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stringlengths 33
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0.59
|
|---|---|---|---|
3n students are sitting in 3 rows of n. The students leave one at a time. All leaving orders are equally likely. Find the probability that there are never two rows where the number of students remaining differs by 2 or more.
|
\frac{6n \cdot (n!)^3}{(3n)!}
|
olympiads
| 0.0625 |
Since \(\left|a+\frac{1}{a}\right| \geqslant 2\) and equality is achieved only when \(|a|=1\), the first equation of the system is equivalent to \(\operatorname{tg} x= \pm 1, \sin y=1\). Then from the second equation, \(\cos z=0\).
|
\left(\frac{\pi}{4} + \frac{\pi n}{2}; \frac{\pi}{2} + 2\pi k; \frac{\pi}{2} + \pi l\right)
|
olympiads
| 0.171875 |
The height of a regular tetrahedron is \( h \). Find its total surface area.
|
\frac{3h^2 \sqrt{3}}{2}
|
olympiads
| 0.390625 |
Provide an example to illustrate that for four different positive integers \( a, b, c, d \), there does not necessarily exist a positive integer \( n \) such that \( a+n, b+n, c+n, d+n \) are pairwise coprime.
|
\text{There does not exist a positive integer } n \text{ such that } a+n, b+n, c+n, \text{ and } d+n \text{ are pairwise relatively prime.}
|
olympiads
| 0.15625 |
Positive real numbers \( u, v, w \) are not equal to 1. Given that \( \log_{u} (vw) + \log_{v} w = 5 \) and \( \log_{v} u + \log_{w} v = 3 \), find the value of \( \log_{w} u \).
|
\frac{4}{5}
|
olympiads
| 0.109375 |
What is the distance? A steamboat, according to one of our officers returning from the East, can travel at a speed of 20 km/h with the current and only 15 km/h against the current. Therefore, the entire journey between two points upstream takes 5 hours longer than downstream. What is the distance between these two points?
|
300 \text{ km}
|
olympiads
| 0.5 |
It is the New Year, and students are making handicrafts to give to the elderly at a senior home. Initially, the arts group students worked on it for one day. Then, an additional 15 students joined them, and together they worked for two more days to complete the task. Assuming every student works at the same rate, and if one student were to do it alone, it would take 60 days to complete the project, how many students are in the arts group?
|
10
|
olympiads
| 0.5 |
Working alone, Alfred and Bill together can complete a certain task in 24 days. If Alfred can only do $\frac{2}{3}$ of what Bill can do, how many days will it take each of them to complete the same task individually?
|
60 \text{ days and } 40 \text{ days}
|
olympiads
| 0.53125 |
Let \( N \) be the set of all positive integers, and let \( f \) be a function from \( N \) to \( N \) such that for any \( s \) and \( t \) in \( N \), the following condition is satisfied:
$$
f\left(t^{2} f(s)\right)=s(f(t))^{2}.
$$
Determine the minimum possible value of \( f(1998) \) among all such functions \( f \).
|
1998
|
olympiads
| 0.140625 |
Find all positive roots of the equation \( x^{x} + x^{1-x} = x + 1 \).
|
x = 1
|
olympiads
| 0.421875 |
Horizontal parallel segments \( AB = 10 \) and \( CD = 15 \) are the bases of trapezoid \( ABCD \). Circle \(\gamma\) of radius 6 has its center within the trapezoid and is tangent to sides \( AB \), \( BC \), and \( DA \). If side \( CD \) cuts out an arc of \(\gamma\) measuring \(120^\circ\), find the area of \(ABCD\).
|
\frac{225}{2}
|
olympiads
| 0.3125 |
Vasya must read an entertaining math book in three days. On the first day, he read half of the book, on the second day, he read one-third of the remaining pages, and on the third day, he read the number of pages equal to half the pages read on the first two days. Did Vasya manage to read the book in three days?
|
Вася успел прочитать книгу за три дня.
|
olympiads
| 0.125 |
Consider the points $1,\frac12,\frac13,...$ on the real axis. Find the smallest value $k \in \mathbb{N}_0$ for which all points above can be covered with 5 **closed** intervals of length $\frac1k$ .
|
k = 10
|
aops_forum
| 0.078125 |
Let \( n \in \mathbb{N} \), where each digit of \( n \) is either 0 or 7, and \( n \) is divisible by 15. Find the smallest natural number \( n \).
|
7770
|
olympiads
| 0.375 |
Given a quadratic function $y$ with respect to $x$ . $y$ takes the values of $-1, -2$ and $1$ for the values of $x=-1,\ 0,\ 1$ respectively.
Find the length of the line segment of the graph of the curve cut by the $x$ -axis.
1956 Tokyo Institute of Technology entrance exam
|
\frac{\sqrt{17}}{2}
|
aops_forum
| 0.1875 |
Find the sum of the squares of the distances from the vertices of a regular $n$-gon inscribed in a circle of radius $R$ to any line passing through the center of the polygon.
|
\frac{1}{2}nR^2
|
olympiads
| 0.265625 |
Calculate: \(\left(2 \frac{2}{3} \times\left(\frac{1}{3}-\frac{1}{11}\right) \div\left(\frac{1}{11}+\frac{1}{5}\right)\right) \div \frac{8}{27} = 7 \underline{1}\).
|
7 \frac{1}{2}
|
olympiads
| 0.296875 |
A king summoned two wise men. He gave the first one 100 blank cards and instructed him to write a positive number on each (the numbers do not have to be different), without showing them to the second wise man. Then, the first wise man can communicate several distinct numbers to the second wise man, each of which is either written on one of the cards or is a sum of the numbers on some cards (without specifying exactly how each number is derived). The second wise man must determine which 100 numbers are written on the cards. If he cannot do this, both will be executed; otherwise, a number of hairs will be plucked from each of their beards equal to the amount of numbers the first wise man communicated. How can the wise men, without colluding, stay alive and lose the minimum number of hairs?
|
101
|
olympiads
| 0.1875 |
In a convex \( n \)-sided polygon, all the diagonals are drawn and no three of them pass through a point. Find a formula for the number of regions formed inside the polygon.
|
1 + \binom{n}{2} - n + \binom{n}{4}
|
olympiads
| 0.25 |
Once, Baba Yaga and Koschei the Deathless were trying to equally divide a magical powder that turns everything into gold. Baba Yaga took some scales and weighed all the powder. The scales showed 6 zolotniks. Then she removed some powder until the scales showed 3 zolotniks. However, Koschei suspected that the scales were lying and weighed the separated portion on the same scales (there were no other scales available). The scales showed 2 zolotniks. Determine the exact weight of the two parts into which Baba Yaga divided the powder. Assume that if the scales lie, they always do so by the same amount.
|
4 \text{ and } 3 \text{ zolotniks}
|
olympiads
| 0.078125 |
Three teams scored 285 points in a competition. If the team from school No. 24 had scored 8 points less, the team from school No. 46 had scored 12 points less, and the team from school No. 12 had scored 7 points less, they all would have scored the same number of points. How many points did the teams from schools No. 24 and No. 12 score together?
|
187
|
olympiads
| 0.578125 |
Find the number of words of length 10 consisting only of the letters "a" and "b" and not containing two consecutive "b"s.
|
144
|
olympiads
| 0.5625 |
If the function \( f(x) = \frac{(\sqrt{1008} x + \sqrt{1009})^{2} + \sin 2018 x}{2016 x^{2} + 2018} \) has a maximum value of \( M \) and a minimum value of \( m \), then \( M + m \) equals \_\_\_\_.
|
1
|
olympiads
| 0.125 |
Given two nonzero numbers, if 1 is added to each of them, and 1 is subtracted from each of them, the sum of the reciprocals of the four resulting numbers will be 0. What number can be obtained if the sum of the original numbers is subtracted from the sum of their reciprocals? Find all possibilities.
|
0
|
olympiads
| 0.28125 |
The distances from point A to point B by the river and by the channel are both equal to 1 km. The speed of the current in the channel is $V$ km/h, while in the river it is $(2V+1)$ km/h. The current in both the river and the channel flows from A to B. If the difference between the travel times of the boat through the channel from B to A and back is added to the travel time of a raft along the river from A to B, the sum is exactly 1 hour. By how many kilometers per hour is the speed of the boat greater than the speed of the current in the channel? The value of $V$ is not given. The answer should be a number.
|
1
|
olympiads
| 0.125 |
Collinear points $A$ , $B$ , and $C$ are given in the Cartesian plane such that $A= (a, 0)$ lies along the x-axis, $B$ lies along the line $y=x$ , $C$ lies along the line $y=2x$ , and $\frac{AB}{BC}=2$ . If $D= (a, a)$ , and the circumcircle of triangle $ADC$ intersects the line $y=x$ again at $E$ , and ray $AE$ intersects $y=2x$ at $F$ , evaluate $\frac{AE}{EF}$ .
|
\frac{\sqrt{2}}{2}
|
aops_forum
| 0.109375 |
What is the remainder on dividing \(1234^{567} + 89^{1011}\) by 12?
|
9
|
olympiads
| 0.296875 |
Determine the angle at which the graph of the curve \( f(x) = e^x - x \) intersects the y-axis.
|
0^
g
|
olympiads
| 0.4375 |
Extend the graph of \( y = \sin x \) over the interval \(\left[0, \frac{\pi}{2}\right)\) to the entire real number line so that it becomes a function with period \(\frac{\pi}{2}\). What is the analytical expression of this new function?
|
\sin \left( x - \frac{k \pi}{2} \right) \quad \text{for } x \in \left[ \frac{k \pi}{2}, \frac{(k+1) \pi}{2}\right) \text{ and } k \in \mathbb{Z}
|
olympiads
| 0.484375 |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively, and $b \neq 1$. Also, both $\frac{C}{A}$ and $\frac{\sin B}{\sin A}$ are roots of the equation $\log_{\sqrt{b}} x = \log_{b}(4x - 4)$. Determine the shape of $\triangle ABC$.
|
Right triangle
|
olympiads
| 0.0625 |
The school plans to organize a movie viewing for the students either on January 4th or January 10th. After finalizing the date, the teacher informs the class leader. However, due to the similarity in pronunciation between "four" and "ten," there is a 10% chance that the class leader hears it incorrectly (mistaking 4 for 10 or 10 for 4). The class leader then informs Xiaoming, who also has a 10% chance of hearing it incorrectly. What is the probability that Xiaoming correctly believes the movie date?
|
82\%
|
olympiads
| 0.375 |
Let $\xi$ and $\eta$ be the lifetimes of a blue and a red lightbulb, respectively. The flashlight fails when the last bulb burns out, i.e., the lifetime of the flashlight is equal to the maximum of $\xi$ and $\eta$. Obviously, $\max (\xi, \eta) \geq \eta$. Proceeding to the expectations: $\operatorname{E} \max (\xi, \eta) \geq \operatorname{E} \eta = 4$. Therefore, the expected lifetime of the flashlight is at least four years.
|
The expected lifetime of the flashlight is at least 4 years
|
olympiads
| 0.0625 |
In the sequence 1, 3, 2, ... each term after the first two is equal to the preceding term minus the term before it, that is, if \( n>2 \), then \( a_n = a_{n-1} - a_{n-2} \). What is the sum of the first one hundred terms of this sequence?
|
5
|
olympiads
| 0.203125 |
Kiril Konstantinovich's age is 48 years, 48 months, 48 weeks, 48 days, and 48 hours. How many full years old is Kiril Konstantinovich?
|
53
|
olympiads
| 0.109375 |
Let \( m \) be a fixed integer greater than 1, and the sequence \( \{x_{n}\} \) be defined as follows:
\[
x_{i}=\left\{
\begin{array}{ll}
2^{i}, & 0 \leq i \leq m-1 ; \\
\sum_{j=1}^{m} x_{i-j}, & i \geq m .
\end{array}
\right.
\]
Find the maximum value of \( k \) such that there are \( k \) consecutive terms in the sequence that are all divisible by \( m \).
|
m-1
|
olympiads
| 0.078125 |
Consider a hexagon with vertices labeled $M$ , $M$ , $A$ , $T$ , $H$ , $S$ in that order. Clayton starts at the $M$ adjacent to $M$ and $A$ , and writes the letter down. Each second, Clayton moves to an adjacent vertex, each with probability $\frac{1}{2}$ , and writes down the corresponding letter. Clayton stops moving when the string he's written down contains the letters $M, A, T$ , and $H$ in that order, not necessarily consecutively (for example, one valid string might be $MAMMSHTH$ .) What is the expected length of the string Clayton wrote?
|
6
|
aops_forum
| 0.0625 |
There are two piles of stones, one containing 15 stones and the other containing 20 stones. Two players take turns in the following game: on each turn, a player can take any number of stones, but only from one pile. The player who cannot make a move loses. Who wins with optimal play?
|
First Player
|
olympiads
| 0.109375 |
How many real numbers \( x \) exist such that the value of the expression \( \sqrt{123 - \sqrt{x}} \) is an integer?
|
12
|
olympiads
| 0.28125 |
In the plane, there are $n$ different lines, each of which intersects all the others, and no three lines pass through the same point. Into how many parts do they divide the plane?
|
F(n) = \frac{n(n + 1)}{2} + 1
|
olympiads
| 0.0625 |
In a test, each question has five answer choices. A top student answers all questions correctly. A failing student answers correctly when they manage to cheat, and otherwise answers randomly (i.e., answers correctly on $1 / 5$ of the questions they did not cheat on). Overall, the failing student answered half of the questions correctly. What fraction of the answers did they manage to cheat on?
|
\frac{3}{8}
|
olympiads
| 0.375 |
Find the derivative.
\[
y = \frac{1}{\sqrt{2}} \ln \left(\sqrt{2} \tan x + \sqrt{1 + 2 \tan^2 x}\right)
\]
|
\frac{1}{\cos^2 x \sqrt{1 + 2 \tan^2 x}}
|
olympiads
| 0.296875 |
Let $n$ be a positive integer. A child builds a wall along a line with $n$ identical cubes. He lays the first cube on the line and at each subsequent step, he lays the next cube either on the ground or on the top of another cube, so that it has a common face with the previous one. How many such distinct walls exist?
|
2^{n-1}
|
aops_forum
| 0.109375 |
Three cones with vertex $A$ touch each other externally, with the first two being identical and the third having an apex angle of $\frac{\pi}{4}$. Each cone internally touches a fourth cone with its vertex at point $A$ and an apex angle of $\frac{3 \pi}{4}$. Find the apex angle of the first two cones. (The apex angle of a cone is defined as the angle between its generatrices in the axial cross-section.)
|
2 \operatorname{arctg} \frac{2}{3}
|
olympiads
| 0.078125 |
Kolya had 10 sheets of paper. On the first step, he chooses one sheet and divides it into two parts. On the second step, he chooses one sheet from those available and divides it into 3 parts, on the third step he chooses one sheet from those available and divides it into 4 parts, and so on. After which step will the number of sheets first exceed 500?
|
31
|
olympiads
| 0.109375 |
Find the functions from $\mathbb{R}$ to $\mathbb{R}$ such that:
$$
f(x+f(x+y))+f(x y)=x+f(x+y)+y f(x)
$$
|
f(x) = x
|
olympiads
| 0.296875 |
The diagonals \( AC \) and \( BD \) of a convex quadrilateral \( ABCD \), which are 3 and 4 respectively, intersect at an angle of \( 75^{\circ} \). What is the sum of the squares of the lengths of the segments connecting the midpoints of opposite sides of the quadrilateral?
|
12.5
|
olympiads
| 0.0625 |
As shown in Figure 3, given that $M$ is a point inside rectangle $ABCD$ with $AB=1$ and $BC=2$, and $t = AM \cdot MC + BM \cdot MD$, find the minimum value of $t$.
|
2
|
olympiads
| 0.125 |
At the beginning of the game, the number 0 is written on the board. Two players take turns. On each turn, a player adds any natural number not exceeding 10 to the written number and writes the result on the board instead of the original number. The winner is the one who first reaches a four-digit number. Which player (the first player or their opponent) has a winning strategy, i.e., a strategy that guarantees their victory regardless of how the opponent plays?
|
The first player (starting player) has a winning strategy.
|
olympiads
| 0.140625 |
If the graph of the function \( g(x) \) is symmetric with the graph of the function \( f(x)=\frac{1-2^{x}}{1+2^{x}} \) about the line \( y = x \), what is the value of \( g\left(\frac{2}{7}\right) \)?
|
\log_{2}\left( \frac{5}{9} \right)
|
olympiads
| 0.3125 |
The vertices of a regular $2012$ -gon are labeled $A_1,A_2,\ldots, A_{2012}$ in some order. It is known that if $k+\ell$ and $m+n$ leave the same remainder when divided by $2012$ , then the chords $A_kA_{\ell}$ and $A_mA_n$ have no common points. Vasya walks around the polygon and sees that the first two vertices are labeled $A_1$ and $A_4$ . How is the tenth vertex labeled?
|
A_{28}
|
aops_forum
| 0.09375 |
Two 7th grade students and several 8th grade students participated in a chess tournament. The two 7th grade students scored 8 points in total, and each 8th grade student scored the same number of points. How many 8th grade students participated in the tournament? (Each participant plays one game against each of the other participants. A win is worth 1 point, a draw is worth 1/2 point, and a loss is worth 0 points.)
|
7 \text{ or } 14
|
olympiads
| 0.140625 |
In the fourth annual Swirled Series, the Oakland Alphas are playing the San Francisco Gammas. The first game is played in San Francisco and succeeding games alternate in location. San Francisco has a 50% chance of winning their home games, while Oakland has a probability of 60% of winning at home. Normally, the series will stretch on forever until one team gets a three-game lead, in which case they are declared the winners. However, after each game in San Francisco, there is a 50% chance of an earthquake, which will cause the series to end with the team that has won more games declared the winner. What is the probability that the Gammas will win?
|
\frac{34}{73}
|
olympiads
| 0.078125 |
A cube with side length $100cm$ is filled with water and has a hole through which the water drains into a cylinder of radius $100cm.$ If the water level in the cube is falling at a rate of $1 \frac{cm}{s} ,$ how fast is the water level in the cylinder rising?
|
\frac{1}{\pi}
|
aops_forum
| 0.21875 |
A pet shop sells cats and two types of birds: ducks and parrots. In the shop, $\tfrac{1}{12}$ of animals are ducks, and $\tfrac{1}{4}$ of birds are ducks. Given that there are $56$ cats in the pet shop, how many ducks are there in the pet shop?
|
7
|
aops_forum
| 0.3125 |
Through a point not lying on a given line, construct a line parallel to the given line using a compass and straightedge.
|
The constructed line MN is parallel to l.
|
olympiads
| 0.203125 |
Neznaika is drawing closed paths inside a $5 \times 8$ rectangle, traveling along the diagonals of $1 \times 2$ rectangles. In the illustration, an example of a path passing through 12 such diagonals is shown. Help Neznaika draw the longest possible path.
|
20
|
olympiads
| 0.125 |
Given a convex quadrilateral \(ABCD\). Point \(M\) is the midpoint of side \(BC\), and point \(N\) is the midpoint of side \(CD\). The segments \(AM\), \(AN\), and \(MN\) divide the quadrilateral into four triangles whose areas, in some order, are consecutive natural numbers. What is the largest possible area that triangle \(ABD\) can have?
|
6
|
olympiads
| 0.0625 |
In a quadrilateral, the diagonals are of lengths \(a\) and \(b\). Find the perimeter of the quadrilateral formed by the midpoints of the sides of the given quadrilateral.
|
a + b
|
olympiads
| 0.203125 |
The non-negative numbers $x,y,z$ satisfy the relation $x + y+ z = 3$ . Find the smallest possible numerical value and the largest possible numerical value for the expression $$ E(x,y, z) = \sqrt{x(y + 3)} + \sqrt{y(z + 3)} + \sqrt{z(x + 3)} . $$
|
3 \leq E(x, y, z) \leq 6
|
aops_forum
| 0.078125 |
Given that the structure of a methane molecule $\mathrm{CH}_{4}$ consists of a central carbon atom surrounded by 4 hydrogen atoms (these four hydrogen atoms form the four vertices of a regular tetrahedron), let $\theta$ be the angle between any two of the four line segments connecting the central carbon atom to the four hydrogen atoms. Determine $\cos \theta$.
|
- \frac{1}{3}
|
olympiads
| 0.0625 |
Given the function \( f(x) = a x - \frac{3}{2} x^2 \) has a maximum value of no more than \( \frac{1}{6} \), and when \( x \in \left[ \frac{1}{4}, \frac{1}{2} \right] \), \( f(x) \geqslant \frac{1}{8} \), find the value of \( a \).
|
1
|
olympiads
| 0.40625 |
Which quantity is greater: five-digit numbers that are not divisible by 5, or those in which neither the first nor the second digit from the left is a five?
|
Еqual
|
olympiads
| 0.078125 |
Using only a ruler, construct a segment \(CD\) that is equal in length and parallel to segment \(AB\) on graph paper.
|
CD = AB
|
olympiads
| 0.40625 |
There are 12 matches, each 2 cm long. Is it possible to form a polygon with an area of 16 cm² using all the matches? (The matches cannot be broken, and all must be used.)
|
16 \text{ cm}^2
|
olympiads
| 0.1875 |
Factorize \(a^{4} + 4 b^{4}\) (this is Sophie Germain's identity).
Hint: We have \(a^{4} + 4 b^{4} = a^{4} + 4 b^{4} + 4 a^{2} b^{2} - 4 a^{2} b^{2}\).
|
(a^2 + 2ab + 2b^2)(a^2 - 2ab + 2b^2)
|
olympiads
| 0.21875 |
A positive integer is thought of. The digit 7 is appended to the end of this number, and from the resulting new number, the square of the original number is subtracted. The remainder is then reduced by 75% of this remainder and the original number is subtracted again. The final result is zero. What number was thought of?
|
7
|
olympiads
| 0.21875 |
The three medians of a triangle divide its angles into six smaller angles, among which exactly \( k \) are greater than \( 30^\circ \). What is the maximum possible value of \( k \)?
|
3
|
olympiads
| 0.46875 |
Construct a circle with the given radius that is tangent to a given line and a given circle. How many solutions does this problem have?
|
0 \; \text{to} \; 8
|
olympiads
| 0.078125 |
If the positive integer \( a \) makes the maximum value of the function \( f(x) = x + \sqrt{13 - 2ax} \) also a positive integer, then that maximum value is ___.
|
7
|
olympiads
| 0.109375 |
You have a lighter and two strings, each of which takes one hour to burn completely when lit from one end (though not necessarily at a uniform rate). How can you measure three quarters of an hour?
|
45 \text{ minutes}
|
olympiads
| 0.09375 |
During a family reunion for the Spring Festival, if the average age of the remaining members, excluding the oldest person, is 18 years old; and if the average age of the remaining members, excluding the youngest person, is 20 years old, it is known that the age difference between the oldest and the youngest person is 40 years. How many people are attending the reunion?
|
21
|
olympiads
| 0.296875 |
Find the limit \(\lim _{x \rightarrow 0} x \cdot \operatorname{ctg} \frac{x}{3}\).
|
3
|
olympiads
| 0.28125 |
The square of the sum of the digits of the number \( A \) is equal to the sum of the digits of the number \( A^2 \). Find all such two-digit numbers \( A \).
|
10, 11, 12, 13, 20, 21, 22, 30, 31
|
olympiads
| 0.078125 |
Let $[n]$ denote the set of integers $\left\{ 1, 2, \ldots, n \right\}$ . We randomly choose a function $f:[n] \to [n]$ , out of the $n^n$ possible functions. We also choose an integer $a$ uniformly at random from $[n]$ . Find the probability that there exist positive integers $b, c \geq 1$ such that $f^b(1) = a$ and $f^c(a) = 1$ . ( $f^k(x)$ denotes the result of applying $f$ to $x$ $k$ times.)
|
\frac{1}{n}
|
aops_forum
| 0.140625 |
In a store, Sasha bought two pens, three notebooks, and one pencil and paid 33 rubles. Dima bought one pen, one notebook, and two pencils and paid 20 rubles. How much did Tanya pay for four pens, five notebooks, and five pencils?
|
73 \text{ рубля}
|
olympiads
| 0.359375 |
Find all integers \( n \) for which the number \( \left|n^{2} - 6n - 27\right| \) is prime.
|
-4, -2, 8, 10
|
olympiads
| 0.359375 |
Find any three vectors with a zero sum such that, by subtracting the third vector from the sum of any two vectors, a vector of length 1 is obtained.
|
\mathbf{a}, \mathbf{b}, \mathbf{c}
|
olympiads
| 0.125 |
An electrician was called to repair a string of four serially connected light bulbs, one of which burnt out. It takes 10 seconds to unscrew any bulb from the string and 10 seconds to screw one in. The time spent on other actions is negligible. What is the minimum amount of time the electrician needs to find the burnt-out bulb, if he has one spare bulb?
|
60 \text{ seconds}
|
olympiads
| 0.109375 |
A bowl contained $10\%$ blue candies and $25\%$ red candies. A bag containing three quarters red candies and one quarter blue candies was added to the bowl. Now the bowl is $16\%$ blue candies. What percentage of the candies in the bowl are now red?
|
45\%
|
aops_forum
| 0.125 |
Find all functions $f: R \to R$ and $g:R \to R$ such that $f(x-f(y))=xf(y)-yf(x)+g(x)$ for all real numbers $x,y$ .
I.Voronovich
|
f(x) = 0 and g(x) = 0 or f(x) = x and g(x) = 0
|
aops_forum
| 0.09375 |
On the coordinate plane, a circle with center \( T(3,3) \) passes through the origin \( O(0,0) \). If \( A \) is a point on the circle such that \( \angle AOT = 45^\circ \) and the area of \( \triangle AOT \) is \( Q \) square units, find the value of \( Q \).
|
9
|
olympiads
| 0.453125 |
As shown, the vertices of the shaded square are the midpoints of each side of the large square $\mathrm{EFGH}$. Semicircles are drawn outward with half of each side of the large square as the diameter, and semicircles are drawn outward with each side of the shaded square as the diameter, forming 8 "crescent" shapes. The total area of these 8 "crescent" shapes is 5 square centimeters. What is the area of the large square EFGH in square centimeters?
|
10 \text{ square cm}
|
olympiads
| 0.0625 |
Find the number of pairs $(a, b)$ of positive integers with the property that the greatest common divisor of $a$ and $ b$ is equal to $1\cdot 2 \cdot 3\cdot ... \cdot50$ , and the least common multiple of $a$ and $ b$ is $1^2 \cdot 2^2 \cdot 3^2\cdot ... \cdot 50^2$ .
|
32768
|
aops_forum
| 0.09375 |
Real numbers \(a\) and \(b\) satisfy the equations \(a^{5} + b^{5} = 3\) and \(a^{15} + b^{15} = 9\). Find the value of the expression \(a^{10} + b^{10}\).
|
5
|
olympiads
| 0.140625 |
How many four-digit numbers divisible by 5 can be formed from the digits \(0, 1, 3, 5, 7\) if each number must not contain repeated digits?
|
42
|
olympiads
| 0.203125 |
An engineering project was initially worked on by 6 people and they completed $\frac{1}{3}$ of the project in 35 days. After that, 6 more people joined to work on the project together. How many days in total did it take to complete the project?
|
70
|
olympiads
| 0.3125 |
One fifth of a swarm of bees is sitting on a Kadamba flower, one third on Silindha flowers. The thrice the difference of the last two numbers went to Kutaja flowers. And there is also one bee flying, attracted by the wonderful fragrance of jasmine and pandanus. Tell me, charming one, how many bees are there in total?
|
15
|
olympiads
| 0.5625 |
On Valentine's Day, each male student in the school gave each female student a valentine card. It turned out that the number of valentine cards was 30 more than the total number of students. How many valentine cards were given?
|
64
|
olympiads
| 0.515625 |
Two-digit numbers whose tens and units digits are reversed, like 18 and 81, are called a "family". Their sum is 99. How many such "families" exist?
|
8
|
olympiads
| 0.359375 |
Find the smallest positive number $\alpha$ such that there exists a positive number $\beta$ satisfying the inequality
$$
\sqrt{1+x}+\sqrt{1-x} \leqslant 2-\frac{x^{\alpha}}{\beta}
$$
for $0 \leqslant x \leqslant 1$.
|
2
|
olympiads
| 0.453125 |
The random variable \(X\) is given by the probability density function \( f(x) = \frac{1}{2} \sin x \) in the interval \((0, \pi)\); outside this interval \(f(x) = 0\). Find the expected value of the random variable \(Y = \varphi(X) = X^2\), by first determining the density function \(g(Y)\) of the variable \(Y\).
|
\frac{\pi^2 - 2}{2}
|
olympiads
| 0.109375 |
In $\triangle ABC$, \( O \) is the circumcenter of $\triangle ABC$, and it satisfies \(\overrightarrow{AO} \cdot \overrightarrow{AB} + \overrightarrow{BO} \cdot \overrightarrow{BC} = \overrightarrow{CO} \cdot \overrightarrow{CA}\). Determine the measure of \(\angle B\).
|
\frac{\pi}{2}
|
olympiads
| 0.140625 |
The sides of a triangle are in geometric progression. What values can the common ratio of the progression take?
|
\frac{\sqrt{5} - 1}{2} \leq q \leq \frac{\sqrt{5} + 1}{2}
|
olympiads
| 0.0625 |
Let $\triangle P_1P_2P_3$ be an equilateral triangle. For each $n\ge 4$ , *Mingmingsan* can set $P_n$ as the circumcenter or orthocenter of $\triangle P_{n-3}P_{n-2}P_{n-1}$ . Find all positive integer $n$ such that *Mingmingsan* has a strategy to make $P_n$ equals to the circumcenter of $\triangle P_1P_2P_3$ .
|
n \equiv 0 \pmod{4}
|
aops_forum
| 0.140625 |
William is biking from his home to his school and back, using the same route. When he travels to school, there is an initial $20^\circ$ incline for $0.5$ kilometers, a flat area for $2$ kilometers, and a $20^\circ$ decline for $1$ kilometer. If William travels at $8$ kilometers per hour during uphill $20^\circ$ sections, $16$ kilometers per hours during flat sections, and $20$ kilometers per hour during downhill $20^\circ$ sections, find the closest integer to the number of minutes it take William to get to school and back.
|
29
|
aops_forum
| 0.421875 |
An integer $ m > 1$ is given. The infinite sequence $ (x_n)_{n\ge 0}$ is defined by $ x_i\equal{}2^i$ for $ i<m$ and $ x_i\equal{}x_{i\minus{}1}\plus{}x_{i\minus{}2}\plus{}\cdots \plus{}x_{i\minus{}m}$ for $ i\ge m$ .
Find the greatest natural number $ k$ such that there exist $ k$ successive terms of this sequence which are divisible by $ m$ .
|
m
|
aops_forum
| 0.140625 |
Find all prime numbers of the form $\tfrac{1}{11} \cdot \underbrace{11\ldots 1}_{2n \textrm{ ones}}$ , where $n$ is a natural number.
|
101
|
aops_forum
| 0.078125 |
Find the smallest positive integer $n$ such that a cube with sides of length $n$ can be divided up into exactly $2007$ smaller cubes, each of whose sides is of integer length.
|
n = 13
|
aops_forum
| 0.140625 |
Compute the limit of the numerical sequence:
\[
\lim _{n \rightarrow \infty} \frac{\sqrt{n+2}-\sqrt[3]{n^{3}+2}}{\sqrt[7]{n+2}-\sqrt[5]{n^{5}+2}}
\]
|
1
|
olympiads
| 0.109375 |
Find the maximum of the function \( f(x, y, z) = 3x + 5y - z \) on the sphere of radius 1, and the points where this maximum is attained.
|
\sqrt{35}
|
olympiads
| 0.359375 |
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