upward, or diagonally right-upward? | 129 | 0.125 |
In the regular tetrahedron \(ABCD\), take the midpoint \(M\) on the altitude \(AH\), and connect \(BM\) and \(CM\). Then \(\angle BMC =\) | 90^\circ | 0.875 |
If \(\sin \theta - \cos \theta = \frac{\sqrt{6} - \sqrt{2}}{2}\), find the value of \(24\left(\sin ^{3} \theta - \cos ^{3} \theta\right)^{2}\). | 12 | 0.625 |
A regular hexagon \(ABCDEF\) has a side length of 2. A laser beam is fired inside the hexagon from point \(A\) and hits line segment \(\overline{BC}\) at point \(G\). The laser then reflects off \(\overline{BC}\) and hits the midpoint of \(\overline{DE}\). Find the length of \(BG\). | \frac{2}{5} | 0.75 |
Numbers 1, 2, 3, ..., 10 are written in a circle in some order. Petya calculated the sums of all triplets of neighboring numbers and wrote the smallest of these sums on the board. What is the largest number that could have been written on the board? | 15 | 0.25 |
Is it possible to place the numbers $1, -1, 0$ in the cells of an $8 \times 8$ square table such that all the sums in each column, each row, and each of the two diagonals are different? | \text{No} | 0.875 |
In the movie "Monkey King: Hero Is Back," there is a scene where Sun Wukong battles mountain demons. Some of the demons are knocked down, such that the number of knocked-down demons is one-third more than the standing demons. After a while, 2 more demons are knocked down, but 10 demons stand back up. At this point, the number of standing demons is one-fourth more than the knocked-down demons. How many demons are standing now? | 35 | 0.25 |
From the digits $1, 2, 3, 4, 5, 6, 7, 8, 9$, nine nine-digit numbers (not necessarily different) are formed with each digit used exactly once in each number. What is the largest number of zeros that the sum of these nine numbers can end with? | 8 | 0.125 |
Determine all prime numbers \( p \) such that \( p^2 - 6 \) and \( p^2 + 6 \) are both prime numbers. | 5 | 0.75 |
Suppose \( x, y \), and \( z \) are real numbers greater than 1 such that
\[
\begin{aligned}
x^{\log _{y} z} & = 2, \\
y^{\log _{y} x} & = 4, \\
z^{\log _{x} y} & = 8 .
\end{aligned}
\]
Compute \(\log _{x} y\). | \sqrt{3} | 0.875 |
Vasya replaced the same digits in two numbers with the same letters, and different digits with different letters. It turned out that the number ZARAZA is divisible by 4, and ALMAZ is divisible by 28. Find the last two digits of the sum ZARAZA + ALMAZ. | 32 | 0.25 |
How many different two-digit numbers can be composed using the digits: a) $1, 2, 3, 4, 5, 6$; b) $0, 1, 2, 3, 4, 5, 6$? | 42 | 0.875 |
How many numbers, divisible by 4 and less than 1000, do not contain any of the digits 6, 7, 8, 9, or 0? | 31 | 0.5 |
In a plane rectangular coordinate system, the coordinates of two vertices of square $OABC$ are $O(0,0)$ and $A(4,3)$, and point $C$ is in the fourth quadrant. Find the coordinates of point $B$. $\qquad$ | (7, -1) | 0.875 |
$O A B C$ is a tetrahedron with $O A, O B$ and $O C$ being mutually perpendicular. Given that $O A = O B = O C = 6x$.
1. If the volume of $O A B C$ is $a x^{3}$, find $a$.
2. If the area of $\triangle A B C$ is $b \sqrt{3} x^{2}$, find $b$.
3. If the distance from $O$ to $\triangle A B C$ is $c \sqrt{3} x$, find $c$.
4. If $\theta$ is the angle of depression from $C$ to the midpoint of $A B$ and $\sin \theta = \frac{\sqrt{d}}{3}$, find $d$. | 6 | 0.75 |
Calculate the limit of the function:
\[
\lim _{x \rightarrow \frac{\pi}{2}}(\cos x+1)^{\sin x}
\] | 1 | 0.875 |
From a natural number, subtract the sum of its digits, then subtract the sum of the digits of the resulting difference. If you continue this process, with which number will the calculations end? | 0 | 0.625 |
Call a positive integer \( N \geq 2 \) "special" if for every \( k \) such that \( 2 \leq k \leq N \), \( N \) can be expressed as a sum of \( k \) positive integers that are relatively prime to \( N \) (although not necessarily relatively prime to each other). How many special integers are there less than 100? | 50 | 0.125 |
In an archery competition held in a certain city, after ranking the scores, the average score of the top seven participants is 3 points lower than the average score of the top four participants. The average score of the top ten participants is 4 points lower than the average score of the top seven participants. How many points more is the sum of the scores of the fifth, sixth, and seventh participants compared to the sum of the scores of the eighth, ninth, and tenth participants? | 28 | 0.75 |
Find the point \( M^{\prime} \) that is symmetric to the point \( M \) with respect to the line.
$$
\begin{aligned}
& M(0, -3, -2) \\
& \frac{x-1}{1} = \frac{y+1.5}{-1} = \frac{z}{1}
\end{aligned}
$$ | M' = (1, 1, 1) | 0.875 |
In the parallelogram \(ABCD\), points \(E\) and \(F\) are located on sides \(AB\) and \(BC\) respectively, and \(M\) is the point of intersection of lines \(AF\) and \(DE\). Given that \(AE = 2BE\) and \(BF = 3CF\), find the ratio \(AM : MF\). | 4:5 | 0.25 |
Five integers were written on the board. By summing them in pairs, the following ten numbers were obtained:
$$
0, 2, 4, 4, 6, 8, 9, 11, 13, 15
$$
What are the five numbers that were written on the board? Is it possible to obtain the following ten numbers by summing five integers in this way:
$$
12, 13, 14, 15, 16, 16, 17, 17, 18, 20?
$$ | \text{No} | 0.125 |
Find the maximum value of the expression \( (\sin 2x + \sin y + \sin 3z)(\cos 2x + \cos y + \cos 3z) \). | \frac{9}{2} | 0.875 |
There is a magical grid paper of dimensions \(2000 \times 70\), initially all cells are grey. A painter starts at a certain cell and paints it red. Every second, the painter takes two steps: one cell to the left and one cell downwards, and paints the cell he lands on red. If the painter stands in the leftmost column and needs to step left, he teleports to the rightmost cell of the same row; if he stands in the bottom row and needs to step downwards, he teleports to the top cell of the same column. After several moves, the painter returns to the cell he started from. How many red cells are there on the grid at that moment? | 14000 | 0.625 |
Two sisters were picking currants: the older one was using a 4-liter bucket, and the younger one was using a 3.5-liter bucket. The older sister was always working faster than the younger one. When the older sister had collected three-quarters of her bucket, and the younger sister had collected more than half of hers, the girls exchanged buckets. Continuing to work at the same speeds as before exchanging buckets, they finished at the same time. How many times faster was the older sister picking berries compared to the younger sister? | 1.5 | 0.75 |
A line parallel to the bases of a trapezoid passes through the intersection point of its diagonals. Find the length of the segment of this line that is enclosed between the non-parallel sides of the trapezoid, given that the lengths of the bases of the trapezoid are 4 cm and 12 cm. | 6 \text{ cm} | 0.625 |
Kelvin the Frog is going to roll three fair ten-sided dice with faces labelled $0,1,2, \ldots, 9$. First he rolls two dice, and finds the sum of the two rolls. Then he rolls the third die. What is the probability that the sum of the first two rolls equals the third roll? | \frac{11}{200} | 0.5 |
In Mexico, ecologists have achieved the passage of a law whereby each car must not be driven at least one day a week (the owner informs the police of the car’s number and "day off" for that car). In a certain family, all adults wish to drive daily (each for their own errands!). How many cars (at minimum) does the family need if there are
a) 5 adults? b) 8 adults? | 10 | 0.625 |
The city's bus network is organized as follows:
1) From each stop, it is possible to reach any other stop without transfer.
2) For each pair of routes, there is exactly one unique stop where passengers can transfer from one route to the other.
3) Each route has exactly three stops.
How many bus routes are in the city? (It is known that there is more than one route.) | 7 | 0.375 |
In a box, there are red and black socks. If two socks are randomly taken from the box, the probability that both of them are red is $1/2$.
a) What is the minimum number of socks that can be in the box?
b) What is the minimum number of socks that can be in the box, given that the number of black socks is even? | 21 | 0.625 |
Solve the inequality \(\sqrt{x-4}+\sqrt{x+1}+\sqrt{2x}-\sqrt{33-x} > 4\). Provide the sum of all its integer solutions in the answer. | 525 | 0.75 |
Let the following system of equations hold for positive numbers \(x, y, z\):
\[ \left\{\begin{array}{l}
x^{2}+x y+y^{2}=48 \\
y^{2}+y z+z^{2}=25 \\
z^{2}+x z+x^{2}=73
\end{array}\right. \]
Find the value of the expression \(x y + y z + x z\). | 40 | 0.375 |
There is an expression written on the board, $\frac{a}{b} \cdot \frac{c}{d} \cdot \frac{e}{f}$, where $a, b, c, d, e, f$ are natural numbers. If the number $a$ is increased by 1, the value of this expression increases by 3. If the number $c$ in the original expression is increased by 1, its value increases by 4. If the number $e$ in the original expression is increased by 1, its value increases by 5. What is the smallest value that the product $b d f$ can take? | 60 | 0.875 |
The settlements Arkadino, Borisovo, and Vadimovo are connected pairwise by straight roads. Adjacent to the road between Arkadino and Borisovo is a square field, one side of which coincides entirely with this road. Adjacent to the road between Borisovo and Vadimovo is a rectangular field, one side of which coincides entirely with this road, and the other side is 4 times longer. Adjacent to the road between Arkadino and Vadimovo is a rectangular forest, one side of which coincides entirely with this road, and the other side is 12 km long. The area of the forest is 45 square kilometers larger than the sum of the areas of the fields. Find the total area of the forest and fields in square kilometers. | 135 | 0.5 |
Two numbers \(x\) and \(y\) are randomly chosen from the interval \([0, 2]\). Find the probability that these numbers satisfy the inequalities \(x^{2} \leq 4y \leq 4x\). | \frac{1}{3} | 0.875 |
Inside the cube \( ABCD A_1 B_1 C_1 D_1 \) is located the center \( O \) of a sphere with a radius of 10. The sphere intersects the face \( A A_1 D_1 D \) in a circle with a radius of 1, the face \( A_1 B_1 C_1 D_1 \) in a circle with a radius of 1, and the face \( C D D_1 C_1 \) in a circle with a radius of 3. Find the length of the segment \( O D_1 \). | 17 | 0.625 |
Find the probability that a randomly selected 8-digit number composed of 0s and 1s has the sum of the digits in even positions equal to the sum of the digits in odd positions. | \frac{35}{128} | 0.875 |
In the sequence $\left\{a_{n}\right\}$, $a_{1}=2$, $a_{2}=-19$, and $a_{n+2}=\left|a_{n+1}\right|-a_{n}$ for all positive integers $n$. Find $a_{2019}$. | 17 | 0.875 |
A regular triangle \(ABC\) is inscribed in a circle. Another, smaller circle is inscribed in the sector bounded by chord \(BC\), and it touches the larger circle at point \(M\) and the chord \(BC\) at point \(K\). Ray \(MK\) intersects the larger circle a second time at point \(N\). Find the length of \(MN\) if the sum of the distances from point \(M\) to the ends of chord \(BC\) is 6. | 6 | 0.25 |
As shown in Figure 18-3, in a regular hexagon consisting of six regions, ornamental plants are to be planted with the requirement that the same type of plant is planted within each region and different types of plants are planted in adjacent regions. Given that there are 4 different types of plants available, how many different planting schemes are possible? | 732 | 0.625 |
A natural number was squared and 600 was subtracted from the result. The same operation was then performed on the resulting number, and so on. What could the original number be, if it is known that after several such operations, the same number was obtained again? | 25 | 0.875 |
In the convex pentagon \(ABCDE\), the angles \(\angle ABC\) and \(\angle CDE\) are \(90^\circ\), \(BC = CD = AE = 1\), and \(AB + ED = 1\). What is the area of the pentagon? | 1 | 0.875 |
What is the last digit of the number: a) $2^{1000}$; b) $3^{1000}$; c) $7^{1000}$? | 1 | 0.25 |
In a certain game, the "magician" asks a person to randomly think of a three-digit number ($abc$), where $a, b, c$ are the digits of this number. They then ask this person to arrange the digits into 5 different numbers: $(acb)$, $(bac)$, $(bca)$, $(cab)$, and $(cba)$. The magician then asks for the sum of these 5 numbers, denoted as $N$. Once the magician knows $N$, they can determine the original number ($abc$).
Given that $N=3194$, assume the role of the "magician" and determine the original number ($abc$). | 358 | 0.25 |
A plane intersects each side of the quadrilateral \(ABCD\) at an internal point. Following a traversing direction, we write down the ratio in which each intersection point divides its corresponding side. What will be the product of the four ratios? | 1 | 0.625 |
In a geometric sequence where the common ratio is greater than 1, what is the maximum number of terms that are integers between 100 and 1000?
| 6 | 0.375 |
The numbers \(a, b, c, d\) belong to the interval \([-11.5, 11.5]\). Find the maximum value of the expression \(a + 2b + c + 2d - ab - bc - cd - da\). | 552 | 0.5 |
Let \( A B C D E F \) be a regular hexagon of area 1. Let \( M \) be the midpoint of \( D E \). Let \( X \) be the intersection of \( A C \) and \( B M \), let \( Y \) be the intersection of \( B F \) and \( A M \), and let \( Z \) be the intersection of \( A C \) and \( B F \). If \([P]\) denotes the area of polygon \( P \) for any polygon \( P \) in the plane, evaluate \([B X C] + [A Y F] + [A B Z] - [M X Z Y]\). | 0 | 0.375 |
Let \( f(x) = \frac{x + a}{x^2 + \frac{1}{2}} \), where \( x \) is a real number and the maximum value of \( f(x) \) is \( \frac{1}{2} \) and the minimum value of \( f(x) \) is \( -1 \). If \( t = f(0) \), find the value of \( t \). | -\frac{1}{2} | 0.625 |
Xiaopang arranges the 50 integers from 1 to 50 in ascending order without any spaces in between. Then, he inserts a "+" sign between each pair of adjacent digits, resulting in an addition expression: \(1+2+3+4+5+6+7+8+9+1+0+1+1+\cdots+4+9+5+0\). Please calculate the sum of this addition expression. The result is ________. | 330 | 0.375 |
Lil Wayne, the rain god, determines the weather. If Lil Wayne makes it rain on any given day, the probability that he makes it rain the next day is 75%. If Lil Wayne doesn't make it rain on one day, the probability that he makes it rain the next day is 25%. He decides not to make it rain today. Find the smallest positive integer \( n \) such that the probability that Lil Wayne makes it rain \( n \) days from today is greater than 49.9%. | 9 | 0.5 |
Simplify:
$$
\frac{3.875 \times \frac{1}{5}+38 \frac{3}{4} \times 0.09-0.155 \div 0.4}{2 \frac{1}{6}+\left[\left(4.32-1.68-1 \frac{8}{25}\right) \times \frac{5}{11}-\frac{2}{7}\right] \div 1 \frac{9}{35}+1 \frac{11}{24}}
$$ | 1 | 0.875 |
The solution set of the inequality \(\log _{a}\left(a-\frac{x^{2}}{2}\right)>\log _{a}(a-x)\) is \(A\), and \(A \cap \mathbf{Z}=\{1\}\). What is the range of values for \(a\)? | (1, +\infty) | 0.375 |
Let \( M \) be a set of \( n \) points on a plane, satisfying the following conditions:
1. There are 7 points in \( M \) that are the vertices of a convex heptagon.
2. For any 5 points in \( M \), if these 5 points are the vertices of a convex pentagon, then the interior of this convex pentagon contains at least one point from \( M \).
Find the minimum value of \( n \). | 11 | 0.125 |
A lieutenant is training recruits in marching drills. Upon arriving at the parade ground, he sees that all the recruits are arranged in several rows, with each row having the same number of soldiers, and that the number of soldiers in each row is 5 more than the number of rows. After finishing the drills, the lieutenant decides to arrange the recruits again but cannot remember how many rows there were initially. So, he orders them to form as many rows as his age. It turns out that each row again has the same number of soldiers, but in each row, there are 4 more soldiers than there were in the original arrangement. How old is the lieutenant? | 24 | 0.875 |
Let the complex number \( z = \cos \theta + \mathrm{i} \sin \theta \) where \( 0^{\circ} \leqslant \theta \leqslant 180^{\circ} \). The complex number \( z \), \( (1+\mathrm{i})z \), and \( 2\bar{z} \) correspond to the points \( P \), \( Q \), and \( R \) on the complex plane, respectively. When \( P \), \( Q \), and \( R \) are not collinear, the fourth vertex of the parallelogram formed with line segments \( PQ \) and \( PR \) as two sides is \( S \). Find the maximum distance from point \( S \) to the origin. | 3 | 0.875 |
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 $n > m \geq 1$. | 106 | 0.625 |
Find the smallest \( n > 2016 \) such that \( 1^{n} + 2^{n} + 3^{n} + 4^{n} \) is not divisible by 10. | 2020 | 0.75 |
If \( a \geq b \geq c \), \( a + b + c = 0 \), and \( x_{1} \), \( x_{2} \) are the two real roots of the quadratic equation \( a x^{2} + b x + c = 0 \), then the sum of the maximum and minimum values of \( \left| x_{1}^{2} - x_{2}^{2} \right| \) is \(\quad\) . | 3 | 0.625 |
Xiaoming puts 127 Go stones into several bags. No matter how many stones (not exceeding 127) a friend wants, Xiaoming can satisfy the request by taking out a few bags. How many bags does Xiaoming need at least? | 7 | 0.625 |
Every month Ivan pays a fixed amount of his salary for a mortgage, and the remaining portion of his salary is spent on current expenses. In December, Ivan paid 40% of his salary for the mortgage. In January, Ivan's salary increased by 9%. By what percentage did the amount spent on current expenses increase in January compared to December? | 15\% | 0.875 |
For \( x, y \in [1,3] \), find the minimum value of the expression
\[
A = \frac{(3xy + x^{2}) \sqrt{3xy + x - 3y} + (3xy + y^{2}) \sqrt{3xy + y - 3x}}{x^{2} y + y^{2} x}
\] | 4 | 0.875 |
The sequence \(\{a_{n}\}\) satisfies \(a_{1}=6\), and for any positive integer \(n\), \(a_{n+1}+n+1=2(a_{n}+1)\). What is the units digit of \(a_{1}+a_{2}+\cdots+a_{2022}\)? | 8 | 0.375 |
Let $\alpha, \beta, \gamma \in \mathbf{R}$ satisfy the equation $\sin \alpha \cdot \cos \beta + |\cos \alpha \cdot \sin \beta| = \sin \alpha \cdot |\cos \alpha| + |\sin \beta| \cdot \cos \beta$. Then, find the minimum value of $(\tan \gamma - \sin \alpha)^2 + (\cot \gamma - \cos \beta)^2$. | 3 - 2\sqrt{2} | 0.375 |
A tank with a mass of $m_{1}=2$ kg rests on a cart with a mass of $m_{2}=10$ kg, which is accelerated with an acceleration of $a=5 \, \text{m/s}^2$. The coefficient of friction between the tank and the cart is $\mu=0.6$. Determine the frictional force acting on the tank from the cart. | 10 \text{N} | 0.125 |
A cylinder \( C \) and a cone \( K \) are circumscribed around a sphere; the volumes of the cylinder and cone are denoted by \( V_{1} \) and \( V_{2} \), respectively. What is the minimum possible value of \( v \) if \( V_{1} = 1 \)? What is the angle at the apex of the axial section of the circumscribed cone \( K \) when its volume is \(\boldsymbol{0}\)? | \frac{4}{3} | 0.25 |
Calculate the limit of the function:
\[ \lim_{x \to 0} \frac{e^{3x} - e^{2x}}{\sin 3x - \tan 2x} \] | 1 | 0.75 |
Find the minimum value of the expression:
\[
\left(\sqrt{2(1+\cos 2x)} - \sqrt{9-\sqrt{7}} \sin x + 1\right) \cdot \left(3 + 2 \sqrt{13 - \sqrt{7}} \cos y - \cos 2y\right)
\]
If the answer is not an integer, round it to the nearest whole number. | -19 | 0.5 |
Determine the maximal size of a set of positive integers with the following properties:
(1) The integers consist of digits from the set \(\{1,2,3,4,5,6\}\).
(2) No digit occurs more than once in the same integer.
(3) The digits in each integer are in increasing order.
(4) Any two integers have at least one digit in common (possibly at different positions).
(5) There is no digit which appears in all the integers. | 32 | 0.125 |
Find the largest number such that each digit, starting from the third one, is equal to the sum of the two preceding digits. | 10112358 | 0.125 |
Let \( x_{1}, x_{2}, \cdots, x_{n} \in \mathbf{R}^{+} \) and define \( S_{n} = \sum_{i=1}^{n}\left(x_{i}+\frac{n-1}{n^{2}} \frac{1}{x_{i}}\right)^{2} \).
1. Find the minimum value of \( S_{n} \).
2. Under the condition \( x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2}=1 \), find the minimum value of \( S_{n} \).
3. Under the condition \( x_{1}+x_{2}+ \cdots+x_{n}=1 \), find the minimum value of \( S_{n} \) and provide a proof. | n | 0.875 |
Birgit has a list of four numbers. Luciano adds these numbers together, three at a time, and gets the sums 415, 442, 396, and 325. What is the sum of Birgit's four numbers? | 526 | 0.875 |
Petya wants to make an unusual die, which, as usual, should have the shape of a cube with dots drawn on its faces (with different numbers of dots on different faces). However, the number of dots on each pair of adjacent faces must differ by at least two (it is allowed to have more than six dots on some faces). How many dots in total are necessary to draw? | 27 | 0.125 |
In a game called "set," all possible four-digit numbers consisting of the digits 1, 2, and 3 (each used exactly once) are considered. A triplet of numbers forms a set if for each digit position, either all three numbers have the same digit, or all three numbers have different digits.
For example, the numbers 1232, 2213, 3221 form a set because in the first position, all three digits (1, 2, and 3) are present; in the second position, only the digit 2 is present; in the third position, all three digits are present; and in the fourth position, all three digits are present. The numbers 1123, 2231, and 3311 do not form a set because in the last position there are two 1s and a 3.
How many total sets exist in the game?
(Note: Rearranging the numbers does not create a new set. For instance, 1232, 2213, and 3221 is considered the same set as 2213, 1232, and 3221.) | 1080 | 0.25 |
Out of 24 matchsticks of the same length, use some of them to form 6 triangles in a plane where each side of the equilateral triangle is one matchstick. Then, use the remaining matchsticks to form squares in the plane where each side of the square is one matchstick. What is the maximum number of such squares that can be formed? | 4 | 0.125 |
What will the inflation be over two years:
$$
\left((1+0,015)^{\wedge} 2-1\right)^{*} 100 \%=3,0225 \%
$$
What will be the real yield of a bank deposit with an extension for the second year:
$$
(1,07 * 1,07 /(1+0,030225)-1) * 100 \%=11,13 \%
$$ | 11.13\% | 0.5 |
Armen paid \$190 to buy movie tickets for a group of \( t \) people, consisting of some adults and some children. Movie tickets cost \$5 for children and \$9 for adults. How many children's tickets did he buy? | 20 | 0.125 |
The numbers $96, 28, 6, 20$ were written on the board. One of them was multiplied, another was divided, another was increased, and another was decreased by the same number. As a result, all the numbers became equal to a single number. What is that number? | 24 | 0.625 |
The line segments \(PQRS\) and \(WXY S\) intersect circle \(C_1\) at points \(P, Q, W\) and \(X\). The line segments intersect circle \(C_2\) at points \(Q, R, X\) and \(Y\). The lengths \(QR\), \(RS\), and \(XY\) are 7, 9, and 18 respectively. The length \(WX\) is six times the length \(YS\). What is the sum of the lengths of \(PS\) and \(WS\)? | 150 | 0.125 |
Given that \(ABCD - A_1B_1C_1D_1\) is a unit cube, determine the distance between the lines \(AC\) and \(A_1D\). | \frac{\sqrt{3}}{3} | 0.375 |
For a positive integer \( n \), define \( s(n) \) as the smallest positive integer \( t \) such that \( n \) is a factor of \( t! \). Compute the number of positive integers \( n \) for which \( s(n) = 13 \). | 792 | 0.375 |
A certain quadratic polynomial is known to have the following properties: its leading coefficient is equal to one, it has integer roots, and its graph (parabola) intersects the line \( y = 2017 \) at two points with integer coordinates. Can the ordinate of the vertex of the parabola be uniquely determined based on this information? | -1016064 | 0.875 |
As shown in Figure 1, a cross-section of cube \(ABCDEFGH\) passes through vertices \(A\), \(C\), and a point \(K\) on edge \(EF\), and divides the cube into two parts with a volume ratio of 3:1. Find the value of \(\frac{EK}{KF}\). | \sqrt{3} | 0.375 |
The classrooms at MIT are each identified with a positive integer (with no leading zeroes). One day, as President Reif walks down the Infinite Corridor, he notices that a digit zero on a room sign has fallen off. Let \( N \) be the original number of the room, and let \( M \) be the room number as shown on the sign.
The smallest interval containing all possible values of \( \frac{M}{N} \) can be expressed as \( \left[\frac{a}{b}, \frac{c}{d}\right) \) where \( a, b, c, d \) are positive integers with \( \operatorname{gcd}(a, b) = \operatorname{gcd}(c, d) = 1 \). Compute \( 1000a + 100b + 10c + d \). | 2031 | 0.5 |
Find the sum of all four-digit numbers that can be formed using the digits \(1, 2, 3, 4, 5\) such that each digit appears at most once. | 399960 | 0.75 |
Investigate for which values of $\alpha>0$, the improper integral $\int_{a}^{b} \frac{d x}{(x-a)^{\alpha}}$ (where $b>a$) converges.
| \alpha < 1 | 0.75 |
Given a natural number \( x = 8^n - 1 \), where \( n \) is a natural number. It is known that \( x \) has exactly three distinct prime divisors, one of which is 31. Find \( x \). | 32767 | 0.75 |
It is known that the only solution to the equation
$$
\pi / 4 = \operatorname{arcctg} 2 + \operatorname{arcctg} 5 + \operatorname{arcctg} 13 + \operatorname{arcctg} 34 + \operatorname{arcctg} 89 + \operatorname{arcctg}(x / 14)
$$
is a natural number. Find it. | 2016 | 0.125 |
Show that we always get a perfect cube when we place \( n \) digits of 9 before the digit 7, \( n \) digits of 0 between the digits 7 and 2, and \( n \) digits of 9 between the digits 2 and 9 in the cube number 729. \( (n=1,2,\ldots) \) | (10^{n+1} - 1)^3 | 0.5 |
In the country of Draconia, there are red, green, and blue dragons. Each dragon has three heads; each head always tells the truth or always lies. Each dragon has at least one head that tells the truth. One day, 530 dragons sat around a round table, and each dragon's heads made the following statements:
- 1st head: "To my left is a green dragon."
- 2nd head: "To my right is a blue dragon."
- 3rd head: "There is no red dragon next to me."
What is the maximum number of red dragons that could have been at the table? | 176 | 0.125 |
Given two linear functions \( f(x) \) and \( g(x) \) such that the graphs \( y=f(x) \) and \( y=g(x) \) are parallel lines, but not parallel to the coordinate axes. Find the minimum value of the function \( (g(x))^2 - 3f(x) \), if the minimum value of the function \( (f(x))^2 - 3g(x) \) is \( \frac{11}{2} \). | -10 | 0.75 |
Joe has a triangle with area \(\sqrt{3}\). What's the smallest perimeter it could have? | 6 | 0.375 |
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 \( ABC \). | \frac{\sqrt{3}}{3} | 0.875 |
a) What is the maximum number of bishops that can be placed on a 1000 by 1000 chessboard such that they do not attack each other?
b) What is the maximum number of knights that can be placed on an 8 by 8 chessboard such that they do not attack each other? | 32 | 0.875 |
Mitya is 11 years older than Shura. When Mitya was as old as Shura is now, he was twice as old as she was then. How old is Mitya? | 33 | 0.875 |
The side length of an equilateral triangle $ABC$ is 4. Point $D$ is the midpoint of side $BC$. A line passes through point $B$ and intersects side $AC$ at point $M$. Perpendiculars $DH$ and $AK$ are dropped from points $D$ and $A$ onto the line $BM$. Calculate the length of segment $AM$ if
$$
A K^{4} - D H^{4} = 15
$$ | 2 | 0.875 |
For any positive integer \( x_{0} \), three sequences \(\{x_{n}\}, \{y_{n}\}, \{z_{n}\}\) are defined as follows:
1. \( y_{0} = 4 \) and \( z_{0} = 1 \);
2. if \( x_{n} \) is even for \( n \geq 0 \), then
\[
x_{n+1} = \frac{x_{n}}{2}, \quad y_{n+1} = 2 y_{n}, \quad z_{n+1} = z_{n};
\]
3. if \( x_{n} \) is odd for \( n \geq 0 \), then
\[
x_{n+1} = x_{n} - \frac{y_{n}}{2} - z_{n}, \quad y_{n+1} = y_{n}, \quad z_{n+1} = y_{n} + z_{n}.
\]
The integer \( x_{0} \) is said to be good if \( x_{n} = 0 \) for some \( n \geq 1 \). Find the number of good integers less than or equal to 1994. | 31 | 0.25 |
The teacher wrote a four-digit number on a piece of paper for Xiaowei to guess. They had four rounds of questions and answers.
Xiaowei: "Is it 8765?"
Teacher: "You guessed two digits correctly, but both are in the wrong positions."
Xiaowei: "Is it 1023?"
Teacher: "You guessed two digits correctly, but both are in the wrong positions."
Xiaowei: "Is it 8642?"
Teacher: "You guessed two digits correctly, and both are in the correct positions."
Xiaowei: "Is it 5430?"
Teacher: "None of the digits are correct."
What is this four-digit number $\qquad$? | 7612 | 0.25 |
Given that \( x = x_{0} \) and \( y = y_{0} \) satisfy the system of equations
\[
\left\{\begin{array}{l}
\frac{x}{3}+\frac{y}{5}=1 \\
\frac{x}{5}+\frac{y}{3}=1
\end{array}\right.
\]
If \( B=\frac{1}{x_{0}}+\frac{1}{y_{0}} \), find the value of \( B \). | \frac{16}{15} | 0.875 |
Does \( n^2 \) have more divisors that are congruent to \( 1 \mod 4 \) or \( 3 \mod 4 \)? | 1 \mod 4 | 0.125 |
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