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A great-grandfather banker left an inheritance to his newborn great-grandson. According to the bank agreement, the amount in the great-grandson's account increases. Each year, on the day after his birthday, the current amount increases by one million rubles more than the previous year. Thus, if the initial amount was zero rubles, after one year it will be +1 million rubles; after 2 years, $1+2$ million rubles; after 3 years, $1+2+3$; and so on. According to the agreement, the process will stop, and the great-grandson will receive the money when the amount in the account is a three-digit number consisting of three identical digits.
How old will the great-grandson be when the conditions of the agreement are fulfilled?
|
36
| 0.875 |
In the sum \(1+3+9+27+81+243+729 \), one can strike out any terms and change some signs in front of the remaining numbers from "+" to "-". Masha wants to get an expression equal to 1 in this way, then (starting from scratch) get an expression equal to 2, then (starting again from scratch) get 3, and so on. Up to what maximum integer will she be able to do this without skipping any numbers?
|
1093
| 0.625 |
Calculate the area of the figure bounded by the curves given by the following equations:
$$
\begin{aligned}
& \left\{\begin{array}{l}
x=16 \cos ^{3} t \\
y=2 \sin ^{3} t
\end{array}\right. \\
& x=2 \quad (x \geq 2)
\end{aligned}
$$
|
4\pi
| 0.375 |
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}
| 0.25 |
In an $8 \times 8$ table, 23 cells are black, and the rest are white. In each white cell, the sum of the black cells located in the same row and the black cells located in the same column is written. Nothing is written in the black cells. What is the maximum value that the sum of the numbers in the entire table can take?
|
234
| 0.125 |
Find the value of the constant \( c \) so that the function \( f(x) = \arctan \frac{2-2x}{1+4x} + c \) is an odd function on the interval \(\left(-\frac{1}{4}, \frac{1}{4}\right) \).
|
-\arctan 2
| 0.5 |
Select 3 numbers from the range 1 to 300 such that their sum is exactly divisible by 3. How many such combinations are possible?
|
1485100
| 0.125 |
Given that \( 0<a<b<c<d<300 \) and the equations:
\[ a + d = b + c \]
\[ bc - ad = 91 \]
Find the number of ordered quadruples of positive integers \((a, b, c, d)\) that satisfy the above conditions.
|
486
| 0.875 |
Let \( A_{10} \) denote the answer to problem 10. Two circles lie in the plane; denote the lengths of the internal and external tangents between these two circles by \( x \) and \( y \), respectively. Given that the product of the radii of these two circles is \( \frac{15}{2} \), and that the distance between their centers is \( A_{10} \), determine \( y^{2} - x^{2} \).
|
30
| 0.75 |
The vertex of a parabola is \( O \) and its focus is \( F \). When a point \( P \) moves along the parabola, find the maximum value of the ratio \( \left|\frac{P O}{P F}\right| \).
|
\frac{2\sqrt{3}}{3}
| 0.875 |
A square-based truncated pyramid is inscribed around a given sphere. What can be the ratio of the truncated pyramid's volume to its surface area?
|
\frac{r}{3}
| 0.75 |
Xiaopang and Xiaoya both have their birthdays in May, and both fall on a Wednesday. Xiaopang's birthday is later, and the sum of their birth dates is 38. What is Xiaopang's birthday in May?
|
26
| 0.5 |
Find the number of ordered triples of nonnegative integers \((a, b, c)\) that satisfy
\[
(a b+1)(b c+1)(c a+1)=84.
\]
|
12
| 0.375 |
The function \( f \) is defined on the set of natural numbers and satisfies the following conditions:
(1) \( f(1)=1 \);
(2) \( f(2n)=f(n) \), \( f(2n+1)=f(2n)+1 \) for \( n \geq 1 \).
Find the maximum value \( u \) of \( f(n) \) when \( 1 \leq n \leq 1989 \) and determine how many values \( n \) (within the same range) satisfy \( f(n)=u \).
|
5
| 0.125 |
Among any \( m \) consecutive natural numbers, if there is always a number whose sum of the digits is a multiple of 6, what is the smallest value of \( m \)?
|
9
| 0.375 |
In the plane, sequentially draw \( n \) segments end-to-end such that the endpoint of the \( n \)-th segment coincides with the starting point of the 1st segment. Each segment is called a "segment". If the starting point of one segment is exactly the endpoint of another segment, these two segments are called adjacent. We stipulate that:
1. Adjacent segments cannot be drawn on the same straight line.
2. Any two non-adjacent segments do not intersect.
A figure satisfying these criteria is called a "simple polyline loop". For instance, a simple polyline loop with ten segments might exactly lie on five straight lines.
If a simple polyline loop’s \( n \) segments exactly lie on six straight lines, find the maximum value of \( n \) and provide an explanation.
|
12
| 0.625 |
If \(a, b, c,\) and \(d\) are consecutive integers, then the sum
\[ ab + ac + ad + bc + bd + cd + 1 \]
is divisible by 12.
|
12
| 0.875 |
As shown in the figure, each small square has a side length of $10 \mathrm{~km}$. There are 2 gas stations in the figure. A car starts from point $A$ and travels along the edges of the small squares to point $B$. If the car needs to refuel every $30 \mathrm{~km}$, how many shortest routes are there for the car to reach point $B$?
|
18
| 0.375 |
The sequence \(\left(x_{n}\right)\) is defined by the recurrence relations
$$
x_{1}=1, \, x_{2}=2, \, x_{n}=\left|x_{n-1}-x_{n-2}\right| \, \text{for} \, n > 2.
$$
Find \(x_{1994}\).
|
0
| 0.75 |
If the function
$$
f(x) = 3 \cos \left(\omega x + \frac{\pi}{6}\right) - \sin \left(\omega x - \frac{\pi}{3}\right) \quad (\omega > 0)
$$
has a minimum positive period of \(\pi\), then the maximum value of \(f(x)\) on the interval \(\left[0, \frac{\pi}{2}\right]\) is \(\qquad\)
|
2\sqrt{3}
| 0.5 |
Find all strictly increasing functions \( f: \mathbb{N} \rightarrow \mathbb{N} \) such that \( f(2) = 2 \) and for all \( m, n \) that are coprime, \( f(mn) = f(m) f(n) \).
|
f(n) = n
| 0.75 |
200 people stand in a circle. Each of them is either a liar or a conformist. Liars always lie. A conformist standing next to two conformists always tells the truth. A conformist standing next to at least one liar can either tell the truth or lie. 100 of the standing people said: "I am a liar," the other 100 said: "I am a conformist." Find the maximum possible number of conformists among these 200 people.
|
150
| 0.625 |
Let the integer sequence \(a_1, a_2, \ldots, a_{10}\) satisfy \(a_{10} = 3a_1\) and \(a_2 + a_8 = 2a_5\), with \(a_{i+1} \in \{1 + a_i, 2 + a_i\}\) for \(i = 1, 2, \ldots, 9\). How many such sequences exist?
|
80
| 0.25 |
During a long voyage of a passenger ship, it was observed that at each dock, a quarter of the passenger composition is renewed, that among the passengers leaving the ship, only one out of ten boarded at the previous dock, and finally, that the ship is always fully loaded.
Determine the proportion of passengers at any time, while the ship is en route, who did not board at either of the two previous docks?
|
\frac{21}{40}
| 0.25 |
Let \( P \) be the intersection point of the directrix \( l \) of an ellipse and its axis of symmetry, and \( F \) be the corresponding focus. \( AB \) is a chord passing through \( F \). Find the maximum value of \( \angle APB \) which equals \( 2 \arctan e \), where \( e \) is the eccentricity.
|
2 \arctan e
| 0.875 |
Find the minimum value of the expression \(\frac{1}{1-x^{2}} + \frac{4}{4-y^{2}}\) under the conditions \(|x| < 1\), \(|y| < 2\), and \(xy = 1\).
|
4
| 0.75 |
The quadratic function \( f(x) = a + bx - x^2 \) satisfies \( f(1+x) = f(1-x) \) for all real numbers \( x \) and \( f(x+m) \) is an increasing function on \( (-\infty, 4] \). Find the range of the real number \( m \).
|
(-\infty, -3]
| 0.125 |
Let $\mathbb{R}$ denote the set of real numbers. Determine all functions $f: \mathbb{R} \rightarrow $\mathbb{R}$ such that
$$
f(x^{2}+f(y))=y+(f(x))^{2}
$$
holds for all $x, y$ in $\mathbb{R}$.
|
f(x) = x
| 0.875 |
ABCD is a parallelogram with an area of 120. K is the midpoint of side AD, and L is the midpoint of side CD. Find the area of triangle BKL.
|
45
| 0.5 |
Write the product of the digits of each natural number from 1 to 2018 (for example, the product of the digits of the number 5 is 5; the product of the digits of the number 72 is \(7 \times 2=14\); the product of the digits of the number 607 is \(6 \times 0 \times 7=0\), etc.). Then find the sum of these 2018 products.
|
184320
| 0.25 |
In a certain country, there are 21 cities, and the government intends to build $n$ roads (all two-way), with each road connecting exactly 2 of the country’s cities. What is the minimum value of $n$ such that, regardless of how the roads are built, it is possible to travel between any 2 cities (possibly passing through intermediate cities)?
|
191
| 0.75 |
(1) In the sequence $\left\{a_{n}\right\}$,
\[a_{1}=1, \quad a_{n+1}=\frac{10+4 a_{n}}{1+a_{n}} \quad (n \in \mathbf{Z}_{+})\]
find the general term formula of the sequence $\left\{a_{n}\right\}$.
(2) Given the sequence $\left\{a_{n}\right\}$ satisfies:
\[a_{1}=2, \quad a_{n+1}=\frac{a_{n}-1}{a_{n}+1} \quad (n \in \mathbf{Z}_{+})\]
find the value of $a_{2019}$.
|
-\frac{1}{2}
| 0.75 |
Let \(\alpha\) and \(\beta\) be a pair of conjugate complex numbers. Given that \(|\alpha - \beta| = 2 \sqrt{3}\), and \(\frac{\alpha}{\beta^{2}}\) is a real number, what is \(|\alpha|\)?
|
2
| 0.875 |
Seven students in a class receive one failing grade every two days of school, while nine other students receive one failing grade every three days. The remaining students in the class never receive failing grades. From Monday to Friday, 30 new failing grades appeared in the class register. How many new failing grades will appear in the class register on Saturday?
|
9
| 0.25 |
In Mezhdugrad, houses stand along one side of the street, with each house having between $1$ and $9$ floors. According to an ancient law of Mezhdugrad, if two houses on the same side of the street have the same number of floors, then, no matter how far apart they are, there must be a house with more floors between them. What is the maximum possible number of houses that can stand on one side of the street in Mezhdugrad?
|
511
| 0.125 |
Point \( O \), lying inside a convex quadrilateral with area \( S \), is reflected symmetrically with respect to the midpoints of its sides.
Find the area of the quadrilateral with vertices at the resulting points.
|
2S
| 0.875 |
In a class, there are \( n \) boys and \( n \) girls (\( n \geq 3 \)). They sat around a round table such that no two boys and no two girls sit next to each other. The teacher has \( 2n \) cards with the numbers \( 1, 2, 3, \ldots, 2n \), each appearing exactly once. He distributes one card to each student in such a way that the number on any girl's card is greater than the number on any boy's card. Then each boy writes down the sum of the numbers on three cards: his own and those of the girls sitting next to him. For which values of \( n \) could all the \( n \) sums obtained be equal?
|
n \text{ is odd}
| 0.125 |
Currently, the exchange rates for the dollar and euro are as follows: $D = 6$ yuan and $E = 7$ yuan. The People's Bank of China determines these exchange rates regardless of market conditions and follows a strategy of approximate currency parity. One bank employee proposed the following scheme for changing the exchange rates. Each year, the exchange rates can be adjusted according to the following four rules: Either change $D$ and $E$ to the pair $(D + E, 2D \pm 1)$, or to the pair $(D + E, 2E \pm 1)$. Moreover, it is prohibited for the dollar and euro rates to be equal at the same time.
For example: From the pair $(6, 7)$, after one year the following pairs are possible: $(13, 11)$, $(11, 13)$, $(13, 15)$, or $(15, 13)$. What is the smallest possible value of the difference between the higher and lower of the simultaneously resulting exchange rates after 101 years?
|
2
| 0.375 |
Let \( a \) and \( b \) be positive whole numbers such that \(\frac{4.5}{11} < \frac{a}{b} < \frac{5}{11} \). Find the fraction \(\frac{a}{b}\) for which the sum \( a+b \) is as small as possible. Justify your answer.
|
\frac{3}{7}
| 0.75 |
Let \( a, b, c, d \) be odd numbers with \( 0 < a < b < c < d \), and \( ad = bc \), \( a+d = 2^k \), \( b+c = 2^m \), where \( k \) and \( m \) are integers. Find the value of \( a \).
|
1
| 0.875 |
Given that $n$ and $k$ are positive integers with $n > k$, and real numbers $a_{1}, a_{2}, \cdots, a_{n} \in (k-1, k)$. Let positive real numbers $x_{1}, x_{2}, \cdots, x_{n}$ satisfy the condition that for any $k$-element subset $I$ of $\{1,2,\cdots, n\}$, the inequality $\sum_{i \in I} x_{i} \leqslant \sum_{i \in I} a_{i}$ holds. Find the maximum value of $x_{1} x_{2} \cdots x_{n}$.
|
a_1 a_2 \cdots a_n
| 0.625 |
Calculate
$$
\sqrt{1+2 \sqrt{1+3 \sqrt{1+\ldots+2017 \sqrt{1+2018 \cdot 2020}}}}
$$
|
3
| 0.125 |
Solve the following system of equations:
$$
\begin{aligned}
x^2 + 7y + 2 &= 2z + 4\sqrt{7x - 3} \\
y^2 + 7z + 2 &= 2x + 4\sqrt{7y - 3} \\
z^2 + 7x + 2 &= 2y + 4\sqrt{7z - 3}
\end{aligned}
$$
|
x = y = z = 1
| 0.375 |
In a family, there are six children. Five of them are respectively 2, 6, 8, 12, and 14 years older than the youngest, and the age of each child is a prime number. How old is the youngest?
|
5
| 0.75 |
Two cyclists set off simultaneously towards each other from points $A$ and $B$ and met 70 km from $A$. Continuing with the same speeds, they reached points $A$ and $B$ and turned back. The second time they met 90 km from $B$. Find the distance from $A$ to $B$.
|
120 \text{ km}
| 0.625 |
In a $5 \times 18$ rectangle, the numbers from 1 to 90 are placed. This results in five rows and eighteen columns. In each column, the median value is chosen, and among the medians, the largest one is selected. What is the minimum possible value that this largest median can take?
Recall that among 99 numbers, the median is such a number that is greater than 49 others and less than 49 others.
|
54
| 0.75 |
Points A and B are 999 km apart, with 1000 milestones along the way. Each milestone indicates the distance to points A and B in the format (distance to A, distance to B), such as (0, 999), (1, 998), (2, 997), ..., (997, 2), (998, 1), (999, 0). How many of these milestones display exactly two different digits on the distances to A and B?
|
40
| 0.25 |
In an $8 \times 8$ chessboard, color the squares using 8 colors such that each square is colored in one color, neighboring squares sharing an edge have different colors, and each row contains all 8 colors. How many different coloring methods are there?
|
8! \times 14833^7
| 0.125 |
The numbers from 1 to 8 are arranged at the vertices of a cube so that the sum of the numbers at any three vertices lying on one face is at least 10. What is the minimal possible sum of the numbers on one face of the cube?
|
16
| 0.25 |
The center of one sphere is on the surface of another sphere with an equal radius. How does the volume of the intersection of the two spheres compare to the volume of one of the spheres?
|
\frac{5}{16}
| 0.875 |
For how many integers \( n \) between 1 and 2005, inclusive, is \( 2 \cdot 6 \cdot 10 \cdots(4n - 2) \) divisible by \( n! \)?
|
2005
| 0.625 |
Given \( f(x) = \max \left| x^3 - a x^2 - b x - c \right| \) for \( 1 \leq x \leq 3 \), find the minimum value of \( f(x) \) as \( a, b, \) and \( c \) range over all real numbers.
|
\frac{1}{4}
| 0.625 |
Let \( n \) be a fixed positive integer. Let \( S \) be any finite collection of at least \( n \) positive reals (not necessarily all distinct). Let \( f(S) = \left( \sum_{a \in S} a \right)^n \), and let \( g(S) \) be the sum of all \( n \)-fold products of the elements of \( S \) (in other words, the \( n \)-th symmetric function). Find \( \sup_S \frac{g(S)}{f(S)} \).
|
\frac{1}{n!}
| 0.625 |
Dodson, Williams, and their horse Bolivar want to reach City B from City A as quickly as possible. Along the road, there are 27 telegraph poles, dividing the whole path into 28 equal intervals. Dodson walks an interval between poles in 9 minutes, Williams in 11 minutes, and either can ride Bolivar to cover the same distance in 3 minutes (Bolivar cannot carry both simultaneously). They depart from City A at the same time, and the journey is considered complete when all three are in City B.
The friends have agreed that Dodson will ride part of the way on Bolivar, then tie Bolivar to one of the telegraph poles, and continue on foot, while Williams will initially walk and then ride Bolivar. At which telegraph pole should Dodson tie Bolivar so that they all reach City B as quickly as possible?
Answer: At the 12th pole, counting from City A.
|
12
| 0.875 |
Let \( n \) be a positive integer such that \(\sqrt{3}\) lies between \(\frac{n+3}{n}\) and \(\frac{n+4}{n+1}\). Determine the value of \( n \).
|
4
| 0.875 |
Among the complex numbers that satisfy the condition \( |z - 5i| \leq 4 \), find the one whose argument has the smallest positive value.
|
2.4 + 1.8 i
| 0.5 |
We start with three piles of coins, initially containing 51, 49, and 5 coins respectively. At each step, we are allowed either to combine two piles into one or to split a pile containing an even number of coins into two equal halves. Is it possible to create three new piles of 52, 48, and 5 coins using these operations?
|
\text{No}
| 0.75 |
Calculate the coefficient of $x^{80}$ in the polynomial $(1+x+x^2+\cdots+x^{80})^3$ after combining like terms.
|
3321
| 0.5 |
João managed to paint the squares of an \( n \times n \) board in black and white so that the intersections of any two rows and any two columns did not consist of squares with the same color. What is the maximum value of \( n \)?
|
4
| 0.375 |
Let \( n \) be a positive integer. Consider the set
\[
S = \left\{\left(x_{1}, x_{2}, \cdots, x_{k}\right) \mid x_{1}, x_{2}, \cdots, x_{k} \in \{0, 1, \cdots, n\}, x_{1} + x_{2} + \cdots + x_{k} > 0 \right\}
\]
which consists of \((n+1)^{k} - 1\) points in \( k \)-dimensional space. Find the minimum number of hyperplanes whose union contains \( S \) but does not include the point \((0, 0, \cdots, 0)\).
|
kn
| 0.375 |
Scientists found a fragment of an ancient mechanics manuscript. It was a piece of a book where the first page was numbered 435, and the last page was numbered with the same digits, but in some different order. How many sheets did this fragment contain?
|
50
| 0.5 |
Let \(ABCD\) be a parallelogram with an area of 1, and let \(M\) be a point on the segment \([BD]\) such that \(MD = 3MB\). Let \(N\) be the point of intersection of the lines \( (AM) \) and \( (CB) \). Calculate the area of the triangle \(MND\).
|
\frac{1}{8}
| 0.375 |
Solve the system in positive numbers:
$$
\begin{cases}x^{y} & =z \\ y^{z} & =x \\ z^{x} & =y\end{cases}
$$
|
x = y = z = 1
| 0.125 |
Let \( n \) be a positive integer and define \(\mathrm{S}_{\mathrm{n}} = \left\{\left(\mathrm{a}_{1}, \mathrm{a}_{2}, \ldots, \mathrm{a}_{2^{n}}\right) \mid \mathrm{a}_{\mathrm{i}} \in\{0,1\}, 1 \leq \mathrm{i} \leq 2^{\mathrm{n}}\right\}\). For \( \mathrm{a}, \mathrm{b} \in \mathrm{S}_{\mathrm{n}} \) where \( a = \left(a_{1}, a_{2}, \ldots, a_{2^{n}}\right) \) and \( b = \left(b_{1}, b_{2}, \ldots, b_{2^{n}}\right) \), define \( d(a, b) = \sum_{i=1}^{2^{n}} \left|a_{i} - b_{i}\right| \). If for any \(a, b \in A\), \( d(a, b) \geq 2^{n-1} \), then \( A \subseteq \mathrm{S}_{\mathrm{n}} \) is called a "good subset". Find the maximum possible value of \( |A| \).
|
2^{n+1}
| 0.625 |
For the smallest value of $a$, the inequality \(\frac{\sqrt[3]{\operatorname{ctg}^{2} x}-\sqrt[3]{\operatorname{tg}^{2} x}}{\sqrt[3]{\sin ^{2} x}-\sqrt[3]{\cos ^{2} x}}<a\) holds for all permissible \( x \in\left(-\frac{3 \pi}{2} ;-\pi\right) \)? If necessary, round your answer to two decimal places.
|
-2.52
| 0.25 |
Let \( A \) be the set of all two-digit positive integers \( n \) for which the number obtained by erasing its last digit is a divisor of \( n \). How many elements does \( A \) have?
|
32
| 0.375 |
In a clock workshop, there are several digital clocks (more than one), displaying time in a 12-hour format (the number of hours on the clock screen ranges from 1 to 12). All clocks run at the same speed but show completely different times: the number of hours on the screen of any two different clocks is different, and the number of minutes as well.
One day, the master added up the number of hours on the screens of all available clocks, then added up the number of minutes on the screens of all available clocks, and remembered the two resulting numbers. After some time, he did the same thing again and found that both the total number of hours and the total number of minutes had decreased by 1. What is the maximum number of digital clocks that could be in the workshop?
|
11
| 0.75 |
Vasya wrote consecutive natural numbers \(N\), \(N+1\), \(N+2\), and \(N+3\) in rectangular boxes. Below each rectangle, he wrote the sum of the digits of the corresponding number in a circle.
The sum of the numbers in the first and second circles equals 200, and the sum of the numbers in the third and fourth circles equals 105. What is the sum of the numbers in the second and third circles?
|
103
| 0.125 |
What are the integers $k$ such that for all real numbers $a, b, c$,
$$
(a+b+c)(ab + bc + ca) + kabc = (a+b)(b+c)(c+a)
$$
|
-1
| 0.5 |
Consider a 99-sided polygon where its edges are initially colored in the sequence red, blue, red, blue, ..., red, blue, yellow, with one color per edge. Given that no two adjacent edges can have the same color, is it possible, through a series of operations where each operation consists of changing the color of exactly one edge, to transform the coloring into the sequence red, blue, red, blue, ..., red, yellow, blue?
|
\text{No}
| 0.5 |
Let \(\mathcal{C}\) be the hyperbola \(y^{2}-x^{2}=1\). Given a point \(P_{0}\) on the \(x\)-axis, we construct a sequence of points \((P_{n})\) on the \(x\)-axis in the following manner: let \(\ell_{n}\) be the line with slope 1 passing through \(P_{n}\), then \(P_{n+1}\) is the orthogonal projection of the point of intersection of \(\ell_{n}\) and \(\mathcal{C}\) onto the \(x\)-axis. (If \(P_{n}=0\), then the sequence simply terminates.)
Let \(N\) be the number of starting positions \(P_{0}\) on the \(x\)-axis such that \(P_{0}=P_{2008}\). Determine the remainder of \(N\) when divided by 2008.
|
254
| 0.75 |
Given $0<a<b<c<d<500$, how many ordered quadruples of integers $(a, b, c, d)$ satisfy $a + d = b + c$ and $bc - ad = 93$?
|
870
| 0.625 |
The factorial of an integer $m$ is denoted as $m!$ and is the product of all positive integers up to $m$. For example, $2!=1 \times 2=2$, and $(3!)!=1 \times 2 \times 3 \times 4 \times 5 \times 6=720$. Given that $((n!)!)!$ is a factor of $(2021!)!$, what is the maximum value of $n$?
|
6
| 0.5 |
Find all values of the parameters \(a, b, c\) for which the system of equations
\[
\left\{
\begin{array}{l}
a x + b y = c \\
b x + c y = a \\
c x + a y = b
\end{array}
\right\}
\]
has at least one negative solution (where \(x, y < 0\)).
|
a + b + c = 0
| 0.875 |
A circle with a radius of 10 cm has rays drawn from a point \(A\) that are tangent to the circle at points \(B\) and \(C\) such that triangle \(ABC\) is equilateral. Find its area.
|
75 \sqrt{3}
| 0.875 |
In triangle \(ABC\), points \(M\) and \(N\) are the midpoints of sides \(AC\) and \(BC\) respectively. It is known that the intersection of the medians of triangle \(AMN\) is the orthocenter of triangle \(ABC\). Find the angle \(ABC\).
|
45^\circ
| 0.875 |
What is the value of \(A^{2}+B^{3}+C^{5}\), given that:
\[
\begin{array}{l}
A=\sqrt[3]{16 \sqrt{2}} \\
B=\sqrt{9 \sqrt[3]{9}} \\
C=\left[(\sqrt[5]{2})^{2}\right]^{2}
\end{array}
\]
|
105
| 0.875 |
Timur and Alexander are counting the trees growing around the house. They move in the same direction but start counting from different trees. How many trees are growing around the house if the tree that Timur counted as the 12th, Alexander counted as the 33rd, and the tree that Timur counted as the 105th, Alexander counted as the 8th?
|
118
| 0.625 |
It is known that the equation \(x^4 - 8x^3 + ax^2 + bx + 16 = 0\) has (with multiplicity) four positive roots. Find \(a - b\).
|
56
| 0.875 |
In a basketball shooting test, each participant needs to make 3 successful shots to pass and can stop once they achieve this, but each participant is allowed a maximum of 5 attempts. Given that a player's probability of making a shot is $\frac{2}{3}$, what is the probability that the player will pass the test?
|
\frac{64}{81}
| 0.5 |
Find a function \( f: \mathbf{Z}_{+} \rightarrow \mathbf{Z}_{+} \) such that for all \( m, n \in \mathbf{Z}_{+} \), the following condition holds:
$$
(n! + f(m)!) \mid (f(n)! + f(m!)).
$$
|
f(n) = n
| 0.75 |
The sequence \( u_0, u_1, u_2, \ldots \) is defined as follows: \( u_0 = 0 \), \( u_1 = 1 \), and \( u_{n+1} \) is the smallest integer \( > u_n \) such that there is no arithmetic progression \( u_i, u_j, u_{n+1} \) with \( i < j < n+1 \). Find \( u_{100} \).
|
981
| 0.125 |
In the diagram, $\triangle PQR$ is right-angled at $P$ and $\angle PRQ=\theta$. A circle with center $P$ is drawn passing through $Q$. The circle intersects $PR$ at $S$ and $QR$ at $T$. If $QT=8$ and $TR=10$, determine the value of $\cos \theta$.
|
\frac{\sqrt{7}}{3}
| 0.875 |
On side \( AB \) of triangle \( ABC \), a point \( K \) is marked, and on the side \( AC \), a point \( M \) is marked. The segments \( BM \) and \( CK \) intersect at point \( P \). It turned out that the angles \( \angle APB \), \( \angle BPC \), and \( \angle CPA \) are each \( 120^\circ \), and the area of the quadrilateral \( AKPM \) is equal to the area of triangle \( BPC \). Find the angle \( \angle BAC \).
|
60^\circ
| 0.875 |
The number of solutions to the equation $\sin |x| = |\cos x|$ in the closed interval $[-10\pi, 10\pi]$ is __.
|
20
| 0.25 |
The image shows a 3x3 grid where each cell contains one of the following characters: 华, 罗, 庚, 杯, 数, 学, 精, 英, and 赛. Each character represents a different number from 1 to 9, and these numbers satisfy the following conditions:
1. The sum of the four numbers in each "田" (four cells in a square) is equal.
2. 华 $\times$ 华 $=$ 英 $\times$ 英 + 赛 $\times$ 赛.
3. 数 > 学
According to the above conditions, find the product of the numbers represented by 华, 杯, and 赛.
|
120
| 0.5 |
There is one three-digit number and two two-digit numbers written on the board. The sum of the numbers containing the digit seven is 208. The sum of the numbers containing the digit three is 76. Find the sum of all three numbers.
|
247
| 0.5 |
Given that \( P \) is a point on a sphere \( O \) with radius \( r \), three mutually perpendicular chords \( PA \), \( PB \), and \( PC \) are drawn through \( P \). If the maximum distance from point \( P \) to the plane \( ABC \) is 1, then \( r \) equals \(\quad\)
|
\frac{3}{2}
| 0.125 |
Find the maximum value of the expression \( x^{2} + y^{2} \) if \( |x-y| \leq 2 \) and \( |3x + y| \leq 6 \).
|
10
| 0.875 |
Find the limit of the function:
$$
\lim _{x \rightarrow 0} \frac{(1+x)^{3}-(1+3 x)}{x+x^{5}}
$$
|
0
| 0.875 |
A $5 \times 5$ square contains a light bulb in each of its cells. Initially, only one light bulb is on. You can change the state of the light bulbs in any $k \times k$ square, with $k > 1$. What are the initial positions of the light bulb that is on that allow turning off all the light bulbs?
|
(3,3)
| 0.25 |
Darya Dmitrievna is preparing a test on number theory. She promised each student to give as many problems as the number of terms they create in the numerical example
$$
a_{1} + a_{2} + \ldots + a_{n} = 2021
$$
where all numbers \( a_{i} \) are natural, greater than 10, and are palindromes (they do not change if their digits are written in reverse order). If a student does not find any such example, they will receive 2021 problems on the test. What is the smallest number of problems a student can receive? (20 points)
|
3
| 0.375 |
If \( f(x) = \frac{25^x}{25^x + P} \) and \( Q = f\left(\frac{1}{25}\right) + f\left(\frac{2}{25}\right) + \cdots + f\left(\frac{24}{25}\right) \), find the value of \( Q \).
|
12
| 0.625 |
Given a positive integer \( n \), find the smallest positive integer \( u_n \) such that for any odd integer \( d \), the number of integers in any \( u_n \) consecutive odd integers that are divisible by \( d \) is at least as many as the number of integers among the odd integers \( 1, 3, 5, \ldots, 2n-1 \) that are divisible by \( d \).
|
u_n = 2n - 1
| 0.5 |
There are two sets of numbers from 1 to 20. All possible sums of two numbers (with each addend taken from a different set) are formed. How many of these sums are divisible by 3?
|
134
| 0.75 |
Ivan Semenovich leaves for work at the same time every day, travels at the same speed, and arrives exactly at 9:00 AM. One day, he overslept and left 40 minutes later than usual. To avoid being late, Ivan Semenovich increased his speed by 60% and arrived at 8:35 AM. By what percentage should he have increased his usual speed to arrive exactly at 9:00 AM?
|
30 \%
| 0.625 |
Let \( M = \{1, 2, \cdots, 2n + 1\} \). \( A \) is a subset of \( M \) with no two elements summing to \( 2n + 2 \). Find the maximum value of \( |A| \).
|
n+1
| 0.75 |
A square with a side length of 100 was cut into two equal rectangles. These rectangles were then placed next to each other as shown in the picture. Find the perimeter of the resulting figure.
|
500
| 0.625 |
Thirty-nine students from seven classes invented 60 problems, with the students from each class inventing the same number of problems (which is not zero), and the students from different classes inventing different numbers of problems. How many students invented one problem each?
|
33
| 0.5 |
Draw the height BH. \(\angle BCH = \angle CBH = 45^\circ\), \(\angle ABH = 60^\circ\). Let BH = HC = \(x\), then BC = \(x \sqrt{2}\), AB = \(2 x\).
\(\frac{AB}{BC} = \frac{BC}{MB} = \sqrt{2}\), \(\angle ABC\) is common. Therefore, \(\triangle MBC \sim \triangle CBA\) by the second criterion of similarity. From this similarity, it follows that \(\angle BMC = 45^\circ\). Then \(\angle AMC = 135^\circ\).
|
135^\circ
| 0.625 |
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