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Given sets \( M = \left\{ x \mid x^{2} - 9 > 0 \right\} \) and \( N = \left\{ x \in \mathbf{Z} \mid x^{2} - 8x + a < 0 \right\} \). If the number of subsets of \( M \cap N \) is 4, then the range of the real number \( a \) is \(\qquad\).
|
[12, 15)
| 0.75 |
In triangle \( \triangle ABC \), \( AB = AC \) and \( \angle A = 100^\circ \), \( I \) is the incenter, and \( D \) is a point on \( AB \) such that \( BD = BI \). Find the measure of \( \angle BCD \).
|
30^\circ
| 0.75 |
Five friends, one of whom had a monkey, bought a bag of nuts, which they planned to divide amongst themselves the next morning. However, during the night, one friend woke up and wanted some nuts. He divided all the nuts in the bag into five equal parts, with one nut left over, which he gave to the monkey, and took his fifth part. Following him, another friend woke up; unaware that someone had already taken nuts, he divided the remaining nuts in the bag again into five parts, with one nut left over, which he gave to the monkey, and took his fifth part. Subsequently, the remaining three friends woke up one after another, each performing the same operation: dividing the remaining nuts into five parts, taking their fifth part, with one nut left over each time, which was given to the monkey. Finally, in the morning, all five friends took out the bag, divided the remaining nuts into five parts, and the one nut left over was once again given to the monkey. Determine the smallest number of nuts in the bag such that this division is possible.
|
15621
| 0.625 |
Let the vertices of a regular 10-gon be \( A_1, A_2, \ldots, A_{10} \) in that order. Show that the length of the segment \( A_1A_4 - A_1A_2 \) is equal to the radius of the circumcircle.
|
A_1A_4 - A_1A_2 = R
| 0.75 |
\( f : \mathbb{R}^2 \rightarrow \mathbb{R}^3 \) (where \( \mathbb{R} \) is the real line) is defined by \( f(x, y) = \left( -\frac{y}{x^2 + 4y^2}, \frac{x}{x^2 + 4y^2}, 0 \right) \). Can we find \( F : \mathbb{R}^3 \rightarrow \mathbb{R}^3 \), such that:
1. If \( F = (F_1, F_2, F_3) \), then \( F_i \) all have continuous partial derivatives for all \( (x, y, z) \neq (0, 0, 0) \);
2. \( \nabla \times F = 0 \) for all \( (x, y, z) \neq 0 \);
3. \( F(x, y, 0) = f(x, y) \)?
|
\text{No}
| 0.25 |
Points \( A \) and \( B \) lie on a circle with center \( O \) and radius 6, and point \( C \) is equidistant from points \( A, B, \) and \( O \). Another circle with center \( Q \) and radius 8 is circumscribed around triangle \( A C O \). Find \( B Q \).
|
10
| 0.625 |
The diagonals of a trapezoid are perpendicular to each other, its height is 4 units, and one of its diagonals is 5 units. What is the area of the trapezoid?
|
\frac{50}{3}
| 0.875 |
Given a polynomial \( P(x) \) with integer coefficients and known values \( P(2) = 3 \) and \( P(3) = 2 \), what is the maximum number of integer solutions that the equation \( P(x) = x \) can have?
|
0
| 0.625 |
If \( c \) boys were all born in June 1990 and the probability that their birthdays are all different is \( \frac{d}{225} \), find \( d \).
|
203
| 0.875 |
14. \( l \) is a tangent line to the hyperbola \( a: \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \quad (a > 0, b > 0) \) at the point \( T \). The line \( l \) intersects the asymptotes of hyperbola \( a \) at points \( A \) and \( B \). The midpoint of the line segment \( AB \) is \( M(2,1) \).
An ellipse \(\frac{x^2}{a_0^2} + \frac{y^2}{b_0^2} = 1 \quad \left(a_0 > 0, b_0 > 0\right)\) has points \( C(0, b_0) \) and \( D(a_0, 0) \), with \( O \) and \( T \) located on opposite sides of the line segment \( CD \). The internal and external angle bisectors of \(\angle TCD\) intersect the \( x \)-axis at points \( E \) and \( F \), respectively. The area of triangle \( \triangle TEF \) is denoted by \( S \).
Find:
(1) The coordinates of point \( T \).
(2) The locus of the midpoint of line segment \( CD \) that minimizes \( S \).
|
1
| 0.25 |
A line of soldiers 1 mile long is jogging. The drill sergeant, in a car, moving at twice their speed, repeatedly drives from the back of the line to the front of the line and back again. When each soldier has marched 15 miles, how much mileage has been added to the car, to the nearest mile?
|
30
| 0.75 |
We say that a number is ascending if each of its digits is greater than the digit to its left. For example, 2568 is ascending, and 175 is not. How many ascending numbers are there between 400 and 600?
|
16
| 0.875 |
Given that the sum of $n$ positive integers $x_{1}, x_{2}, \cdots, x_{n}$ is 2016. If these $n$ numbers can be divided into 32 groups with equal sums as well as into 63 groups with equal sums, find the minimum value of $n$.
|
94
| 0.125 |
You are trapped in a room with only one exit, a long hallway with a series of doors and land mines. To get out, you must open all the doors and disarm all the mines. In the room is a panel with 3 buttons, which contains an instruction manual. The red button arms a mine, the yellow button disarms two mines and closes a door, and the green button opens two doors. Initially, 3 doors are closed, and 3 mines are armed. The manual warns that attempting to disarm two mines or open two doors when only one is armed/closed will reset the system to its initial state. What is the minimum number of buttons you must push to get out?
|
9
| 0.25 |
How many positive numbers are there among the 2014 terms of the sequence: \( \sin 1^{\circ}, \sin 10^{\circ}, \sin 100^{\circ}, \sin 1000^{\circ}, \ldots ? \)
|
3
| 0.75 |
Find the intersection point of the line and the plane.
$$
\begin{aligned}
& \frac{x-1}{-1}=\frac{y+5}{4}=\frac{z-1}{2} \\
& x-3y+7z-24=0
\end{aligned}
$$
|
(0, -1, 3)
| 0.875 |
Given \( z_{1}=x+\sqrt{5}+y i \) and \( z_{2}=x-\sqrt{5}+y i \), where \( x, y \in \mathbf{R} \) and \(\left|z_{1}\right|+\left|z_{2}\right|=6 \), find the product of the maximum and minimum values of \( f(x, y)=|2 x-3 y-12| \).
|
72
| 0.875 |
In a plane, 4 points \( A_{1}, A_{2}, A_{3}, A_{4} \) are given, and the distance between any two of them is at least 1. What is the maximum possible number of line segments \( A_{i}A_{j} \) with a length of 1 that can be drawn between these points?
|
5
| 0.5 |
50 students with blond hair, brown hair, and red hair are sitting around a round table. It is known that in any group of students sitting consecutively, there is at least one student with brown hair between any two students with blond hair, and at least one student with red hair between any two students with brown hair. What is the minimum number of students with red hair that can be sitting at this table?
|
17
| 0.5 |
Petya's favorite TV game is called "Lottery on the Couch." During the game, viewers can send SMS messages with three-digit numbers containing only the digits 1, 2, 3, and 4. At the end of the game, the host announces a three-digit number, also consisting only of these digits. An SMS is considered a winning one if the number in it differs from the host's number by no more than one digit (for example, if the host announces the number 423, then messages 443 and 123 are winning, but 243 and 224 are not).
Petya wants to send as few messages as possible so that at least one is guaranteed to be winning. How many SMS messages does he need to send?
|
8
| 0.375 |
In a circle \(\omega\) centered at \(O\), \(AA'\) and \(BB'\) are diameters perpendicular to each other such that the points \(A, B, A', B'\) are arranged in an anticlockwise direction in this order. Let \(P\) be a point on the minor arc \(A'B'\) such that \(AP\) intersects \(BB'\) at \(D\) and \(BP\) intersects \(AA'\) at \(C\). Suppose the area of the quadrilateral \(ABCD\) is 100. Find the radius of \(\omega\).
|
10
| 0.875 |
Let \( n \in \mathbb{N}^{*} \). Show that for all \( a, b > 0 \), we have:
\[
\left(1+\frac{a}{b}\right)^{n}+\left(1+\frac{b}{a}\right)^{n} \geqslant 2^{n+1}
\]
When does equality hold?
|
a=b
| 0.25 |
Compute the definite integral:
$$
\int_{0}^{2 \pi} \sin ^{6}\left(\frac{x}{4}\right) \cos ^{2}\left(\frac{x}{4}\right) d x
$$
|
\frac{5\pi}{64}
| 0.375 |
It is known that the lengths of the sides of a triangle are consecutive natural numbers, and the radius of its inscribed circle is 4. Find the radius of the circumcircle of this triangle.
|
\frac{65}{8}
| 0.875 |
In triangle \(ABC\), point \(N\) is taken on side \(AB\), and point \(M\) is taken on side \(AC\). Segments \(CN\) and \(BM\) intersect at point \(O\). The ratio \(AN:NB = 2:3\) and \(BO:OM = 5:2\). Find the ratio \(CO:ON\).
|
5:2
| 0.875 |
A team of five people is laying a 30-meter pipe into the ground. To prepare the trench, they decided to dig in turns with the condition that each person digs for the amount of time it takes the other four together to dig one-third of the trench. The 30-meter trench was completed when the last team member had worked their allotted time. How many meters of the trench could they have dug in the same time if they all worked together?
|
80 \text{m}
| 0.5 |
Given a segment \( AB \) of fixed length 3 with endpoints moving on the parabola \( y^2 = x \), find the shortest distance from the midpoint \( M \) of segment \( AB \) to the y-axis.
|
\frac{5}{4}
| 0.75 |
Calculate the lengths of the arcs of the curves given by the equations in polar coordinates.
$$
\rho=8 \cos \varphi, 0 \leq \varphi \leq \frac{\pi}{4}
$$
|
2\pi
| 0.875 |
The sports event lasted for \( n \) days; \( N \) sets of medals were awarded; on the 1st day, 1 set of medals and \( \frac{1}{7} \) of the remaining sets were awarded; on the 2nd day, 2 sets of medals and \( \frac{1}{7} \) of the remaining sets were awarded; on the next to last, \( (n-1) \)th day, \( (n-1) \) sets of medals and \( \frac{1}{7} \) of all remaining sets were awarded; finally, on the last day, the remaining \( n \) sets of medals were awarded. How many days did the sports event last and how many sets of medals were awarded?
|
36
| 0.375 |
The minimum positive period of the function \( f(x) = |\sin(2x) + \sin(3x) + \sin(4x)| \) is equal to...
|
2 \pi
| 0.5 |
Let \( p = a^b + b^a \). If \( a \), \( b \), and \( p \) are all prime, what is the value of \( p \)?
|
17
| 0.875 |
Find the area of the figure defined on the coordinate plane by the system
$$
\left\{\begin{array}{l}
\sqrt{1-x}+2 x \geqslant 0 \\
-1-x^{2} \leqslant y \leq 2+\sqrt{x}
\end{array}\right.
$$
|
4
| 0.375 |
To reach the Solovyov family's dacha from the station, one must first travel 3 km on the highway and then 2 km on a path. Upon arriving at the station, the mother called her son Vasya at the dacha and asked him to meet her on his bicycle. They started moving towards each other at the same time. The mother walks at a constant speed of 4 km/h, while Vasya rides at a speed of 20 km/h on the path and 22 km/h on the highway. At what distance from the station did Vasya meet his mother? Give the answer in meters.
|
800 \text{ meters}
| 0.75 |
In a tournament, each participant plays a match against every other participant. The winner of a match earns 1 point, the loser 0 points, and if the match is a draw, both players earn half a point. At the end of the tournament, the participants are ranked according to their score (in the case of a tie, the order is arbitrary). It is noted that each participant won half of their points against the last ten ranked players. How many people participated in the tournament?
|
25
| 0.75 |
The figure $ABCDE$ is a star pentagon (each line segment of the pentagon has 2 vertices on one side and 1 vertex on the other side). What is the sum of the angles $A B C, B C D, C D E, D E A, E A B$?
|
180^\circ
| 0.5 |
What are the last three digits of \(2003^N\), where \(N = 2002^{2001}\)?
|
241
| 0.25 |
Consider the equation
$$
(x-1)(x-2) \ldots(x-2016) = (x-1)(x-2) \ldots(x-2016)
$$
written on a board. What is the smallest integer $k$ such that we can erase $k$ factors among these 4032 factors so that there is at least one factor remaining on each side of the equation and the equation no longer has any real solutions?
|
2016
| 0.375 |
Let the real numbers \(a\) and \(b\) such that the equation \(a x^{3} - x^{2} + b x - 1 = 0\) has three positive real roots. For all real numbers \(a\) and \(b\) that meet the conditions, find the minimum value of \(P = \frac{5 a^{2} - 3 a b + 2}{a^{2}(b - a)}\).
|
12 \sqrt{3}
| 0.5 |
Different three-digit numbers are formed using the digits 1, 3, and 5, where all the digits in each number are distinct. Find the sum of all such three-digit numbers.
|
1998
| 0.875 |
The side of a triangle is \(a\). Find the segment that connects the midpoints of the medians drawn to the two other sides.
|
\frac{a}{4}
| 0.75 |
Find all values of the parameters \(a, b, c\) for which the system of equations
\[
\left\{
\begin{array}{l}
a x + b y = c \\
b x + c y = a \\
c x + a y = b
\end{array}
\right\}
\]
has at least one negative solution (where \(x, y < 0\)).
|
a + b + c = 0
| 0.875 |
In a certain year, a specific date was never a Sunday in any month. Determine this date.
|
31
| 0.5 |
A digit is inserted between the digits of a two-digit number to form a three-digit number. Some two-digit numbers, when a certain digit is inserted in between, become three-digit numbers that are $k$ times the original two-digit number (where $k$ is a positive integer). What is the maximum value of $k$?
|
19
| 0.5 |
Given a tetrahedron \( S-ABC \), point \( A_1 \) is the centroid of \( \triangle SBC \). Point \( G \) is on the segment \( AA_1 \) such that \(\frac{|AG|}{|GA_1|}=3\). Line \( SG \) intersects the plane of \( \triangle ABC \) at point \( M \). Determine \(\frac{|A_1 M|}{|AS|} = \quad \).
|
\frac{1}{3}
| 0.875 |
This century will mark the 200th anniversary of the birth of the famous Russian mathematician Pafnuty Lvovich Chebyshev, a native of Kaluga province. The sum of the digits in the hundreds and thousands places of the year he was born is 3 times the sum of the digits in the units and tens places, and the digit in the tens place is greater than the digit in the units place. Determine the year of birth of P.L. Chebyshev, given that he was born and died in the same century and lived for 73 years.
|
1821
| 0.75 |
Two players are playing a game. One of them thinks of a sequence of integers ($x_{1}, x_{2}, \ldots, x_{n}$), which are single-digit numbers, both positive and negative. The other player is allowed to ask for the value of the sum $a_{1} x_{1}+\ldots+a_{n} x_{n}$, where $(a_{1}, \ldots, a_{n})$ is any chosen set of coefficients. What is the minimum number of questions needed for the guesser to determine the thought-of sequence?
|
1
| 0.125 |
Arrange the numbers 1 to 9 in a row from left to right such that every three consecutive numbers form a three-digit number that is a multiple of 3. How many arrangements are there?
|
1296
| 0.125 |
In a right triangle, the sine of the smaller angle is $\frac{1}{3}$. A line is drawn perpendicular to the hypotenuse, dividing the triangle into two equal areas. In what ratio does this line divide the hypotenuse?
|
2:1
| 0.125 |
Find the minimum value of the function \( f(x)=\cos 3x + 4 \cos 2x + 8 \cos x \) for \( x \in \mathbb{R} \).
|
-5
| 0.75 |
Construct the cross-section of a triangular pyramid \( A B C D \) with a plane passing through the midpoints \( M \) and \( N \) of edges \( A C \) and \( B D \) and the point \( K \) on edge \( C D \), for which \( C K: K D = 1: 2 \). In what ratio does this plane divide edge \( A B \)?
|
1:2
| 0.625 |
A permutation of $\{1, 2, \ldots, n\}$ is chosen at random. What is the average number of fixed points?
|
1
| 0.75 |
Let \( A, B, C \) be distinct points on a circle \( \odot O \) such that \(\angle AOB = 120^\circ\). Point \( C \) lies on the minor arc \( \overparen{AB} \) (and \( C \) is not coincident with \( A \) or \( B \)). If \(\overrightarrow{OC} = \lambda \overrightarrow{OA} + \mu \overrightarrow{OB}\) (\( \lambda, \mu \in \mathbb{R} \)), what is the range of values for \( \lambda + \mu \)?
|
(1, 2]
| 0.625 |
Given the function \( f(x)=\{\begin{array}{ll}x+\frac{1}{2} & 0 \leqslant x \leqslant \frac{1}{2}, \\ 2(1-x) & \frac{1}{2}<x \leqslant 1,\end{array} \), define \( f_{n}(x)=\underbrace{f(f(\cdots f}_{n \uparrow 1}(x) \cdots)), n \in \mathbf{N}^{*} \). Find the value of \( f_{2006}\left(\frac{2}{15}\right) \).
|
\frac{19}{30}
| 0.625 |
Find the range of the function \( f(x) = g(g^2(x)) \), where \( g(x) = \frac{3}{x^2 - 4x + 5} \).
|
[ \frac{3}{50}, 3]
| 0.5 |
Laura has 2010 lights and 2010 switches, with each switch controlling a different light. She wants to determine which switch corresponds to which light. To achieve this, Charlie will operate the switches. Each time, Charlie presses some switches simultaneously, and the number of lit lights is the same as the number of switches pressed (note: after each operation, the switches return to their original state).
Questions:
(1) What is the maximum number of different operations Charlie can perform so that Laura can accurately determine the correspondence between the switches and lights?
(2) If Laura operates the switches herself, what is the minimum number of operations needed to determine the correspondence between the switches and lights?
|
11
| 0.375 |
A line passing through the origin with a positive slope intersects the ellipse \(\frac{x^{2}}{4} + y^{2} = 1\) at points \(E\) and \(F\). Given points \(A(2,0)\) and \(B(0,1)\), find the maximum area of the quadrilateral \(AEBF\).
|
2\sqrt{2}
| 0.875 |
In a city, there are 9 bus stops and several buses. Any two buses have at most one common stop. Each bus stops at exactly three stops. What is the maximum number of buses that can be in the city?
|
12
| 0.375 |
Solve the inequality in integers:
$$
\frac{1}{\sqrt{x-2y+z+1}} + \frac{2}{\sqrt{2x-y+3z-1}} + \frac{3}{\sqrt{3y-3x-4z+3}} > x^2 - 4x + 3
$$
|
(3, 1, -1)
| 0.25 |
The sequence \(101, 104, 116, \cdots\) is given by the general term \(a_{n} = 100 + n^{2}\), where \(n = 1, 2, 3, \cdots\). For each \(n\), let \(d_{n}\) denote the greatest common divisor of \(a_{n}\) and \(a_{n+1}\). Find the maximum value of \(d_{n}\) for all positive integers \(n\).
|
401
| 0.625 |
Cat food is sold in large and small packages (with more food in the large package than in the small one). One large package and four small packages are enough to feed a cat for exactly two weeks. Is one large package and three small packages necessarily enough to feed the cat for 11 days?
|
Yes
| 0.875 |
In front of an elevator are people weighing 130, 60, 61, 65, 68, 70, 79, 81, 83, 87, 90, 91, and 95 kg. The elevator has a capacity of 175 kg. What is the minimum number of trips the elevator must make so that all the people can be transported?
|
7
| 0.625 |
Calculate \(\sin (\alpha+\beta)\), given \(\sin \alpha+\cos \beta=\frac{1}{4}\) and \(\cos \alpha+\sin \beta=-\frac{8}{5}\).
|
\frac{249}{800}
| 0.75 |
Anya calls a date beautiful if all 6 digits in its recording are different. For example, 19.04.23 is a beautiful date, but 19.02.23 and 01.06.23 are not.
a) How many beautiful dates will there be in April 2023?
b) How many beautiful dates will there be in the entire year of 2023?
|
30
| 0.25 |
Find all real numbers \( a \) such that there exist non-negative real numbers \( x_1, x_2, x_3, x_4, x_5 \) satisfying
\[
\sum_{k=1}^{5} k x_k = a, \quad \sum_{k=1}^{5} k^3 \cdot x_k = a^2, \quad \sum_{k=1}^{5} k^5 x_k = a^3.
\]
|
0, 1, 4, 9, 16, 25
| 0.875 |
What is the smallest eight-digit positive integer that has exactly four digits which are 4?
|
10004444
| 0.125 |
The house number. A person mentioned that his friend's house is located on a long street (where the houses on the side of the street with his friend's house are numbered consecutively: $1, 2, 3, \ldots$), and that the sum of the house numbers from the beginning of the street to his friend's house matches the sum of the house numbers from his friend's house to the end of the street. It is also known that on the side of the street where his friend's house is located, there are more than 50 but fewer than 500 houses.
What is the house number where the storyteller's friend lives?
|
204
| 0.5 |
In the diagram, point \( P \) is inside quadrilateral \( ABCD \). Also, \( DA = DP = DC \) and \( AP = AB \). If \(\angle ADP = \angle CDP = 2x^\circ\), \(\angle BAP = (x+5)^\circ\), and \(\angle BPC = (10x-5)^\circ\), what is the value of \( x \)?
|
13
| 0.75 |
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 |
The four-digit number $\overline{a b c d}$ is divisible by 3, and $a, b, c$ are permutations of three consecutive integers. How many such four-digit numbers are there?
|
184
| 0.25 |
In a right-angled triangle $ABC$ with the right angle at $C$, the angle bisector $BD$ and the altitude $CH$ are drawn. A perpendicular $CK$ is dropped from vertex $C$ to the angle bisector $BD$. Find the angle $HCK$ if $BK: KD = 3:1$.
|
30^\circ
| 0.625 |
If three numbers are chosen simultaneously from the integers $1,2,\dots,14$ in such a way that the absolute value of the difference between any two numbers is not less than 3, how many different ways can this be done?
|
120
| 0.5 |
In a group of 8 people, each person knows exactly 6 others. In how many ways can you choose four people such that every pair among the four knows each other? (Assume that if person A knows person B, then person B also knows person A, and people do not know themselves.)
|
16
| 0.625 |
Given the numbers \(\log _{\left(\frac{x}{2}-1\right)^{2}}\left(\frac{x}{2}-\frac{1}{4}\right)\), \(\log _{\sqrt{x-\frac{11}{4}}}\left(\frac{x}{2}-1\right)\), \(\log _{\frac{x}{2}-\frac{1}{4}}\left(x-\frac{11}{4}\right)^{2}\). For which \(x\) are two of these numbers equal, and the third one greater than them by 1?
|
5
| 0.75 |
Consider the line \( l: y = kx + m \) (where \( k \) and \( m \) are integers) which intersects the ellipse \( \frac{x^2}{16} + \frac{y^2}{12} = 1 \) at two distinct points \( A \) and \( B \), and the hyperbola \( \frac{x^2}{4} - \frac{y^2}{12} = 1 \) at two distinct points \( C \) and \( D \). Determine whether there exists a line \( l \) such that the vector \( \overrightarrow{AC} + \overrightarrow{BD} = 0 \). If such a line exists, how many such lines are there? If not, provide an explanation.
|
9
| 0.5 |
Let \( p \) be a prime number and \( q \) a prime divisor of the number \( 1 + p + \ldots + p^{p-1} \). Show that \( q \equiv 1 \pmod{p} \).
|
q \equiv 1 \pmod{p}
| 0.625 |
Find a number of the form \(2^{l} 3^{m}\) if the sum of all its divisors is 403.
|
144
| 0.375 |
A heap of stones has a total weight of 100 kg, where the weight of each stone does not exceed 2 kg. Choose some stones from this heap in any combination and find the difference between the sum of these chosen stones' weights and 10 kg. Denote by $d$ the minimum value of the absolute value of these differences. Find the maximum value of $d$ among all heaps of stones satisfying the above conditions.
|
\frac{10}{11}
| 0.125 |
Calculate the area of the figure bounded by the ellipse given by the parametric equations \(x=4 \cos t\) and \(y=3 \sin t\).
|
12\pi
| 0.875 |
The domain of the function \( f(x) \) is \( (0,1) \), and the function is defined as follows:
\[
f(x)=\begin{cases}
x, & \text{if } x \text{ is an irrational number}, \\
\frac{p+1}{q}, & \text{if } x=\frac{p}{q}, \; p, q \in \mathbf{N}^{*}, \; (p, q) = 1, \; p < q.
\end{cases}
\]
Find the maximum value of \( f(x) \) in the interval \(\left( \frac{7}{8}, \frac{8}{9} \right) \).
|
\frac{16}{17}
| 0.25 |
Let \( p \) and \( q \) be two prime numbers such that \( q \) divides \( 3^p - 2^p \). Show that \( p \) divides \( q - 1 \).
|
p \mid q - 1
| 0.5 |
Cat food is sold in large and small packages (with more food in the large package than in the small one). One large package and four small packages are enough to feed a cat for exactly two weeks. Is one large package and three small packages necessarily enough to feed the cat for 11 days?
|
Yes
| 0.875 |
Given that \( T = \sin 50^{\circ} \times \left(S + \sqrt{3} \times \tan 10^{\circ}\right) \), find the value of \( T \).
|
1
| 0.75 |
Find the maximum possible volume of a cylinder inscribed in a cone with a height of 27 and a base radius of 9.
|
324\pi
| 0.625 |
Anya, Borya, and Vasya took the same 6-question test, where each question could be answered with "yes" or "no". The answers are shown in the table:
| Question # | 1 | 2 | 3 | 4 | 5 | 6 |
| :--- | :--- | :--- | :--- | :--- | :--- | :--- |
| Anya | no | yes | no | yes | no | yes |
| Borya | no | no | yes | yes | no | yes |
| Vasya | yes | no | no | no | yes | no |
It turned out that Anya had two incorrect answers, and Borya had only two correct answers. How many incorrect answers does Vasya have?
|
3
| 0.875 |
A bag contains 15 balls, marked with the 15 numbers $2^{0}, 2^{1}, 2^{2}, \ldots, 2^{14}$ respectively. Each ball is either red or blue, and there is at least one ball of each color. Let $a$ be the sum of the numbers on all red balls, $b$ be the sum of the numbers on all blue balls, and $d$ be the H.C.F. of $a$ and $b$. Find the greatest possible value of $d$.
|
4681
| 0.5 |
The random variable \( X \) is defined by the distribution function
$$
F(x)= \begin{cases}
0 & \text{if } x \leq 0 \\
\frac{(1-\cos x)}{2} & \text{if } 0 < x \leq \pi \\
1 & \text{if } x > \pi
\end{cases}
$$
Find the probability density function of the variable \( X \). Calculate the probability that the random variable \( X \) takes values in the interval \( (\pi / 3, \pi / 2) \).
|
\frac{1}{4}
| 0.875 |
Two motorcyclists start racing from the same point and at the same time, one with a speed of 80 km/h and the other with a speed of 60 km/h. Half an hour later, a third motorcyclist starts from the same point in the same direction. Find the speed of the third motorcyclist, given that he catches up with the first motorcyclist 1 hour and 15 minutes later than he catches up with the second motorcyclist.
|
100 \text{ km/h}
| 0.875 |
Find all values of the parameter \(a\) for which the quadratic trinomial \(\frac{1}{3} x^2 + \left(a+\frac{1}{2}\right) x + \left(a^2 + a\right)\) has two roots, the sum of the cubes of which is exactly 3 times their product. In your answer, specify the largest of such \(a\).
|
-\frac{1}{4}
| 0.75 |
What is the minimum number of points that must be marked inside a convex $n$-gon such that any triangle whose vertices are vertices of the $n$-gon contains at least one marked point inside it?
|
n-2
| 0.5 |
In the diagram, the area of triangle $\triangle ABC$ is 100 square centimeters, and the area of triangle $\triangle ABD$ is 72 square centimeters. Point $M$ is the midpoint of side $CD$, and $\angle MHB = 90^{\circ}$. Given that $AB = 20$ centimeters, find the length of $MH$ in centimeters.
|
8.6
| 0.625 |
In a convex quadrilateral \(ABCD\), side \(AB\) is equal to diagonal \(BD\), \(\angle A=65^\circ\), \(\angle B=80^\circ\), and \(\angle C=75^\circ\). What is \(\angle CAD\) (in degrees)?
|
15^\circ
| 0.25 |
Given that \( m \in \{11, 13, 15, 17, 19\} \) and \( n \in \{2000, 2001, \cdots, 2019\} \), what is the probability that the units digit of \( m^n \) is 1?
|
\frac{2}{5}
| 0.875 |
By definition, a polygon is regular if all its angles and sides are equal. Points \( A, B, C, D \) are consecutive vertices of a regular polygon (in that order). It is known that the angle \( ABD = 135^\circ \). How many vertices does this polygon have?
|
12
| 0.375 |
One mole of an ideal gas undergoes a closed cycle in which:
$1-2$ - isobaric process, where the volume increases by 4 times;
$2-3$ - isothermal process, where the pressure increases;
$3-1$ - process in which the gas is compressed according to the law $T=\gamma V^{2}$.
Determine by what factor the volume in state 3 exceeds the initial volume in state 1.
|
2
| 0.75 |
For which values of \(a\) does the equation \(|x-3| = a x - 1\) have two solutions? Enter the midpoint of the interval of parameter \(a\) in the provided field. Round the answer to three significant digits according to rounding rules and enter it in the provided field.
|
0.667
| 0.75 |
Let's call a number greater than 25 semi-prime if it is the sum of two distinct prime numbers. What is the largest number of consecutive natural numbers that can be semi-prime?
|
5
| 0.25 |
If \( P \) is the circumcenter of \( \triangle ABC \), and
\[
\overrightarrow{P A}+\overrightarrow{P B}+\lambda \overrightarrow{P C}=\mathbf{0}, \quad \text{with} \quad \angle C=120^{\circ},
\]
determine the value of the real number \( \lambda \).
|
-1
| 0.125 |
There are 36 students at the Multiples Obfuscation Program, including a singleton, a pair of identical twins, a set of identical triplets, a set of identical quadruplets, and so on, up to a set of identical octuplets. Two students look the same if and only if they are from the same identical multiple. Nithya the teaching assistant encounters a random student in the morning and a random student in the afternoon (both chosen uniformly and independently), and the two look the same. What is the probability that they are actually the same person?
|
\frac{3}{17}
| 0.5 |
Compute the limit of the function:
$$
\lim _{x \rightarrow \pi} \frac{\ln (2+\cos x)}{\left(3^{\sin x}-1\right)^{2}}
$$
|
\frac{1}{2 \ln^2 3}
| 0.25 |
Suppose that a function \( M(n) \), where \( n \) is a positive integer, is defined by
\[
M(n)=\left\{
\begin{array}{ll}
n - 10 & \text{if } n > 100 \\
M(M(n + 11)) & \text{if } n \leq 100
\end{array}
\right.
\]
How many solutions does the equation \( M(n) = 91 \) have?
|
101
| 0.625 |
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