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Compute the value of the infinite series
$$
\sum_{n=2}^{\infty} \frac{n^{4}+3 n^{2}+10 n+10}{2^{n} \cdot \left(n^{4}+4\right)}
$$
|
\frac{11}{10}
| 0.375 |
A fishing boat fishes in the territorial waters of a foreign country without permission. Each time the net is cast, it causes an equivalent-valued loss in the country's fish resources. The probability that the boat will be detained by the foreign coast guard during any net casting is $1/k$, where $k$ is a natural number specific to the country. Assume that the events of being detained or not during each net casting are independent of previous fishing activities. If the boat is detained by the coast guard, all previously caught fish are confiscated, and future fishing in those waters is prohibited. The captain plans to leave the foreign territorial waters after casting the net $n$ times.
Given that there is always a possibility of being detained by the foreign coast guard, making the fishing profit a random variable, determine the value of $n$ that maximizes the expected value of the fishing profit.
|
n = k-1
| 0.5 |
Find all functions \( f: \mathbb{R} \to \mathbb{R} \) such that for all real numbers \( x \) and \( y \), the following equation holds:
$$
f(y f(x+y) + f(x)) = 4x + 2y f(x+y)
$$
|
f(x) = 2x
| 0.75 |
We are considering numbers that can be expressed as \(10 \times n + 1\), where \( \mathbf{n} \) is a positive integer. For example: \(11 = 10 \times 1 + 1\) and \(331 = 10 \times 33 + 1\). If such a number cannot be expressed as the product of two smaller numbers of the form \(10 \times n + 1\) (these two smaller numbers can be equal), we call this number a "Zhonghuan number". For example, \(341 = 11 \times 31\), since it can be expressed as the product of two numbers of the form \(10 \times n + 1\), it is not a "Zhonghuan number". However, 11 cannot be expressed as the product of two smaller numbers of the form \(10 \times n + 1\), so it is a "Zhonghuan number". How many "Zhonghuan numbers" are there among \(11, 21, 31, \ldots, 991\)?
|
87
| 0.625 |
On a ruler, three marks are indicated: 0, 2, and 5. How can you use this ruler to measure a segment equal to 6?
|
6
| 0.5 |
Let \( P \) be a polynomial with real coefficients such that
$$
\forall x \in \mathbb{R}, P(x) \geq 0.
$$
Show that there exist two polynomials \( A \) and \( B \) with real coefficients such that \( P = A^{2} + B^{2} \).
|
P = A^2 + B^2
| 0.75 |
Real numbers \(a, b, c\) and positive number \(\lambda\) make \(f(x) = x^3 + ax^2 + bx + c\) have three real roots \(x_1, x_2, x_3\), and satisfy:
1. \(x_2 - x_1 = \lambda\);
2. \(x_3 > \frac{1}{2}(x_1 + x_2)\).
Find the maximum value of \(\frac{2a^3 + 27c - 9ab}{\lambda^3}\).
|
\frac{3\sqrt{3}}{2}
| 0.875 |
For each natural number from 1 to 999, Damir subtracted the last digit from the first digit and wrote all the resulting 1000 differences on the board. For example, for the number 7, Damir wrote 0; for the number 105, he wrote (-4); for the number 61, he wrote 5.
What is the sum of all the numbers on the board?
|
495
| 0.75 |
In the pyramid \( A B C D \), the angle \( A B C \) is \( \alpha \). The orthogonal projection of point \( D \) onto the plane \( A B C \) is point \( B \). Find the angle between the planes \( A B D \) and \( C B D \).
|
\alpha
| 0.75 |
A circle is constructed on the side $BC$ of triangle $ABC$ as its diameter, and it intersects segment $AB$ at point $D$. Find the ratio of the areas of triangles $ABC$ and $BCD$, given that $AC = 15$, $BC = 20$, and $\angle ABC = \angle ACD$.
|
\frac{25}{16}
| 0.375 |
Let \(m\) be a positive integer, and let \(T\) denote the set of all subsets of \(\{1, 2, \ldots, m\}\). Call a subset \(S\) of \(T\) \(\delta\)-good if for all \(s_1, s_2 \in S\), \(s_1 \neq s_2\), \(|\Delta(s_1, s_2)| \geq \delta m\), where \(\Delta\) denotes symmetric difference (the symmetric difference of two sets is the set of elements that are in exactly one of the two sets). Find the largest possible integer \(s\) such that there exists an integer \(m\) and a \(\frac{1024}{2047}\)-good set of size \(s\).
|
2048
| 0.375 |
Find the largest 5-digit number \( A \) that satisfies the following conditions:
1. Its 4th digit is greater than its 5th digit.
2. Its 3rd digit is greater than the sum of its 4th and 5th digits.
3. Its 2nd digit is greater than the sum of its 3rd, 4th, and 5th digits.
4. Its 1st digit is greater than the sum of all other digits.
(from the 43rd Moscow Mathematical Olympiad, 1980)
|
95210
| 0.75 |
A regular hexagon is drawn on the plane with a side length of 1. Using only a ruler, construct a segment whose length is $\sqrt{7}$.
|
\sqrt{7}
| 0.875 |
In the permutation \(a_{1}, a_{2}, a_{3}, a_{4}, a_{5}\) of \(1, 2, 3, 4, 5\), how many permutations are there that satisfy \(a_{1} < a_{2}, a_{2} > a_{3}, a_{3} < a_{4}, a_{4} > a_{5}\)?
|
16
| 0.875 |
Egor wrote a number on the board and encrypted it according to the rules of letter puzzles (different letters correspond to different digits, identical letters - identical digits). The result was the word "GWATEMALA". How many different numbers could Egor have originally written if his number was divisible by 5?
|
114240
| 0.375 |
Given that the real numbers \( x \) and \( y \) satisfy \( 4x^2 - 5xy + 4y^2 = 5 \), let \( S = x^2 + y^2 \). Determine the value of \( \frac{1}{S_{\max}} + \frac{1}{S_{\min}} \).
|
\frac{8}{5}
| 0.875 |
Given an infinite grid where each cell is either red or blue, such that in any \(2 \times 3\) rectangle exactly two cells are red, determine how many red cells are in a \(9 \times 11\) rectangle.
|
33
| 0.25 |
Find the flux of the vector field
$$
\vec{a} = x \vec{i} + y \vec{j} + z \vec{k}
$$
through the part of the surface
$$
x^2 + y^2 = 1
$$
bounded by the planes \( z = 0 \) and \( z = 2 \). (The normal vector is outward to the closed surface formed by these surfaces).
|
4\pi
| 0.375 |
Find the largest three-digit number that is equal to the sum of its digits and the square of twice the sum of its digits.
|
915
| 0.375 |
Consider a \(2 \times 3\) grid where each entry is one of 0, 1, and 2. For how many such grids is the sum of the numbers in every row and in every column a multiple of 3? One valid grid is shown below:
$$
\left[\begin{array}{lll}
1 & 2 & 0 \\
2 & 1 & 0
\end{array}\right]
$$
|
9
| 0.5 |
When Vitya was a first grader, he had a set of 12 number cards: two cards with the number 1, two cards with the number 2, and so on up to the number 6. Vitya placed them on the table in a random order from left to right, and then removed the first 1, the first 2, the first 3, and so on. For example, if Vitya initially had the sequence 434653625112, the resulting sequence would be 436512. What is the probability that the sequence left on the table is 123456?
|
\frac{1}{720}
| 0.125 |
Given that positive integers \( a, b, c \) satisfy the equation \( a^{2} b + b^{2} c + a c^{2} + a + b + c = 2(a b + b c + a c) \), find \( \frac{c^{2017}}{a^{2016} + b^{2018}} \).
|
\frac{1}{2}
| 0.875 |
On the sides \( AB, BC \), and \( AC \) of triangle \( ABC \), points \( K, L \), and \( M \) are taken respectively, such that \( AK:KB = 2:3 \), \( BL:LC = 1:2 \), and \( CM:MA = 3:1 \). In what ratio does the segment \( KL \) divide the segment \( BM \)?
|
1:1
| 0.625 |
A permutation of $\{1, \ldots, n\}$ is drawn at random. How many fixed points does it have on average?
|
1
| 0.75 |
Adam the Ant started at the left-hand end of a pole and crawled $\frac{2}{3}$ of its length. Benny the Beetle started at the right-hand end of the same pole and crawled $\frac{3}{4}$ of its length. What fraction of the length of the pole are Adam and Benny now apart?
|
\frac{5}{12}
| 0.875 |
For \(x = \frac{\pi}{2n}\), find the value of the sum
$$
\cos ^{2}(x)+\cos ^{2}(2 x)+\cos ^{2}(3 x)+\ldots+\cos ^{2}(n x)
$$
|
\frac{n-1}{2}
| 0.375 |
The following numbers are written on a wall: \(1, 3, 4, 6, 8, 9, 11, 12, 16\). Four of these numbers were written by Vova, four were written by Dima, and one of the numbers is the house number of the local police officer. The officer discovered that the sum of the numbers written by Vova is three times the sum of the numbers written by Dima. Calculate the house number.
|
6
| 0.75 |
Find the smallest natural number that has exactly 12 different natural divisors, the largest prime divisor of which is the number 101, and the last digit is zero.
|
2020
| 0.625 |
In the base of a regular triangular prism is a triangle \( ABC \) with a side length of \( a \). Points \( A_{1} \), \( B_{1} \), and \( C_{1} \) are taken on the lateral edges, and their distances from the base plane are \( \frac{a}{2} \), \( a \), and \( \frac{3a}{2} \) respectively. Find the angle between the planes \( ABC \) and \( A_{1} B_{1} C_{1} \).
|
45^{\circ}
| 0.75 |
Find all odd integers \( y \) such that there exists an integer \( x \) satisfying
\[ x^{2} + 2 y^{2} = y x^{2} + y + 1. \]
|
y = 1
| 0.75 |
Given a triangle \(ABC\) with \(AB=4\), \(BC=4\), and \(AC=1\). From point \(A\), an angle bisector is drawn that intersects the circumcircle of this triangle at point \(D\). Find the length \(DI\), where \(I\) is the incenter of triangle \(ABC\).
|
\frac{8}{3}
| 0.125 |
Several boys and girls are seated around a round table. It is known that to the left of exactly 7 girls sit girls, and to the left of 12 girls sit boys. It is also known that for 75% of the boys, girls sit to their right. How many people are seated at the table?
|
35
| 0.625 |
A cylinder with a volume of 9 is inscribed in a cone. The plane of the upper base of this cylinder truncates the original cone, forming a frustum with a volume of 63. Find the volume of the original cone.
|
64
| 0.75 |
Denote by \(\mathbb{N}\) the set of all positive integers. Find all functions \(f: \mathbb{N} \rightarrow \mathbb{N}\) such that for all positive integers \(m\) and \(n\), the integer \(f(m) + f(n) - mn\) is nonzero and divides \(mf(m) + nf(n)\).
|
f(n) = n^2
| 0.875 |
How many two-digit numbers have the tens digit greater than the units digit?
|
45
| 0.75 |
1500 products are packed into 15 boxes for shipment. After sealing the boxes, it is discovered that one product was not packed into any box. Due to strict sealing requirements, boxes cannot be reopened for inspection. Someone suggests using a scale to weigh the boxes to find which box is missing a product (because the box missing a product will be lighter). Normally, it would take 14 weighings to find the box missing the product. Is there a way to reduce the number of weighings and still determine which box is missing a product?
|
4
| 0.125 |
Given a convex quadrilateral \(ABCD\) with \(\angle C = 57^{\circ}\), \(\sin \angle A + \sin \angle B = \sqrt{2}\), and \(\cos \angle A + \cos \angle B = 2 - \sqrt{2}\), find the measure of angle \(D\) in degrees.
|
168^\circ
| 0.875 |
The vertex \( A \) of the acute triangle \( ABC \) is equidistant from the circumcenter \( O \) and the orthocenter \( H \). Determine all possible values for the measure of angle \( A \).
|
60^\circ
| 0.875 |
There are three cars, $A$, $B$, and $C$, starting from the same location and traveling along the same road. They each catch up to a cyclist, Xiao Long, who is traveling in the same direction, at 5 minutes, 10 minutes, and 12 minutes respectively. Given that the speeds of $A$ and $B$ are 24 km/h and 20 km/h respectively, find the speed of car $C$, in km/h.
|
\frac{58}{3}
| 0.875 |
Given that \(a - b = 2 + \sqrt{3}\) and \(b - c = 2 - \sqrt{3}\), find the value of \(a^2 + b^2 + c^2 - ab - bc - ca\).
|
15
| 0.25 |
Arrange in ascending order the numbers that each contain exactly once each of the digits $1, 2, \ldots, 9$. Which number is in the 100,000th position?
|
358926471
| 0.25 |
Given the parabola \( y^{2} = 4ax \) where \( 0 < a < 1 \) with a focus at \( F \). Taking \( A(a + 4, 0) \) as the center of a circle, and \(|AF|\) as the radius, draw a circle above the x-axis that intersects the parabola at two distinct points \( M \) and \( N \). Let \( P \) be the midpoint of \( MN \).
(1) Find the value of \( |MF| + |NF| \).
(2) Does a value of \( a \) exist such that \( |MF| \), \( |PF| \), and \( |NF| \) form an arithmetic sequence? If such a value exists, find \( a \); if it does not, provide an explanation.
|
8
| 0.625 |
As shown in the diagram, \( AB \) is a path 28 meters long, and \( M \) is the midpoint of \( AB \). A dog starts running from a point on the left side of \( M \) along the path. The first run is 10 meters, the second run is 14 meters; the odd-numbered runs are 10 meters each, and the even-numbered runs are 14 meters each. The dog changes direction as follows: if after finishing a run \( M \) is to its right, it runs to the right next; if \( M \) is to its left, it runs to the left next. After running 20 times, the dog is 1 meter to the left of point \( B \). How far was the dog from point \( A \) when it started?
|
7
| 0.25 |
ABCDEF is a six-digit number. All of its digits are different and arranged in ascending order from left to right. This number is a perfect square.
Determine what this number is.
|
134689
| 0.5 |
For how many ordered pairs of positive integers \((a, b)\) is \(1 < a + b < 22\)?
|
210
| 0.875 |
Find the pairs of positive integers \((a, b)\) such that \(\frac{a b^{2}}{a+b}\) is a prime number, where \(a \neq b\).
|
(6,2)
| 0.875 |
Simplify the following expressions:
$$
\frac{a^{3}+b^{3}}{(a-b)^{2}+ab}, \quad \frac{x^{2}-4ax+4a^{2}}{x^{2}-4a^{2}}, \quad \frac{xy-2x-3y+6}{xy-2x}
$$
|
\frac{x-3}{x}
| 0.375 |
As shown in Figure 1, a line segment of length 1 is divided into two parts, $x$ and $y$. Then, the segment of length $x$ is bent into a semicircular arc $ACB$, and the segment of length $y$ is folded into a rectangle $ABDE$ along three sides $(BD, DE, EA)$, forming a closed "curved shape" $ACBDEA$. What is the maximum area of this curved shape?
|
\frac{1}{2(\pi+4)}
| 0.625 |
Father has left several identical gold coins to his children. According to his will, the oldest child receives one coin and one-seventh of the remaining coins, the next child receives two coins and one-seventh of the remaining coins, the third child receives three coins and one-seventh of the remaining coins, and so on through the youngest child. If every child inherits an integer number of coins, find the number of children and the number of coins.
|
36
| 0.25 |
There were two decks of 36 cards each on the table. The first deck was shuffled and placed on top of the second deck. Then, for each card in the first deck, the number of cards between it and the same card from the second deck was counted (i.e., the number of cards between the sevens of hearts, between the queens of spades, etc.). What is the sum of these 36 counted numbers?
|
1260
| 0.375 |
Given that the positive real numbers \( u, v, \) and \( w \) are all not equal to 1, if \(\log _{u} (v w)+\log _{v} w=5\) and \(\log _{v} u+\log _{w} v=3\), then find the value of \(\log _{w} u\).
|
\frac{4}{5}
| 0.875 |
Consider all possible 100-digit numbers where each digit is either 1 or 2. For each number, compute the remainder when divided by 1024. How many different remainders are there?
|
1024
| 0.25 |
After walking $\frac{4}{9}$ of the bridge's length, a pedestrian notices a car catching up from behind, which has not yet entered the bridge. The pedestrian then turns back and meets the car at the start of the bridge. If the pedestrian had continued walking, the car would have caught up with them at the end of the bridge. Find the ratio of the car's speed to the pedestrian's speed.
|
9
| 0.75 |
Find natural numbers \(a, b, c\) such that for any \(n \in \mathbf{N}\) with \(n > 2\), the following inequality holds:
\[ b - \frac{c}{(n-2)!} < \frac{2^{3}-a}{2!} + \frac{3^{3}-a}{3!} + \cdots + \frac{n^{3}-a}{n!} < b. \]
|
a = 5, b = 9, c = 4
| 0.625 |
Decode the equation ${ }_{* *}+{ }_{* * *}={ }_{* * * *}$, given that both addends and the sum remain unchanged when read from right to left.
|
22 + 979 = 1001
| 0.5 |
A repunit is a positive integer, all of whose digits are 1's. Let \( a_{1} < a_{2} < a_{3} < \ldots \) be a list of all the positive integers that can be expressed as the sum of distinct repunits. Compute \( a_{111} \).
|
1223456
| 0.125 |
The following shows the graph of \( y = p x^{2} + 5 x + p \). The points \( A = (0, -2) \), \( B = \left(\frac{1}{2}, 0\right) \), \( C = (2, 0) \), and \( O = (0, 0) \) are given.
Find the value of \( p \).
|
-2
| 0.875 |
Find the equation of the line passing through the point \( M(-1, 4) \) and perpendicular to the line \( x - 2y + 4 = 0 \).
|
2x + y - 2 = 0
| 0.625 |
If real numbers $\alpha, \beta, \gamma$ form a geometric sequence with a common ratio of 2, and $\sin \alpha, \sin \beta, \sin \gamma$ also form a geometric sequence, then $\cos \alpha = $ ?
|
-\frac{1}{2}
| 0.875 |
We will call a natural number interesting if all its digits are different, and the sum of any two adjacent digits is the square of a natural number. Find the largest interesting number.
|
6310972
| 0.375 |
If \( x^{3} - 3 \sqrt{2} x^{2} + 6 x - 2 \sqrt{2} - 8 = 0 \), find the value of \( x^{5} - 41 x^{2} + 2012 \).
|
1998
| 0.75 |
The numbers \(a_1, a_2, a_3, a_4,\) and \(a_5\) form a geometric progression. Among them, there are both rational and irrational numbers. What is the maximum number of terms in this progression that can be rational numbers?
|
3
| 0.875 |
Each face of a hexahedron and each face of a regular octahedron are equilateral triangles with side length \(a\). The ratio of the radii of the inscribed spheres (inradii) of the two polyhedra is a reduced fraction \(\frac{m}{n}\). What is the product \(m \cdot n\)?
|
6
| 0.25 |
How many six-digit numbers exist that have three even and three odd digits?
|
281250
| 0.25 |
Is it possible to fill an $n \times n$ table with the numbers $-1, 0,$ and $1$ such that the sums of all rows, all columns, and both main diagonals are different? The main diagonals of the table are the diagonals drawn from the top-left corner to the bottom-right corner and from the top-right corner to the bottom-left corner.
|
\text{No}
| 0.75 |
In the sequence $\left\{a_{n}\right\}$, $a_{0} = 1$, $a_{1} = 2$, and $n(n+1) a_{n+1} = n(n-1) a_{n} - (n-2) a_{n-1}$ for $n \geq 1$. Find $a_{n}$ for $n \geq 2$.
|
\frac{1}{n!}
| 0.5 |
Karlson filled a conical glass with lemonade and drank half of it by height (measured from the surface of the liquid to the top of the cone), and the other half was finished by Kid. By what factor did Karlson drink more lemonade than Kid?
|
7
| 0.75 |
Eli, Joy, Paul, and Sam want to form a company, and the company will have 16 shares to split among the 4 people. The following constraints are imposed:
- Every person must get a positive integer number of shares, and all 16 shares must be given out.
- No one person can have more shares than the other three people combined.
Assuming that shares are indistinguishable, but people are distinguishable, in how many ways can the shares be given out?
|
315
| 0.25 |
Integers \(1, 2, \ldots, n\) are written (in some order) on the circumference of a circle. What is the smallest possible sum of moduli of the differences of neighboring numbers?
|
2n - 2
| 0.125 |
We call the "last digit of a number" the digit that is most to the right. For example, the last digit of 2014 is the digit 4.
a) What is the last digit of \(11^{11}\)?
b) What is the last digit of \(9^{9}\)? And what is the last digit of \(9219^{9219}\)?
c) What is the last digit of \(2014^{2014}\)?
|
6
| 0.875 |
The sequence \( x_{n} \) has its first two elements as \( x_{1}=1001 \) and \( x_{2}=1003 \). For \( n \geq 1 \), the recurrence relation is given by:
\[ x_{n+2}=\frac{x_{n+1}-2004}{x_{n}}. \]
What is the sum of the first 2004 terms of the sequence?
|
1338004
| 0.5 |
The Red Sox play the Yankees in a best-of-seven series that ends as soon as one team wins four games. Suppose that the probability that the Red Sox win Game \( n \) is \(\frac{n-1}{6}\). What is the probability that the Red Sox will win the series?
|
\frac{1}{2}
| 0.125 |
What is the probability that in a group of 13 people, at least two were born in September? (For simplicity, assume that each month is equally likely.)
|
0.296
| 0.125 |
Buratino buried two ingots in the Field of Miracles - one gold and one silver. On days with good weather, the gold ingot increases by 30%, and the silver ingot increases by 20%. On days with bad weather, the gold ingot decreases by 30%, and the silver ingot decreases by 20%. After one week, it turns out that one ingot increased and the other decreased. How many days of good weather were there?
|
4
| 0.5 |
In the diagram, there is a karting track layout. Start and finish are at point $A$, but the go-kart driver can return to point $A$ any number of times and resume the lap.
The path from $A$ to $B$ and vice versa takes one minute. The loop also takes one minute to complete. The loop can only be traversed counterclockwise (the arrows indicate the possible direction of travel). The driver does not turn back halfway and does not stop. The racing duration is 10 minutes. Find the number of possible different routes (sequences of segments traversed).
|
34
| 0.25 |
The function \( g \) defined on the set of integers satisfies the following conditions:
1) \( g(1) - 1 > 0 \)
2) \( g(x) g(y) + x + y + xy = g(x+y) + x g(y) + y g(x) \) for any \( x, y \in \mathbb{Z} \)
3) \( 3 g(x+1) = g(x) + 2x + 3 \) for any \( x \in \mathbb{Z} \).
Find \( g(-6) \).
|
723
| 0.875 |
Find all real functions such that
$$
(x+y)(f(x)-f(y))=(x-y) f(x+y)
$$
|
f(x) = ax^2 + bx
| 0.5 |
In a circle, mutually perpendicular diameters $AB$ and $CD$ are drawn, and a point $M$ is taken arbitrarily on the arc $AC$. Find $MB + MD$ if $MA + MC = a$.
|
a(1 + \sqrt{2})
| 0.25 |
Xiaoming and Xiaoliang are two stamp enthusiasts. Xiaoming exchanges two stamps with a face value of 1 yuan 6 jiao each (according to the face value of the stamps) for Xiaoliang's stamps with a face value of 2 jiao each. Before the exchange, the number of stamps Xiaoliang had was 5 times the number Xiaoming had. After the exchange, the number of stamps Xiaoliang has is 3 times the number Xiaoming has. Find the total number of stamps they have.
|
168
| 0.875 |
What is the area of the set of points $P(x ; y)$ in the right-angled coordinate system that satisfy the condition $|x+y|+|x-y| \leq 4?$
|
16
| 0.5 |
In the number \(2016^{* * * *} 02 *\), each of the 5 asterisks needs to be replaced with any of the digits \(0, 2, 4, 6, 7, 8\) (digits can be repeated) so that the resulting 11-digit number is divisible by 6. In how many ways can this be done?
|
2160
| 0.625 |
Milan has a bag of 2020 red balls and 2021 green balls. He repeatedly draws 2 balls out of the bag uniformly at random. If they are the same color, he changes them both to the opposite color and returns them to the bag. If they are different colors, he discards them. Eventually, the bag has 1 ball left. Let \( p \) be the probability that it is green. Compute \( \lfloor 2021 p \rfloor \).
|
2021
| 0.5 |
On the coordinate plane, the graphs of three reduced quadratic polynomials intersect the y-axis at the points $-15,-6,-27$ respectively. For each of the polynomials, the coefficient at $x$ is a natural number, and the larger root is a prime number. Find the sum of all roots of these polynomials.
|
-9
| 0.75 |
Find the smallest positive integer that cannot be expressed in the form $\frac{2^{a}-2^{b}}{2^{c}-2^{d}}$, where $a$, $b$, $c$, and $d$ are all positive integers.
|
11
| 0.625 |
The first brigade of workers is paving a section of road, while the second brigade, which has 6 more workers, is paving another section of road that is three times larger in area. The productivity of all workers is the same. What is the minimum number of workers that could have been in the first brigade if they completed their work faster? If there are no solutions, then the answer should be 0.
|
4
| 0.875 |
Calculate the limit of the function:
$$
\lim _{x \rightarrow 3} \frac{\ln (2 x-5)}{e^{\sin \pi x}-1}
$$
|
-\frac{2}{\pi}
| 0.875 |
In a class of 28 students, 4 prizes are distributed. How many ways can this be done,
a) if the prizes are identical and each student can receive at most one prize?
b) if the prizes are identical and each student can receive more than one prize?
c) if the prizes are distinct and each student can receive at most one prize?
d) if the prizes are distinct and each student can receive more than one prize?
|
614656
| 0.5 |
Determine all functions from \(\mathbb{R}\) to \(\mathbb{R}\) satisfying:
\[ \forall(x, y) \in \mathbb{R}^{2}, f(x+y) = x f(x) + y f(y) \]
|
f(x) = 0
| 0.875 |
There are 10,001 students at a university. Some students join together to form several clubs (a student may belong to different clubs). Some clubs join together to form several societies (a club may belong to different societies). There are a total of \( k \) societies. Suppose that the following conditions hold:
1. Each pair of students are in exactly one club.
2. For each student and each society, the student is in exactly one club of the society.
3. Each club has an odd number of students. In addition, a club with \( 2m + 1 \) students (where \( m \) is a positive integer) is in exactly \( m \) societies.
Find all possible values of \( k \).
|
5000
| 0.625 |
Find the largest natural number that cannot be expressed as the sum of two composite numbers.
|
11
| 0.75 |
The circus arena is illuminated by \( n \) different spotlights. Each spotlight illuminates a certain convex figure. It is known that if any one spotlight is turned off, the arena will still be fully illuminated, but if any two spotlights are turned off, the arena will no longer be fully illuminated. For what values of \( n \) is this possible?
|
n \geq 2
| 0.75 |
Given \(\alpha\) and \(\beta\) satisfy the equations
\[
\begin{array}{c}
\alpha^{3}-3 \alpha^{2}+5 \alpha-4=0, \\
\beta^{3}-3 \beta^{2}+5 \beta-2=0 .
\end{array}
\]
find \(\alpha + \beta\).
|
2
| 0.875 |
Katya correctly solves a problem with a probability of $4 / 5$, and the magic pen solves a problem correctly without Katya's help with a probability of $1 / 2$. In a test containing 20 problems, solving 13 correctly is enough to get a "good" grade. How many problems does Katya need to solve on her own and how many should she leave to the magic pen to ensure that the expected number of correct answers is at least 13?
|
10
| 0.625 |
There are 200 computers in a computer center, some of which are connected by cables in pairs, with a total of 345 cables used. We call a "cluster" a set of computers such that any computer in this set can send a signal to all others through the cables. Initially, all computers formed one cluster. However, one night an evil hacker cut several cables, resulting in 8 clusters. Find the maximum possible number of cables that were cut.
|
153
| 0.875 |
Given a sequence of real numbers \( a_{1}, a_{2}, a_{3}, \ldots \) that satisfy
1) \( a_{1}=\frac{1}{2} \), and
2) \( a_{1} + a_{2} + \ldots + a_{k} = k^{2} a_{k} \), for \( k \geq 2 \).
Determine the value of \( a_{100} \).
|
\frac{1}{10100}
| 0.75 |
Let $n$ be an integer such that $1 \leq n \leq 30$. Ariane and Bérénice are playing the following game. First, Ariane chooses $n$ distinct numbers from the integers $1, 2, \ldots, 30$. Then, based on Ariane's choices, Bérénice chooses an integer $d \geq 2$. Bérénice wins if $d$ divides at least two of the numbers chosen by Ariane. Otherwise, Ariane wins.
For which values of $n$ can Ariane ensure she will win?
|
1 \leq n \leq 11
| 0.5 |
From the 2015 positive integers 1, 2, ... , 2015, select $k$ numbers such that the sum of any two different selected numbers is not a multiple of 50. Find the maximum value of $k$.
|
977
| 0.125 |
Vanya wrote the number 1347 on the board.
- Look! - Petya noticed. - In this number, each of the last two digits is equal to the sum of the two preceding digits.
- Exactly! - Vasya agreed.
- Can you write the largest four-digit number with this property?
Help Vasya solve Petya's task.
|
9099
| 0.625 |
Given a square $ABCD$ with side length $a$, vertex $A$ lies in plane $\beta$, and the other vertices are on the same side of plane $\beta$. The distances from points $B$ and $D$ to plane $\beta$ are $1$ and $2$, respectively. If the angle between plane $ABCD$ and plane $\beta$ is $30^\circ$, find $a$.
|
2 \sqrt{5}
| 0.25 |
Five friends - Katya, Polina, Alena, Lena, and Sveta - gather every day in the park after buying ice cream at the shop around the corner. One day, they had a conversation:
Polina: I was standing next to Alena.
Alena: I was the first in line!
Lena: No one was behind me.
Katya: There were five people in front of me.
Sveta: There was only one person behind me.
The girls are friends, so they do not lie to each other. How many people were there between Katya and Polina?
|
0
| 0.375 |
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