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stringlengths 18
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At the namesake festival, 45 Alexanders, 122 Borises, 27 Vasily, and several Gennady attended. At the beginning of the festival, all of them lined up so that no two people with the same name stood next to each other. What is the minimum number of Gennadys that could have attended the festival?
|
49
| 0.5 |
Anton, Borya, Vasya, and Grisha met. It is known that each of them is either from the tribe of knights (who always tell the truth) or from the tribe of liars (who always lie). Anton said that he and Grisha are from different tribes. Borya and Vasya called each other liars. Grisha claimed that among the four of them, there are at least two knights. How many knights are there actually?
|
1
| 0.75 |
Find the number of four-digit numbers, composed of the digits 1, 2, 3, 4, 5, 6, 7 (each digit can be used no more than once), that are divisible by 15.
|
36
| 0.125 |
A ball thrown vertically upwards has its height above the ground expressed as a quadratic function with respect to its time of motion. Xiaohong throws two balls vertically upwards one after the other, with a 1-second interval between them. Assume the initial height above the ground for both balls is the same, and each reaches the same maximum height 1.1 seconds after being thrown. If the first ball's height matches the second ball's height at $t$ seconds after the first ball is thrown, determine $t = \qquad$ .
|
1.6
| 0.875 |
Gru and the Minions plan to make money through cryptocurrency mining. They chose Ethereum as one of the most stable and promising currencies. They bought a system unit for 9499 rubles and two graphics cards for 31431 rubles each. The power consumption of the system unit is 120 W, and for each graphics card, it is 125 W. The mining speed for one graphics card is 32 million hashes per second, allowing it to earn 0.00877 Ethereum per day. 1 Ethereum equals 27790.37 rubles. How many days will it take for the team's investment to pay off, considering electricity costs of 5.38 rubles per kWh? (20 points)
|
165
| 0.625 |
Three sportsmen called Primus, Secundus, and Tertius take part in a race every day. Primus wears the number '1' on his shirt, Secundus wears '2', and Tertius wears '3'.
On Saturday, Primus wins, Secundus is second, and Tertius is third. Using their shirt numbers this result is recorded as '123'.
On Sunday, Primus starts the race in the lead with Secundus in second. During Sunday's race:
- Primus and Secundus change places exactly 9 times,
- Secundus and Tertius change places exactly 10 times,
- Primus and Tertius change places exactly 11 times.
How will Sunday's result be recorded?
|
231
| 0.25 |
Two congruent parallelograms have equal sides parallel to each other. In both parallelograms, we mark the vertices, the midpoints of the sides, and the intersection point of the diagonals. We choose one point from the marked points of the first parallelogram and another point from the marked points of the second parallelogram in every possible way, and construct the midpoint of the segment connecting the two chosen points.
How many different midpoints do we get in this way?
|
25
| 0.375 |
The base of the quadrilateral pyramid \( S A B C D \) is a square \( A B C D \), and \( S A \) is the height of the pyramid. Let \( M \) and \( N \) be the midpoints of the edges \( S C \) and \( A D \), respectively. What is the maximum possible area of the triangle \( B S A \) if \( M N = 3 \)?
|
9
| 0.875 |
The denominators of two irreducible fractions are 600 and 700. What is the smallest possible value of the denominator of their sum (when written as an irreducible fraction)?
|
168
| 0.625 |
Given the sequence \( a_{n} = 1 + n^{3} \) where the sequence is \(\{2, 9, 28, 65, \ldots\} \) and \( \delta_{n} = \operatorname{gcd}(a_{n+1}, a_{n}) \), find the maximum value that \(\delta_{n}\) can take.
|
7
| 0.625 |
Let $\mathbb{R}_{\geq 0}$ denote the set of nonnegative real numbers. Find all functions $f: \mathbb{R}_{\geq 0} \rightarrow \mathbb{R}_{\geq 0}$ such that, for all $x, y \in \mathbb{R}_{\geq 0}$,
$$
f\left(\frac{x+f(x)}{2}+y\right)=2 x-f(x)+f(f(y))
$$
and
$$
(f(x)-f(y))(x-y) \geq 0
$$
|
f(x) = x
| 0.875 |
The dividend is six times larger than the divisor, and the divisor is four times larger than the quotient. Find the dividend.
|
144
| 0.875 |
Find the integer solutions of the equation
$$
x^{4}+y^{4}=3 x^{3} y
$$
|
(0,0)
| 0.75 |
Suppose the three sides of a triangular field are all integers, and its area is equal to the perimeter. What is the largest possible area of the field?
|
60
| 0.25 |
Let \( N_{0} \) be the set of non-negative integers, and \( f: N_{0} \rightarrow N_{0} \) be a function such that \( f(0)=0 \) and for any \( n \in N_{0} \), \( [f(2n+1)]^{2} - [f(2n)]^{2} = 6f(n) + 1 \) and \( f(2n) > f(n) \). Determine how many elements in \( f(N_{0}) \) are less than 2004.
|
128
| 0.625 |
The sequence is defined recursively:
\[ x_{0} = 0, \quad x_{n+1} = \frac{(n^2 + n + 1) x_{n} + 1}{n^2 + n + 1 - x_{n}}. \]
Find \( x_{8453} \).
|
8453
| 0.875 |
In triangle \(ABC\), angle \(B\) is \(120^\circ\) and \(AB = 2BC\). The perpendicular bisector of side \(AB\) intersects \(AC\) at point \(D\). Find the ratio \(AD : DC\).
|
2:3
| 0.375 |
Write the five-digit number "12345" 403 times to form a 2015-digit number "$123451234512345 \cdots$". Starting from the left, sequentially delete all digits located in the odd positions of this number to form a new number. Repeat this process of deleting all digits in the odd positions of the newly formed number until only one digit remains. What is the last remaining digit?
|
4
| 0.125 |
Petya ascended a moving escalator, counting 75 steps, and then descended the same escalator (i.e., moving against the direction of the escalator), counting 150 steps. While descending, Petya walked three times faster than when ascending. How many steps does the escalator have when it is stopped?
|
120
| 0.875 |
Calculate: $(10 \times 19 \times 20 \times 53 \times 100 + 601) \div 13 = \ ?$
|
1549277
| 0.375 |
Anton wrote three natural numbers \(a\), \(b\), and \(c\) on the board. Ira drew three rectangles with dimensions \(a \times b\), \(a \times c\), and \(b \times c\) on the board. It turned out that the difference in areas of some pair of rectangles is 1, and the difference in areas of another pair of rectangles is 49. What can \(a + b + c\) equal? List all possible options.
|
16
| 0.75 |
Given the polynomial \( P(x) = x^5 - x^2 + 1 \) with roots \( r_1, r_2, \ldots, r_5 \), and another polynomial \( Q(x) = x^2 + 1 \), find \( Q(r_1) Q(r_2) Q(r_3) Q(r_4) Q(r_5) \).
|
5
| 0.25 |
Petya glued a paper cube and wrote numbers from 1 to 6 on its faces such that the sums of the numbers on any two opposite faces are equal. Vasya wants to cut this cube to obtain the net shown in the figure. He tries to make the sums of the numbers in the horizontal and vertical directions on this net as close as possible. What is the smallest positive difference he can get, regardless of how Petya arranged the numbers?
|
1
| 0.625 |
On 2016 cards, the numbers from 1 to 2016 were written (each number exactly once). Then \( k \) cards were taken. What is the smallest \( k \) such that among them there will be two cards with numbers whose square root difference is less than 1?
|
45
| 0.875 |
Given that $F$ is the right focus of the ellipse $\frac{x^{2}}{1+a^{2}} + y^{2} = 1$ for $a > 0$, $M(m, 0)$ and $N(0, n)$ are points on the $x$-axis and $y$-axis respectively, and they satisfy $\overrightarrow{M N} \cdot \overrightarrow{N F} = 0$. Let point $P$ satisfy $\overrightarrow{O M} = 2 \overrightarrow{O N} + \overrightarrow{P O}$.
(1) Find the locus $C$ of point $P$.
(2) For any line passing through point $F$ that intersects the locus $C$ at points $A$ and $B$, let lines $O A$ and $O B$ intersect the line $x = -a$ at points $S$ and $T$ respectively (with $O$ being the origin). Determine whether $\overrightarrow{F S} \cdot \overrightarrow{F T}$ is a constant value. If it is, find this constant; if not, provide an explanation.
|
0
| 0.625 |
The purchase of a book cost 1 ruble plus an additional one-third of the price of the book. What is the price of the book?
|
1.5 \text{ rubles}
| 0.25 |
Given \( n \) points on a plane where any three points form the vertices of a right triangle, determine the maximum value of \( n \).
|
4
| 0.375 |
A certain conference is attended by \(12k\) people (\(k\) is a positive integer), and each person knows exactly \(3k+6\) other attendees. Assume that for any two people, the number of people who know both of them is the same. Determine the number of people attending this conference.
(IMC 36th Preliminary Problem)
|
36
| 0.625 |
Find the largest four-digit number in which all the digits are different and which is divisible by any of its digits (don't forget to explain why it is the largest).
|
9864
| 0.125 |
The pages in a book are numbered as follows: the first sheet contains two pages (numbered 1 and 2), the second sheet contains the next two pages (numbered 3 and 4), and so on. A mischievous boy named Petya tore out several consecutive sheets: the first torn-out page has the number 185, and the number of the last torn-out page consists of the same digits but in a different order. How many sheets did Petya tear out?
|
167
| 0.75 |
We assembled a large cube from \( k^{3} \) congruent small cubes. What are those values of \( k \) for which it is possible to color the small cubes with two colors in such a way that each small cube has exactly two neighbors of the same color? (Two small cubes are considered neighbors if they share a face.)
|
k \text{ is even}
| 0.375 |
In the number $2016^{* * * *} 02 * *$, each of the six asterisks must be replaced with any of the digits $0, 2, 4, 5, 7, 9$ (digits may be repeated) so that the resulting 12-digit number is divisible by 15. How many ways can this be done?
|
5184
| 0.125 |
The domain of the function $f$ is the set of real numbers. For each number $a$, it holds that if $a<x<a+100$, then $a \leq f(x) \leq a+100$. Determine the function $f$.
|
f(x) = x
| 0.875 |
What is the smallest positive integer \( n \) which cannot be written in any of the following forms?
- \( n = 1 + 2 + \cdots + k \) for a positive integer \( k \).
- \( n = p^k \) for a prime number \( p \) and integer \( k \).
- \( n = p + 1 \) for a prime number \( p \).
|
22
| 0.875 |
Two dice are thrown. Find the probability that the sum of the points on the rolled faces is even and at least one of the dice shows a six.
|
\frac{5}{36}
| 0.875 |
Oleg writes a real number above each column of a blank $50 \times 50$ grid (outside the grid) and a real number to the left of each row (outside the grid). It is known that all 100 real numbers he writes are distinct, and there are exactly 50 rational numbers and 50 irrational numbers among them. Then, Oleg writes the sum of the number to the left of the row and the number above the column in each cell of the grid (creating an addition table). What is the maximum number of rational numbers that can appear in this grid?
|
1250
| 0.125 |
The worm consists of a white head and several segments, as shown in the figure.
When the worm is born, it has a head and one white segment. Each day, a new segment is added to the worm in one of the following ways:
- Either a white segment splits into a white and a gray segment.
- Or a gray segment splits into a gray and a white segment.
(On the fourth day, the worm reaches maturity and stops growing - its body consists of a head and four segments.)
How many different color variants of mature worms of this species can exist?
Hint: What could a two-day-old worm look like?
|
4
| 0.25 |
Given that \( a \) and \( b \) are integers, and \(\frac{127}{a}-\frac{16}{b}=1\). What is the maximum value of \( b \)?
|
2016
| 0.5 |
Calculate: \((1000 + 15 + 314) \times (201 + 360 + 110) + (1000 - 201 - 360 - 110) \times (15 + 314) =\) .
|
1000000
| 0.375 |
Let \( n \geq 2 \) be a fixed integer. Find the least constant \( C \) such that the inequality
\[ \sum_{i<j} x_{i} x_{j}\left(x_{i}^{2}+x_{j}^{2}\right) \leq C\left(\sum_{i} x_{i}\right)^{4} \]
holds for every \( x_{1}, \ldots, x_{n} \geq 0 \) (the sum on the left consists of \(\binom{n}{2}\) summands). For this constant \( \bar{C} \), characterize the instances of equality.
|
\frac{1}{8}
| 0.875 |
The sum
$$
\frac{1}{1 \times 2 \times 3}+\frac{1}{2 \times 3 \times 4}+\frac{1}{3 \times 4 \times 5}+\cdots+\frac{1}{100 \times 101 \times 102}
$$
can be expressed as $\frac{a}{b}$, a fraction in its simplest form. Find $a+b$.
|
12877
| 0.375 |
On a $6 \times 6$ grid, place Go pieces in all squares, with one piece in each square. The number of white pieces in each row must be different from one another, and the number of white pieces in each column must be the same. How many black Go pieces are there in total on the $6 \times 6$ grid?
A. 18
B. 14
C. 12
D. 10
|
18
| 0.75 |
Let \( n \geqslant 4, \alpha_{1}, \alpha_{2}, \cdots, \alpha_{n} ; \beta_{1}, \beta_{2}, \cdots, \beta_{n} \) be two sets of real numbers, satisfying \( \sum_{j=1}^{n} \alpha_{j}^{2}<1 \) and \( \sum_{j=1}^{n} \beta_{j}^{2}<1 \). Define
\[ A^{2}=1-\sum_{j=1}^{n} \alpha_{j}^{2}, \, B^{2}=1-\sum_{j=1}^{n} \beta_{j}^{2}, \, W=\frac{1}{2}\left(1-\sum_{j=1}^{n} \alpha_{j} \beta_{j}\right)^{2} . \]
Find all real numbers \(\lambda\) such that the equation \( x^{n}+\lambda\left(x^{n-1}+\cdots+x^{3}+W x^{2}+A B x+1\right)=0 \) has only real roots.
|
0
| 0.75 |
For which \( n \) can we find a cyclic shift \( a_1, a_2, \ldots, a_n \) of \( 1, 2, 3, \ldots, n \) (i.e., \( i, i+1, i+2, \ldots, n, 1, 2, \ldots, i-1 \) for some \( i \)) and a permutation \( b_1, b_2, \ldots, b_n \) of \( 1, 2, 3, \ldots, n \) such that \( 1 + a_1 + b_1 = 2 + a_2 + b_2 = \ldots = n + a_n + b_n \)?
|
n \text{ is odd}
| 0.5 |
Let \( g(x) = \ln(2^x + 1) \). Then \( g(-4) - g(-3) + g(-2) - g(-1) + g(1) - g(2) + g(3) - g(4) = \) ?
|
-2 \ln 2
| 0.5 |
The non-negative reals \(a, b, c, d\) satisfy \(a^5 + b^5 \leq 1\) and \(c^5 + d^5 \leq 1\). Show that \(a^2c^3 + b^2d^3 \leq 1\).
|
1
| 0.125 |
Place 6 points inside a rectangle of dimensions \( 4 \times 3 \). Show that there are at least two points whose distance is less than or equal to \( \sqrt{5} \).
|
\sqrt{5}
| 0.625 |
When I saw Eleonora, I found her very pretty. After a brief trivial conversation, I told her my age and asked how old she was. She answered:
- When you were as old as I am now, you were three times older than me. And when I will be three times older than I am now, together our ages will sum up to exactly a century.
How old is this capricious lady?
|
15
| 0.75 |
Find all integers \( n \) for which \( n^2 + 20n + 11 \) is a perfect square.
|
35
| 0.25 |
Piercarlo chooses \( n \) integers from 1 to 1000 inclusive. None of his integers is prime, and no two of them share a factor greater than 1. What is the greatest possible value of \( n \)?
|
12
| 0.625 |
The picture shows several circles connected by segments. Tanya chooses a natural number \( n \) and places different natural numbers in the circles so that the following property holds for all these numbers:
If numbers \( a \) and \( b \) are not connected by a segment, then the sum \( a^2 + b^2 \) must be coprime with \( n \). If they are connected, then the numbers \( a^2 + b^2 \) and \( n \) must have a common natural divisor greater than 1. What is the smallest \( n \) for which such an arrangement exists?
|
65
| 0.125 |
As shown in the figure, a truck travels from $A$ to $C$ at a speed of 50 km/h, while at the same time, delivery person A and delivery person B set out from $B$ to travel to $A$ and $C$ respectively, with both traveling at the same speed. If A travels 3 km before meeting the truck, and 12 minutes later, the truck and B arrive at $C$ simultaneously, then what is the distance between cities $A$ and $C$ in kilometers?
|
17.5 \text{ km}
| 0.375 |
In the city of liars and knights, there are 366 inhabitants, each born on a different day of a leap year. All inhabitants of the city answered two questions. To the question "Were you born in February?" 100 people answered affirmatively, and to the question "Were you born on the 30th day?" 60 people answered affirmatively. How many knights were born in February?
|
29
| 0.25 |
Gavrila saw a squirrel in the window, sitting on a tree branch directly opposite him at a distance of 3 meters 75 centimeters. He decided to feed the creature and threw a nut horizontally at a speed of 5 m/s directly towards the squirrel. Will the squirrel be able to catch the nut if it can jump with great speed in any direction for a distance of 1 meter 70 centimeters? Assume that the acceleration due to gravity is $g = 10 \mathrm{~m} / \mathrm{s}^{2}$ and neglect air resistance.
|
\text{No}
| 0.875 |
Given the sets
$$
\begin{array}{l}
A=\{2,0,1,7\}, \\
B=\{x \mid x=a b, a, b \in A\} .
\end{array}
$$
determine the number of elements in set $B$.
|
7
| 0.875 |
Mr. Canada chooses a positive real \( a \) uniformly at random from \((0,1]\), chooses a positive real \( b \) uniformly at random from \((0,1]\), and then sets \( c = \frac{a}{a+b} \). What is the probability that \( c \) lies between \(\frac{1}{4}\) and \(\frac{3}{4}\)?
|
\frac{2}{3}
| 0.875 |
Calculate the value of
$$
I_{m}=\cot ^{2} \frac{\pi}{2 m+1}+\cdots+\cot ^{2} \frac{m \pi}{2 m+1}.
$$
|
\frac{m(2m-1)}{3}
| 0.75 |
From 24 identical wooden cubes, a "pipe" was glued - a cube $3 \times 3 \times 3$ with the "core" of three cubes removed. Can a diagonal be drawn in each square on the surface of the "pipe" to form a closed path that does not pass through any vertex more than once?
|
\text{No}
| 0.75 |
If \( a, b, c, d, e \) are consecutive positive integers, where \( a < b < c < d < e \), such that \( b+c+d \) is a perfect square and \( a+b+c+d+e \) is a perfect cube, what is the minimum value of \( c \)?
|
675
| 0.625 |
In the given set of numbers, one number is equal to the average of all the numbers, the largest number is 7 greater than the average, the smallest number is 7 less than the average, and most numbers in the set have below-average values.
What is the smallest number of numbers that can be in the set?
|
7
| 0.75 |
Given a triangle \( \triangle ABC \) with area 1, and side length \( a \) opposite to angle \( A \), find the minimum value of \( a^2 + \frac{1}{\sin A} \).
|
3
| 0.75 |
On a line, the points $A, B, C, D$ are marked in that order. Point $M$ is the midpoint of segment $AC$, and point $N$ is the midpoint of segment $BD$. Find the length of segment $MN$, given that $AD=68$ and $BC=26$.
|
21
| 0.875 |
A covered rectangular football field with a length of 90 m and a width of 60 m is being designed to be illuminated by four floodlights, each hanging from some point on the ceiling. Each floodlight illuminates a circle, with a radius equal to the height at which the floodlight is hanging. Determine the minimally possible height of the ceiling, such that the following conditions are met: every point on the football field is illuminated by at least one floodlight, and the height of the ceiling must be a multiple of 0.1 m (for example, 19.2 m, 26 m, 31.9 m, etc.).
|
27.1 \text{ m}
| 0.375 |
Given positive real numbers \(a_{1}, a_{2}, \cdots, a_{n}\) and non-negative real numbers \(b_{1}, b_{2}, \cdots, b_{n}\) satisfying:
1. \(a_{1} + a_{2} + \cdots + a_{n} + b_{1} + b_{2} + \cdots + b_{n} = n\)
2. \(a_{1} a_{2} \cdots a_{n} + b_{1} b_{2} \cdots b_{n} = \frac{1}{2}\)
Find the maximum value of \(a_{1} a_{2} \cdots a_{n} \left( \frac{b_{1}}{a_{1}} + \frac{b_{2}}{a_{2}} + \cdots + \frac{b_{n}}{a_{n}} \right)\).
|
\frac{1}{2}
| 0.875 |
Given that \( \tan \left(\frac{\pi}{4}+\alpha\right)=2 \), find the value of \( \frac{1}{2 \sin \alpha \cos \alpha+\cos ^{2} \alpha} \).
|
\frac{2}{3}
| 0.875 |
We build a $4 \times 4 \times 4$ cube out of sugar cubes. How many different rectangular parallelepipeds can the sugar cubes determine, if the rectangular parallelepipeds differ in at least one sugar cube?
|
1000
| 0.5 |
What is the smallest number of distinct integers needed so that among them one can select both a geometric progression and an arithmetic progression of length 5?
|
6
| 0.125 |
Famous skater Tony Hawk rides a skateboard (segment $AB$) in a ramp, which is a semicircle with a diameter $PQ$. Point $M$ is the midpoint of the skateboard, and $C$ is the foot of the perpendicular dropped from point $A$ to the diameter $PQ$. What values can the angle $\angle ACM$ take if the angular measure of arc $AB$ is $24^{\circ}$?
|
12^\circ
| 0.875 |
Given that \( p \) is a prime number and \( r \) is the remainder when \( p \) is divided by 210, if \( r \) is a composite number that can be expressed as the sum of two perfect squares, find \( r \).
|
r = 169
| 0.625 |
a) What is the maximum number of queens that can be placed on a chessboard with 64 squares such that no two queens threaten each other?
b) The same question for a chessboard with \( n^2 \) squares.
|
n
| 0.125 |
A semicircle with radius 2021 has diameter \( AB \) and center \( O \). Points \( C \) and \( D \) lie on the semicircle such that \(\angle AOC < \angle AOD = 90^{\circ} \). A circle of radius \( r \) is inscribed in the sector bounded by \( OA \) and \( OC \) and is tangent to the semicircle at \( E \). If \( CD = CE \), compute \(\lfloor r \rfloor\).
|
673
| 0.625 |
Can a circle be circumscribed around the quadrilateral \( A B C D \) if \( \angle A D C=30^{\circ} \), \( A B=3 \), \( B C=4 \), and \( A C=6 \)?
|
\text{No}
| 0.875 |
In the 100-digit number 12345678901234...7890, all digits in odd positions were crossed out; in the resulting 50-digit number, all digits in odd positions were crossed out again, and so on. The crossing out continued as long as there were digits to cross out. Which digit was crossed out last?
|
4
| 0.25 |
Given a right triangular prism \(ABC-A_{1}B_{1}C_{1}\) with the base being a right triangle, \(\angle ACB = 90^{\circ}\), \(AC = 6\), \(BC = CC_{1} = \sqrt{2}\), and \(P\) is a moving point on \(BC_{1}\), find the minimum value of \(CP + PA_{1}\).
|
5\sqrt{2}
| 0.125 |
Let the function \( f(z) \) (where \( z \) is a complex number) satisfy \( f(f(z)) = (z \bar{z} - z - \bar{z})^{2} \). If \( f(1) = 0 \), then what is \( |f(\mathrm{i}) - 1| \)?
|
1
| 0.875 |
Find the sum of natural numbers from 1 to 3000 inclusive, which have common divisors with the number 3000 that are greater than 1.
|
3301500
| 0.125 |
A large rectangle is divided into four identical smaller rectangles by slicing parallel to one of its sides. The perimeter of the large rectangle is 18 metres more than the perimeter of each of the smaller rectangles. The area of the large rectangle is 18 square metres more than the area of each of the smaller rectangles. What is the perimeter in metres of the large rectangle?
|
28
| 0.75 |
Let \( x \) and \( y \) be positive numbers whose sum is equal to 2. Find the maximum value of the expression \( x^2 y^2 (x^2 + y^2) \).
|
2
| 0.75 |
On the edge \(AD\) and the diagonal \(A_1C\) of the parallelepiped \(ABCDA_1B_1C_1D_1\), points \(M\) and \(N\) are taken respectively, such that the line \(MN\) is parallel to the plane \(BDC_1\) and \(AM:AD = 1:5\). Find the ratio \(CN:CA_1\).
|
\frac{3}{5}
| 0.625 |
Points \( M \), \( N \), and \( K \) are located on the lateral edges \( AA_1 \), \( BB_1 \), and \( CC_1 \) of the triangular prism \( ABC A_1 B_1 C_1 \) such that \( \frac{AM}{AA_1} = \frac{2}{3} \), \( \frac{BN}{BB_1} = \frac{3}{5} \), \( \frac{CK}{CC_1} = \frac{4}{7} \). Point \( P \) belongs to the prism. Find the maximum possible volume of the pyramid \( MNKP \), if the volume of the prism is 27.
|
6
| 0.25 |
Given that $\sin \alpha+\sin \beta+\sin \gamma=0$ and $\cos \alpha+\cos \beta+\cos \gamma=0$, find the value of $\cos ^{2} \alpha+\cos ^{2} \beta+ \cos ^{2} \gamma$.
|
\frac{3}{2}
| 0.875 |
Is it possible to place natural numbers from 1 to 42 (each exactly once) in a $6 \times 7$ (6 rows and 7 columns) rectangular table such that the sum of numbers in each $1 \times 2$ vertical rectangle is even?
|
\text{No}
| 0.75 |
In the following addition problem, eight different letters each represent one of the digits from 2 to 9. The same letters represent the same digits, and different letters represent different digits. Find $\overline{\mathrm{NINE}} = \quad$
$$
\begin{array}{r}
O N E \\
T W O \\
+\quad S I X \\
\hline N I N E
\end{array}
$$
|
2526
| 0.375 |
All 25 students of the 7th grade "A" class participated in a quiz consisting of three rounds. In each round, each participant scored a certain number of points. It is known that in each round, as well as in the sum of all three rounds, all participants scored different amounts of points.
Kolia from 7th grade "A" finished third in the first round, fourth in the second, and fifth in the third round of the quiz. What is the lowest possible overall position Kolia could achieve among all his classmates based on the total points from all three rounds?
|
10
| 0.375 |
A point $M$ is taken on the side $AB$ of the trapezoid $ABCD$ such that $AM: BM = 2: 3$. A point $N$ is taken on the opposite side $CD$ such that the segment $MN$ divides the trapezoid into parts, one of which has an area three times greater than the other. Find the ratio $CN: DN$ if $BC: AD = 1: 2$.
|
3:29
| 0.125 |
A number is said to be TOP if it has 5 digits and when the product of the 1st and 5th digits is equal to the sum of the 2nd, 3rd, and 4th digits. For example, 12,338 is TOP because it has 5 digits and $1 \cdot 8 = 2 + 3 + 3$.
a) What is the value of $a$ such that $23,4a8$ is TOP?
b) How many TOP numbers end with 2 and start with 1?
c) How many TOP numbers start with 9?
|
112
| 0.25 |
Given the sequence \(\left\{a_{n}\right\}\) where \(a_{1}=7\) and \(\frac{a_{n+1}}{a_{n}}=a_{n}+2\) for \(n=1,2,3,\cdots\), find the smallest positive integer \(n\) such that \(a_{n}>4^{2018}\).
|
12
| 0.75 |
Given that \( x \) and \( y \) are real numbers such that \( |x| + x + y = 10 \) and \( |y| + x - y = 10 \), if \( P = x + y \), find the value of \( P \).
|
4
| 0.875 |
Find all three-digit numbers \(\overline{abc}\) that satisfy \(\overline{abc} = (a + b + c)^3\).
|
512
| 0.5 |
If the edge length of the cube $A_{1} A_{2} A_{3} A_{4}-B_{1} B_{2} B_{3} B_{4}$ is 1, then the number of elements in the set $\left\{x \mid x=\overrightarrow{A_{1} B_{1}} \cdot \overrightarrow{A_{i} B_{j}}, i \in\{1,2,3,4\}, j \in\{1,2,3,4\}\right\}$ is $\qquad$.
|
1
| 0.375 |
On the island of Friends and Foes, every citizen is either a Friend (who always tells the truth) or a Foe (who always lies). Seven citizens are sitting in a circle. Each declares "I am sitting between two Foes". How many Friends are there in the circle?
|
3
| 0.75 |
In a table containing $A$ columns and 100 rows, natural numbers from 1 to $100 \cdot A$ are written by rows in ascending order, starting from the first row. The number 31 is in the fifth row. In which row is the number 100?
|
15
| 0.875 |
Let the function \( f(x) \) be a differentiable function defined on the interval \( (-\infty, 0) \), with its derivative denoted as \( f'(x) \), and satisfying the inequality \( 2 f(x) + x f'(x) > x^2 \). Then, find the solution set of the inequality:
\[ (x+2017)^{2} f(x+2017) - f(-1) > 0. \]
|
(-\infty, -2018)
| 0.875 |
At a mathematics competition, three problems were given: $A$, $B$, and $C$. There were 25 students who each solved at least one problem. Among the students who did not solve problem $A$, twice as many solved $B$ as solved $C$. One more student solved only problem $A$ than the number of those who also solved problem $A$. Half of the students who solved only one problem did not solve $A$. How many students solved only problem $B$?
|
6
| 0.875 |
For the set \( S = \left\{ \left( a_1, a_2, \cdots, a_5 \right) \mid a_i = 0 \text{ or } 1, i = 1, 2, \cdots, 5 \right\} \), the distance between any two elements \( A = \left( a_1, a_2, \cdots, a_5 \right) \) and \( B = \left( b_1, b_2, \cdots, b_5 \right) \) is defined as:
\[ d(A, B) = \left| a_1 - b_1 \right| + \cdots + \left| a_5 - b_5 \right| \]
Find a subset \( T \) of \( S \) such that the distance between any two elements in \( T \) is greater than 2. What is the maximum number of elements that subset \( T \) can contain? Provide a proof for your conclusion.
|
4
| 0.875 |
The arithmetic mean of three two-digit natural numbers \( x, y, z \) is 60. What is the maximum value that the expression \( \frac{x + y}{z} \) can take?
|
17
| 0.875 |
Twenty phones are connected by wires in such a way that each wire connects two phones, each pair of phones is connected by at most one wire, and no more than two wires extend from any one phone. The wires need to be colored (each wire entirely with one color) such that the wires extending from each phone are of different colors. What is the minimum number of colors needed for such a coloring?
|
3
| 0.75 |
Leon has cards with digits from 1 to 7. How many ways are there to combine these cards into two three-digit numbers (one card will not be used) so that each of them is divisible by 9?
|
36
| 0.125 |
Find the value of \( a > 1 \) for which the equation \( a^x = \log_a x \) has a unique solution.
|
a = e^{1/e}
| 0.625 |
In square ABCD, diagonals AC and BD meet at point E. Point F is on CD such that ∠CAF = ∠FAD. If AF intersects ED at point G, and EG = 24 cm, find the length of CF.
|
48 \text{ cm}
| 0.125 |
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