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stringlengths 33
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|---|---|---|---|
Anya wants to buy ice cream that costs 19 rubles. She has two 10-ruble coins, two 5-ruble coins, and one 2-ruble coin in her pocket. Anya randomly picks three coins from her pocket without looking. Find the probability that the selected coins will be enough to pay for the ice cream.
|
0.4
|
olympiads
| 0.171875
|
Given the function \( f(x) = \frac{\sqrt{2} \sin \left(x + \frac{\pi}{4}\right) + 2x^2 + x}{2x^2 + \cos x} \), let \( M \) be its maximum value and \( m \) be its minimum value. Find the value of \( M + m \).
|
2
|
olympiads
| 0.0625
|
On an island, there are knights who always tell the truth and liars who always lie. Before a friendly match, 30 islanders gathered wearing shirts with numbers on them—random natural numbers. Each of them said, "I have a shirt with an odd number." After that, they exchanged shirts, and each said, "I have a shirt with an even number." How many knights participated in the exchange?
|
15
|
olympiads
| 0.46875
|
Determine all real $ x$ satisfying the equation \[ \sqrt[5]{x^3 \plus{} 2x} \equal{} \sqrt[3]{x^5\minus{}2x}.\] Odd roots for negative radicands shall be included in the discussion.
|
0
|
aops_forum
| 0.234375
|
By how much did the dollar exchange rate change over the course of 2014 (from January 1, 2014, to December 31, 2014)? Provide the answer in rubles, rounded to the nearest whole number (answer - whole number).
|
24
|
olympiads
| 0.0625
|
The nonzero numbers \(x\) and \(y\) satisfy the inequalities \(x^{4} - y^{4} > x\) and \(y^{4} - x^{4} > y\). What sign can the product \(xy\) have (indicate all possibilities)?
|
positive
|
olympiads
| 0.078125
|
It is given that \(a, b, c\) are three real numbers such that the roots of the equation \(x^{2} + 3x - 1 = 0\) also satisfy the equation \(x^{4} + a x^{2} + b x + c = 0\). Find the value of \(a + b + 4c + 100\).
|
93
|
olympiads
| 0.0625
|
In an arithmetic sequence \(\left\{a_{n}\right\}\), if \(\frac{a_{11}}{a_{10}} < -1\), and its partial sum \(S_{n}\) has a maximum value, then when \(S_{n}\) attains its smallest positive value, \(n = \) \(\qquad\).
|
19
|
olympiads
| 0.0625
|
We define the ridiculous numbers recursively as follows:
[list=a]
[*]1 is a ridiculous number.
[*]If $a$ is a ridiculous number, then $\sqrt{a}$ and $1+\sqrt{a}$ are also ridiculous numbers.
[/list]
A closed interval $I$ is ``boring'' if
- $I$ contains no ridiculous numbers, and
- There exists an interval $[b,c]$ containing $I$ for which $b$ and $c$ are both ridiculous numbers.
The smallest non-negative $l$ such that there does not exist a boring interval with length $l$ can be represented in the form $\dfrac{a + b\sqrt{c}}{d}$ where $a, b, c, d$ are integers, $\gcd(a, b, d) = 1$ , and no integer square greater than 1 divides $c$ . What is $a + b + c + d$ ?
|
9
|
aops_forum
| 0.0625
|
The function \( f(x) = \frac{x^{2}}{8} + x \cos x + \cos (2x) \) (for \( x \in \mathbf{R} \)) has a minimum value of ___
|
-1
|
olympiads
| 0.109375
|
One fine spring evening, I went to the Longchamp racetrack. I bet on the first horse and doubled the amount of money I had with me. Inspired by this example, I bet 60 francs on the second horse and lost it completely. However, thanks to the third horse, I doubled my cash again. Nevertheless, I lost 60 francs on the fourth horse. The fifth horse allowed me to double the remaining amount once more. But the sixth horse, on which I also bet 60 francs, proved fatal for me - after this race, I was left with nothing from the original amount.
With how much money did I go to the racetrack?
|
52.5 \text{ francs}
|
olympiads
| 0.421875
|
There is a mathematical operator "$\odot$" that satisfies the following equations: $2 \odot 4=8, 4 \odot 6=14, 5 \odot 3=13, 8 \odot 7=23$. According to this rule, what is $9 \odot 3=?$
|
21
|
olympiads
| 0.078125
|
Let $M$ be on segment $ BC$ of $\vartriangle ABC$ so that $AM = 3$ , $BM = 4$ , and $CM = 5$ . Find the largest possible area of $\vartriangle ABC$ .
|
\frac{27}{2}
|
aops_forum
| 0.078125
|
Find all values of the parameter \( a \) for which the system
\[
\left\{\begin{array}{l}
x = -|y-\sqrt{a}| + 6 - \sqrt{a} \\
(|x|-8)^2 + (|y|-15)^2 = 289
\end{array}\right.
\]
has exactly three solutions.
|
a \in \{ 9, 121, \left(\frac{17\sqrt{2}-1}{2}\right)^2 \}.
|
olympiads
| 0.0625
|
Nadya wants to cut a sheet of paper into 48 identical rectangles for a game. What is the minimum number of cuts she needs to make, given that the pieces of paper can be rearranged but not folded, and Nadya is able to cut any number of layers of paper simultaneously? (Each cut is a straight line from edge to edge of the piece.)
|
6
|
olympiads
| 0.0625
|
In a tetrahedron \(ABCD\), the faces \(ABC\) and \(ABD\) have areas \(p\) and \(q\) respectively, and form an angle \(\alpha\) between them. Find the area of the cross-section passing through the edge \(AB\) and the center of the sphere inscribed in the tetrahedron.
|
\frac{2pq \cos(\alpha/2)}{p + q}
|
olympiads
| 0.0625
|
Let $ABCD$ be a square, such that the length of its sides are integers. This square is divided in $89$ smaller squares, $88$ squares that have sides with length $1$ , and $1$ square that has sides with length $n$ , where $n$ is an integer larger than $1$ . Find all possible lengths for the sides of $ABCD$ .
|
13 \text{ and } 23
|
aops_forum
| 0.265625
|
When Harold Tompkins arrived at the station this morning, he found that he had little money with him. He spent half of his money on a ticket and 5 cents on candies. Then, he spent half of what was left plus 10 cents on a newspaper upon leaving the train. Next, half of the remaining money was spent on a bus fare, and he gave 15 cents to a beggar standing at the club door. Consequently, he now only has 5 cents left. How much money did he take from home?
|
2 dollars 10 cents
|
olympiads
| 0.140625
|
What is the maximum number of natural numbers that can be written in a row such that the sum of any three consecutive numbers is even, and the sum of any four consecutive numbers is odd?
|
5
|
olympiads
| 0.28125
|
In trapezoid \(ABCD\), the bases \(AD = 16\) and \(BC = 9\) are given. On the extension of \(BC\), a point \(M\) is chosen such that \(CM = 3.2\).
In what ratio does line \(AM\) divide the area of trapezoid \(ABCD\)?
|
8:15
|
olympiads
| 0.09375
|
The numbers $12,14,37,65$ are one of the solutions of the equation $xy-xz+yt=182$ . What number corresponds to which letter?
|
x = 12, y = 37, z = 65, t = 14
|
aops_forum
| 0.09375
|
There is a group of students with less than 100 people. If they are arranged in rows of 9, there are 4 students left. If they are arranged in rows of 7, there are 3 students left. How many students are there at most in this group?
|
94
|
olympiads
| 0.53125
|
Three businessmen - Smith, Robinson, and Jones - live in the Leeds-Sheffield area. Three railroad workers with the same last names also live in the area. Businessman Robinson and a conductor live in Sheffield, businessman Jones and a stoker live in Leeds, and businessman Smith and the railroad engineer live halfway between Leeds and Sheffield. The conductor’s namesake earns $10,000 a year, and the engineer earns exactly 1/3 of what the businessman who lives closest to him earns. Finally, railroad worker Smith beats the stoker at billiards.
What is the last name of the engineer?
|
Smith
|
olympiads
| 0.375
|
In an arithmetic sequence $\left\{a_{n}\right\}$, given $\left|a_{5}\right|=\left|a_{11}\right|$ and $d>0$, find the positive integer value of $n$ that minimizes the sum of the first $n$ terms, $S_{n}$.
|
7 \text{ or } 8
|
olympiads
| 0.28125
|
When \( 1999 \) is divided by 7, the remainder is \( R \). Find the value of \( R \).
|
1
|
olympiads
| 0.0625
|
Two brothers had tickets to a stadium located 10 km from their home. Initially, they planned to walk to the stadium. However, they changed their plan and decided to use a bicycle. They agreed that one would start on the bicycle and the other would walk simultaneously. After covering part of the distance, the first brother would leave the bicycle, and the second brother would ride the bicycle after reaching it, continuing until he caught up with the first brother at the entrance of the stadium. How much time do the brothers save compared to their initial plan to walk the entire way, given that each brother covers each kilometer 12 minutes faster on the bicycle than on foot?
|
1 \text{ hour}
|
olympiads
| 0.09375
|
Using the numbers 3, 0, and 8, how many unique three-digit numbers can be formed without repeating any digits?
|
4
|
olympiads
| 0.234375
|
Find the largest term of the sequence \( x_{n} = \frac{n-1}{n^{2}+1} \).
|
0.2
|
olympiads
| 0.5
|
In triangle \(ABC\), points \(E\) and \(D\) are on sides \(AB\) and \(BC\) such that segments \(AD\) and \(CE\) are equal, \(\angle BAD = \angle ECA\), and \(\angle ADC = \angle BEC\). Find the angles of the triangle.
|
60^
|
olympiads
| 0.109375
|
There were 100 cards lying face up with the white side on the table. Each card has one white side and one black side. Kostya flipped 50 cards, then Tanya flipped 60 cards, and after that Olya flipped 70 cards. As a result, all 100 cards ended up lying with the black side up. How many cards were flipped three times?
|
40
|
olympiads
| 0.078125
|
Find all prime numbers $p$ , for which there exist $x, y \in \mathbb{Q}^+$ and $n \in \mathbb{N}$ , satisfying $x+y+\frac{p}{x}+\frac{p}{y}=3n$ .
|
p
|
aops_forum
| 0.0625
|
The perimeter of a rectangle is 126 cm, and the difference between its sides is 37 cm. Find the area of the rectangle.
|
650 \, \text{cm}^2
|
olympiads
| 0.203125
|
A rectangle $ABEF$ is drawn on the leg $AB$ of a right triangle $ABC$ , whose apex $F$ is on the leg $AC$ . Let $X$ be the intersection of the diagonal of the rectangle $AE$ and the hypotenuse $BC$ of the triangle. In what ratio does point $X$ divide the hypotenuse $BC$ if it is known that $| AC | = 3 | AB |$ and $| AF | = 2 | AB |$ ?
|
2:3
|
aops_forum
| 0.078125
|
Find the smallest natural number \( N \) such that \( N+2 \) is divisible by 2, \( N+3 \) is divisible by 3, \ldots, \( N+10 \) is divisible by 10.
|
2520
|
olympiads
| 0.28125
|
Solve the system of equations:
\[
\begin{cases}
y^{2}+xy=15 \\
x^{2}+xy=10
\end{cases}
\]
|
(2, 3), (-2, -3)
|
olympiads
| 0.09375
|
$A, B, C, D, E$ participated in a dart competition, and only one person hit the center of the dartboard, but it is unknown who did it.
$A$ said, "Either I did not hit it, or $C$ hit it."
$B$ said, "It was not $E$ who hit it."
$C$ said, "If it was not $D$ who hit it, then it must have been $B$ who hit it."
$D$ said, "Neither I hit it, nor did $B$ hit it."
$E$ said, "Neither $C$ hit it, nor did $A$ hit it."
We know that exactly two of the five people are telling the truth. Determine who hit the center of the dartboard.
|
E
|
olympiads
| 0.0625
|
A bag contains counters, of which ten are coloured blue and \( Y \) are coloured yellow. Two yellow counters and some more blue counters are then added to the bag. The proportion of yellow counters in the bag remains unchanged before and after the additional counters are placed into the bag.
Find all possible values of \( Y \).
|
1, 2, 4, 5, 10, 20
|
olympiads
| 0.234375
|
What is the sum of all of the distinct prime factors of \( 25^{3} - 27^{2} \)?
|
28
|
olympiads
| 0.453125
|
The lengths of the three sides of a right triangle form a geometric sequence. The sine of the smallest of the angles in the triangle is $\tfrac{m+\sqrt{n}}{k}$ where $m$ , $n$ , and $k$ are integers, and $k$ is not divisible by the square of any prime. Find $m + n + k$ .
|
6
|
aops_forum
| 0.140625
|
Find the ratio \(\frac{b^{2}}{a c}\) given that one of the roots of the equation \(a x^{2} + b x + c = 0\) is 4 times the other root \((a \neq 0, c \neq 0)\).
|
\frac{25}{4}
|
olympiads
| 0.421875
|
On the side $BC$ of triangle $ABC$, a point $M$ is marked such that $AB = BM$ and $AM = MC$. It is known that angle $B$ is five times the angle $C$. Find the angles of the triangle.
|
\angle A = 60^\circ, \angle B = 100^\circ, \angle C = 20^\circ
|
olympiads
| 0.0625
|
Find \( n > 1 \) such that using stamp denominations of \( n \) and \( n+2 \), it is possible to obtain any value \( \geq 2n + 2 \).
|
3
|
olympiads
| 0.171875
|
Find all positive integers \( n > 1 \) such that \( 2^{2} + 3^{2} + \cdots + n^{2} = p^{k} \), where \( p \) is a prime and \( k \) is a positive integer.
|
2, 3, 4, 7
|
olympiads
| 0.0625
|
Ivan set off from point A to point B on a tractor, and Peter set off for the same destination in a Mercedes. Peter arrived at point B, waited 10 minutes, and then called Ivan to find out that Ivan had only covered one-third of the distance and was passing by a café. Peter drove to meet Ivan but didn't see him; he arrived at the café, spent half an hour having a snack, and then drove back to point B. Eventually, Peter and Ivan arrived at point B simultaneously. How much time did Ivan spend on the entire journey, given that both traveled at constant speeds?
|
1 \text{ hour and } 15 \text{ minutes}
|
olympiads
| 0.09375
|
On 2016 cards, the numbers from 1 to 2016 were written (each number exactly once). Then \( k \) cards were taken. What is the smallest \( k \) such that among them there will be two cards with numbers whose square root difference is less than 1?
|
45
|
olympiads
| 0.09375
|
Xiao Gang goes to buy milk and finds that it's on special offer that day: each bag costs 2.5 yuan, and there is a "buy two, get one free" promotion. Xiao Gang has 30 yuan. What is the maximum number of bags of milk he can buy?
|
18
|
olympiads
| 0.484375
|
You have an unmarked ruler and a special tool that allows you to measure the distance between any two points and mark this distance on any given line from any starting point on that line. How can you use these tools and a pencil to divide a given segment into two equal parts?
|
M \text{ is the midpoint of } AB
|
olympiads
| 0.1875
|
The circles \(\omega_{1}\) and \(\omega_{2}\) intersect at points \(A\) and \(B\), and a circle centered at point \(O\) encompasses the circles \(\omega_{1}\) and \(\omega_{2}\), touching them at points \(C\) and \(D\) respectively. It turns out that points \(A\), \(C\), and \(D\) are collinear. Find the angle \(A B O\).
|
90^\circ
|
olympiads
| 0.28125
|
18 students went to catch butterflies. Each student caught a maximum of 3 butterflies, catching a total of 32 butterflies. The number of students who caught 1 butterfly is 5 more than the number of students who caught 2 butterflies and 2 more than the number of students who caught 3 butterflies. How many students did not catch any butterflies?
|
1
|
olympiads
| 0.25
|
In the figure below, identical squares \(ABCD\) and \(XYZW\) overlap each other in such a way that the vertex \(A\) is at the center of \(XYZW\) and the line segment \(AB\) divides the line segment \(YZ\) in the ratio \(1:2\). If the ratio of the area of \(XYZW\) to the overlapped region is \(c:1\), determine the value of \(c\).
|
4
|
olympiads
| 0.390625
|
\(n\) cards are dealt to two players. Each player has at least one card. How many possible hands can the first player have?
|
2^n - 2
|
olympiads
| 0.109375
|
Assume a random number selector can only choose one number from $1, 2, \cdots, 9$ and makes these selections with equal probability. Determine the probability that the product of the $n$ chosen numbers is divisible by 10 after $n$ selections $(n > 1)$.
|
1 - \frac{8^n + 5^n - 4^n}{9^n}
|
olympiads
| 0.28125
|
Four children were discussing the answer to a problem.
Kolia said: "The number is 9."
Roman said: "The number is a prime number."
Katya said: "The number is even."
Natasha said: "The number is divisible by 15."
One boy and one girl answered correctly, while the other two were wrong. What is the actual answer to the problem?
|
2
|
olympiads
| 0.15625
|
Find the derivative of the given order.
\[ y = (x^3 + 3) e^{4x + 3}, y^{(IV)} = ? \]
|
(256x^3 + 768x^2 + 576x + 864)e^{4x + 3}
|
olympiads
| 0.21875
|
Alex picks his favorite point \((x, y)\) in the first quadrant on the unit circle \(x^{2}+y^{2}=1\), such that a ray from the origin through \((x, y)\) is \(\theta\) radians counterclockwise from the positive \(x\)-axis. He then computes \(\cos ^{-1}\left(\frac{4x + 3y}{5}\right)\) and is surprised to get \(\theta\). What is \(\tan (\theta)\)?
|
\frac{1}{3}
|
olympiads
| 0.421875
|
Find all prime numbers $p$ such that the number $$ 3^p+4^p+5^p+9^p-98 $$ has at most $6$ positive divisors.
|
2, 3
|
aops_forum
| 0.09375
|
In the rectangular coordinate system, a circle $\Omega$ is drawn with its center at the focus of the parabola $\Gamma: y^{2}=6x$ and is tangent to the directrix of $\Gamma$. Find the area of the circle $\Omega$.
|
9 \pi
|
olympiads
| 0.515625
|
Let \( P \) be a 2023-sided polygon. All but one side have length 1. What is the maximum possible area of \( P \)?
|
\frac{1011}{2} \cot \left(\frac{\pi}{4044}\right)
|
olympiads
| 0.265625
|
A group of pirates (raiders, sailors, and cabin boys) divided 200 gold and 600 silver coins among themselves. Each raider received 5 gold and 10 silver coins, each sailor received 3 gold and 8 silver coins, and each cabin boy received 1 gold and 6 silver coins. How many pirates were there altogether?
|
80
|
olympiads
| 0.421875
|
Given the function \( f(x) = x^3 \), the tangent to the curve at the point \( (a_k, f(a_k)) \) (where \( k \in \mathbf{N}^{*} \)) intersects the x-axis at \( (a_{k+1}, 0) \). If \( a_1 = 1 \), find the value of \[ \frac{f\left(\sqrt[3]{a_1}\right) + f\left(\sqrt[3]{a_2}\right) + \cdots + f\left(\sqrt[3]{a_{10}}\right)}{1 - \left(\frac{2}{3}\right)^{10}}. \]
|
3
|
olympiads
| 0.078125
|
The length of a rectangle was decreased by 10%, and the width was decreased by 20%. Consequently, the perimeter of the rectangle decreased by 12%. By what percentage will the perimeter of the rectangle decrease if its length is decreased by 20% and its width is decreased by 10%?
|
18\%
|
olympiads
| 0.203125
|
A test field used for experimenting with a new variety of rice has an area of 40 mu. A portion of the field is planted with the new variety, while the other portion is planted with the old variety (the planting areas are not necessarily equal) for the purpose of comparing results. The old variety yields 500 kilograms per mu; of the new variety, 75% was unsuccessful, yielding only 400 kilograms per mu, but the remaining 25% was successful, yielding 800 kilograms per mu. Calculate the total amount of rice produced by this test field in kilograms.
|
20000 \text{ kg}
|
olympiads
| 0.578125
|
In a convex quadrilateral inscribed around a circle, the products of opposite sides are equal. The angle between a side and one of the diagonals is $20^{\circ}$. Find the angle between this side and the other diagonal.
|
70^
|
olympiads
| 0.0625
|
Yasha and Grisha are playing a game: first, they take turns naming a number from 1 to 105 (Grisha goes first, and the numbers must be different). Then each counts the number of different rectangles with integer sides whose perimeter equals the named number. The player with the higher number of rectangles wins. What number should Grisha name to win? Rectangles that differ only by rotation are considered the same. For example, rectangles $2 \times 3$ and $3 \times 2$ are considered the same.
|
104
|
olympiads
| 0.109375
|
On the New Year's table, there are 4 glasses arranged in a row: the first and third glasses contain orange juice, while the second and fourth glasses are empty. In anticipation of guests, Valya absentmindedly and randomly pours juice from one glass to another. Each time, she can take a full glass and pour its entire contents into one of the two empty glasses.
Find the expected number of pourings needed to achieve the opposite situation: the first and third glasses are empty, while the second and fourth glasses are full.
|
6
|
olympiads
| 0.0625
|
João and Maria received 3 chocolate bars of size $5 \times 3$ divided into $1 \times 1$ squares. They decide to play a game. João takes one of the bars and cuts it into two smaller rectangular bars along one of the lines dividing the squares of the bar. Then, Maria takes any one of the bars and also divides it using one of the already marked lines on it. They continue cutting the bars alternately, and the winner is the one who, after their move, leaves only $1 \times 1$ squares as pieces. Who wins the game?
|
Maria
|
olympiads
| 0.234375
|
Real numbers \( a \) and \( b \) are such that \( a^3 + b^3 = 1 - 3ab \). Find all possible values that the sum \( a + b \) can take.
|
1 \text{ and } -2
|
olympiads
| 0.1875
|
Determine all numbers $x, y$ and $z$ satisfying the system of equations $$ \begin{cases} x^2 + yz = 1 y^2 - xz = 0 z^2 + xy = 1\end{cases} $$
|
x = y = z = \pm \frac{\sqrt{2}}{2}
|
aops_forum
| 0.09375
|
If integer \( x \) satisfies \( x \geq 3+\sqrt{3+\sqrt{3+\sqrt{3+\sqrt{3+\sqrt{3}}}}} \), find the minimum value of \( x \).
|
6
|
olympiads
| 0.296875
|
There is a sequence of numbers: \(1, 3, 8, 22, 60, 164, 448, \cdots\). The first number is 1, the second number is 3, and from the third number onwards, each number is exactly twice the sum of the previous two numbers. What is the remainder when the 2000th number in this sequence is divided by 9?
|
3
|
olympiads
| 0.578125
|
Given \( m \in \mathbb{R} \), find the equation of the locus of the intersection points of the two lines \( mx - y + 1 = 0 \) and \( x - my - 1 = 0 \).
|
x - y = 0 ext{ or } x - y + 1 = 0
|
olympiads
| 0.109375
|
Let \(\mathbb{R}_{>0}\) be the set of positive real numbers. Find all functions \( f: \mathbb{R}_{>0} \rightarrow \mathbb{R}_{>0} \) such that, for every \( x \in \mathbb{R}_{>0} \), there exists a unique \( y \in \mathbb{R}_{>0} \) satisfying
\[
x f(y) + y f(x) \leq 2 .
\]
|
f(x) = \frac{1}{x}
|
olympiads
| 0.3125
|
Triangle $ABC$ has $AB = 4$ , $AC = 5$ , and $BC = 6$ . An angle bisector is drawn from angle $A$ , and meets $BC$ at $M$ . What is the nearest integer to $100 \frac{AM}{CM}$ ?
|
80
|
aops_forum
| 0.15625
|
On a plane, there are fixed points \(A\) and \(B\). For any \(k\) points \(P_1, P_2, \ldots, P_k\) on this plane, there are two points \(P_i\) and \(P_j\) (\(i, j \in \{1,2,3,\ldots,k\}\) and \(i \neq j\)) that satisfy \[\left|\sin \angle A P_i B - \sin \angle A P_j B\right| \leq \frac{1}{1992}\]. What is the minimum value of \(k\)?
|
1993
|
olympiads
| 0.40625
|
\(\log_{4} \log_{2} x + \log_{2} \log_{4} x = 2\).
|
16
|
olympiads
| 0.296875
|
For what integer value of $k$ is one of the roots of the equation $4x^{2} - (3k + 2)x + (k^{2} - 1) = 0$ three times smaller than the other?
|
k = 2
|
olympiads
| 0.109375
|
Cauchy and Bunyakovsky loved to play darts in the evenings. However, their dartboard had unequal sectors, so the probabilities of hitting different sectors were not the same. One day, Cauchy threw a dart and hit a sector. Bunyakovsky goes next. What is more likely: that Bunyakovsky will hit the same sector as Cauchy or the next sector in a clockwise direction?
|
The probability that Bunyakovsky hits the same sector is higher.
|
olympiads
| 0.0625
|
Find all the positive integers \( n \) such that \( n-1 \) divides \( n^3 + 4 \).
|
2 \text{ and } 6
|
olympiads
| 0.28125
|
A car is traveling at a speed of 60 km/h. By how much should it increase its speed to travel one kilometer half a minute faster?
|
60 \text{ km/h}
|
olympiads
| 0.390625
|
When dividing a number by 72, the remainder is 68. How will the quotient change, and what will be the remainder, if the same number is divided by 24?
|
20
|
olympiads
| 0.46875
|
In the regular triangular pyramid \(P-ABC\), point \(M\) is a moving point inside (including the boundary) \(\triangle ABC\), and the distances from point \(M\) to the three lateral faces \(PAB\), \(PBC\), and \(PCA\) are in an arithmetic sequence. Then the locus of point \(M\) is \(\quad\) .
|
a line segment
|
olympiads
| 0.140625
|
Square $ABCD$ and rectangle $BEFG$ are placed as shown in the figure. $AG = CE = 2$ cm. What is the difference in the area between square $ABCD$ and rectangle $BEFG$ in square centimeters?
|
4
|
olympiads
| 0.15625
|
Find the coordinates of point $A$ equidistant from points $B$ and $C$.
$A(0 ; y ; 0)$
$B(-2 ; 8 ; 10)$
$C(6 ; 11 ; -2)$
|
A(0; -\frac{7}{6}; 0)
|
olympiads
| 0.078125
|
Find all integers $n$ greater than or equal to $4$ that satisfy the following conditions:
- Take an arbitrary convex $n$ -gon $P$ on the coordinate plane whose vertices are lattice points (points whose coordinates are both integers). There are a total of $n$ triangles that share two sides with $P$ . Let $S_1, S_2, \ldots, S_n$ be their areas, and let $S$ be the area of $P$ . Then, the greatest common divisor of $2S_1, 2S_2, \ldots, 2S_n$ divides the integer $2S$ .
|
n = 4 } or n = 5
|
aops_forum
| 0.0625
|
The larger base of a trapezoid is twice the length of its smaller base. A line parallel to the bases is drawn through the point of intersection of the diagonals. Find the ratio of the height of each of the two resulting trapezoids to the height of the original trapezoid.
|
\frac{1}{3}, \frac{2}{3}
|
olympiads
| 0.078125
|
Line \( l_1 \) passes through the point \( P(3, 2) \) and has an inclination angle of \( \arctan \frac{3}{4} \). If \( l_1 \) intersects line \( l_2: x - 2y + 11 = 0 \) at point \( Q \), find the distance \( |PQ| \).
|
25
|
olympiads
| 0.3125
|
What is the minimum force needed to press down on a cube with a volume of 10 cm$^{3}$, floating in water, so that it is fully submerged? The density of the cube's material is $500 \mathrm{kg/m}^{3}$, the density of water is $1000 \mathrm{kg/m}^{3}$, and the acceleration due to gravity should be taken as $10 \mathrm{m/s}^{2}$. Provide the answer in SI units.
|
0.05 \text{ N}
|
olympiads
| 0.375
|
A pot contains $3 \pi$ liters of water taken at a temperature of $t=0{ }^{\circ} C$ and brought to a boil in 12 minutes. After this, without removing the pot from the stove, ice at a temperature of $t=0{ }^{\circ} C$ is added. The next time the water begins to boil is after 15 minutes. Determine the mass of the added ice. The specific heat capacity of water is $c_{B}=4200 \, \text{J} / \text{kg} \cdot { }^{\circ} \mathrm{C}$, the specific heat of fusion of ice is $\lambda=3.3 \times 10^{5} \, \text{J} / \text{kg}$, the density of water is $\rho = 1000 \, \text{kg} / \text{m}^{3}$.
Answer: 2.1 kg
|
2.1 \, \text{kg}
|
olympiads
| 0.3125
|
Someone said to their friend: "Give me 100 rupees, and I'll be twice as rich as you," to which the latter replied: "If you give me just 10 rupees, I'll be six times richer than you." The question is: how much money did each person have?
|
x = 40, \quad y = 170
|
olympiads
| 0.5
|
Along the southern shore of a boundless sea stretches an archipelago of an infinite number of islands. The islands are connected by an endless chain of bridges, and each island is connected by a bridge to the shore. In the event of a strong earthquake, each bridge independently has a probability $p=0.5$ of being destroyed. What is the probability that after a strong earthquake, one can travel from the first island to the shore using the remaining intact bridges?
|
\frac{2}{3}
|
olympiads
| 0.0625
|
A chessboard \(8 \times 8\) was cut into several equal parts such that all white squares remained uncut, while each black square was cut. How many parts could there be?
|
2, 4, 8, 16, 32
|
olympiads
| 0.359375
|
Beginner millionaire Bill buys a bouquet of 7 roses for $20. Then, he can sell a bouquet of 5 roses for $20 per bouquet. How many bouquets does he need to buy to "earn" a difference of $1000?
|
125
|
olympiads
| 0.125
|
A $150 \times 324 \times 375$ rectangular prism is composed of unit cubes. How many unit cubes does a diagonal of this rectangular prism pass through?
|
768
|
olympiads
| 0.0625
|
In testing a ship, the following table shows the relationship between speed \(v\) (knots) and power \(H\) (horsepower):
\begin{tabular}{|c|c|c|c|}
\hline
\(H\) & 300 & 780 & 1420 \\
\hline
\(v\) & 5 & 7 & 9 \\
\hline
\end{tabular}
Assuming that the relationship between \(H\) and \(v\) is a quadratic function, find the power of the ship at a speed of 6 knots.
|
520 \text{ horsepower}
|
olympiads
| 0.515625
|
A diagonal of a regular 2006-gon \( P \) is called good if its endpoints divide the boundary of \( P \) into two parts, each containing an odd number of sides. The sides of \( P \) are also called good. Let \( P \) be divided into triangles by 2003 diagonals, none of which have common interior points. What is the greatest number of isosceles triangles, each of which has two good sides, that such a division can have?
|
1003
|
olympiads
| 0.296875
|
Yesterday, Nikita bought several pens: black ones for 9 rubles each and blue ones for 4 rubles each. Today, he went to the same store and discovered that the prices had changed: black pens now cost 4 rubles each, and blue pens now cost 9 rubles each. Seeing this, Nikita said regretfully: "If I were buying the same pens today, I would have saved 49 rubles." Is he correct?
|
Nikita is mistaken.
|
olympiads
| 0.078125
|
A number between 2013 and 2156, when divided by 5, 11, and 13, yields the same remainder. What is the maximum remainder?
|
4
|
olympiads
| 0.171875
|
For any \( t > 0 \), consider the sum \( S = r_{1}^{2} + r_{2}^{2} + r_{3}^{2} + r_{4}^{2} \), where \( r_{1}, r_{2}, r_{3}, r_{4} \) are the roots of the polynomial
\[ P_{t}[X] = \frac{1}{t} X^{4} + \left(1 - \frac{10}{t}\right) X^{3} - 2 X^{2} + \sqrt[3]{2 t} X + \arctan (t) \]
What is the minimum value of \( |S| \) and for which \( t \) is it reached?
|
36 at t = 8
|
olympiads
| 0.078125
|
There are $n \ge 2$ houses on the northern side of a street. Going from the west to the east, the houses are numbered from 1 to $n$ . The number of each house is shown on a plate. One day the inhabitants of the street make fun of the postman by shuffling their number plates in the following way: for each pair of neighbouring houses, the currnet number plates are swapped exactly once during the day.
How many different sequences of number plates are possible at the end of the day?
|
2^{n-1}
|
aops_forum
| 0.09375
|
Let $p,q$ and $s{}$ be prime numbers such that $2^sq =p^y-1$ where $y > 1.$ Find all possible values of $p.$
|
3 \text{ and } 5
|
aops_forum
| 0.140625
|
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