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stringlengths 33
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|---|---|---|---|
In the morning, a lot of people gathered at the railway platform waiting for the train. One-tenth of all waiting passengers left on the first train, one-seventh of the remaining passengers left on the second train, and one-fifth of the remaining passengers left on the third train. How many passengers were originally on the platform if there were 216 passengers left after the third train departed?
|
350
|
olympiads
| 0.21875
|
Find all integers \( n \) such that \( 5n - 7 \), \( 6n + 1 \), and \( 20 - 3n \) are all prime numbers.
|
6
|
olympiads
| 0.09375
|
A point moves along the arc of a circle and then transitions onto a straight line; what should be the position of the straight line relative to the arc for this transition to occur smoothly, without any jolt?
|
The straight line must be tangent to the circular arc.
|
olympiads
| 0.375
|
A cyclist planned to travel from point \( A \) to point \( B \) in 5 hours, moving at a constant speed. He rode at the planned speed until the halfway point and then increased his speed by 25%. How much time did the entire trip take?
|
4 \text{ hours} 30 \text{ minutes}
|
olympiads
| 0.296875
|
Let \( P \) be a point on the ellipse \(\frac{x^{2}}{100} + \frac{y^{2}}{36} = 1\), and let \( F_{1} \) and \( F_{2} \) be the foci. If the area of the triangle \(\triangle P F_{1} F_{2}\) is 36, then \(\angle F_{1} P F_{2} =\) ____ .
|
90^{\circ}
|
olympiads
| 0.140625
|
Given the set of positive integers
$$
A=\left\{a_{1}, a_{2}, \cdots, a_{1000}\right\},
$$
where $ a_{1} < a_{2} < \cdots < a_{1000} \leq 2017 $. If for any $ 1 \leq i, j \leq 1000$, $ i+j \in A $ implies $ a_{i} + a_{j} \in A $, determine the number of sets $ A $ that satisfy these conditions.
|
2^{17}
|
olympiads
| 0.078125
|
Let the set \( T = \{0, 1, \dots, 6\} \),
$$
M = \left\{\left.\frac{a_1}{7}+\frac{a_2}{7^2}+\frac{a_3}{7^3}+\frac{a_4}{7^4} \right\rvert\, a_i \in T, i=1,2,3,4\right\}.
$$
If the elements of the set \( M \) are arranged in decreasing order, what is the 2015th number?
|
\frac{386}{2401}
|
olympiads
| 0.234375
|
The number of edges of a convex polyhedron is 99. What is the greatest number of edges a plane, not passing through its vertices, can intersect?
|
66
|
olympiads
| 0.0625
|
Let \(a\) be an integer such that \(a \neq 1\). Given that the equation \((a-1) x^{2} - m x + a = 0\) has two roots which are positive integers, find the value of \(m\).
|
3
|
olympiads
| 0.390625
|
In a certain football invitational tournament, 16 cities participate, with each city sending two teams, Team A and Team B. According to the competition rules, after several days of matches, it was found that aside from Team A from city $A$, the number of matches already played by each of the other teams was different. Find the number of matches already played by Team B from city $A$.
|
15
|
olympiads
| 0.1875
|
Given real numbers \( x, y, z \geq 0 \) such that
\[ x + y + z = 30 \]
and
\[ 3x + y - z = 50 \]
find the range of \( T = 5x + 4y + 2z \).
|
[120, 130]
|
olympiads
| 0.078125
|
Among the permutations of integers $1, 2, \cdots, n$, how many permutations are there such that each number is either greater than all the numbers preceding it or less than all the numbers preceding it?
|
2^{n-1}
|
olympiads
| 0.09375
|
From 24 matches, a figure has been formed in the shape of a $3 \times 3$ square (see the figure), with the side length of each small square equal to the length of a matchstick. What is the minimum number of matchsticks that can be removed so that no whole $1 \times 1$ square formed by the matchsticks remains?
|
5
|
olympiads
| 0.078125
|
Students are in classroom with $n$ rows. In each row there are $m$ tables. It's given that $m,n \geq 3$ . At each table there is exactly one student. We call neighbours of the student students sitting one place right, left to him, in front of him and behind him. Each student shook hands with his neighbours. In the end there were $252$ handshakes. How many students were in the classroom?
|
72
|
aops_forum
| 0.0625
|
The number $11 \ldots 122 \ldots 2$ (comprising 100 ones followed by 100 twos) can be represented as a product of two consecutive natural numbers.
|
33\ldots3 \times 33\ldots34
|
olympiads
| 0.09375
|
In \(\triangle ABC\),
$$
\begin{array}{l}
z = \frac{\sqrt{65}}{5} \sin \frac{A+B}{2} + \mathrm{i} \cos \frac{A-B}{2}, \\
|z| = \frac{3 \sqrt{5}}{5}.
\end{array}
$$
Find the maximum value of \(\angle C\).
|
\pi - \arctan \frac{12}{5}
|
olympiads
| 0.125
|
There are 9 people in a club. Every day, three of them went to a cafe together, while the others did not go to the cafe. After 360 days, it turned out that any two people in the club had been to the cafe together the same number of times. What is that number?
|
30
|
olympiads
| 0.125
|
How many different grid squares can be selected in an $n \times n$ grid such that their sides are parallel to the sides of the grid?
|
\frac{n(n + 1)(2n + 1)}{6}
|
olympiads
| 0.125
|
Gari is seated in a jeep, and at the moment, has one 10-peso coin, two 5-peso coins, and six 1-peso coins in his pocket. If he picks four coins at random from his pocket, what is the probability that these will be enough to pay for his jeepney fare of 8 pesos?
|
\frac{37}{42}
|
olympiads
| 0.078125
|
$O$ and $I$ are the circumcentre and incentre of $\vartriangle ABC$ respectively. Suppose $O$ lies in the interior of $\vartriangle ABC$ and $I$ lies on the circle passing through $B, O$ , and $C$ . What is the magnitude of $\angle B AC$ in degrees?
|
60^ ext{\circ}
|
aops_forum
| 0.265625
|
Find all positive integers \( m \) and \( n \) so that for any \( x \) and \( y \) in the interval \([m, n]\), the value of \(\frac{5}{x}+\frac{7}{y}\) will also be in \([m, n]\).
|
(1,12), (2,6), (3,4)
|
olympiads
| 0.0625
|
In the plane, two points $A$ and $B$ are given, with the distance between them being $d$. Construct a square such that points $A$ and $B$ lie on its boundary, in a way that minimizes the sum of the distances from point $A$ to the vertices of the square. What is this minimum sum?
|
(1 + \sqrt{2})d
|
olympiads
| 0.203125
|
Determine all positive integer solutions \(x, y\) to the equation \(x^2 - 2 \times y! = 2021\).
|
(45,2)
|
olympiads
| 0.21875
|
Given a function \( f(x) \) such that \( f(1)=2 \), and
\[
f(x+1) = \frac{1 + f(x)}{1 - f(x)}
\]
holds for any \( x \) in its domain, compute \( f(2016) \).
|
\frac{1}{3}
|
olympiads
| 0.203125
|
Find all primes \( p \) such that there exist integers \( a, b, c \), and \( k \) satisfying the equations
\[
\begin{aligned}
& a^{2}+b^{2}+c^{2}=p \\
& a^{4}+b^{4}+c^{4}=k p .
\end{aligned}
\]
|
2, 3
|
olympiads
| 0.125
|
Let the points \( A(1,1), B, \) and \( C \) be on the ellipse \( x^{2} + 3y^{2} = 4 \). Find the equation of the line \( BC \) such that the area of \( \triangle ABC \) is maximized.
|
x + 3y + 2 = 0
|
olympiads
| 0.0625
|
Find the least number $ c$ satisfyng the condition $\sum_{i=1}^n {x_i}^2\leq cn$
and all real numbers $x_1,x_2,...,x_n$ are greater than or equal to $-1$ such that $\sum_{i=1}^n {x_i}^3=0$
|
1
|
aops_forum
| 0.484375
|
Solve the equation \(x^{3}-3 y^{3}-9 z^{3}=0\) in integers.
|
(0, 0, 0)
|
olympiads
| 0.203125
|
A right triangle $ABC$ with hypotenuse $AB$ is inscribed in a circle. Point $D$ is taken on the longer leg $BC$ such that $AC = BD$, and point $E$ is the midpoint of the arc $AB$ that contains point $C$. Find the angle $DEC$.
|
90^ extcirc}
|
olympiads
| 0.15625
|
An exhibition organizer split the exhibition "I Build, You Build, We Build" into two parts. To gauge the reactions of visitors, each visitor filled out a simple questionnaire upon leaving. The following interesting facts emerged from the survey:
- $96\%$ of visitors who liked the first part also liked the second part.
- $60\%$ of visitors who liked the second part also liked the first part.
- $59\%$ of visitors did not like either the first part or the second part.
What percentage of all visitors stated that they liked the first part of the exhibition?
|
25\%
|
olympiads
| 0.140625
|
Given a fixed triangle \( A B C \). Let \( D \) be an arbitrary point in the plane of the triangle that does not coincide with its vertices. A circle with center at \( D \) passing through \( A \) intersects lines \( A B \) and \( A C \) at points \( A_{b} \) and \( A_{c} \) respectively. Similarly defined are points \( B_{a}, B_{c}, C_{a} \), and \( C_{b} \). A point \( D \) is called good if the points \( A_{b}, A_{c}, B_{a}, B_{c}, C_{a} \), and \( C_{b} \) lie on a single circle.
How many points can be good for a given triangle \( A B C \)?
|
2, 3 \text{ or } 4
|
olympiads
| 0.203125
|
Chim Tu has four different colored T-shirts and can wear an outfit consisting of three or four T-shirts worn in a specific order. Two outfits are distinct if the sets of T-shirts used are different or if the sets of T-shirts used are the same but the order in which they are worn is different. Given that Chim Tu changes his outfit every three days and never wears the same outfit twice, how many days of winter can Chim Tu survive without repeating an outfit?
|
144
|
olympiads
| 0.1875
|
The sum of the lengths of the diagonals of a rhombus is \( m \), and its area is \( S \). Find the side length of the rhombus.
|
\frac{\sqrt{m^2 - 4S}}{2}
|
olympiads
| 0.328125
|
Find the cosine of the angle between vectors $\overrightarrow{A B}$ and $\overrightarrow{A C}$.
$A(2, 3, 2), B(-1, -3, -1), C(-3, -7, -3)$
|
\cos(\overrightarrow{AB}, \overrightarrow{AC}) = 1
|
olympiads
| 0.21875
|
Let $p$ be a prime number, $p \ge 5$ , and $k$ be a digit in the $p$ -adic representation of positive integers. Find the maximal length of a non constant arithmetic progression whose terms do not contain the digit $k$ in their $p$ -adic representation.
|
p - 1
|
aops_forum
| 0.453125
|
What are the integers \( p \) such that \( p \), \( p+2 \), and \( p+4 \) are prime numbers?
|
3
|
olympiads
| 0.421875
|
Given that the perimeter of a rectangular paper $ABCD$ is 10, and it is cut twice parallel to its length and width into 9 smaller rectangles of unequal sizes, what is the total perimeter of these 9 rectangles?
|
30
|
olympiads
| 0.203125
|
Calculate: $7 \frac{1}{3}-\left(2.4+1 \frac{2}{3} \times 4\right) \div 1 \frac{7}{10}=$
|
2
|
olympiads
| 0.109375
|
An elevator containing 9 passengers can stop at ten different floors. The passengers exit in groups of two, three, and four people. In how many ways can this happen?
|
\frac{10!}{4}
|
olympiads
| 0.390625
|
The work team was working at a rate fast enough to process $1250$ items in ten hours. But after working for six hours, the team was given an additional $150$ items to process. By what percent does the team need to increase its rate so that it can still complete its work within the ten hours?
|
30\%
|
aops_forum
| 0.25
|
Determine the number of $2021$ -tuples of positive integers such that the number $3$ is an element of the tuple and consecutive elements of the tuple differ by at most $1$ .
|
3^{2021} - 2^{2021}
|
aops_forum
| 0.0625
|
The decimal representations of natural numbers are written consecutively, starting from one, up to some number $n$ inclusive: $12345678910111213 \ldots(n)$
Does there exist such a $n$ such that in this sequence all ten digits appear the same number of times?
|
There does not exist such an } n.
|
olympiads
| 0.09375
|
For the largest possible $n$, can you create two bi-infinite sequences $A$ and $B$ such that any segment of length $n$ from sequence $B$ is contained in $A$, where $A$ has a period of 1995, but $B$ does not have this property (it is either not periodic or has a different period)?
Comment: The sequences can consist of arbitrary symbols. The problem refers to the minimal period.
|
1995
|
olympiads
| 0.53125
|
Let $a, b, c, d$ be distinct non-zero real numbers satisfying the following two conditions: $ac = bd$ and $\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}= 4$ .
Determine the largest possible value of the expression $\frac{a}{c}+\frac{c}{a}+\frac{b}{d}+\frac{d}{b}$ .
|
4
|
aops_forum
| 0.265625
|
What will be the length of the strip if a cubic kilometer is cut into cubic meters and laid out in a single line?
|
1,000,000 \text{ km}
|
olympiads
| 0.125
|
Three runners $-X$, $Y$, and $Z$ are participating in a race. $Z$ was delayed at the start and started last, while $Y$ started second. During the race, $Z$ swapped places with other participants 6 times, and $X$ swapped places 5 times. It is known that $Y$ finished before $X$. In what order did they finish?
|
Y finishes first, X finishes second, Z finishes third
|
olympiads
| 0.109375
|
There are 11 empty boxes. In one move, a player can put one coin in each of any 10 boxes. Two people play, taking turns. The winner is the player after whose move there will be 21 coins in any one of the boxes. Which player has a winning strategy?
|
Second Player Wins
|
olympiads
| 0.0625
|
From a uniform straight rod, a piece with a length of $s=80 \mathrm{~cm}$ was cut. By how much did the center of gravity of the rod shift as a result?
|
40 \ \mathrm{cm}
|
olympiads
| 0.375
|
Given the numbers $\alpha$ and $\beta$ satisfying the following equations:
$$
\alpha^{3}-3 \alpha^{2}+5 \alpha=1,
$$
$$
\beta^{3}-3 \beta^{2}+5 \beta=5,
$$
find the value of $\alpha + \beta$.
|
2
|
olympiads
| 0.140625
|
Giuseppe has a sheet of plywood that measures $22 \times 15$. Giuseppe wants to cut out as many rectangular pieces of $3 \times 5$ as possible from it. How can he do this?
|
22
|
olympiads
| 0.15625
|
Given a sequence $\left\{a_{n}\right\}$ where $a_{1}=1$ and $n \cdot a_{n+1}=2\left(a_{1}+a_{2}+\cdots+a_{n}\right)$ for $n \in \mathbf{N}_{+}$, find the general term formula for the sequence.
|
a_n = n
|
olympiads
| 0.21875
|
Xiao Ha, Xiao Shi, and Xiao Qi each have some bones. Xiao Ha has 2 more bones than twice the number of bones Xiao Shi has. Xiao Shi has 3 more bones than three times the number of bones Xiao Qi has. Xiao Ha has 5 fewer bones than seven times the number of bones Xiao Qi has. How many bones do they have in total?
|
141
|
olympiads
| 0.46875
|
Let \( MA = 1 = a \) and \( MC = 2 = b \). The angle \(\angle AMC\) is equal to \(120^\circ\), so by the cosine rule:
\[ AC^2 = a^2 + b^2 + ab \]
In the figure, equal angles are marked with the same number. Triangles \(ANM\) and \(BNC\) are similar (due to two equal angles), so \(\frac{BC}{AM} = \frac{NC}{MN}\). Triangles \(CNM\) and \(BNA\) are similar (due to two equal angles), so \(\frac{AB}{MC} = \frac{AN}{MN}\).
Combining these ratios:
\[
\frac{BC}{AM} + \frac{AB}{MC} = \frac{AN}{MN} + \frac{NC}{MN} = \frac{AC}{MN} \rightarrow \frac{1}{AM} + \frac{1}{MC} = \frac{1}{MN}
\]
\[
\frac{1}{a} + \frac{1}{b} = \frac{1}{MN} \rightarrow MN = \frac{ab}{a+b}
\]
|
\sqrt{7} \text{ and } \frac{2}{3}
|
olympiads
| 0.296875
|
Find the integral \( \int \frac{d x}{\sin^{2} x \cos^{2} x} \).
|
-\cot x + \tan x + C
|
olympiads
| 0.25
|
A traveler visited a village where each person either always tells the truth or always lies. The villagers stood in a circle, and each one told the traveler whether their right-hand neighbor was truthful or deceitful. Based on these statements, the traveler uniquely determined the proportion of truthful villagers. Determine this proportion.
|
\frac{1}{2}
|
olympiads
| 0.453125
|
There are five gifts priced at 2 yuan, 5 yuan, 8 yuan, 11 yuan, and 14 yuan, and five boxes priced at 1 yuan, 3 yuan, 5 yuan, 7 yuan, and 9 yuan. Each gift is paired with one box. How many different total prices are possible?
|
19
|
olympiads
| 0.453125
|
Let \( A \) be the sum of the digits of the number \( 16^{16} \), and \( B \) be the sum of the digits of the number \( A \). Find the sum of the digits of the number \( B \) without calculating \( 16^{16} \).
|
7
|
olympiads
| 0.1875
|
Given a sequence \(\left\{a_{n}\right\}\) with the general term
$$
a_{n}=2^{n}+3^{n}+6^{n}-1 \quad (n \in \mathbf{Z}_{+}),
$$
find the positive integers that are coprime with every term of this sequence.
|
1
|
olympiads
| 0.125
|
A cafe has 3 tables and 5 individual counter seats. People enter in groups of size between 1 and 4, and groups never share a table. A group of more than 1 will always try to sit at a table, but will sit in counter seats if no tables are available. Conversely, a group of 1 will always try to sit at the counter first. One morning, \( M \) groups consisting of a total of \( N \) people enter and sit down. Then, a single person walks in and realizes that all the tables and counter seats are occupied by some person or group. What is the minimum possible value of \( M + N \)?
|
16
|
olympiads
| 0.078125
|
You are tossing an unbiased coin. The last $ 28 $ consecutive flips have all resulted in heads. Let $ x $ be the expected number of additional tosses you must make before you get $ 60 $ consecutive heads. Find the sum of all distinct prime factors in $ x $ .
|
5
|
aops_forum
| 0.078125
|
Let \(\tan \alpha\) and \(\tan \beta\) be two solutions of the equation \(x^{2}-3x-3=0\). Find the value of
$$
\left|\sin^2(\alpha+\beta) - 3 \sin(\alpha+\beta) \cos(\alpha+\beta) - 3 \cos^2(\alpha+\beta)\right|.
$$
(Note: \(|x|\) denotes the absolute value of \(x\).)
|
3
|
olympiads
| 0.15625
|
Given two distinct circles in the plane, let \( n \) be the number of common tangent lines that can be drawn to these two circles. What are all possible values of \( n \)?
|
0, 2, 3, 4
|
olympiads
| 0.15625
|
Given quadrilateral \(ABCD\) with an area of 45, diagonals \(AC\) and \(BD\) intersect at point \(P\). Points \(M\) and \(N\) are located on sides \(AB\) and \(CD\) respectively such that \(MB = \frac{1}{3} AB\), \(BP = \frac{3}{5} BD\), \(NC = \frac{2}{3} DC\), and \(PC = \frac{2}{3} AC\). Find the area of quadrilateral \(MBCN\).
|
\frac{79}{3}
|
olympiads
| 0.09375
|
Let \(\Omega\) be a circle of radius 8 centered at point \(O\), and let \(M\) be a point on \(\Omega\). Let \(S\) be the set of points \(P\) such that \(P\) is contained within \(\Omega\), or such that there exists some rectangle \(ABCD\) containing \(P\) whose center is on \(\Omega\) with \(AB = 4\), \(BC = 5\), and \(BC \parallel OM\). Find the area of \(S\).
|
164 + 64\pi
|
olympiads
| 0.203125
|
A circle is randomly chosen in a circle of radius $1$ in the sense that a point is randomly chosen for its center, then a radius is chosen at random so that the new circle is contained in the original circle. What is the probability that the new circle contains the center of the original circle?
|
\frac{1}{4}
|
aops_forum
| 0.125
|
Solve the equation
$$
1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\cdots}} \cdots \frac{1}{x}}=x
$$
where the fractional sign on the left is repeated \(n\) times.
|
\frac{1 + \sqrt{5}}{2}
|
olympiads
| 0.5625
|
For every permutation \( S \) of \( (1,2, \cdots, n) \) with \( n \geq 2 \), let \( f(S) \) be the minimum absolute difference between any two adjacent elements in \( S \). Find the maximum value of \( f(S) \).
|
\left\lfloor \frac{n}{2} \right\rfloor
|
olympiads
| 0.078125
|
For which natural numbers \( n \) and \( k \) do the inequalities \( \left|x_{1}\right|+\ldots+\left|x_{k}\right| \leqslant n \) and \( \left|y_{1}\right|+\ldots+\left|y_{n}\right| \leqslant k \) have the same number of integer solutions \( \left(x_{1}, \ldots, x_{k}\right) \) and \( \left(y_{1}, \ldots, y_{n}\right) \)?
|
For any natural n and k
|
olympiads
| 0.0625
|
Compute the limit of the function:
$$\lim _{x \rightarrow \frac{\pi}{2}} \frac{2+\cos x \cdot \sin \frac{2}{2 x-\pi}}{3+2 x \sin x}$$
|
\frac{2}{3+\pi}
|
olympiads
| 0.21875
|
First, the boat traveled 10 km downstream, and then twice that distance on a lake the river flows into. The entire trip took 1 hour. Find the boat's own speed, given that the river's current speed is 7 km/h.
|
28 \text{ km/h}
|
olympiads
| 0.34375
|
A businessman initially invested $2000 in his business. Every 3 years, he increased his capital by 50%. What was his capital after 18 years?
|
22781.25
|
olympiads
| 0.1875
|
On a sphere with a radius of 2, there are three circles each with a radius of 1, and each circle touches the other two. Find the radius of a circle, smaller than the given ones, that is also situated on the sphere and touches each of the given circles.
|
1 - \sqrt{\frac{2}{3}}
|
olympiads
| 0.078125
|
Given that \( a, b \), and \( c \) are natural numbers, and that \(\mathrm{LCM}(a, b)=945\) and \(\mathrm{LCM}(b, c)=525\), what could \(\mathrm{LCM}(a, c)\) equal?
|
675 or 4725
|
olympiads
| 0.140625
|
Let $\{x_1, x_2, x_3, ..., x_n\}$ be a set of $n$ distinct positive integers, such that the sum of any $3$ of them is a prime number. What is the maximum value of $n$ ?
|
4
|
aops_forum
| 0.265625
|
Given that \( n \) is a positive integer, the real number \( x \) satisfies
$$
|1 - | 2 - |3 - \cdots |(n-1) - |n - x||\cdots||| = x.
$$
Determine the value of \( x \).
|
\frac{1}{2}
|
olympiads
| 0.0625
|
Find the integers $n$ such that the fraction $\frac{3n + 10}{5n + 16}$ is in its simplest form.
|
n est impair
|
olympiads
| 0.125
|
The base side of a regular triangular prism is equal to 1. Find the lateral edge of the prism, given that a sphere can be inscribed in it.
|
\frac{\sqrt{3}}{3}
|
olympiads
| 0.109375
|
A magician and his assistant have a stack of cards, all with the same back and one of 2017 possible colors on the front (with one million cards for each color). The trick is as follows: the magician leaves the room, and the audience picks $n$ cards and places them face up in a row on the table. The assistant then turns over $n-1$ of these cards, leaving only one card face up. The magician re-enters the room, observes the cards on the table, and must guess the color of the face-down card. Find the minimum value of $n$ such that the magician and the assistant can perform this trick using a predetermined strategy.
|
2018
|
olympiads
| 0.28125
|
A compact disc has the shape of a circle with a 5-inch diameter and a 1-inch-diameter circular hole in the center. Assuming the capacity of the CD is proportional to its area, how many inches would need to be added to the outer diameter to double the capacity?
|
2 ext{ inches}
|
olympiads
| 0.265625
|
In a physical education class, 25 students from class 5"B" stood in a line. Each student is either an honor student who always tells the truth or a troublemaker who always lies.
Honor student Vlad took the 13th position. All the students, except Vlad, said: "There are exactly 6 troublemakers between me and Vlad." How many troublemakers are in the line in total?
|
12
|
olympiads
| 0.171875
|
The sum of the squares of two numbers is 20, and their product is 8. Find these numbers.
|
2 ext{ and } 4
|
olympiads
| 0.265625
|
Find the radius of the circumcircle of a right triangle if the radius of the incircle of this triangle is $3 \mathrm{~cm}$ and one of the legs is $10 \mathrm{~cm}$.
|
7.25 \, \text{cm}
|
olympiads
| 0.15625
|
Find the flux of the vector field
$$
\vec{a}=x \vec{i}+(y+z) \vec{j}+(z-y) \vec{k}
$$
through the surface
$$
x^{2}+y^{2}+z^{2}=9
$$
cut by the plane $z=0 \quad(z \geq 0)$ (outward normal to the closed surface formed by these surfaces).
|
54 \pi
|
olympiads
| 0.34375
|
Participants of a summer physics-mathematics camp for schoolchildren received either an orange or a purple T-shirt as a gift. The number of participants in the physics group who received an orange T-shirt is equal to the number of participants in the mathematics group who received a purple T-shirt. Which group is larger - participants in the mathematics group, or those who received orange T-shirts?
|
\text{There are an equal number of participants in the mathematics group as there are who received an orange T-shirt.}
|
olympiads
| 0.171875
|
Calculate the limit of the function:
$$
\lim _{x \rightarrow 0}(1-\ln (\cos x))^{\frac{1}{\tan^2 x}}
$$
|
e^{\frac{1}{2}}
|
olympiads
| 0.171875
|
Let $p > 2$ be a fixed prime number. Find all functions $f: \mathbb Z \to \mathbb Z_p$ , where the $\mathbb Z_p$ denotes the set $\{0, 1, \ldots , p-1\}$ , such that $p$ divides $f(f(n))- f(n+1) + 1$ and $f(n+p) = f(n)$ for all integers $n$ .
|
g(x) = x
|
aops_forum
| 0.109375
|
At 12 o'clock, the angle between the hour hand and the minute hand is 0 degrees. After that, at what time do the hour hand and the minute hand form a 90-degree angle for the 6th time? (12-hour format)
|
3:00
|
olympiads
| 0.0625
|
On an island, kangaroos are always either grey or red. One day, the number of grey kangaroos increased by 28% while the number of red kangaroos decreased by 28%. The ratios of the two types of kangaroos were exactly reversed. By what percentage did the total number of kangaroos change?
|
-4\%
|
olympiads
| 0.0625
|
The king summons two wizards to the palace. He asks Wizard A to write 100 positive real numbers on a card (duplicates allowed) and keep it secret from Wizard B. Wizard B must then accurately write down all 100 positive real numbers; if he fails, both wizards will be executed. Wizard A is allowed to provide a series of numbers to Wizard B, where each number is either one of the 100 positive real numbers or a sum of several of those numbers. However, Wizard A cannot indicate which numbers are the individual real numbers and which are the sums. The king decides to pull an equal number of beard hairs from both wizards according to the number of numbers in the series. Without the wizards being able to communicate beforehand, how many beard hairs will the wizards need to lose at minimum to avoid execution?
|
101
|
olympiads
| 0.140625
|
A pen costs $11$ € and a notebook costs $13$ €. Find the number of ways in which a person can spend exactly $1000$ € to buy pens and notebooks.
|
7
|
aops_forum
| 0.140625
|
Right triangle \( XYZ \), with hypotenuse \( YZ \), has an incircle of radius \(\frac{3}{8}\) and one leg of length 3. Find the area of the triangle.
|
\frac{21}{16}
|
olympiads
| 0.0625
|
$N$ is an integer whose representation in base $b$ is $777$ . Find the smallest integer $b$ for which $N$ is the fourth power of an integer.
|
18
|
aops_forum
| 0.125
|
The side of a rhombus is 8 cm, and its acute angle is $30^{\circ}$. Find the radius of the inscribed circle.
|
2 ext{ cm}
|
olympiads
| 0.265625
|
The diagonal of a parallelogram, of length \(b\), is perpendicular to the side of the parallelogram, of length \(a\). Find the length of the other diagonal of the parallelogram.
|
\sqrt{4a^2 + b^2}
|
olympiads
| 0.078125
|
Write the canonical equations of the line.
$$
\begin{aligned}
& 2 x-3 y-2 z+6=0 \\
& x-3 y+z+3=0
\end{aligned}
$$
|
\frac{x + 3}{9} = \frac{y}{4} = \frac{z}{3}
|
olympiads
| 0.0625
|
Solve the equation \(12 \sin x - 5 \cos x = 13\).
$$
12 \sin x - 5 \cos x = 13
$$
|
x = \frac{\pi}{2} + \operatorname{arctan}\left( \frac{5}{12} \right) + 2k\pi
|
olympiads
| 0.09375
|
Given a cyclic quadrilateral \(ABCD\). Rays \(AB\) and \(DC\) intersect at point \(E\), and rays \(DA\) and \(CB\) intersect at point \(F\). Ray \(BA\) intersects the circumcircle of triangle \(DEF\) at point \(L\), and ray \(BC\) intersects the same circle at point \(K\). The length of segment \(LK\) is 5, and \(\angle EBC = 15^\circ\). Find the radius of the circumcircle of triangle \(EFK\).
|
5
|
olympiads
| 0.140625
|
In an addition problem with two numbers, the first addend is 2000 less than the sum, and the sum is 6 more than the second addend.
Reconstruct the problem.
|
6 + 2000 = 2006
|
olympiads
| 0.515625
|
Into how many parts do the planes of the faces of a tetrahedron divide the space?
|
15
|
olympiads
| 0.171875
|
In triangle \(ABC\), points \(M\) and \(N\) are chosen on sides \(AB\) and \(AC\) respectively. It is known that \(\angle ABC = 70^\circ\), \(\angle ACB = 50^\circ\), \(\angle ABN = 20^\circ\), and \(\angle ACM = 10^\circ\). Find \(\angle NMC\).
|
\angle NMC = 30^
\circ
|
olympiads
| 0.140625
|
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