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The plane is divided by parallel lines that are a distance of $2a$ apart. A needle of length $2l$ $(l<a)$ is randomly thrown onto the plane. Find the probability that the needle will intersect any of the lines.
|
\frac{2l}{\pi a}
| 0.5 |
In a store, there are 9 headphones, 13 computer mice, and 5 keyboards for sale. Besides these, there are also 4 sets of "keyboard and mouse" and 5 sets of "headphones and mouse." How many ways can you buy three items: headphones, a keyboard, and a mouse? Answer: 646.
|
646
| 0.75 |
A rook has traveled across an \( n \times n \) board, visiting each cell exactly once, where each of its moves was exactly one cell. The cells are numbered from 1 to \( n^2 \) in the order of the rook's path. Let \( M \) be the maximum difference between the numbers of adjacent (by side) cells. What is the smallest possible value of \( M \)?
|
2n - 1
| 0.125 |
On the radius $A O$ of a circle with center $O$, a point $M$ is chosen. On one side of $A O$, points $B$ and $C$ are chosen on the circle such that $\angle A M B = \angle O M C = \alpha$. Find the length of $B C$ if the radius of the circle is 12 and $\cos \alpha = \frac{5}{6}$.
|
20
| 0.25 |
Put ping pong balls in 10 boxes. The number of balls in each box must not be less than 11, must not be 17, must not be a multiple of 6, and must be different from each other. What is the minimum number of ping pong balls needed?
|
174
| 0.75 |
Given a plane $\alpha$ and a triangle $ABC$ whose plane is not parallel to the plane $\alpha$, find a point $S$ in space such that the lines $SA, SB, SC$ intersecting the plane $\alpha$ produce points $A', B', C'$ such that triangle $A'B'C'$ is congruent to another given triangle $MNP$.
|
S
| 0.5 |
Determine the smallest possible positive integer \( n \) with the following property: For all positive integers \( x, y, \) and \( z \) with \( x \mid y^{3} \), \( y \mid z^{3} \), and \( z \mid x^{3} \), it also holds that \( x y z \mid (x+y+z)^{n} \).
|
n = 13
| 0.375 |
Given the real number \( x \) satisfies \( 20 \sin x = 22 \cos x \), find the largest integer not greater than the real number \( \left( \frac{1}{\sin x \cos x} - 1 \right)^7 \).
|
1
| 0.625 |
Let \( k \) and \( n \) be integers such that \( 1 \leq k \leq n \). What is the maximum number of \( k \)-element subsets of the set \(\{1, 2, \ldots, n\}\) such that for any two of these subsets, one is composed of the \( k \) smallest elements of their union?
|
n - k + 1
| 0.25 |
In trapezoid \(ABCD\), the bases \(AD\) and \(BC\) are equal to 8 and 18, respectively. It is known that the circumcircle of triangle \(ABD\) is tangent to lines \(BC\) and \(CD\). Find the perimeter of the trapezoid.
|
56
| 0.125 |
The committee consists of a chairperson, a deputy chairperson, and five more members. In how many ways can the members of the committee distribute the duties of the chairperson and the deputy chairperson among themselves?
|
42
| 0.875 |
Given \( n \) points in the plane, show that it is possible to choose at least \( \sqrt{n} \) of them such that they do not form an equilateral triangle.
|
\sqrt{n}
| 0.875 |
In the polynomial \((1-z)^{b_{1}} \cdot (1-z^{2})^{b_{2}} \cdot (1-z^{3})^{b_{3}} \cdots (1-z^{32})^{b_{32}}\), where \(b_{i} (i=1,2, \cdots, 32)\) are positive integers, this polynomial has the following remarkable property: after expanding it and removing the terms with \(z\) of degree higher than 32, exactly \(1-2z\) remains. Determine \(b_{32}\) (the answer can be expressed as the difference of two powers of 2).
|
2^{27} - 2^{11}
| 0.125 |
On a birthday card printed with April 29, a child inserts two positive integers $x$ and $y$ between the 4 and the 29, forming a five-digit number $\overline{4 x y 29}$. This number is the square of the integer $T$ that corresponds to the child's birthday: $\overline{4 x y 29} = T^{2}$. What is the integer $T$ that corresponds to the child's birthday?
|
223
| 0.875 |
Two people are tossing a coin: one tossed it 10 times, and the other tossed it 11 times.
What is the probability that the second person's coin landed on heads more times than the first person's coin?
|
\frac{1}{2}
| 0.5 |
A box contains two white socks, three blue socks, and four grey socks. Three of the socks have holes in them, but Rachel does not know what colour these socks are. She takes one sock at a time from the box without looking. How many socks must she take to be certain she has a pair of socks of the same colour without holes?
|
7
| 0.75 |
Given the sequence $\left\{a_{n}\right\}$ with the partial sum $S_{n}=2 a_{n}-1$ for $n=1,2,\cdots$, and the sequence $\left\{b_{n}\right\}$ that satisfies $b_{1}=3$ and $b_{k+1}=a_{k}+b_{k}$ for $k=1,2,\cdots$, find the partial sum of the sequence $\left\{b_{n}\right\}$.
|
2^n + 2n - 1
| 0.625 |
In a $3 \times 3$ grid (each cell is a $1 \times 1$ square), place two identical chess pieces, with at most one piece per cell. There are ___ different ways to arrange the pieces (if two arrangements can overlap by rotation, they are considered the same arrangement).
|
10
| 0.625 |
Susie thinks of a positive integer \( n \). She notices that, when she divides 2023 by \( n \), she is left with a remainder of 43. Find how many possible values of \( n \) there are.
|
19
| 0.625 |
There is a uniformly growing grassland. If 20 cows are grazed, they will just finish eating all the grass in 60 days. If 30 cows are grazed, they will just finish eating all the grass in 35 days. Now, 6 cows are grazing on the grassland. After a month, 10 more cows are added. How many more days will it take for all the grass to be eaten?
|
84
| 0.875 |
Grain warehouses A and B each originally stored a certain number of full bags of grain. If 90 bags are transferred from warehouse A to warehouse B, the number of bags in warehouse B will be twice the number in warehouse A. If an unspecified number of bags are transferred from warehouse B to warehouse A, the number of bags in warehouse A will be six times the number in warehouse B. What is the minimum number of bags originally stored in warehouse A?
|
153
| 0.5 |
Let \( N \geqslant 2 \) be a natural number. What is the sum of all fractions of the form \( \frac{1}{mn} \), where \( m \) and \( n \) are coprime natural numbers such that \( 1 \leqslant m < n \leqslant N \) and \( m+n > N \)?
|
\frac{1}{2}
| 0.75 |
Find all \( x \in [1,2) \) such that for any positive integer \( n \), the value of \( \left\lfloor 2^n x \right\rfloor \mod 4 \) is either 1 or 2.
|
\frac{4}{3}
| 0.875 |
The last two digits of $\left[\frac{10^{93}}{10^{31}+3}\right]$ are $\qquad$ (where [x] denotes the greatest integer less than or equal to $x$).
|
08
| 0.625 |
In a middle school math competition, there were three problems labeled A, B, and C. Among the 25 students who participated, each student solved at least one problem. Among the students who did not solve problem A, the number who solved problem B was twice the number of those who solved problem C. The number of students who solved only problem A was one more than the number of students who solved A along with any other problem. Among the students who solved only one problem, half did not solve problem A. How many students solved only problem B?
|
6
| 0.5 |
Different positive 3-digit integers are formed from the five digits \(1, 2, 3, 5, 7\), and repetitions of the digits are allowed. As an example, such positive 3-digit integers include 352, 577, 111, etc. Find the sum of all the distinct positive 3-digit integers formed in this way.
|
49950
| 0.75 |
On the radius \( AO \) of a circle with center \( O \), a point \( M \) is selected. On one side of \( AO \) on the circle, points \( B \) and \( C \) are chosen such that \( \angle AMB = \angle OMC = \alpha \). Find the length of \( BC \) if the radius of the circle is 10 and \( \sin \alpha = \frac{\sqrt{21}}{5} \).
|
8
| 0.125 |
Given the sequence \(\left\{a_{n}\right\}\) such that \(a_{1} = 0\) and \(a_{n+1} = a_{n} + 1 + 2 \sqrt{1+a_{n}}\) for \(n = 1, 2, \ldots\), find \(a_{n}\).
|
a_n = n^2 - 1
| 0.875 |
On his birthday, the last guest to arrive was Yana, who gave Andrey a ball, and the second last was Eduard, who gave him a calculator. While using the calculator, Andrey noticed that the product of the total number of gifts he received and the number of gifts he had before Eduard arrived is exactly 16 more than the product of his age and the number of gifts he had before Yana arrived. How many gifts does Andrey have?
|
18
| 0.875 |
Find the smallest positive integer \( m \) such that the equation regarding \( x, y, \) and \( z \):
\[ 2^x + 3^y - 5^z = 2m \]
has no positive integer solutions.
|
11
| 0.5 |
Let \( ABC \) be a right triangle with hypotenuse \( AC \). Let \( B' \) be the reflection of point \( B \) across \( AC \), and let \( C' \) be the reflection of \( C \) across \( AB' \). Find the ratio of \([BCB']\) to \([BC'B']\).
|
1
| 0.75 |
To determine the number of positive divisors of a number, you simply factor it as powers of distinct primes and multiply the successors of the exponents. For example, \(2016=2^{5} \cdot 3^{2} \cdot 5^{1}\) has \((5+1)(2+1)(1+1)=36\) positive divisors. Consider the number \(n=2^{7} \cdot 3^{4}\).
a) Determine the number of positive divisors of \(n^{2}\).
b) How many divisors of \(n^{2}\) are greater than \(n\)?
c) How many divisors of \(n^{2}\) are greater than \(n\) and are not multiples of \(n\)?
|
28
| 0.625 |
Find all natural numbers that are 5 times greater than their last digit.
|
25
| 0.75 |
Given that \(O\) is the origin of coordinates, the curve \(C_1: x^2 - y^2 = 1\) intersects the curve \(C_2: y^2 = 2px\) at points \(M\) and \(N\). If the circumcircle of \(\triangle OMN\) passes through the point \(P\left(\frac{7}{2}, 0\right)\), then the equation of the curve \(C_2\) is ____.
|
y^2 = \frac{3}{2} x
| 0.5 |
There are exactly 120 ways to color five cells in a \(5 \times 5\) grid such that each row and each column has exactly one colored cell.
There are exactly 96 ways to color five cells in a \(5 \times 5\) grid, excluding a corner cell, such that each row and each column has exactly one colored cell.
How many ways are there to color five cells in a \(5 \times 5\) grid, excluding two corner cells, such that each row and each column has exactly one colored cell?
|
78
| 0.625 |
Find the maximum value of the expression \(\cos (x-y)\), given that \(\sin x - \sin y = \frac{3}{4}\).
|
\frac{23}{32}
| 0.75 |
In a singing competition, the participants were Rooster, Crow, and Cuckoo. Each jury member voted for one of the three singers. The Woodpecker counted that there were 59 judges, with a total of 15 votes for Rooster and Crow combined, 18 votes for Crow and Cuckoo combined, and 20 votes for Cuckoo and Rooster combined. The Woodpecker is bad at counting, but each of the four numbers he named differs from the correct number by no more than 13. How many judges voted for Crow?
|
13
| 0.875 |
There are 29 students in a class: some are honor students who always tell the truth, and some are troublemakers who always lie.
All the students in this class sat at a round table.
- Several students said: "There is exactly one troublemaker next to me."
- All other students said: "There are exactly two troublemakers next to me."
What is the minimum number of troublemakers that can be in the class?
|
10
| 0.75 |
a) How many four-digit numbers have a sum of their digits that is even?
b) A two-digit number with distinct and non-zero digits is called beautiful if the tens digit is greater than the units digit. How many beautiful numbers exist?
c) How many even four-digit numbers can we form using the digits $0,1,2,3,4,5$ without repeating any digit?
d) What is the average of all five-digit numbers that can be formed using each of the digits $1,3,5,7,$ and $8$ exactly once?
|
53332.8
| 0.375 |
Let \( S \) be a subset of the set \(\{1, 2, 3, \ldots, 2015\} \) such that for any two elements \( a, b \in S \), the difference \( a - b \) does not divide the sum \( a + b \). Find the maximum possible size of \( S \).
|
672
| 0.5 |
Let \( p_{i} \) be the \(i^{\text {th }}\) prime number; for example, \( p_{1}=2, p_{2}=3 \), and \( p_{3}=5 \). For each prime number, construct the point \( Q_{i}\left(p_{i}, 0\right) \). Suppose \( A \) has coordinates \((0,2)\). Determine the sum of the areas of the triangles \(\triangle A Q_{1} Q_{2}, \triangle A Q_{2} Q_{3}, \triangle A Q_{3} Q_{4}, \triangle A Q_{4} Q_{5}, \triangle A Q_{5} Q_{6}, \) and \(\triangle A Q_{6} Q_{7}\).
|
15
| 0.875 |
Given a triangle \(ABC\). An incircle is inscribed in it, touching the sides \(AB\), \(AC\), and \(BC\) at the points \(C_1\), \(B_1\), \(A_1\) respectively. Find the radius of the excircle \(w\), which touches the side \(AB\) at the point \(D\), the extension of the side \(BC\) at the point \(E\), and the extension of the side \(AC\) at the point \(G\). It is known that \(CE = 6\), the radius of the incircle is 1, and \(CB_1 = 1\).
|
6
| 0.25 |
How many square columns are there where the edge length measured in cm is an integer, and the surface area measured in $\mathrm{cm}^{2}$ is equal to the volume measured in $\mathrm{cm}^{3}$?
|
4
| 0.625 |
Given that $x$ is a four-digit number and the sum of its digits is $y$. If the value of $\frac{x}{y}$ is minimized, what is $x$?
|
1099
| 0.75 |
You are given an unlimited supply of red, blue, and yellow cards to form a hand. Each card has a point value and your score is the sum of the point values of those cards. The point values are as follows: the value of each red card is 1, the value of each blue card is equal to twice the number of red cards, and the value of each yellow card is equal to three times the number of blue cards. What is the maximum score you can get with fifteen cards?
|
168
| 0.625 |
Let the set \( M = \{1, 2, \cdots, 1000\} \). For any non-empty subset \( X \) of \( M \), let \( a_X \) represent the sum of the largest and smallest numbers in \( X \). What is the arithmetic mean of all such \( a_X \)?
|
1001
| 0.375 |
Let \(ABC\) be a triangle with \(AB = 2\), \(CA = 3\), and \(BC = 4\). Let \(D\) be the point diametrically opposite \(A\) on the circumcircle of \(\triangle ABC\), and let \(E\) lie on line \(AD\) such that \(D\) is the midpoint of \(\overline{AE}\). Line \(l\) passes through \(E\) perpendicular to \(\overline{AE}\), and \(F\) and \(G\) are the intersections of the extensions of \(\overline{AB}\) and \(\overline{AC}\) with \(l\). Compute \(FG\).
|
\frac{1024}{45}
| 0.875 |
We define \( a \star b = a \times a - b \times b \). Find the value of \( 3 \star 2 + 4 \star 3 + 5 \star 4 + \cdots + 20 \star 19 \).
|
396
| 0.875 |
Let \(ABC\) be a triangle and \(O\) be the center of its circumcircle. Let \(P\) and \(Q\) be points on the sides \(AC\) and \(AB\), respectively. Let \(K, L\), and \(M\) be the midpoints of the segments \(BP, CQ\), and \(PQ\), respectively, and let \(\Gamma\) be the circle passing through \(K, L\), and \(M\). Suppose that the line \(PQ\) is tangent to the circle \(\Gamma\).
Show that \(OP = OQ\).
|
OP = OQ
| 0.875 |
29 boys and 15 girls attended a ball. Some boys danced with some of the girls (no more than once with each pair). After the ball, each person told their parents how many times they danced. What is the maximum number of different numbers the children could have mentioned?
|
29
| 0.25 |
Define: \(\triangle a = a + (a + 1) + (a + 2) + \cdots + (2a - 2) + (2a - 1)\). For example: \(\triangle 5 = 5 + 6 + 7 + 8 + 9\). What is the result of \(\triangle 1 + \triangle 2 + \triangle 3 + \cdots + \triangle 19 + \triangle 20\)?
|
4200
| 0.5 |
Find the sum of all four-digit natural numbers composed of the digits 3, 6, and 9.
|
539946
| 0.375 |
Before the Christmas concert, students offered 60 handmade items for sale, allowing each customer to choose their own price. The entire proceeds were for charity. At the start of the concert, students calculated the average revenue per sold item, which resulted in an exact whole number. Since they had not yet sold all 60 items, they continued to offer them after the concert. Seven more items were bought for a total of 2505 Kč, increasing the average revenue per sold item to exactly 130 Kč. How many items remained unsold?
|
24
| 0.875 |
Two people, A and B, depart simultaneously from points A and B, respectively, and travel towards each other, meeting at point C. If at the start, person A increases their speed by $\frac{1}{4}$ and person B increases their speed by 10 km/h, and they still meet at point C, determine the original speed of person B in km/h.
|
40 \text{ km/h}
| 0.875 |
Given $f(x)$ is a function defined on $\mathbf{R}$, and for any real number $x$, it satisfies $2 f(x) + f\left(x^{2}-1\right) = 1$. Find the value of $f(\sqrt{2})$.
|
\frac{1}{3}
| 0.875 |
A square with a side of 75 mm was cut by two parallel cuts into three rectangles. It turned out that the perimeter of one of these rectangles is half the sum of the perimeters of the other two. What is the perimeter of this rectangle? Provide the answer in centimeters.
|
20 \ \text{cm}
| 0.625 |
If \(\frac{1}{9}\) of 60 is 5, what is \(\frac{1}{20}\) of 80?
|
4
| 0.5 |
The Absent-Minded Scientist commutes to work on weekdays using the Ring Line of the Moscow Metro from "Taganskaya" station to "Kievskaya" station, and back in the evening. When he enters the station, he boards the first arriving train. It is known that in both directions, trains run at approximately equal intervals. The train traveling via the northern route (through "Belorusskaya") takes 17 minutes to go from "Kievskaya" to "Taganskaya" or vice versa, while the train traveling via the southern route (through "Paveletskaya") takes 11 minutes.
According to his long-term observations, the train traveling counterclockwise arrives at "Kievskaya" on average 1 minute and 15 seconds after the clockwise train arrives. The same is true for "Taganskaya." Moreover, the scientist notes that his average travel time from home to work is 1 minute less than his average travel time from work to home. Find the expected interval between trains traveling in one direction.
|
3 \text{ minutes}
| 0.125 |
Calculate the limit of the function:
$$\lim _{x \rightarrow 0} \frac{5^{2 x}-2^{3 x}}{\sin x+\sin x^{2}}$$
|
\ln \frac{25}{8}
| 0.5 |
A light ray falls at an angle $\alpha=60^{\circ}$ on the front surface of a plane-parallel glass plate. The refractive index of the glass is $n=1.6$. By what angle from the direction of the incident ray is the ray, reflected from the back surface of the plate and exiting back through the front surface, deviated?
|
60^\circ
| 0.125 |
Let \( x \) be the number of 7th graders who together scored \( n \) points in the tournament. The number of 8th graders participating in the tournament is \( 10 \times x \), and they scored a total of \( 4.5 \times n \) points. Thus, the total number of students participating in the tournament is \( 11 \times x \), and they scored \( 5.5 \times n \) points collectively. The total number of points scored by all participants is equal to the number of games played. We obtain the equation:
$$
\begin{gathered}
5.5 \times n = \frac{11 \times x \times (11x - 1)}{2} \\
n = x \times (11x - 1)
\end{gathered}
$$
Each 7th grader played \( 11x - 1 \) games (since there are \( 11x \) participants in total). Therefore, \( x \) 7th graders could collectively score \( n \) points, i.e., \( x \times (11x - 1) \) points, only if each of them won all their games. This is possible only when \( x = 1 \) (since two 7th graders cannot simultaneously win against each other).
|
1
| 0.75 |
Given a rectangle \(ABCD\). A circle intersects the side \(AB\) at points \(K\) and \(L\). Find the length of segment \(MN\) if \(AK = 10\), \(KL = 17\), \(DN = 7\).
|
23
| 0.25 |
Find all integers $n$ such that $n^{4} + 6n^{3} + 11n^{2} + 3n + 31$ is a perfect square.
|
n = 10
| 0.375 |
Let \( r_{1}, r_{2}, \cdots, r_{20} \) be the roots of the polynomial \( x^{20}-7x^{3}+1 \). If \(\frac{1}{r_{1}^{2}+1}+\frac{1}{r_{2}^{2}+1}+\cdots+\frac{1}{r_{20}^{2}+1} \) can be expressed in the form \( \frac{m}{n} \) (with \( m \) and \( n \) coprime), find the value of \( m+n \).
|
240
| 0.75 |
Find all functions \( g: \mathbb{N} \rightarrow \mathbb{N} \) such that \( (g(n)+m)(g(m)+n) \) is a perfect square for all \( n, m \).
|
g(n) = n + c
| 0.625 |
A circle on a plane divides the plane into 2 parts. A circle and a line can divide the plane into a maximum of 4 parts. A circle and 2 lines can divide the plane into a maximum of 8 parts. A circle and 5 lines can divide the plane into a maximum of how many parts?
|
26
| 0.125 |
In the pyramid $ABCD$, the edges are given as follows: $AB = 7$, $BC = 8$, $CD = 4$. Find the edge $DA$ given that the lines $AC$ and $BD$ are perpendicular.
|
1
| 0.625 |
Given a cube \( A B C D A_{1} B_{1} C_{1} D_{1} \) with edge length 1. A line \( l \) passes through the point \( E \), the midpoint of edge \( C_{1} D_{1} \), and intersects the lines \( A D_{1} \) and \( A_{1} B \). Find the distance from point \( E \) to the point of intersection of line \( l \) with the line \( A_{1} B \).
|
1.5
| 0.125 |
Integrate the equation
$$
\left(y^{3}-2 x y\right) dx+\left(3 x y^{2}-x^{2}\right) dy=0
$$
|
y^3 x - x^2 y = C
| 0.875 |
Find the area of a triangle if two of its medians are equal to $\frac{15}{7}$ and $\sqrt{21}$, and the cosine of the angle between them is $\frac{2}{5}$.
|
6
| 0.5 |
A parallelepiped \( ABCD A_{1} B_{1} C_{1} D_{1} \) is given. A point \( X \) is chosen on edge \( A_{1} D_{1} \) and a point \( Y \) is chosen on edge \( BC \). It is known that \( A_{1}X = 5 \), \( BY = 3 \), and \( B_{1} C_{1} = 14 \). The plane \( C_{1}XY \) intersects the ray \( DA \) at point \( Z \). Find \( DZ \).
|
20
| 0.5 |
A novice gardener planted daisies, buttercups, and marguerites in their garden. When they sprouted, it turned out that there were 5 times more daisies than non-daisies and 5 times fewer buttercups than non-buttercups. What fraction of the sprouted plants are marguerites?
|
0
| 0.125 |
Find all real polynomials \( p(x) \) such that \( 1 + p(x) \equiv \frac{p(x-1) + p(x+1)}{2} \).
|
p(x) = x^2 + bx + c
| 0.75 |
What is the three-digit (integer) number that is 12 times the sum of its digits?
|
108
| 0.625 |
Let \( ABC \) be a triangle, and let \( M \) be the midpoint of side \( AB \). If \( AB \) is 17 units long and \( CM \) is 8 units long, find the maximum possible value of the area of \( ABC \).
|
68
| 0.75 |
Given a triangle \(ABC\) such that the median, altitude, and angle bisector from \(A\) divide the angle \(\widehat{A}\) into four equal angles \(\alpha\). Express all the angles in the figure in terms of \(\alpha\) and calculate \(\alpha\).
|
22.5^\circ
| 0.125 |
In an $11 \times 11$ square, the central cell is colored black. Maxim found the largest rectangular grid area that is entirely within the square and does not contain the black cell. How many cells are in that rectangle?
|
55
| 0.125 |
As shown in the figure, the area of rectangle $ABCD$ is $56 \mathrm{~cm}^{2}$. Given that $BE=3 \mathrm{~cm}$ and $DF=2 \mathrm{~cm}$, please find the area of triangle $AEF$.
|
25 \text{ cm}^2
| 0.375 |
Figure 0 consists of a square with side length 18. For each integer \(n \geq 0\), Figure \(n+1\) consists of Figure \(n\) with the addition of two new squares constructed on each of the squares that were added in Figure \(n\). The side length of the squares added in Figure \(n+1\) is \(\frac{2}{3}\) of the side length of the smallest square(s) in Figure \(n\). Define \(A_{n}\) to be the area of Figure \(n\) for each integer \(n \geq 0\). What is the smallest positive integer \(M\) with the property that \(A_{n}<M\) for all integers \(n \geq 0\)?
|
2916
| 0.75 |
Given constants \( a \) and \( b \) with \( a \neq 0 \), the function \( f(x) = \frac{x}{a x + 3} \) satisfies \( f(2) = 1 \) and \( f(x) = x \) has a unique solution for \( x \).
1. Find the values of \( a \) and \( b \).
2. Given \( x_1 = 1 \) and \( x_n = f(x_{n-1}) \) for \( n = 2, 3, \ldots \), find the general formula for the sequence \( \{x_n\} \).
|
x_n = \frac{2}{n+1}
| 0.625 |
Anya calls a date beautiful if all 6 digits in its representation are different. For example, 19.04.23 is a beautiful date, but 19.02.23 and 01.06.23 are not. How many beautiful dates are there in the year 2023?
|
30
| 0.25 |
The sequence \(\left\{\alpha_{n}\right\}\) is an arithmetic sequence with common difference \(\beta\). The sequence \(\left\{\sin \alpha_{n}\right\}\) is a geometric sequence with common ratio \(q\). Given \(\alpha_{1}\) and \(\beta\) are real numbers, find \(q\).
|
\pm 1
| 0.125 |
A number \( A \) consisting of eight non-zero digits is added to a seven-digit number consisting of identical digits, resulting in an eight-digit number \( B \). It turns out that \( B \) can be obtained by permuting some of the digits of \( A \). What digit can \( A \) start with if the last digit of \( B \) is 5?
|
5
| 0.125 |
Find those prime numbers $p$ for which the number $p^{2}+11$ has exactly 6 positive divisors.
|
3
| 0.375 |
Let \( a_{1}=1, a_{2}=2 \) and for all \( n \geq 2 \), \[ a_{n+1}=\frac{2 n}{n+1} a_{n}-\frac{n-1}{n+1} a_{n-1}. \] It is known that \( a_{n} > 2 + \frac{2009}{2010} \) for all \( n \geq m \), where \( m \) is a positive integer. Find the least value of \( m \).
|
4021
| 0.875 |
Given a cube \( ABCD A_1 B_1 C_1 D_1 \) with an edge length of 1. A sphere passes through vertices \( A \) and \( C \) and the midpoints \( F \) and \( E \) of edges \( B_1 C_1 \) and \( C_1 D_1 \) respectively. Find the radius \( R \) of this sphere.
|
\frac{\sqrt{41}}{8}
| 0.625 |
Calculate: \(\left(2 \frac{2}{3} \times\left(\frac{1}{3}-\frac{1}{11}\right) \div\left(\frac{1}{11}+\frac{1}{5}\right)\right) \div \frac{8}{27} = 7 \underline{1}\).
|
7 \frac{1}{2}
| 0.375 |
Find a five-digit number that has the following property: when multiplied by 9, the result is a number represented by the same digits but in reverse order.
|
10989
| 0.5 |
In triangle \( ABC \) with sides \( AB = c \), \( BC = a \), and \( AC = b \), a median \( BM \) is drawn. Incircles are inscribed in triangles \( ABM \) and \( BCM \). Find the distance between the points where these incircles touch the median \( BM \).
|
\frac{|a-c|}{2}
| 0.25 |
The base of the rectangular parallelepiped $A B C D A_{1} B_{1} C_{1} D_{1}$ is a square $A B C D$. Find the maximum possible angle between the line $B D_{1}$ and the plane $B D C_{1}$.
|
\arcsin \frac{1}{3}
| 0.375 |
Person A and person B start from locations A and B respectively, walking towards each other at speeds of 65 meters per minute and 55 meters per minute simultaneously. They meet after 10 minutes. What is the distance between A and B in meters? Also, what is the distance of the meeting point from the midpoint between A and B in meters?
|
50
| 0.5 |
A fair coin is flipped ten times. Let $\frac{i}{j}$ be the probability that heads do not appear consecutively, where $i$ and $j$ are coprime. Find $i + j$.
|
73
| 0.625 |
In a sequence of positive integers, an inversion is a pair of positions where the element at the leftmost position is greater than the element at the rightmost position. For example, the sequence 2, 5, 3, 1, 3 has 5 inversions: between the first and the fourth position, between the second position and all the positions to its right, and finally between the third and the fourth position. Among all sequences of positive integers whose elements sum to $n$, what is the maximum possible number of inversions if
a) $n=7$ ?
b) $n=2019$ ?
Note: The sequences of positive integers considered in this problem can have more than 5 elements.
|
509545
| 0.25 |
Point \( D \) lies on the side \( AC \) of triangle \( ABC \). A circle with diameter \( BD \) intersects sides \( AB \) and \( BC \) at points \( P \) and \( T \) respectively. Points \( M \) and \( N \) are the midpoints of the segments \( AD \) and \( CD \) respectively. It is known that \( PM \| TN \).
a) Find the angle \( ABC \).
b) Additionally, it is known that \( MP = 1 \), \( NT = \frac{3}{2} \), and \( BD = \sqrt{5} \). Find the area of triangle \( ABC \).
|
5
| 0.375 |
Three sisters, whose average age is 10, all have different ages. The average age of one pair of the sisters is 11, while the average age of a different pair is 12. What is the age of the eldest sister?
|
16
| 0.875 |
Suppose \( m \) and \( n \) are integers with \( 0 < m < n \). Let \( P = (m, n) \), \( Q = (n, m) \), and \( O = (0,0) \). For how many pairs of \( m \) and \( n \) will the area of triangle \( OPQ \) be equal to 2024?
|
6
| 0.5 |
How many zeros are at the end of the product \( s(1) \cdot s(2) \cdot \ldots \cdot s(100) \), where \( s(n) \) denotes the sum of the digits of the natural number \( n \)?
|
19
| 0.75 |
There are 5 different lines on a plane, forming $m$ intersection points. How many different values can $m$ take?
|
9
| 0.25 |
In square \( ABCD \), \( E \) and \( F \) are the midpoints of \( AB \) and \( BC \) respectively. \( AF \) and \( CE \) intersect at \( G \), and \( AF \) and \( DE \) intersect at \( H \). Determine the ratio \( AH : HG : GF \).
|
6:4:5
| 0.875 |
Mat is digging a hole. Pat asks him how deep the hole will be. Mat responds with a riddle: "I am $90 \mathrm{~cm}$ tall and I have currently dug half the hole. When I finish digging the entire hole, the top of my head will be as far below the ground as it is above the ground now." How deep will the hole be when finished?
|
120 \text{ cm}
| 0.25 |
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