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stringlengths 33
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|---|---|---|---|
There are 196 students numbered from 1 to 196 arranged in a line. Students at odd-numbered positions (1, 3, 5, ...) leave the line. The remaining students are renumbered starting from 1 in order. Then, again, students at odd-numbered positions leave the line. This process repeats until only one student remains. What was the initial number of this last remaining student?
|
128
|
olympiads
| 0.0625
|
Scott stacks golfballs to make a pyramid. The first layer, or base, of the pyramid is a square of golfballs and rests on a flat table. Each golfball, above the first layer, rests in a pocket formed by four golfballs in the layer below. Each layer, including the first layer, is completely filled. The four triangular faces of the pyramid include a total of exactly 145 different golfballs. How many layers does this pyramid have?
|
9 \text{ layers}
|
olympiads
| 0.0625
|
There are three pastures full of grass. The first pasture is 33 acres and can feed 22 cows for 27 days. The second pasture is 28 acres and can feed 17 cows for 42 days. How many cows can the third pasture, which is 10 acres, feed for 3 days (assuming the grass grows at a uniform rate and each acre produces the same amount of grass)?
|
20
|
olympiads
| 0.0625
|
Fili and Kili are playing chess. Besides the chessboard, they have one rook, which they placed in the bottom right corner. They take turns making moves with the rook, but are only allowed to move it up or to the left (any number of squares). The player who cannot make a move loses. Kili moves first. Who will win with optimal play?
|
Fili wins
|
olympiads
| 0.28125
|
Numbers 1447, 1005, and 1231 share certain characteristics: each number is a four-digit number starting with 1, and each number contains exactly two identical digits. How many such four-digit numbers are there in total?
|
432
|
olympiads
| 0.0625
|
Let $N$ be the number of ordered triples $(a,b,c) \in \{1, \ldots, 2016\}^{3}$ such that $a^{2} + b^{2} + c^{2} \equiv 0 \pmod{2017}$ . What are the last three digits of $N$ ?
|
000
|
aops_forum
| 0.109375
|
Solve the equation: \(\lfloor 20x + 23 \rfloor = 20 + 23x\). Recall that \(\lfloor a \rfloor\) denotes the floor function of \(a\), which is the greatest integer less than or equal to \(a\).
|
\frac{16}{23}, \frac{17}{23}, \frac{18}{23}, \frac{19}{23}, \frac{20}{23}, \frac{21}{23}, \frac{22}{23}, 1
|
olympiads
| 0.234375
|
Márcia is in a store buying a recorder she has wanted for a long time. When the cashier registers the price, she exclaims: "It's not possible, you have recorded the number backwards, you swapped the order of two digits, I remember that last week it cost less than 50 reais!" The cashier responds: "I'm sorry, but yesterday all our items had a 20% increase." What is the new price of the recorder?
|
54 \text{ reais}
|
olympiads
| 0.078125
|
In the given five-pointed star, the characters "华", "罗", "庚", "金", "杯" at the vertices represent the numbers 1 to 5, with each character representing a different number. The sum of the numbers at the endpoints of each line segment is exactly five consecutive natural numbers. If "杯" represents the number 1, determine the number represented by "华".
|
3 \text{ or } 4
|
olympiads
| 0.25
|
There are three piles of stones. Sisyphus moves one stone from one pile to another. For each move, he receives from Zeus a number of coins equal to the difference between the number of stones in the pile to which he moves the stone and the number of stones in the pile from which he takes the stone (the transferred stone itself is not counted). If this difference is negative, Sisyphus returns the corresponding amount to Zeus. (If Sisyphus cannot pay, the generous Zeus allows him to move the stones on credit.) At some point, it turns out that all the stones are in the same piles where they were initially. What is the maximum total earnings of Sisyphus by this moment?
|
0
|
olympiads
| 0.0625
|
In \(\triangle ABC\), let \(AB = c\), \(BC = a\), and \(AC = b\). Suppose that \(\frac{b}{c - a} - \frac{a}{b + c} = 1\), find the value of the greatest angle of \(\triangle ABC\) in degrees.
|
120^
|
olympiads
| 0.0625
|
For which base of logarithms is the equality $\log 1 = 0$ true?
|
any base
|
olympiads
| 0.140625
|
Calculate the limit of the function:
$$\lim _{x \rightarrow 3} \frac{\sqrt[3]{5+x}-2}{\sin \pi x}$$
|
- rac{1}{12 ext{ π}}
|
olympiads
| 0.25
|
Let x; y; z be real numbers, satisfying the relations $x \ge 20$ $y \ge 40$ $z \ge 1675$ x + y + z = 2015
Find the greatest value of the product P = $xy z$
|
\frac{721480000}{27}
|
aops_forum
| 0.546875
|
A torpedo boat is anchored 9 km from the nearest point on the shore. A messenger needs to be sent from the boat to a camp located 15 km along the shore from the nearest point of the boat. If the messenger travels on foot at a speed of 5 km/h and rows at a speed of 4 km/h, at which point on the shore should he land to reach the camp in the shortest possible time?
|
12
|
olympiads
| 0.09375
|
Let \( A = \{1, -1, \mathrm{i}, -\mathrm{i} \} \) (where \( \mathrm{i} \) is the imaginary unit), and \( f(x) \) be a function whose domain and range are both \( A \). Given that for any \( x, y \in A \), \( f(x y) = f(x) f(y) \), how many such functions \( f(x) \) satisfy this condition?
|
2
|
olympiads
| 0.1875
|
The Mad Hatter's clock gains 15 minutes per hour, while the March Hare's clock loses 10 minutes per hour. One day, they set their clocks according to the Dormouse's clock, which is stopped and always shows 12:00, and agreed to meet at 5 o'clock in the evening for their traditional five-o'clock tea. How long will the Mad Hatter wait for the March Hare if both arrive exactly at 17:00 according to their own clocks?
|
2 ext{ hours}
|
olympiads
| 0.09375
|
In triangle \(ABC\), a median \(AM\) is drawn (point \(M\) lies on side \(BC\)). It is known that the angle \(\angle CAM\) is \(30^\circ\) and side \(AC\) is 2. Find the distance from point \(B\) to line \(AC\).
|
1
|
olympiads
| 0.0625
|
A circle has a radius of 4 units, and a point \( P \) is situated outside the circle. A line through \( P \) intersects the circle at points \( A \) and \( B \). If \( P A = 4 \) units and \( P B = 6 \) units, how far is \( P \) from the center of the circle?
|
2 \sqrt{10}
|
olympiads
| 0.375
|
The maximum value of the function \( f(x)=\frac{2 \sin x \cos x}{1+\sin x+\cos x} \) is _____.
|
\sqrt{2} - 1
|
olympiads
| 0.0625
|
In the triangular prism \( S-ABC \), it is known that \( AB = AC \) and \( SB = SC \). Find the angle between line \( SA \) and line \( BC \).
|
\frac{\pi}{2}
|
olympiads
| 0.203125
|
Given a positive integer \(n > 1\). Let \(a_{1}, a_{2}, \cdots, a_{n}\) be a permutation of \(1, 2, \cdots, n\). If \(i < j\) and \(a_{i} < a_{j}\), then \(\left(a_{i}, a_{j}\right)\) is called an ascending pair. \(X\) is the number of ascending pairs in \(a_{1}, a_{2}, \cdots, a_{n}\). Find \(E(X)\).
|
\frac{n(n-1)}{4}
|
olympiads
| 0.125
|
The soup prepared by tourists on the seashore during a successful fishing trip was good, but due to their lack of culinary experience, they added too little salt, so additional salt had to be added at the table. The next time, for the same amount of soup, they put in twice as much salt as the first time, but even then they had to add salt at the table, although using half the amount of salt compared to the first time. What fraction of the necessary amount of salt was added to the soup by the cook the first time?
|
\frac{1}{3}
|
olympiads
| 0.421875
|
Lisa drove from her home to Shangri-La. In the first hour, she drove $\frac{1}{3}$ of the total distance. In the second hour, she drove $\frac{1}{2}$ of the remaining distance. In the third hour, she drove $\frac{1}{10}$ less than what she drove in the first hour. At this point, she still has 9 kilometers left to reach Shangri-La. The distance between Lisa's home and Shangri-La is $\qquad$ kilometers.
|
90
|
olympiads
| 0.09375
|
There are 100 points on a circle that are about to be colored in two colors: red or blue. Find the largest number $k$ such that no matter how I select and color $k$ points, you can always color the remaining $100-k$ points such that you can connect 50 pairs of points of the same color with lines in a way such that no two lines intersect.
|
k = 50
|
aops_forum
| 0.4375
|
How many three-digit natural numbers have an even number of distinct natural divisors?
|
878
|
olympiads
| 0.3125
|
In Figure 1, \( BC \) is the diameter of the circle. \( A \) is a point on the circle, \( AB \) and \( AC \) are line segments, and \( AD \) is a line segment perpendicular to \( BC \). If \( BD = 1 \), \( DC = 4 \), and \( AD = a \), find the value of \( a \).
|
2
|
olympiads
| 0.125
|
Find all functions \( f: \mathbb{Z}^* \rightarrow \mathbb{R} \) that satisfy the equation
\[ f(n+m) + f(n-m) \equiv f(3n) \]
for \( n, m \in \mathbb{Z}^* \) (where \(\mathbb{Z}^*\) is the set of non-negative integers) and \( n \geq m \).
|
f(n) \equiv 0
|
olympiads
| 0.375
|
On a table, there are three cones standing on their bases and touching each other. The heights of the cones are equal, and the radii of their bases are $2r$, $3r$, and $10r$. A sphere of radius 2 is placed on the table, touching all the cones. It turns out that the center of the sphere is equidistant from all the points of contact with the cones. Find $r$.
|
r = 1
|
olympiads
| 0.0625
|
Find the maximum value of the expression \( x + y \), where \( x \) and \( y \) are integer solutions of the equation \( 3x^{2} + 5y^{2} = 345 \).
|
13
|
olympiads
| 0.296875
|
How many fractions between \(\frac{1}{6}\) and \(\frac{1}{3}\) inclusive can be written with a denominator of 15?
|
3
|
olympiads
| 0.1875
|
Let $n$ be a given positive integer. Solve the system
\[x_1 + x_2^2 + x_3^3 + \cdots + x_n^n = n,\]
\[x_1 + 2x_2 + 3x_3 + \cdots + nx_n = \frac{n(n+1)}{2}\]
in the set of nonnegative real numbers.
|
(1, 1, \, \ldots, 1)
|
aops_forum
| 0.0625
|
Two isosceles triangles are given with equal perimeters. The base of the second triangle is 15% larger than the base of the first, and the leg of the second triangle is 5% smaller than the leg of the first triangle. Find the ratio of the sides of the first triangle.
|
\frac{a}{b} = \frac{2}{3}
|
olympiads
| 0.15625
|
In a right triangle \( \triangle ABC \) with a right angle at \( B \), the angle bisector of angle \( A \) intersects side \( BC \) at point \( D \). It is known that \( BD = 4 \) and \( DC = 6 \). Find the area of triangle \( ADC \).
|
\frac{60\sqrt{13}}{13}
|
olympiads
| 0.078125
|
A square is inscribed in a right triangle with legs of lengths 6 and 8, such that the square shares the right angle with the triangle. Find the side length of the square.
|
24/7
|
olympiads
| 0.21875
|
Three journalists observed a person during breakfast and made the following notes:
Jules: "First he drank whiskey, then ate duck with oranges and dessert. Finally, he drank coffee."
Jacques: "He did not drink an aperitif. He ate pie and 'Belle Helene' pear."
Jim: "First he drank whiskey, then ate pie and strawberry sherbet. He finished breakfast with a cup of coffee."
(Note: All information from one journalist is completely false, another made only one false statement, and the third never lies.)
Based on these conflicting reports, determine what the person had for dessert.
|
ext{strawberry sherbet}
|
olympiads
| 0.140625
|
Calculate the following sum given \(xyz = 1\):
\[
\frac{1}{1+x+xy}+\frac{1}{1+y+yz}+\frac{1}{1+z+zx}
\]
|
1
|
olympiads
| 0.09375
|
Given a triangle with sides 12 cm, 15 cm, and 18 cm. A circle is drawn that touches both of the shorter sides and has its center on the longest side. Find the segments into which the center of the circle divides the longest side of the triangle.
|
8 \\text{cm} \\text{and} \\ 10 \\text{cm}
|
olympiads
| 0.125
|
Petya has a 3x3 grid. He places pieces in the cells according to the following rules:
- No more than one piece can be placed in each cell;
- A piece can be placed in an empty cell if the total number of pieces already present in the corresponding row and column is even (0 is considered an even number).
What is the maximum number of pieces Petya can place?
|
9
|
olympiads
| 0.0625
|
Find all triples $ \left(p,x,y\right)$ such that $ p^x\equal{}y^4\plus{}4$ , where $ p$ is a prime and $ x$ and $ y$ are natural numbers.
|
(5, 1, 1)
|
aops_forum
| 0.078125
|
Fill integers $1, 2, 3, \cdots, 2016^{2}$ into a $2016 \times 2016$ grid, with each cell containing one number, and all numbers being different. Connect the centers of any two cells with a vector, with the direction from the smaller number to the bigger number. If the sum of numbers in each row and each column of the grid is equal, find the sum of all such vectors.
|
0
|
olympiads
| 0.140625
|
Find all $(m,n) \in \mathbb{Z}^2$ that we can color each unit square of $m \times n$ with the colors black and white that for each unit square number of unit squares that have the same color with it and have at least one common vertex (including itself) is even.
|
(m, n) \in \mathbb{Z}^2 \text{ such that } 2 \mid mn
|
aops_forum
| 0.078125
|
Let $m$ be a positive integer, and let $a_0, a_1,\ldots,a_m$ be a sequence of reals such that $a_0=37$ , $a_1=72$ , $a_m=0$ , and \[a_{k+1}=a_{k-1}-\frac{3}{a_k}\] for $k=1,2, \dots, m-1$ . Find $m$ .
|
889
|
aops_forum
| 0.421875
|
Two workers dug a trench in two hours. Following this, the first worker became tired and started working three times slower, while the second worker got motivated and started working three times faster, resulting in them taking one hour to dig another trench. By how many times was the second worker's initial productivity greater than the first worker's initial productivity?
|
\frac{5}{3}
|
olympiads
| 0.28125
|
Prince Accelerated invited Princess Slow on a journey to his castle. After waiting for a long time, he set out to meet her. After two days of traveling, he met her at one-fifth of her journey. Together, they continued traveling twice as fast as the princess traveled alone. They arrived at the prince's castle on the second Saturday after they met. On which day did they meet, given that the princess started her journey from her castle on a Friday?
|
Wednesday
|
olympiads
| 0.171875
|
Find all triples of natural numbers for which the following condition is met: the product of any two of them, increased by one, is divisible by the remaining number.
|
(1, 1, 1)
|
olympiads
| 0.203125
|
A ball rolled into a pool and floated on the water. Its highest point was $2 \mathrm{~cm}$ above the water surface. The diameter of the circle marked by the water surface on the ball was $8 \mathrm{~cm}$.
Determine the diameter of the ball.
(L. Hozová)
|
10 ext{ cm}
|
olympiads
| 0.203125
|
Šárka declared:
"We are three sisters; I am the youngest, Líba is three years older, and Eliška is eight years older. Our mom likes to hear that all of us (including her) have an average age of 21 years. When I was born, my mom was already 29."
How many years ago was Šárka born?
|
11
|
olympiads
| 0.21875
|
Find the smallest integer value of \( a \) for which the system of equations
$$
\left\{\begin{array}{l}
\frac{y}{a-\sqrt{x}-1}=4 \\
y=\frac{\sqrt{x}+5}{\sqrt{x}+1}
\end{array}\right.
$$
has a unique solution.
|
3
|
olympiads
| 0.1875
|
Find all triples of prime numbers \( p, q, r \) such that the fourth power of each of them, decreased by 1, is divisible by the product of the other two.
|
\{2, 3, 5\}
|
olympiads
| 0.09375
|
A chord divides the circle in the ratio 7:11. Find the inscribed angles subtended by this chord.
|
70^{\circ}, 110^{\circ}
|
olympiads
| 0.21875
|
It is known that the numbers EGGPLANT and TOAD are divisible by 3. What is the remainder when the number CLAN is divided by 3? (Letters denote digits, identical letters represent identical digits, different letters represent different digits).
|
0
|
olympiads
| 0.359375
|
Given the sequences \( \left\{a_{n}\right\} \) and \( \left\{b_{n}\right\} \) such that
\[
\begin{array}{l}
a_{1} = -1, \quad b_{1} = 2, \\
a_{n+1} = -b_{n}, \quad b_{n+1} = 2a_{n} - 3b_{n} \quad (n \in \mathbb{Z}_{+}).
\end{array}
\]
Find the value of \( b_{2015} + b_{2016} \).
|
-3 \times 2^{2015}
|
olympiads
| 0.0625
|
Let the functions \( f(\alpha, x) \) and \( g(\alpha) \) be defined as
\[ f(\alpha, x)=\frac{\left(\frac{x}{2}\right)^{\alpha}}{x-1} \]
\[ g(\alpha)=\left.\frac{d^{4} f}{d x^{4}}\right|_{x=2} \]
Then \( g(\alpha) \) is a polynomial in \( \alpha \). Find the leading coefficient of \( g(\alpha) \).
|
\frac{1}{16}
|
olympiads
| 0.125
|
Find the sum of the real roots of the polynomial \[ \prod_{k=1}^{100} \left( x^2-11x+k \right) = \left( x^2-11x+1 \right)\left( x^2-11x+2 \right)\dots\left(x^2-11x+100\right). \]
|
330
|
aops_forum
| 0.078125
|
Given that \( x \) and \( y \) are real numbers, find the minimum value of the function \( z = \sqrt{x^2 + y^2 - 2x - 2y + 2} + \sqrt{x^2 + y^2 - 4y + 4} \). Also, specify the real values of \( x \) and \( y \) when this minimum value is achieved.
|
\sqrt{2}
|
olympiads
| 0.171875
|
If the first digit of a two-digit natural number \( N \) is multiplied by 4, and the second digit is multiplied by 2, then the sum of the resulting numbers equals \( \frac{N}{2} \). Find \( N \) (list all solutions).
|
32, 64, 96
|
olympiads
| 0.59375
|
A large candle is $119$ centimeters tall. It is designed to burn down more quickly when it is first lit and more slowly as it approaches its bottom. Specifically, the candle takes $10$ seconds to burn down the first centimeter from the top, $20$ seconds to burn down the second centimeter, and $10k$ seconds to burn down the $k$ -th centimeter. Suppose it takes $T$ seconds for the candle to burn down completely. Then $\tfrac{T}{2}$ seconds after it is lit, the candle's height in centimeters will be $h$ . Find $10h$ .
|
350
|
aops_forum
| 0.453125
|
Explain what the statement " \(\sqrt{2}\) cannot be computed exactly, it can only be approximated" means.
|
√2 cannot be exactly computed, it can only be approximated.
|
olympiads
| 0.390625
|
The price of a product has increased by 40%. By what percentage should we decrease our consumption of this product if we can only spend 12% more money on purchasing it?
|
20\%
|
olympiads
| 0.546875
|
How many zeros are at the end of the product of \(2.5 \times 6 \times 10 \times 25 \times 7 \times 75 \times 94\)?
|
3
|
olympiads
| 0.203125
|
As shown in the figure, \( P \) is a point inside the square \( ABCD \), and \(\angle APB = 90^\circ\). Diagonals \( AC \) and \( BD \) intersect at \( O \). Given that \( AP = 3 \text{ cm} \) and \( BP = 5 \text{ cm} \), find the area of triangle \( OBP \).
|
2.5
|
olympiads
| 0.171875
|
In the coordinate plane, find the area of the region bounded by the curve $ C: y\equal{}\frac{x\plus{}1}{x^2\plus{}1}$ and the line $ L: y\equal{}1$ .
|
1 - \frac{\pi}{4} - \frac{\ln(2)}{2}
|
aops_forum
| 0.15625
|
The trapezoid $ABCD$ is divided by the line segment $CE$ into a triangle and a parallelogram, as shown in the diagram. Point $F$ is the midpoint of the segment $CE$, the line $DF$ passes through the midpoint of the segment $BE$, and the area of the triangle $CDE$ is $3 \text{ cm}^{2}$.
Determine the area of the trapezoid $ABCD$.
|
12 \text{ cm}^2
|
olympiads
| 0.09375
|
Determine all positive integer values of \( n \) for which \( 2^n + 1 \) is divisible by 3.
|
n \text{ is odd}
|
olympiads
| 0.5
|
If one and a half chickens lay one and a half eggs in one and a half days, how many chickens plus half a chicken, laying eggs one and a half times faster, will lay ten and a half eggs in one and a half weeks?
|
10.5\ \text{eggs}
|
olympiads
| 0.078125
|
Given that point \( O \) is inside triangle \( \triangle ABC \), and \( 3 \overrightarrow{AB} + 2 \overrightarrow{BC} + \overrightarrow{CA} = 4 \overrightarrow{AO} \), let the area of \( \triangle ABC \) be \( S_1 \) and the area of \( \triangle OBC \) be \( S_2 \). Find the value of \( \frac{S_1}{S_2} \).
|
2
|
olympiads
| 0.078125
|
The diagram shows a solid with six triangular faces and five vertices. Andrew wants to write an integer at each of the vertices so that the sum of the numbers at the three vertices of each face is the same. He has already written the numbers 1 and 5 as shown. What is the sum of the other three numbers he will write?
|
11
|
olympiads
| 0.078125
|
On an island, there are knights, liars, and followers; each one knows who is who among them. All 2018 island inhabitants were lined up and each was asked to answer "Yes" or "No" to the question: "Are there more knights than liars on the island?" The inhabitants answered one by one in such a way that the others could hear. Knights told the truth, liars lied. Each follower answered the same as the majority of those who had answered before them, and if the number of "Yes" and "No" answers was equal, they could give either answer. It turned out that there were exactly 1009 "Yes" answers. What is the maximum number of followers that could be among the island inhabitants?
|
1009
|
olympiads
| 0.1875
|
Calculate the limit of the numerical sequence:
$$\lim_{n \rightarrow \infty} \left( \frac{3n^2 - 5n}{3n^2 - 5n + 7} \right)^{n+1}$$
|
1
|
olympiads
| 0.5
|
Compute the maximum number of sides of a polygon that is the cross-section of a regular hexagonal prism.
|
8
|
olympiads
| 0.0625
|
Find the Green's function for the boundary value problem
$$
y^{\prime \prime}(x) + k^{2} y = 0, \quad y(0)=y(1)=0
$$
|
G(x, \xi)
|
olympiads
| 0.09375
|
Solve the following equation in the set of integer pairs:
$$
(x+2)^{4}-x^{4}=y^{3} \text {. }
$$
|
(-1, 0)
|
olympiads
| 0.109375
|
January 1st of a certain non-leap year fell on a Saturday. How many Fridays are there in this year?
|
52
|
olympiads
| 0.09375
|
Find the radius of the circumscribed circle around an isosceles trapezoid with bases of lengths 2 cm and 14 cm, and a lateral side of 10 cm.
|
5\sqrt{2}
|
olympiads
| 0.0625
|
How many non-empty subsets of \(\{1,2,3,4,5,6,7,8\}\) have exactly \(k\) elements and do not contain the element \(k\) for some \(k = 1, 2, \ldots, 8\)?
|
127
|
olympiads
| 0.15625
|
Let \( 0^{\circ} < \alpha < 45^{\circ} \). If \(\sin \alpha \cos \alpha = \frac{3 \sqrt{7}}{16}\) and \(A = \sin \alpha\), find the value of \(A\).
|
\frac{\sqrt{7}}{4}
|
olympiads
| 0.140625
|
Three points \( A \), \( B \), and \( C \) are randomly selected on the unit circle. Find the probability that the side lengths of triangle \( \triangle ABC \) do not exceed \( \sqrt{3} \).
|
\frac{1}{3}
|
olympiads
| 0.0625
|
If \(\frac{6 \sqrt{3}}{3 \sqrt{2} + 2 \sqrt{3}} = 3 \sqrt{\alpha} - 6\), determine the value of \(\alpha\).
|
6
|
olympiads
| 0.265625
|
The numbers 1447, 1005, and 1231 have something in common: each is a four-digit number beginning with 1 that has exactly two identical digits. How many such numbers are there?
|
432
|
aops_forum
| 0.203125
|
Within a cube with edge length 6, there is a regular tetrahedron with edge length \( x \) that can rotate freely inside the cube. What is the maximum value of \( x \)?
|
2\sqrt{6}
|
olympiads
| 0.125
|
In $\triangle ABC$, $\angle C=90^{\circ}$, $\angle B=30^{\circ}$, $AC=2$, $M$ is the midpoint of $AB$. Fold $\triangle ACM$ along $CM$ such that the distance between $A$ and $B$ becomes $2\sqrt{2}$. Find the distance from point $M$ to plane $ABC$.
|
1
|
olympiads
| 0.0625
|
Let be given an integer $n\ge 2$ and a positive real number $p$ . Find the maximum of
\[\displaystyle\sum_{i=1}^{n-1} x_ix_{i+1},\]
where $x_i$ are non-negative real numbers with sum $p$ .
|
\frac{p^2}{4}
|
aops_forum
| 0.109375
|
Seven dwarfs are seated around a round table, each with a mug in front of them. Some of these mugs contain milk. One of the dwarfs pours all of their milk into the mugs of the others in equal amounts. Then, the dwarf to their right does the same, followed by the next dwarf to the right, and so on. After the seventh and final dwarf has poured their milk into the others' mugs, each mug contains the same amount of milk as it did initially. Altogether, the mugs contain 3 liters of milk. How much milk was originally in each mug?
|
\left( \frac{6}{7}, \frac{5}{7}, \frac{4}{7}, \frac{3}{7}, \frac{2}{7}, \frac{1}{7}, 0 \right)
|
olympiads
| 0.34375
|
Amy, Bruce, Chris, Donna, and Eve had a race. They each made one true and one false statement about the order in which they finished:
- Amy: Bruce came second and I finished in third place.
- Bruce: I finished second and Eve was fourth.
- Chris: I won and Donna came second.
- Donna: I was third and Chris came last.
- Eve: I came fourth and Amy won.
In which order did the participants finish?
|
Bruce, Donna, Amy, Eve, Chris
|
olympiads
| 0.171875
|
23 cans of paint weighing 25 kg each were purchased to paint the floor of a gym. The gym measures 65 meters in length and 32 meters in width. Will there be enough paint if 250 grams of paint are required per square meter?
|
The paint is sufficient.
|
olympiads
| 0.234375
|
Segments \( AM \) and \( BH \) are the median and altitude, respectively, of an acute-angled triangle \( ABC \). It is known that \( AH = 1 \) and \( 2 \angle MAC = \angle MCA \). Find the side \( BC \).
|
2
|
olympiads
| 0.1875
|
In a room, there are knights, who always tell the truth, and liars, who always lie (both are definitely present). They were asked the question: "How many liars are there in the room?" All possible answers from 1 to 200 were given (some possibly multiple times). How many liars could there actually be?
|
199 \text{ or } 200
|
olympiads
| 0.265625
|
Find all functions \(f, g: (0, +\infty) \rightarrow (0, +\infty)\) such that for all \(x \in \mathbf{R}^{-}\), the following holds:
$$
f(g(x)) = \frac{x}{x f(x) - 2}, \quad \text{and} \quad g(f(x)) = \frac{x}{x g(x) - 2}.
$$
|
f(x) = g(x) = \frac{3}{x}
|
olympiads
| 0.0625
|
Around a round table with $2n$ seats, $n$ girls and $n$ boys are seated. How many possible ways are there to do this while respecting an alternating girl-boy pattern?
|
2(n!)^2
|
olympiads
| 0.15625
|
As shown in the figure, Rourou's garden is a large rectangle composed of 4 square plots of land and 1 small rectangular pool. If the area of each plot of land is 20 square meters and the length of the garden is 9 meters, what is the perimeter of the pool (the shaded area in the figure)? ___ meters.
|
18
|
olympiads
| 0.0625
|
The base of the pyramid is a right triangle with a hypotenuse equal to 6 and an acute angle of $15^{\circ}$. All lateral edges are inclined to the plane of the base at an angle of $45^{\circ}$. Find the volume of the pyramid.
|
4.5
|
olympiads
| 0.09375
|
Oleg gathered a bag of coins. Sasha counted them and found that if all the coins are divided into five equal piles, there will be two leftover coins. Additionally, if divided into four equal piles, there will be one leftover coin. However, the coins can be divided into three equal piles without any leftovers. What is the smallest number of coins Oleg could have?
|
57
|
olympiads
| 0.15625
|
Find the largest integer \( N \) such that both \( N + 496 \) and \( N + 224 \) are perfect squares.
|
4265
|
olympiads
| 0.203125
|
Construct a mean proportional between two given lines without circumscribing circles and drawing perpendiculars.
|
\sqrt{a \cdot b}
|
olympiads
| 0.171875
|
Given that both $\alpha$ and $\cos(\alpha \pi)$ are rational numbers, determine all possible values of $\cos(\alpha \pi)$.
|
0, \pm 1, \pm \frac{1}{2}
|
olympiads
| 0.203125
|
Find all integers \(x, y, z\) such that
\[
3 x^{2} + 7 y^{2} = z^{4}
\]
|
(x, y, z) = (0, 0, 0)
|
olympiads
| 0.34375
|
Through the points \( R \) and \( E \), located on the sides \( AB \) and \( AD \) of the parallelogram \( ABCD \), such that \( AR = \frac{2}{3} AB \) and \( AE = \frac{1}{3} AD \), a line is drawn. Find the ratio of the area of the parallelogram to the area of the resulting triangle.
|
9
|
olympiads
| 0.21875
|
Given a square with a side length of 8 cm. Each side of the square is divided by a point into two segments with lengths of 2 cm and 6 cm. Find the area of the quadrilateral whose vertices are these points.
|
32 \, \text{cm}^2 \, \text{, } \, 40 \, \text{cm}^2 \, \text{, } \, 24 \, \text{cm}^2
|
olympiads
| 0.15625
|
Given a regular hexagon \( A B C D E F \) with a side length of 1, calculate \((\overrightarrow{A B}+\overrightarrow{D C}) \cdot(\overrightarrow{A D}+\overrightarrow{B E})\).
|
-3
|
olympiads
| 0.078125
|
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