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|---|---|---|
The number of lattice points (points with integer coordinates) inside the region bounded by the right branch of the hyperbola \( x^{2}-y^{2}=1 \) and the line \( x=100 \), excluding the boundary, is \(\qquad\) .
|
9800
| 0.5 |
Suppose complex numbers \( z_{1}, z_{2} \) satisfy \( \left|z_{1}\right| = \left|z_{1} + z_{2}\right| = 3 \), and \( \left|z_{1} - z_{2}\right| = 3 \sqrt{3} \). Then \(\log _{3}\left|\left(z_{1} \overline{z_{2}}\right)^{2000} + \left(\overline{z_{1}} z_{2}\right)^{2000}\right|\) equals ______.
|
4000
| 0.875 |
What is the smallest square number whose first five digits are 4 and the sixth digit is 5?
|
666667
| 0.125 |
Given that \( f(x) \) is a periodic function on \(\mathbf{R}\) with the smallest positive period of 2, and when \( 0 \leq x < 2 \), \( f(x) = x^3 - x \), determine the number of points where the graph of the function \( f(x) \) intersects the x-axis in the interval \([0, 6]\).
|
7
| 0.875 |
Find the smallest natural number that ends with the digit 6 such that moving this digit to the front increases the number exactly fourfold.
|
153846
| 0.875 |
Little children were eating candies. Each ate 7 candies less than all the others combined but still more than one candy.
How many candies were eaten in total?
|
21 \text{ candies}
| 0.75 |
A die is rolled three times. The first result is $a$, the second is $b$, and the third is $c$. Assuming each number from 1 to 6 is equally likely to appear:
1. Calculate the probability that $a b + 2 c \geq a b c$.
2. Calculate the probability that $a b + 2 c$ and $2 a b c$ are coprime.
|
\frac{13}{72}
| 0.125 |
What is the maximum number of cells that can be colored on a $6 \times 6$ board such that no four colored cells can be selected whose centers form a rectangle with sides parallel to the sides of the board?
|
16
| 0.25 |
Given that \(\log _{2} a \cdot \log _{2} b=1\) (where \(a>1\) and \(b>1\)), find the minimum value of \(a b\).
|
4
| 0.875 |
The remainder when a certain natural number \( n \) is divided by 22 is 7, and the remainder when \( n \) is divided by 33 is 18. Find the remainder when \( n \) is divided by 66.
|
51
| 0.875 |
The graphs \( y = 2 \cos 3x + 1 \) and \( y = - \cos 2x \) intersect at many points. Two of these points, \( P \) and \( Q \), have \( x \)-coordinates between \(\frac{17 \pi}{4}\) and \(\frac{21 \pi}{4}\). The line through \( P \) and \( Q \) intersects the \( x \)-axis at \( B \) and the \( y \)-axis at \( A \). If \( O \) is the origin, what is the area of \( \triangle BOA \)?
|
\frac{361\pi}{8}
| 0.875 |
In triangle $ABC$, the angles $\angle B = 30^{\circ}$ and $\angle A = 90^{\circ}$ are known. Point $K$ is marked on side $AC$, and points $L$ and $M$ are marked on side $BC$ such that $KL = KM$ (point $L$ lies on segment $BM$).
Find the length of segment $LM$ if it is known that $AK = 4$, $BL = 31$, and $MC = 3$.
|
14
| 0.625 |
There is a unique quadruple of positive integers \((a, b, c, k)\) such that \(c\) is not a perfect square and \(a + \sqrt{b + \sqrt{c}}\) is a root of the polynomial \(x^4 - 20x^3 + 108x^2 - kx + 9\). Compute \(c\).
|
7
| 0.75 |
Which members of the sequence 101, 10101, 1010101, ... are prime?
|
101
| 0.375 |
For which values of the parameter \( a \) does the equation
$$
5^{x^{2}+2ax+a^{2}} = ax^{2}+2a^{2}x+a^{3}+a^{2}-6a+6
$$
have exactly one solution?
|
a = 1
| 0.5 |
Consider all functions \( f: \mathbb{Z} \rightarrow \mathbb{Z} \) satisfying
\[ f(f(x)+2x+20) = 15. \]
Call an integer \( n \) good if \( f(n) \) can take any integer value. In other words, if we fix \( n \), for any integer \( m \), there exists a function \( f \) such that \( f(n) = m \). Find the sum of all good integers \( x \).
|
-35
| 0.25 |
Find the smallest term in the following sequence:
\[ a_{1} = 1993^{1094^{1995}}, \]
\[ a_{n+1} = \begin{cases}
\frac{1}{2} a_{n}, & \text{if } a_{n} \text{ is even}, \\
a_{n} + 7, & \text{if } a_{n} \text{ is odd}.
\end{cases} \]
|
1
| 0.375 |
Point $M$ lies on the leg $AC$ of the right triangle $ABC$ with a right angle at $C$, such that $AM=2$ and $MC=6$. Segment $MH$ is the altitude of triangle $AMB$. Point $D$ is located on the line $MH$ such that the angle $ADB$ is $90^{\circ}$, and points $C$ and $D$ lie on the same side of line $AB$. Find the length of segment $DC$ if the tangent of the angle $ACH$ is $1 / 7$. (16 points)
|
4
| 0.625 |
Solve the system of inequalities
$$
\left\{\begin{array}{l}
|2 x-3| \leqslant 3 \\
\frac{1}{x}<1
\end{array}\right.
$$
|
(1, 3]
| 0.75 |
There are 26 stamp collectors from different countries who want to exchange the latest commemorative stamps of each country through mutual correspondence. To ensure that each of the 26 individuals ends up possessing a complete set of the latest commemorative stamps from all 26 countries, at least how many letters need to be exchanged?
|
50
| 0.625 |
Ken is the best sugar cube retailer in the nation. Trevor, who loves sugar, is coming over to make an order. Ken knows Trevor cannot afford more than 127 sugar cubes, but might ask for any number of cubes less than or equal to that. Ken prepares seven cups of cubes, with which he can satisfy any order Trevor might make. How many cubes are in the cup with the most sugar?
|
64
| 0.875 |
The function \( f(x) \) satisfies \( f(x) + f(1-x) = \frac{1}{2} \) for all \( x \in \mathbb{R} \).
(1) Let the sequence \( \{a_n\} \) be defined as:
\[ a_n = f(0) + f\left(\frac{1}{n}\right) + f\left(\frac{2}{n}\right) + \cdots + f\left(\frac{n-1}{n}\right) + f(1) \]
Is the sequence \( \{a_n\} \) an arithmetic sequence? Please provide a proof.
(2) Let
\[ b_n = \frac{4}{4a_n - 1}, \quad T_n = b_1^2 + b_2^2 + b_3^2 + \cdots + b_n^2, \quad S_n = 32 - \frac{16}{n} \]
Compare the values of \( T_n \) and \( S_n \).
|
T_n \leq S_n
| 0.75 |
Given that \(ab = 1000\) with \(a > 1\) and \(b > 1\), what is the maximum value of \(\sqrt{1 + \lg a} + \sqrt{1 + \lg b}\)?
|
\sqrt{10}
| 0.875 |
In the pentagon \( U V W X Y \), \(\angle U = \angle V = 90^\circ \), \( U Y = V W \), and \( Y X = X W \). Four equally spaced points are marked between \( U \) and \( V \), and perpendiculars are drawn through each point. The dark shaded region has an area of \( 13 \mathrm{~cm}^2 \) and the light shaded region has an area of \( 10 \mathrm{~cm}^2 \). What is the area, in \(\mathrm{cm}^2\), of the entire pentagon?
A) 45
B) 47
C) 49
D) 58
E) 60
|
45
| 0.5 |
Find all pairs \((a, b)\) of integers such that \(\sqrt{2010+2 \sqrt{2009}}\) is a solution of the quadratic equation \(x^{2} + ax + b = 0\).
|
( - 2 , - 2008 )
| 0.875 |
At the World Meteorological Conference, each participant successively announced the average monthly temperature in their hometown. Meanwhile, all others recorded the product of the temperatures in their and the speaking participant's hometown. In total, 42 positive and 48 negative numbers were recorded. What is the minimum number of times a positive temperature could have been announced?
|
4
| 0.75 |
On a $4 \times 4$ board, numbers from 1 to 16 must be placed in the squares, without repetition, so that the sum of the numbers in each row, column, and diagonal is the same. We call this sum the Magic Sum.
a) What is the Magic Sum for this board?
b) If the sum of the squares marked with $X$ on the board below is 34, what is the sum of the squares marked with $Y$?
| $Y$ | | | $Y$ |
| :--- | :--- | :--- | :--- |
| | $X$ | $X$ | |
| | $X$ | $X$ | |
| $Y$ | | | $Y$ |
c) If we fill the board with consecutive natural numbers from $k$ to $(k+15)$, so that the Magic Sum is 50, what is the value of $k$?
|
5
| 0.375 |
In tetrahedron \(ABCD\), \(AD = 2\sqrt{3}\), \(\angle BAC = 60^\circ\), \(\angle BAD = \angle CAD = 45^\circ\). If a sphere that is tangent to plane \(ABC\) at point \(D\) and is internally tangent to the circumsphere of the tetrahedron has a radius of 1, find the radius of the circumsphere of tetrahedron \(ABCD\).
|
3
| 0.625 |
263. Military Puzzle. An officer ordered the soldiers to form 12 rows with 11 people in each row in such a way that he could stand at a point equidistant from each row.
"Sir, there are only 120 of us," said one of the soldiers.
Is it possible to fulfill the officer's order?
|
Yes
| 0.75 |
We construct pyramids perpendicular to each face of a given cube. What is the volume of the resulting solid if all its vertices lie on a spherical surface and the length of each edge of the cube is $a$?
|
a^3 \sqrt{3}
| 0.375 |
$E$ is the midpoint of side $BC$ of parallelogram $ABCD$. Line $AE$ intersects the diagonal $BD$ at point $G$. If the area of triangle $\triangle BEG$ is 1, find the area of parallelogram $ABCD$.
|
12
| 0.875 |
When the product
$$
\left(2021 x^{2021}+2020 x^{2020}+\cdots+3 x^{3}+2 x^{2}+x\right)\left(x^{2021}-x^{2020}+\cdots+x^{3}-x^{2}+x-1\right)
$$
is expanded and simplified, what is the coefficient of \(x^{2021}\)?
|
-1011
| 0.5 |
Determine the value of \( a \) such that the polynomial \( x^{n} - a x^{n-1} + a x - 1 \) is divisible by \( (x-1)^{2} \).
|
\frac{n}{n-2}
| 0.75 |
The germination rate of the seeds of a given plant is $90\%$. Find the probability that out of four planted seeds: a) three will germinate; b) at least three will germinate.
|
0.9477
| 0.875 |
How many digits does the number \(2^{100}\) have? What are its last three digits? (Give the answers without calculating the power directly or using logarithms!) If necessary, how could the power be quickly calculated?
|
376
| 0.25 |
On the extensions of the medians \(A K\), \(B L\), and \(C M\) of triangle \(A B C\), points \(P\), \(Q\), and \(R\) are taken such that \(K P = \frac{1}{2} A K\), \(L Q = \frac{1}{2} B L\), and \(M R = \frac{1}{2} C M\). Find the area of triangle \(P Q R\) if the area of triangle \(A B C\) is 1.
|
\frac{25}{16}
| 0.75 |
A group of toddlers in a kindergarten collectively has 90 teeth. Any two toddlers together have no more than 9 teeth. What is the minimum number of toddlers that can be in the group?
|
23
| 0.5 |
Find the distance from point $M_{0}$ to the plane passing through the three points $M_{1}$, $M_{2}$, $M_{3}$.
$M_{1}(1, 2, -3)$
$M_{2}(1, 0, 1)$
$M_{3}(-2, -1, 6)$
$M_{0}(3, -2, -9)$
|
2\sqrt{6}
| 0.625 |
Suppose that \( x, y \), and \( z \) are complex numbers of equal magnitude that satisfy
\[ x + y + z = -\frac{\sqrt{3}}{2} - i \sqrt{5} \]
and
\[ x y z = \sqrt{3} + i \sqrt{5}. \]
If \( x = x_{1} + i x_{2}, y = y_{1} + i y_{2} \), and \( z = z_{1} + i z_{2} \) for real \( x_{1}, x_{2}, y_{1}, y_{2}, z_{1} \), and \( z_{2} \), then
\[
\left(x_{1} x_{2} + y_{1} y_{2} + z_{1} z_{2}\right)^{2}
\]
can be written as \(\frac{a}{b}\) for relatively prime positive integers \( a \) and \( b \). Compute \( 100a + b \).
|
1516
| 0.125 |
Suppose that \( f_{1}(x)=\frac{1}{2-x} \). For each positive integer \( n \geq 2 \), define \( f_{n}(x)=f_{1}\left(f_{n-1}(x)\right) \) for all real numbers \( x \) in the domain of \( f_{1}\left(f_{n-1}(x)\right) \). The value of \( f_{2019}(4) \) can be written as \( \frac{a}{b} \) where \( a \) and \( b \) are positive integers with no common divisor larger than 1. What is \( (a, b) \)?
|
(6053, 6056)
| 0.75 |
"That" and "this," and half of "that" and "this" - how many percent is this of three-quarters of "that" and "this"?
|
200\%
| 0.875 |
Let \( a, b, c \) be positive integers such that \( \frac{1}{a} + \frac{1}{b} = \frac{1}{c} \) and \( \operatorname{gcd}(a, b, c) = 1 \). Suppose \( a + b \leq 2011 \). Determine the largest possible value of \( a + b \).
|
1936
| 0.75 |
The hares are sawing a log again, but this time both ends of the log are fixed. Ten intermediate pieces have fallen, while two end pieces remain fixed. How many cuts did the hares make?
|
11
| 0.375 |
A convex quadrilateral has three sides measuring $1 \mathrm{~cm}$, $4 \mathrm{~cm}$, and $8 \mathrm{~cm}$. Its diagonals are perpendicular to each other. What is the length of the fourth side?
|
7
| 0.875 |
Given a tetrahedron $ABCD$ with an internal point $P$, what is the minimum number of its edges that appear obtuse when viewed from point $P$?
|
3
| 0.125 |
There are eight ways to evaluate the expression " $\pm 2^{11} \pm 2^{5} \pm 2$ ". When these eight values are listed in decreasing order, what is the third value in the list?
|
2018
| 0.625 |
We have $3^{2k}$ identical-looking coins, one of which is counterfeit and weighs slightly less than a real one. Additionally, we have three two-pan balances. It is known that two of the balances are accurate, while one is faulty (the result shown by the faulty balance has no connection to the actual weights of the coins placed on them and can be either correct or arbitrarily incorrect for different weighings). We do not know which balance is faulty. How can we identify the counterfeit coin in $3k + 1$ weighings?
|
3k+1
| 0.875 |
Given the hyperbola \( P : \frac{x^{2}}{9}-\frac{y^{2}}{16}=1 \) with left and right foci \( B \) and \( C \), point \( A \) lies on \( P \). \( I \) is the incenter of triangle \( ABC \) and the line \( AI \) passes through the point \( (1,0) \). If \( \overrightarrow{A I}=x \overrightarrow{A B}+y \overrightarrow{A C} \), then \( x+y \) equals ____.
|
\frac{3}{4}
| 0.5 |
Find all functions \( f: \mathbf{Q} \rightarrow \mathbf{Q} \) (where \( \mathbf{Q} \) is the set of rational numbers) such that:
1. \( f(1) = 2 \)
2. For any \( x, y \in \mathbf{Q} \),
\[ f(xy) = f(x) f(y) - f(x + y) + 1 \]
|
f(x) = x + 1
| 0.875 |
As shown in the figure, the hyperbola function \( y = \frac{k}{x} \) where \( k > 0 \), passes through the midpoint \( D \) of the hypotenuse \( OB \) of the right triangle \( OAB \) and intersects the leg \( AB \) at point \( C \). If the area of \( \triangle OBC \) is 3, then find the value of \( k \).
|
2
| 0.625 |
Find the maximum constant \( k \) such that for all real numbers \( a, b, c, d \) in the interval \([0,1]\), the inequality \( a^2 b + b^2 c + c^2 d + d^2 a + 4 \geq k(a^3 + b^3 + c^3 + d^3) \) holds.
|
2
| 0.875 |
What is the solution of the equation \( 24 \div (3 \div 2) = (24 \div 3) \div m \)?
|
\frac{1}{2}
| 0.875 |
Let \( x = \frac{1 - \sqrt{3}}{1 + \sqrt{3}} \) and \( y = \frac{1 + \sqrt{3}}{1 - \sqrt{3}} \). If \( b = 2x^2 - 3xy + 2y^2 \), find the value of \( b \).
|
25
| 0.75 |
In a certain kingdom, the king has decided to build 25 new towns on 13 uninhabited islands so that on each island there will be at least one town. Direct ferry connections will be established between any pair of new towns which are on different islands. Determine the least possible number of these connections.
|
222
| 0.625 |
In an exam, 153 people scored no more than 30 points, with an average score of 24 points. 59 people scored no less than 80 points, with an average score of 92 points. The average score of those who scored more than 30 points is 62 points. The average score of those who scored less than 80 points is 54 points. How many people participated in this exam?
|
1007
| 0.75 |
An institute has 8 classes, each with 40 students. Three students are selected in such a way that each selected student is from a different class. What is the probability that a particular student will be among the selected ones?
|
\frac{3}{320}
| 0.875 |
In a circle with center $O$, a chord $AB$ intersects the diameter at point $M$ and forms an angle of $60^\circ$ with the diameter. Find $OM$ if $AM = 10$ cm and $BM = 4$ cm.
|
6 \text{ cm}
| 0.75 |
The non-negative integers \(a, b, c, d\) are such that
\[ a b + b c + c d + d a = 707 \]
What is the smallest possible value of the sum \(a + b + c + d\)?
|
108
| 0.5 |
On a horizontal plane, three points are 100 meters, 200 meters, and 300 meters away from the base point of an antenna. The sum of the angles of elevation to the antenna from these three points is $90^{\circ}$. What is the height of the antenna?
|
100
| 0.875 |
Fedya was 7 kopecks short of buying a portion of ice cream, and Masha was only 1 kopeck short. However, even when they combined all their money, it was still not enough to buy one portion of ice cream. How much did one portion of ice cream cost?
|
7
| 0.75 |
Four people, A, B, C, and D, move into four rooms numbered 1, 2, 3, and 4, with one person per room. Given that B does not live in room 2, and B and C must live in adjacent rooms, how many possible arrangements are there?
|
8
| 0.875 |
In the sequence \(\{a_n\}\), \(a_1 = 1\), \(a_2 = 3\), and \(a_{n+2} = |a_{n+1} - a_n|\) for \(n \in \mathbf{Z}_{+}\). What is \(a_{2014}\)?
|
1
| 0.5 |
As shown in Figure 1-10, 5 beads are strung together to make a necklace using a string. If there are 3 different colors of beads available, how many different necklaces can be made?
|
39
| 0.75 |
If \( p \) is a real constant such that the roots of the equation \( x^{3} - 6p x^{2} + 5p x + 88 = 0 \) form an arithmetic sequence, find \( p \).
|
2
| 0.875 |
Let \( n > 1 \) and \( m \) be natural numbers. A parliament consists of \( m n \) members who have formed \( 2 n \) commissions, so that:
1. Each commission consists of \( m \) deputies.
2. Each Member of Parliament is a member of exactly 2 commissions.
3. No two commissions have more than one joint member.
Determine the largest possible value of \( m \) as a function of \( n \) so that this is possible.
|
2n - 1
| 0.625 |
Given that point \( P \) is a moving point on the line \( l: kx + y + 4 = 0 \) (where \( k > 0 \)), and \( PA \) and \( PB \) are two tangents to the circle \( C: x^2 + y^2 - 2y = 0 \), with \( A \) and \( B \) being the points of tangency. If the minimum area of the quadrilateral \( PACB \) is 2, find the value of \( k \).
|
2
| 0.75 |
Calculate: \( 7 \frac{4480}{8333} \div \frac{21934}{25909} \div 1 \frac{18556}{35255} = \)
|
\frac{35}{6}
| 0.375 |
Given the equation \( x y = 6(x + y) \), find all positive integer solutions \(\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right), \cdots,\left(x_{n}, y_{n}\right)\) and compute \(\sum_{k=1}^{n}\left(x_{k}+y_{k}\right)\).
|
290
| 0.5 |
Let \( m \) be the largest positive integer such that for every positive integer \( n \leqslant m \), the following inequalities hold:
\[
\frac{2n + 1}{3n + 8} < \frac{\sqrt{5} - 1}{2} < \frac{n + 7}{2n + 1}
\]
What is the value of the positive integer \( m \)?
|
27
| 0.375 |
Let \( f(x) \) be an increasing continuous function defined on the interval \([0, 3]\), and let \( g(x) \) be its inverse function, such that \( g(x) > f(x) \) for all positive \( x \) where both functions are defined. Additionally, \( f(0) = 0 \) and \( f(3) = 2 \).
The area under the graph of \( f(x) \) on the interval \([0, 3]\) is 2. Find the area of the figure bounded by the graphs of \( f(x) \) and \( g(x) \), as well as the segment connecting the points \((3, 2)\) and \((2, 3)\).
|
4.5
| 0.625 |
Two swimmers do their training in a rectangular quarry. The first swimmer finds it convenient to start at one corner of the quarry, so he swims along the diagonal to the opposite corner and back. The second swimmer finds it convenient to start from a point that divides one of the sides of the quarry in the ratio \(2018:2019\). He swims around a quadrilateral, visiting one point on each shore, and returns to the starting point. Can the second swimmer select points on the three other shores in such a way that his path is shorter than that of the first swimmer? What is the minimum value that the ratio of the length of the longer path to the shorter path can take?
|
1
| 0.625 |
How can you measure 50 cm from a string, which is $2 / 3$ meters long, without any measuring instruments?
|
50 \text{ cm}
| 0.875 |
The numbers from 1 to 9 are written in an arbitrary order. A move is to reverse the order of any block of consecutive increasing or decreasing numbers. For example, a move changes 91 653 2748 to 91 356 2748. Show that at most 12 moves are needed to arrange the numbers in increasing order.
|
12
| 0.875 |
Let \( f(a, b, c) = \frac{1}{\sqrt{1+2a}} + \frac{1}{\sqrt{1+2b}} + \frac{1}{\sqrt{1+2c}} \), where \( a, b, c > 0 \) and \( abc = 1 \). Find the minimum value of the constant \( \lambda \) such that \( f(a, b, c) < \lambda \) always holds.
|
2
| 0.625 |
If \( f(x) = a_0 + a_1 x + a_2 x^2 + \cdots + a_4 \cos x^{4028} \) is the expansion of \( \left(x^2 + x + 2\right)^{2014} \), then find the value of
\[
2a_0 - a_1 - a_2 + 2a_3 - a_4 - a_5 + \cdots + 2a_{4020} - a_{4027} - a_{4028}
\]
.
|
2
| 0.625 |
In a convex pentagon \(ABCDE\), \(\angle A = 60^\circ\), and the other angles are equal to each other. It is known that \(AB = 6\), \(CD = 4\), and \(EA = 7\). Find the distance from point \(A\) to the line \(CD\).
|
\frac{9\sqrt{3}}{2}
| 0.125 |
For any real sequence $\left\{x_{n}\right\}$, define the sequence $\left\{y_{n}\right\}$ as follows:
$$
y_{1} = x_{1}, \quad y_{n+1} = x_{n+1} - \left(\sum_{i=1}^{n} x_{i}^{2}\right)^{\frac{1}{2}} \quad (n \geqslant 1).
$$
Find the smallest positive number $\lambda$ such that for any real sequence $\left\{x_{n}\right\}$ and any positive integer $m$, the following inequality holds:
$$
\frac{1}{m} \sum_{i=1}^{m} x_{i}^{2} \leqslant \sum_{i=1}^{m} \lambda^{m-i} y_{i}^{2}.
$$
|
2
| 0.625 |
A line passing through the focus \( F \) of a parabola intersects the parabola at points \( A \) and \( B \). If the projections of \( A \) and \( B \) onto the directrix of the parabola are \( A_1 \) and \( B_1 \) respectively, what is the measure of \( \angle A_1 F B_1 \)?
|
90^\circ
| 0.875 |
8 people are seated in two rows with 4 people in each row. Among them, 2 specific people must sit in the front row, and 1 specific person must sit in the back row.
|
5760
| 0.75 |
As shown in the figure, in the square $ABCD$, an inscribed circle $\odot O$ is tangent to $AB$ at point $Q$ and to $CD$ at point $P$. Connect $PA$ and $PB$ to form an isosceles triangle $\triangle PAB$. Rotate these three shapes (the square, circle, and isosceles triangle) around their common axis of symmetry $PQ$ for one complete turn to form a cylinder, sphere, and cone. Find the ratio of their volumes.
|
3:2:1
| 0.875 |
The function \( g \) is defined on the set of triples of integers and takes real values. It is known that for any four integers \( a, b, c \), and \( n \), the following equalities hold: \( g(n a, n b, n c) = n \cdot g(a, b, c) \), \( g(a+n, b+n, c+n) = g(a, b, c) + n \), \( g(a, b, c) = g(c, b, a) \). Find \( g(14, 15, 16) \).
|
15
| 0.75 |
An ant starts at the point \((1,0)\). Each minute, it walks from its current position to one of the four adjacent lattice points until it reaches a point \((x, y)\) with \(|x|+|y| \geq 2\). What is the probability that the ant ends at the point \((1,1)\)?
|
\frac{7}{24}
| 0.75 |
Find the sum of the squares of the distances from the vertices of a regular $n$-gon, inscribed in a circle of radius $R$, to an arbitrary line passing through the center of the polygon.
|
\frac{nR^2}{2}
| 0.5 |
Given positive numbers \( a \) and \( b \) that satisfy \( 2 + \log_{2} a = 3 + \log_{3} b = \log_{6}(a+b) \), find the value of \( \frac{1}{a} + \frac{1}{b} \).
|
108
| 0.875 |
Calculate the value of the following integral:
$$
\int_{0}^{\pi / 2^{n+1}} \sin x \cdot \cos x \cdot \cos 2 x \cdot \cos 2^{2} x \cdot \ldots \cdot \cos 2^{n-1} x \, dx
$$
|
\frac{1}{2^{2n}}
| 0.125 |
Given the quadratic function \( f(x) = a x^{2} + b x + c \) where \( a, b, c \in \mathbf{R}_{+} \), if the function has real roots, determine the maximum value of \( \min \left\{\frac{b+c}{a}, \frac{c+a}{b}, \frac{a+b}{c}\right\} \).
|
\frac{5}{4}
| 0.875 |
Three tired cowboys entered a saloon and hung their hats on a bison horn at the entrance. Late at night, when the cowboys were leaving, they were unable to distinguish one hat from another and randomly took three hats. Find the probability that none of them took their own hat.
|
\frac{1}{3}
| 0.875 |
Given a cube \( ABCD-A_{1}B_{1}C_{1}D_{1} \), find the value of \( \cos \theta \), where \( \theta \) is the angle between the planes \( AB_{1}D_{1} \) and \( A_{1}BD \), with \( 0^{\circ} \leq \theta \leq 90^{\circ} \).
|
\frac{1}{3}
| 0.75 |
Find the radius of a circle that is tangent to two concentric circles with radii 3 and 5.
|
1
| 0.75 |
Given that the number of positive divisors of 3600 is \(m\), and the number of positive divisors of 36 is \(n\), find the value of \(\frac{m}{n}\).
|
5
| 0.75 |
Density is the ratio of an object's mass to its volume. The volume after compaction is $V_{2} = 0.8 V_{1}$. Given that the mass did not change due to compaction, the density after compaction is $\rho_{2} = \frac{1}{0.8}\rho_{1} = 1.25\rho_{1}$, which means it increased by $25\%$.
|
25\%
| 0.75 |
The parabola with equation \(y = -\frac{1}{4} x^2 + 5x - 21\) has its vertex at point \(A\) and crosses the \(x\)-axis at \(B(b, 0)\) and \(F(f, 0)\) where \(b < f\). A second parabola with its vertex at \(B\) passes through \(A\) and crosses the \(y\)-axis at \(D\). What are the coordinates of \(D\)?
|
(0, 9)
| 0.875 |
There are 40 students in a class. Of these, 22 are involved in physical education clubs, 16 in a mathematics club, and 20 in artistic performance clubs. Additionally, 8 students are involved in both physical education and mathematics clubs, 6 in both artistic performance and mathematics clubs, 10 in both physical education and artistic performance clubs, and 2 in physical education, artistic performance, and mathematics clubs. How many students participate in only one club? How many do not participate in any clubs at all?
|
4
| 0.125 |
In trapezoid \(A B C D\), the base \(A D\) is four times larger than \(B C\). A line passing through the midpoint of diagonal \(B D\) and parallel to \(A B\) intersects side \(C D\) at point \(K\). Find the ratio \(D K : K C\).
|
2:1
| 0.125 |
Given that \( A \) and \( B \) are two subsets of \(\{1, 2, \ldots, 100\}\) such that \( |A| = |B| \), \( A \cap B = \emptyset \), and for any \( x \in A \), \( 2x + 2 \in B \). Find the maximum value of \( |A \cup B| \).
|
66
| 0.25 |
On one side of a right angle with vertex at point \( O \), two points \( A \) and \( B \) are taken, where \( O A = a \) and \( O B = b \). Find the radius of the circle passing through points \( A \) and \( B \) and tangent to the other side of the angle.
|
\frac{a + b}{2}
| 0.875 |
Using each of the nine digits exactly once, form prime numbers (numbers that are divisible only by 1 and themselves) such that their sum is minimized.
|
207
| 0.125 |
Given a convex quadrilateral \(ABCD\) with side \(AD\) equal to 3. The diagonals \(AC\) and \(BD\) intersect at point \(E\), and it is known that the areas of triangles \(ABE\) and \(DCE\) are both equal to 1. Find the side \(BC\), given that the area of \(ABCD\) does not exceed 4.
|
3
| 0.75 |
Two buildings in a residential area are 220 meters apart. If 10 trees are planted at equal intervals between them, what is the distance between the 1st tree and the 6th tree?
|
100 \text{ meters}
| 0.75 |
The mother squirrel collects pine nuts, gathering 20 each day on sunny days and only 12 each day on rainy days. Over several consecutive days, the squirrel collected a total of 112 pine nuts, with an average of 14 pine nuts per day. How many of these days were rainy?
|
6
| 0.875 |
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