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stringlengths 33
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float64 0.06
0.59
|
|---|---|---|---|
The perimeter of a rectangle is 48 cm, and its length is 2 cm more than its width. Find the area of this rectangle.
|
143 \, \text{cm}^2
|
olympiads
| 0.140625
|
Given the point \( A(0,1) \) and the curve \( C: y = \log_a x \) which always passes through point \( B \), if \( P \) is a moving point on the curve \( C \) and the minimum value of \( \overrightarrow{AB} \cdot \overrightarrow{AP} \) is 2, then the real number \( a = \) _______.
|
e
|
olympiads
| 0.109375
|
Andrey, Borya, Vitya, and Gena are playing on a $100 \times 2019$ board (100 rows, 2019 columns). They take turns in the following order: Andrey first, then Borya, followed by Vitya, and finally Gena, then Andrey again, and so on. On each turn, a player must color two uncolored cells that form a rectangle of two cells. Andrey and Borya color vertical rectangles $2 \times 1$, while Vitya and Gena color horizontal rectangles $1 \times 2$. The player who cannot make a move loses. Which three players can collude and play in such a way that the remaining player is guaranteed to lose? (It is sufficient to provide one such group of three players.)
|
\text{{Andrey, Borya, and Gena}}
|
olympiads
| 0.09375
|
Compute
$$
\int_{L} \frac{60 e^{z}}{z(z+3)(z+4)(z+5)} d z
$$
where \( L \) is a unit circle centered at the origin.
|
2 \pi i
|
olympiads
| 0.109375
|
Which values of \( x \) satisfy the inequalities \( x^{2} - 12x + 32 > 0 \) and \( x^{2} - 13x + 22 < 0 \) simultaneously?
|
2 < x < 4
|
olympiads
| 0.09375
|
Whom should you interrogate first? The statements of three suspects contradict each other: Smith accuses Brown of lying, Brown - Jones, and Jones says not to believe either Brown or Smith. Whom would you, as an investigator, interrogate first?
|
Jones
|
olympiads
| 0.25
|
Let \( A \) be any \( k \)-element subset of the set \(\{1, 2, 3, 4, \ldots, 100\}\). Determine the minimum value of \( k \) such that we can always guarantee the existence of two numbers \( a \) and \( b \) in \( A \) such that \( |a - b| \leq 4 \).
|
21
|
olympiads
| 0.125
|
Find the mass of the body $\Omega$ with density $\mu=z$, bounded by the surfaces
$$
x^{2} + y^{2} = 4, \quad z=0, \quad z=\frac{x^{2} + y^{2}}{2}
$$
|
\frac{16\pi}{3}
|
olympiads
| 0.078125
|
In a cylinder with a base radius of 6, there are two spheres each with a radius of 6, and the distance between their centers is 13. If a plane is tangent to both spheres and intersects the cylindrical surface, forming an ellipse, what is the sum of the lengths of the major and minor axes of this ellipse? ( ).
|
25
|
olympiads
| 0.0625
|
Find the maximum value that the expression \(a e k - a f h + b f g - b d k + c d h - c e g\) can take, given that each of the numbers \(a, b, c, d, e, f, g, h, k\) equals \(\pm 1\).
|
4
|
olympiads
| 0.0625
|
Unit squares \(ABCD\) and \(EFGH\) have centers \(O_1\) and \(O_2\) respectively, and are originally situated such that \(B\) and \(E\) are at the same position and \(C\) and \(H\) are at the same position. The squares then rotate clockwise about their centers at the rate of one revolution per hour. After 5 minutes, what is the area of the intersection of the two squares?
|
\frac{2 - \sqrt{3}}{4}
|
olympiads
| 0.0625
|
The common external tangent of two externally tangent circles makes an angle \( \alpha \) with the line joining their centers. Find the ratio of the radii of these circles.
|
\cot^2 \left( \frac{\pi}{4} - \frac{\alpha}{2} \right)
|
olympiads
| 0.140625
|
Let \(ABCDE\) be a convex pentagon such that \(\angle ABC = \angle ACD = \angle ADE = 90^\circ\) and \(AB = BC = CD = DE = 1\). Compute the length \(AE\).
|
2
|
olympiads
| 0.328125
|
A store has a safe with the password $3854 \square 942$. The fifth digit of the password is forgotten, but it is known that the password is the product of $5678 \times 6789$. What should be filled in the blank $\square$?
|
7
|
olympiads
| 0.109375
|
Dodson, Williams, and their horse Bolivar want to reach City B from City A as quickly as possible. Along the road, there are 27 telegraph poles, dividing the whole path into 28 equal intervals. Dodson walks an interval between poles in 9 minutes, Williams in 11 minutes, and either can ride Bolivar to cover the same distance in 3 minutes (Bolivar cannot carry both simultaneously). They depart from City A at the same time, and the journey is considered complete when all three are in City B.
The friends have agreed that Dodson will ride part of the way on Bolivar, then tie Bolivar to one of the telegraph poles, and continue on foot, while Williams will initially walk and then ride Bolivar. At which telegraph pole should Dodson tie Bolivar so that they all reach City B as quickly as possible?
Answer: At the 12th pole, counting from City A.
|
12
|
olympiads
| 0.5625
|
For which integer values of \( x \) is the number \( \frac{5x + 2}{17} \) an integer?
|
3, 20, 37, \ldots, -14, -31, -48, \ldots
|
olympiads
| 0.15625
|
Find all functions $f:\mathbb{N} \rightarrow \mathbb{N}$ such that for every prime number $p$ and natural number $x$ , $$ \{ x,f(x),\cdots f^{p-1}(x) \} $$ is a complete residue system modulo $p$ . With $f^{k+1}(x)=f(f^k(x))$ for every natural number $k$ and $f^1(x)=f(x)$ .
|
f(x) = x + 1
|
aops_forum
| 0.328125
|
Find a function \( f: \mathbf{R}_{+} \rightarrow \mathbf{R}_{+} \) that satisfies the condition: For any three distinct positive real numbers \( a, b, \) and \( c \), the three line segments of lengths \( a, b, \) and \( c \) can form a triangle if and only if the three line segments of lengths \( f(a), f(b), \) and \( f(c) \) can also form a triangle.
|
f(x) = cx \ \text{where} \ c \ \text{is an arbitrary positive real number.
|
olympiads
| 0.0625
|
The perimeter of a rhombus is $100 \text{ cm}$, and the sum of its diagonals is $62 \text{ cm}$. What are the lengths of the diagonals?
|
48 \, \text{cm} \, \text{and} \, 14 \, \text{cm}
|
olympiads
| 0.328125
|
Given that \(a, b \in \mathbb{R}\), and the equation \(x^{4} + a x^{3} + 2 x^{2} + b x + 1 = 0\) has a real root, find the minimum possible value of \(a^2 + b^2\).
|
8
|
olympiads
| 0.078125
|
Vera, Nadia, and Lyuba have dresses in three different colors: pink, purple, and turquoise. Their hats are the same three colors. Only Vera's dress and hat are the same color. Nadia's dress and hat are not pink, and Lyuba's hat is purple. Indicate the color of the dress and hat for each girl.
|
\text{Vera: Pink dress and pink hat, Nadia: Purple dress and turquoise hat, Lyuba: Turquoise dress and purple hat}
|
olympiads
| 0.078125
|
In one box, there are 15 blue balls, and in another, there are 12 white balls. Two players take turns playing. On each turn, a player is allowed to take either 3 blue balls or 2 white balls. The winner is the one who takes the last balls. How should the starting player play to win?
|
First player should take 2 white balls on the first move
|
olympiads
| 0.078125
|
Let \( A \), \( B \), and \( C \) be the remainders when the polynomial \( P(x) \) is divided by \( x-a \), \( x-b \), and \( x-c \), respectively.
Find the remainder when the same polynomial is divided by the product \( (x-a)(x-b)(x-c) \).
|
A\frac{(x-b)(x-c)}{(a-b)(a-c)} + B\frac{(x-a)(x-c)}{(b-a)(b-c)} + C\frac{(x-a)(x-b)}{(c-a)(c-b)}
|
olympiads
| 0.1875
|
Given the set \( S = \{1, 2, 3, \ldots, 2000, 2001\} \), if a subset \( T \) of \( S \) has the property that for any three elements \( x, y, z \) in \( T \), \( x + y \neq z \), what is the maximum number of elements that \( T \) can have?
|
1001
|
olympiads
| 0.234375
|
There are some bullfinches in a pet store. One of the children exclaimed, "There are more than fifty bullfinches!" Another replied, "Don't worry, there are fewer than fifty bullfinches." The mother added, "At least there is one!" The father concluded, "Only one of your statements is true." Can you determine how many bullfinches are in the store, knowing that a bullfinch was purchased?
|
50
|
olympiads
| 0.171875
|
There are several bags of apples. Each bag contains either 12 or 6 apples. How many apples are there in total, if it is known that the total number of apples is not less than 70 and not more than 80? List all possible options.
|
72 \text{ and } 78
|
olympiads
| 0.265625
|
Marty and three other people took a math test. Everyone got a non-negative integer score. The average score was 20. Marty was told the average score and concluded that everyone else scored below average. What was the minimum possible score Marty could have gotten in order to definitively reach this conclusion?
|
61
|
olympiads
| 0.0625
|
If a particular year has 53 Fridays and 53 Saturdays, what day of the week is March 1st of that year?
|
Tuesday
|
olympiads
| 0.109375
|
A sequence of numbers is arranged in a line, and its pattern is as follows: the first two numbers are both 1. From the third number onward, each number is the sum of the previous two numbers: $1, 1, 2, 3, 5, 8, 13, 21, 34, 55$. How many even numbers are there among the first 100 numbers in this sequence (including the 100th number)?
|
33
|
olympiads
| 0.125
|
Find all real numbers \( a \) and \( b \) such that \( (x-1)^{2} \) divides \( a x^{4} + b x^{3} + 1 \).
|
a = 3 and b = -4
|
olympiads
| 0.28125
|
One autumn day, a Scattered Scientist glanced at his antique wall clock and saw three flies asleep on the dial. The first one was exactly at the 12 o'clock mark, and the other two were precisely at the 2 o'clock and 5 o'clock marks. The Scientist measured and determined that the hour hand posed no threat to the flies, but the minute hand would sweep them all away one by one. Find the probability that exactly two of the three flies will be swept away by the minute hand precisely 40 minutes after the Scientist noticed the flies.
|
\frac{1}{2}
|
olympiads
| 0.0625
|
What is the minimum number of points that need to be marked inside a convex $n$-gon so that each triangle with vertices at the vertices of this $n$-gon contains at least one marked point inside?
|
n-2
|
olympiads
| 0.09375
|
Let $ n \geq 3$ and consider a set $ E$ of $ 2n \minus{} 1$ distinct points on a circle. Suppose that exactly $ k$ of these points are to be colored black. Such a coloring is **good** if there is at least one pair of black points such that the interior of one of the arcs between them contains exactly $ n$ points from $ E$ . Find the smallest value of $ k$ so that every such coloring of $ k$ points of $ E$ is good.
|
n
|
aops_forum
| 0.09375
|
Find all natural integer solutions to the equation \((n+2)! - (n+1)! - n! = n^2 + n^4\).
|
3
|
olympiads
| 0.09375
|
The centroid of a triangle is $S$, its orthocenter is $M$, the center of its incircle is $O$, and the center of its circumcircle is $K$. If any two of these points coincide, under which conditions does it follow that the triangle is equilateral?
|
The triangle is equilateral if any two of S, M, O, or K coincide.
|
olympiads
| 0.109375
|
In which stars does the number 2011 appear? Position all the numbers that appear on the referred stars.
|
2003, 2004, 2005, 2006, 2007, 2008, 2009, 2010, 2011, 2012, 2013
|
olympiads
| 0.21875
|
How many integers from 1 to 1000 are divisible by 4 but not by 3 and not by 10?
|
133
|
olympiads
| 0.515625
|
Rita the painter rolls a fair $6\text{-sided die}$ that has $3$ red sides, $2$ yellow sides, and $1$ blue side. Rita rolls the die twice and mixes the colors that the die rolled. What is the probability that she has mixed the color purple?
|
\frac{1}{6}
|
aops_forum
| 0.5625
|
In a sequence of 99 consecutive natural numbers, the largest number is 25.5 times the smallest number. What is the average of these 99 natural numbers?
|
53
|
olympiads
| 0.484375
|
Let $a=256$ . Find the unique real number $x>a^2$ such that
\[\log_a \log_a \log_a x = \log_{a^2} \log_{a^2} \log_{a^2} x.\]
|
2^{32}
|
aops_forum
| 0.125
|
Find the smallest number, written using only ones, that is divisible by a number consisting of one hundred threes (\(333 \ldots 33\)).
|
\underbrace{11 \ldots 1}_{300}
|
olympiads
| 0.0625
|
Eva writes consecutive natural numbers: 1234567891011. Which digit is written in the 2009th position?
|
0
|
olympiads
| 0.171875
|
Given three non-zero vectors $\bar{a}, \bar{b}, \bar{c}$, each pair of which is non-collinear. Find their sum, if $(\bar{a}+\bar{b}) \| \bar{c}$ and $(\bar{b}+\bar{c}) \| \bar{a}$.
|
\overline{0}
|
olympiads
| 0.125
|
Solve the Cauchy problem for the equation \(y'' = 1 + x + x^2 + x^3\), given that \(y = 1\) and \(y' = 1\) when \(x = 0\).
|
y = \frac{x^2}{2} + \frac{x^3}{6} + \frac{x^4}{12} + \frac{x^5}{20} + x + 1
|
olympiads
| 0.46875
|
Given that \(\frac{\sin (\alpha+\beta)}{\sin (\alpha-\beta)}=3\), find the value of \(\frac{\tan \alpha}{\tan \beta}\).
|
2
|
olympiads
| 0.34375
|
Find the largest integer \( n \) such that there exists a set of \( n \) points in the plane where, for any choice of three of them, some two are unit distance apart.
|
7
|
olympiads
| 0.09375
|
Given that \(\tan \alpha + \cot \alpha = 4\), find \(\sqrt{\sec^2 \alpha + \csc^2 \alpha - \frac{1}{2} \sec \alpha \csc \alpha}\).
|
\sqrt{14}
|
olympiads
| 0.15625
|
In the decimal representation of \( A \), the digits appear in (strictly) increasing order from left to right. What is the sum of the digits of \( 9A \)?
|
9
|
olympiads
| 0.21875
|
A pedestrian left city $A$ at noon heading towards city $B$. A cyclist left city $A$ at a later time and caught up with the pedestrian at 1 PM, then immediately turned back. After returning to city $A$, the cyclist turned around again and met the pedestrian at city $B$ at 4 PM, at the same time as the pedestrian.
By what factor is the cyclist's speed greater than the pedestrian's speed?
|
\frac{5}{3}
|
olympiads
| 0.0625
|
Find the number of squares in the sequence given by $ a_0\equal{}91$ and $ a_{n\plus{}1}\equal{}10a_n\plus{}(\minus{}1)^n$ for $ n \ge 0.$
|
0
|
aops_forum
| 0.109375
|
The number 2017 has 7 ones and 4 zeros in binary representation. When will the nearest year come where the binary representation of the year has no more ones than zeros? (Enter the year.)
|
2048
|
olympiads
| 0.0625
|
Find all functions from the set of positive integers to the set of positive integers such that:
$$
m^{2} + f(n) \mid m f(m) + n
$$
for all \( m, n > 0 \).
|
f(n) = n \text{ for all } n \in \mathbb{N}^{*}
|
olympiads
| 0.203125
|
Let \( n \) be a positive integer not less than 3, written on the blackboard as the initial value. Player A and Player B take turns modifying the number on the blackboard. Player A can add 3 or subtract 1 on each turn, while Player B can add 2 or subtract 2 on each turn. The game ends, and the player wins when someone writes a number \( k \) such that \( |k - n| \geq n \). If a player writes a number that has already appeared in the current game, the game is a draw. In the case of a draw, a new game starts with the initial value of the previous game decreased by 1. Player B decides who goes first in the first game, and for subsequent games (if any), the player who was supposed to go next in the previous game starts first. The question is: Is there a guaranteed winning strategy?
|
甲 必胜
|
olympiads
| 0.109375
|
Seven cards numbered $1$ through $7$ lay stacked in a pile in ascending order from top to bottom ( $1$ on top, $7$ on bottom). A shuffle involves picking a random card *of the six not currently on top*, and putting it on top. The relative order of all the other cards remains unchanged. Find the probability that, after $10$ shuffles, $6$ is higher in the pile than $3$ .
|
\frac{3^{10} - 2^{10}}{2 \cdot 3^{10}}
|
aops_forum
| 0.40625
|
Let $A$ and $B$ be sets such that there are exactly $144$ sets which are subsets of either $A$ or $B$ . Determine the number of elements $A \cup B$ has.
|
8
|
aops_forum
| 0.0625
|
Majka studied multi-digit numbers in which odd and even digits alternate regularly. She called numbers starting with an odd digit "funny" and those starting with an even digit "happy." (For example, the number 32387 is funny, and the number 4529 is happy.)
Among three-digit numbers, determine whether there are more funny or happy numbers, and by how much.
|
25
|
olympiads
| 0.171875
|
For a positive integer $n$ , let $d(n)$ be the number of all positive divisors of $n$ . Find all positive integers $n$ such that $d(n)^3=4n$ .
|
2, 128, 2000
|
aops_forum
| 0.078125
|
Find the general solution of the equation \( y^{\prime \prime \prime} = \sin x + \cos x \).
|
y =
\cos x - \sin x + \frac{C_1 x^2}{2} + C_2 x + C_3
|
olympiads
| 0.203125
|
Let \( S = \{-3, -2, 1, 2, 3, 4\} \). Two distinct numbers \( a \) and \( b \) are randomly chosen from \( S \). Let \( g(a, b) \) be the minimum value of the function \( f(x) = x^2 - (a+b)x + ab \) with respect to the variable \( x \). Find the maximum value of \( g(a, b) \).
|
-\frac{1}{4}
|
olympiads
| 0.21875
|
There are 16 members on the Height-Measurement Matching Team. Each member was asked, "How many other people on the team - not counting yourself - are exactly the same height as you?" The answers included six 1's, six 2's, and three 3's. What was the sixteenth answer? (Assume that everyone answered truthfully.)
|
3
|
olympiads
| 0.0625
|
For positive integer $n$ , define $S_n$ to be the minimum value of the sum \[ \sum_{k=1}^n \sqrt{(2k-1)^2+a_k^2}, \] where $a_1,a_2,\ldots,a_n$ are positive real numbers whose sum is 17. There is a unique positive integer $n$ for which $S_n$ is also an integer. Find this $n$ .
|
12
|
aops_forum
| 0.296875
|
A ball is thrown from the surface of the Earth at an angle of $30^{\circ}$ with an initial speed of $v_{0} = 20 \, \mathrm{m/s}$. Neglecting air resistance, in how much time will the velocity vector of the ball turn by $60^{\circ}$? The acceleration due to gravity is $g = 10 \, \mathrm{m/s}^{2}$.
|
2 \, \text{s}
|
olympiads
| 0.125
|
Determine all triangles such that the lengths of the three sides and its area are given by four consecutive natural numbers.
|
3, 4, 5
|
aops_forum
| 0.171875
|
The line \( l \) is given by the images of its points \( M \) and \( N \) (with their projections). The plane \( \beta \) is given by its trace \( b \) and a point \( B \) (with its projection). Construct the image of the intersection point of the plane \( \beta \) with the line \( l \).
|
P
|
olympiads
| 0.09375
|
Find the sum of all the integral values of \( x \) that satisfy
$$
\sqrt{x+3-4 \sqrt{x-1}}+\sqrt{x+8-6 \sqrt{x-1}}=1
$$
|
45
|
olympiads
| 0.0625
|
In a carriage, any $m(m \geqslant 3)$ passengers have a unique common friend (if person A is person B's friend, person B is also person A's friend, and no one is their own friend). How many friends does the person with the most friends have in this carriage?
|
m
|
olympiads
| 0.0625
|
Starting from 37, adding 5 before each previous term, forms the following sequence:
\[37,537,5537,55537,555537,...\]
How many prime numbers are there in this sequence?
|
1
|
aops_forum
| 0.578125
|
Calculate the area of the figure bounded by the asteroid given by the parametric equations \( x = 4 \cos^3 t, y = 4 \sin^3 t \).
|
S = 6\pi
|
olympiads
| 0.1875
|
Determine the sum of the following series:
$$
1 + 2 \cdot x + 3 \cdot x^{2} + \ldots + n \cdot x^{n-1} + (n+1) \cdot x^{n}
$$
What is the sum if $x < 1$ and $n$ is infinitely large?
|
\frac{1}{(1-x)^2}
|
olympiads
| 0.421875
|
Tanya was 16 years old 19 months ago, and Misha will be 19 years old in 16 months. Who is older? Explain your answer.
|
Misha is older
|
olympiads
| 0.125
|
How many parallelograms can be placed inside a regular hexagon with a side length of 3?
|
12
|
olympiads
| 0.1875
|
Find the value of \(\cot \left(\cot ^{-1} 2+\cot ^{-1} 3+\cot ^{-1} 4+\cot ^{-1} 5\right)\).
|
\frac{5}{14}
|
olympiads
| 0.28125
|
Let \( s(n) \) denote the sum of the digits of the natural number \( n \). Solve the equation \( n + s(n) = 2018 \).
|
2008
|
olympiads
| 0.0625
|
In triangle \(ABC\), the incenter, the foot of the altitude dropped to side \(AB\), and the excenter opposite \(C\), which touches side \(AB\) and the extensions of the other two sides, were marked. After that, the triangle itself was erased. Reconstruct the triangle.
|
ABC
|
olympiads
| 0.0625
|
For what values of the number \(a\) do the three graphs \(y = ax + a\), \(y = x\), and \(y = 2 - 2ax\) intersect at a single point?
|
a = \frac{1}{2} \text{ and } a = -2
|
olympiads
| 0.265625
|
If \( a, b, c \) are positive integers and satisfy \( c=(a+b i)^{3}-107 i \), where \( i^{2}=-1 \), find \( c \).
|
198
|
olympiads
| 0.453125
|
A square is divided into 8 smaller squares, with 7 of the smaller squares having a side length of 2. What is the side length of the original square? $\qquad$ .
|
8
|
olympiads
| 0.0625
|
Given \(\log _{4}(x+2y) + \log _{4}(x-2y) = 1\), find the minimum value of \(|x| - |y|\).
|
\sqrt{3}
|
olympiads
| 0.0625
|
Find two prime numbers such that both their sum and their difference are also prime numbers.
|
2 \text{ and } 5
|
olympiads
| 0.25
|
Each of the ten cards has a real number written on it. For every non-empty subset of these cards, the sum of all the numbers written on the cards in that subset is calculated. It is known that not all of the obtained sums turned out to be integers. What is the largest possible number of integer sums that could have resulted?
|
511
|
olympiads
| 0.171875
|
Determine all positive integers \( n \) such that \( 3^n + 1 \) is divisible by \( n^2 \).
|
1
|
olympiads
| 0.1875
|
Find the real value of \(x\) such that \(x^{3} + 3x^{2} + 3x + 7 = 0\).
|
-1 - \sqrt[3]{6}
|
olympiads
| 0.3125
|
Find the Wronskian determinant for the functions: \( y_{1}(x) = \sin x \),
\[ y_{2}(x) = \sin \left( x + \frac{\pi}{8} \right), \quad y_{3}(x) = \sin \left( x - \frac{\pi}{8} \right) \]
|
0
|
olympiads
| 0.5
|
Find the number of permutations of \( n \) distinct elements \( a_{1}, a_{2}, \cdots, a_{n} \) such that \( a_{1} \) and \( a_{2} \) are not adjacent.
|
(n-2)(n-1)!
|
olympiads
| 0.421875
|
A person has 13 pieces of a gold chain containing 80 links. Separating one link costs 1 cent, and attaching a new one - 2 cents.
What is the minimum amount needed to form a closed chain from these pieces?
Remember, larger and smaller links must alternate.
|
30 ext{ cents}
|
olympiads
| 0.09375
|
Let the set \( M = \{1, 2, \cdots, 100\} \), and let \( A \) be a subset of \( M \) that contains at least one cube number. Determine the number of such subsets \( A \).
|
2^{100} - 2^{96}
|
olympiads
| 0.46875
|
Let the set $\boldsymbol{A}$ consist of all three-, five-, seven-, and nine-digit numbers that use decimal digits $1, 2, \ldots, n$ (not necessarily distinct), and the set $\boldsymbol{B}$ consist of all two-, four-, six-, and eight-digit numbers that use decimal digits $1, 2, \ldots, m$ (not necessarily distinct). For which $m$ will the number of elements in $\boldsymbol{B}$ be at least as many as in $\boldsymbol{A}$, if $n=6$?
|
m = 8 \text{ or } m = 9
|
olympiads
| 0.265625
|
Given that $\boldsymbol{a}$ and $\boldsymbol{b}$ are unit vectors and $|3 \boldsymbol{a} + 4 \boldsymbol{b}| = |4 \boldsymbol{a} - 3 \boldsymbol{b}|$, and that $|\boldsymbol{c}| = 2$, find the maximum value of $|\boldsymbol{a} + \boldsymbol{b} - \boldsymbol{c}|$.
|
\sqrt{2} + 2
|
olympiads
| 0.0625
|
The ratio of the radius of the circle inscribed in an isosceles triangle to the radius of the circle circumscribed around this triangle is $k$. Find the angle at the base of the triangle.
|
\alpha = \arccos{k}
|
olympiads
| 0.078125
|
A positive integer is conceived. The digit 7 is appended to its right. The square of the conceived number is then subtracted from this new number. The remainder is then reduced by 75% of itself and the conceived number is also subtracted from this result. The final result is zero. What is the conceived number?
|
7
|
olympiads
| 0.28125
|
How many solutions in natural numbers \(x, y\) does the system of equations have?
$$
\left\{\begin{array}{l}
\text{GCD}(x, y) = 20! \\
\text{LCM}(x, y) = 30!
\end{array}\right.
$$
(where \(n! = 1 \cdot 2 \cdot 3 \cdot \ldots \cdot n\))
|
1024
|
olympiads
| 0.0625
|
Solve the equation \(\left(\frac{7}{4} - 3 \cos 2x\right) \cdot |1 + 2 \cos 2x| = \sin x (\sin x + \sin 5x)\).
|
x= \pm\frac{\pi}{6} + \frac{k \pi}{2}, \; k \in \mathbb{Z}
|
olympiads
| 0.078125
|
Given a complex number \( z = \cos \theta + i \sin \theta \) (where \( 0^\circ \leq \theta \leq 180^\circ \)), the corresponding points of the complex numbers \( z \), \( (1+i)z \), and \( 2z \) on the complex plane are \( P \), \( Q \), and \( R \) respectively. When \( P \), \( Q \), and \( R \) are not collinear, the fourth vertex of the parallelogram formed by line segments \( PQ \) and \( PR \) is \( S \). Find the maximum distance from point \( S \) to the origin.
|
3
|
olympiads
| 0.0625
|
A motorboat departed from the pier simultaneously with a raft and traveled 40/3 km downstream. Without stopping, it turned around and went upstream. After traveling 28/3 km, it met the raft. If the river current speed is 4 km/h, what is the motorboat's own speed?
|
\frac{68}{3} \text{ km/h}
|
olympiads
| 0.453125
|
Find all pairs of positive numbers \(a\) and \(b\) for which the numbers \(\sqrt{ab}\), \(\frac{a+b}{2}\), and \(\sqrt{\frac{a^2 + b^2}{2}}\) can form an arithmetic progression.
|
a = b
|
olympiads
| 0.1875
|
Find the real number \(\alpha\) such that the curve \(f(x)=e^{x}\) is tangent to the curve \(g(x)=\alpha x^{2}\).
|
\frac{e^2}{4}
|
olympiads
| 0.25
|
Find \(\lim _{x \rightarrow 0}[(x-\sin x) \ln x]\).
|
0
|
olympiads
| 0.484375
|
A positive integer $x$ is randomly generated with equal probability from the set $\{1, 2, \cdots, 2016\}$. What is the probability that the sum of the digits of $x$ in binary representation does not exceed 8?
|
\frac{655}{672}
|
olympiads
| 0.09375
|
Find all sets of four positive real numbers \((a, b, c, d)\) satisfying
\[
\begin{array}{l}
a + b + c + d = 1, \\
\text{and} \\
\max \left\{\frac{a^{2}}{b}, \frac{b^{2}}{a}\right\} \max \left\{\frac{c^{2}}{d}, \frac{d^{2}}{c}\right\} = (\min \{a+b, c+d\})^{4}.
\end{array}
\]
|
\left(\frac{1}{4}, \frac{1}{4}, \frac{1}{4}, \frac{1}{4}\right)
|
olympiads
| 0.171875
|
Let \( d \) be the greatest common divisor (GCD) of eight natural numbers whose sum is 595. What is the greatest possible value of \( d \)?
|
35
|
olympiads
| 0.25
|
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