problem
stringlengths 18
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|---|---|---|
Find the natural number \( n \) that is the product of the primes \( p, q, \) and \( r \), given that
\[
r - q = 2p \quad \text{and} \quad rq + p^2 = 676
\]
|
2001
| 0.625 |
Given that the operation "※" satisfies the following equations:
\(4 ※ 2 = 14\),
\(5 ※ 3 = 22\),
\(3 ※ 5 = 4\),
\(7 ※ 18 = 31\).
According to these rules, what is \(6 ※ 8\)?
|
28
| 0.75 |
In a regular tetrahedron \( ABCD \) with side length \( \sqrt{2} \), it is known that \( \overrightarrow{AP} = \frac{1}{2} \overrightarrow{AB} \), \( \overrightarrow{AQ} = \frac{1}{3} \overrightarrow{AC} \), and \( \overrightarrow{AR} = \frac{1}{4} \overrightarrow{AD} \). If point \( K \) is the centroid of \( \triangle BCD \), then what is the volume of the tetrahedron \( KPQR \)?
|
\frac{1}{36}
| 0.5 |
As shown in Figure 1, in triangle \( \triangle ABC \), the circumcenter is \( O \) and the incenter is \( I \). It is given that \( OI \perp AI \) and \( AB = 10 \), \( AC = 18 \). Find the length of \( BC \).
|
14
| 0.125 |
A white ball is added to an urn initially containing two balls, then one ball is drawn from the urn at random. Find the probability that the drawn ball is white, assuming all initial compositions of the balls (in terms of color) are equally likely.
|
\frac{2}{3}
| 0.75 |
Using the digits $1, 2, 3, 4$ to form 6-digit numbers where each digit can appear no more than 2 times, how many different 6-digit numbers can be formed?
|
1440
| 0.125 |
Find the number of integers between 1 and 200 inclusive whose distinct prime divisors sum to 16.
|
6
| 0.125 |
The diagonals of quadrilateral \(ABCD\) intersect at point \(O\). It is known that \(AB = BC = CD\), \(AO = 8\), and \(\angle BOC = 120^\circ\). What is the length of \(DO\)?
|
DO = 8
| 0.625 |
A sphere with radius \( \frac{3}{2} \) has its center at point \( N \). From point \( K \), which is at a distance of \( \frac{3 \sqrt{5}}{2} \) from the center of the sphere, two lines \( K L \) and \( K M \) are drawn, tangent to the sphere at points \( L \) and \( M \) respectively. Find the volume of the pyramid \( K L M N \), given that \( M L = 2 \).
|
1
| 0.375 |
Let the real numbers \(x_{1}, x_{2}, \cdots, x_{1997}\) satisfy the following two conditions:
1. \(-\frac{1}{\sqrt{3}} \leqslant x_{i} \leqslant \sqrt{3}\) for \(i=1,2,\cdots,1997\)
2. \(x_{1} + x_{2} + \cdots + x_{1997} = -318 \sqrt{3}\)
Find the maximum value of \(x_{1}^{12} + x_{2}^{12} + \cdots + x_{1997}^{12}\), and give a reason for your answer.
|
189548
| 0.375 |
The teacher asked the students to calculate \(\overline{AB} . C + D . E\). Xiao Hu accidentally missed the decimal point in \(D . E\), getting an incorrect result of 39.6; while Da Hu mistakenly saw the addition sign as a multiplication sign, getting an incorrect result of 36.9. What should the correct calculation result be?
|
26.1
| 0.75 |
Given the set \( A=\{1,2, \cdots, 10\} \). Define a function \( f: A \rightarrow A \) that satisfies:
(1) For any \( x, y \in A \), if \( x \leq y \), then \( f(x) \leq f(y) \);
(2) \( f(3)=4 \).
How many such functions \( f \) are there?
|
17160
| 0.25 |
In a frame of dimensions \(8 \times 8\) with a width of 2 cells, there are a total of 48 cells.
How many cells are in a frame of dimensions \(254 \times 254\) with a width of 2 cells?
|
2016
| 0.875 |
As shown in the figure, given that the area of rectangle $ADEF$ is 16, the area of triangle $ADB$ is 3, and the area of triangle $ACF$ is 4, find the area of triangle $ABC$.
|
6.5
| 0.125 |
How many points of a body must be fixed at least for the entire body to remain stationary?
|
3
| 0.625 |
Given that sets \( A \), \( B \), \( C \) are subsets of \(\{1, 2, \ldots, 2020\}\), and \( A \subseteq C \), \( B \subseteq C \), find the number of ordered triples \((A, B, C)\).
|
5^{2020}
| 0.75 |
Highlight the solution \( (0,0) \) by dividing both parts of the first equation by \( xy \) and the second by \( x^{2} y^{2} \). We get:
\[
\left\{\begin{array}{c}\left(\frac{x}{y}+\frac{y}{x}\right)(x+y)=15 \\ \left(\frac{x^{2}}{y^{2}}+\frac{y^{2}}{x^{2}}\right)\left(x^{2}+y^{2}\right)=85\end{array}\right.
\]
Let \( x + y = a \ and \ \frac{x}{y} + \frac{y}{x} = b \), then the first equation will be:
\[
ab = 15
\]
Considering that:
\[
\frac{x^{2}}{y^{2}} + \frac{y^{2}}{x^{2}} = b^{2} - 2 \ and \ xy = \frac{a^{2}}{2+b}
\]
Then, the second equation can be written as:
\[
(b^{2} - 2) a^{2} b = 85(2 + b)
\]
Substitute the expression for \( a \) from the first equation into this, we get:
\[
\frac{3}{b}(b^{2} - 2) 15 = 17(2 + b) \Rightarrow 14b^{2} - 17b - 45 = 0
\]
Then:
\[
b_{1} = -\frac{9}{7}, \ b_{2} = \frac{5}{2}
\]
Then:
\[
a_{1} = \frac{15}{b} = -\frac{15}{\frac{9}{7}} = -\frac{35}{3}, \ a_{2} = 6
\]
We have two systems:
1) \(\left\{\begin{array}{l}x + y = 6 \\ \frac{x}{y} + \frac{y}{x} = \frac{5}{2}\end{array}\right.\)
2) \(\left\{\begin{array}{l}x + y = -\frac{35}{3} \\ \frac{x}{y} + \frac{y}{x} = -\frac{9}{7}\end{array}\right.\)
Solve the first system:
\(\left\{\begin{array}{c}x + y = 6 \\ x^{2} + y^{2} = \frac{5}{2} xy\end{array} \Rightarrow \left\{\begin{array}{c} x + y = 6 \\ (x + y)^{2} = 4.5xy\end{array}\Rightarrow \left\{\begin{array}{c}x + y = 6 \\ 4.5xy = 36\end{array}\right.\right.\right.\)
\(\left\{\begin{array}{c}x + y = 6 \\ xy = 8\end{array}\right.\)
By Vieta's theorem:
\[
x = 2, \ y = 4 \ or \ x = 4, \ y = 2
\]
Solve the second system:
\(\left\{\begin{array}{l}x + y = -\frac{35}{3} \\ \frac{x}{y} + \frac{y}{x} + \frac{9}{7} = 0\ \left(\frac{y}{x}\right)^{2}+\frac{9}{7}\left(\frac{y}{x}\right)+1=0 \quad D<0\right.\}\) has no solutions.
|
(2, 4), (4, 2)
| 0.75 |
We assume in this exercise and the following ones that $a b c = 1$. Show that
$$
\frac{1}{a^{3}(b+c)}+\frac{1}{b^{3}(c+a)}+\frac{1}{c^{3}(a+b)} \geq \frac{3}{2}
$$
|
\frac{3}{2}
| 0.75 |
A hospital has the following employees:
- Sara Dores da Costa: rheumatologist
- Iná Lemos: pulmonologist
- Ester Elisa: nurse
- Ema Thomas: traumatologist
- Ana Lisa: psychoanalyst
- Inácio Filho: obstetrician
a) In how many ways can the employees form a line?
b) In how many ways can the same employees sit at a round table? Remember that at a round table, if everyone moves to the seat to their left, the table looks the same!
c) And in how many ways can the employees form a committee consisting of a president, vice-president, and substitute?
|
120
| 0.5 |
What is the largest integer for which each pair of consecutive digits is a square?
|
81649
| 0.625 |
How many roots does the equation
$$
\overbrace{f(f(\ldots f}^{10 \text{ times } f}(x) \ldots)) + \frac{1}{2} = 0
$$
have, where \( f(x) = |x| - 1 \)?
|
20
| 0.5 |
The motion of the fluid is defined by the complex potential \( f(z) = \ln \operatorname{sh} \pi z \). Find the flow rate \( N_{L} \) through the circle \( 2|z| = 3 \) and the circulation \( \Gamma_{L} \) around it.
|
\Gamma_{L} = 0
| 0.125 |
Calculate:
$$
\left(10^{4}-9^{4}+8^{4}-7^{4}+\cdots+2^{4}-1^{4}\right)+\left(10^{2}+9^{2}+5 \times 8^{2}+5 \times 7^{2}+9 \times 6^{2}+9 \times 5^{2}+13 \times 4^{2}+13 \times 3^{2}\right) =
$$
|
7615
| 0.5 |
Fill each blank with a number not equal to 1, such that the equation holds true. The number of different ways to fill the blanks is $\qquad$.
$$
[\mathrm{A} \times(\overline{1 \mathrm{~B}}+\mathrm{C})]^{2}=\overline{9 \mathrm{DE} 5}
$$
|
8
| 0.875 |
The value of the product \(\cos \frac{\pi}{15} \cos \frac{2 \pi}{15} \cos \frac{3 \pi}{15} \cdots \cos \frac{7 \pi}{15}\).
|
\frac{1}{128}
| 0.75 |
Let \( \triangle ABC \) be an acute triangle with \( BC = 5 \). \( E \) is a point on \( AC \) such that \( BE \perp AC \). \( F \) is a point on \( AB \) such that \( AF = BF \). Moreover, \( BE = CF = 4 \). Find the area of the triangle.
|
8 \sqrt{3} - 6
| 0.375 |
In a triangle, two sides are given. For what value of the angle between them will the area of the triangle be the greatest?
|
90^\circ
| 0.875 |
A coin is flipped 2021 times. What is the probability that an even number of tails will result?
|
\frac{1}{2}
| 0.875 |
What amount can a worker, who is a tax resident, expect after paying personal income tax if they are credited with 45000 for their work?
|
39150
| 0.125 |
Given a chessboard, it is allowed to recolor all the cells of any horizontal or vertical line in a different color. Is it possible to obtain a board with exactly one black cell?
|
\text{No}
| 0.5 |
What is the angular measure of an arc if the radius drawn to its endpoint forms an angle of $40^{\circ}$ with its chord?
|
100^\circ
| 0.75 |
From Moscow to city \( N \), a passenger can travel by train, taking 20 hours. If the passenger waits for a flight (waiting will take more than 5 hours after the train departs), they will reach city \( N \) in 10 hours, including the waiting time. By how many times is the plane’s speed greater than the train’s speed, given that the plane will be above this train 8/9 hours after departure from the airport and will have traveled the same number of kilometers as the train by that time?
|
10
| 0.75 |
Let \( A B C D \) be a parallelogram such that \( \angle B A D = 60^\circ \). Let \( K \) and \( L \) be the midpoints of \( B C \) and \( C D \), respectively. Assuming that \( A B K L \) is a cyclic quadrilateral, find \( \angle A B D \).
|
75^\circ
| 0.125 |
On the unit square \(ABCD\), the points \(A_1, B_1, C_1, D_1\) are marked such that \(AA_1 = BB_1 = CC_1 = DD_1 = \frac{1}{5}\). The points \(A\) and \(C_1\), \(B\) and \(D_1\), \(C\) and \(A_1\), \(D\) and \(B_1\) are then connected. What is the area of the small quadrilateral formed by these intersecting lines?
|
\frac{1}{41}
| 0.5 |
Find all ordered pairs \((m, n)\) of integers such that \(4^m - 4^n = 255\).
|
(4,0)
| 0.625 |
A string of 33 pearls has its middle pearl as the largest and most valuable. The values of the remaining pearls decrease by $3000 \mathrm{Ft}$ per pearl towards one end and by $4500 \mathrm{Ft}$ per pearl towards the other end. How much is the middle pearl worth if the total value of the string is 25 times the value of the fourth pearl from the middle on the more expensive side?
|
90000
| 0.375 |
A table consists of several rows of numbers. From the second row onwards, each number in a row is equal to the sum of the two numbers directly above it. The last row contains only one number. The first row is made up of the first 100 positive integers arranged in ascending order. What is the number in the last row? (You may use exponential notation to express the number).
|
101 \times 2^{98}
| 0.375 |
Given a positive integer \(n > 1\). Let \(a_{1}, a_{2}, \cdots, a_{n}\) be a permutation of \(1, 2, \cdots, n\). If \(i < j\) and \(a_{i} < a_{j}\), then \(\left(a_{i}, a_{j}\right)\) is called an ascending pair. \(X\) is the number of ascending pairs in \(a_{1}, a_{2}, \cdots, a_{n}\). Find \(E(X)\).
|
\frac{n(n-1)}{4}
| 0.875 |
What is the smallest positive odd number that has the same number of divisors as 360?
|
3465
| 0.25 |
As shown in Diagram 2, in the cube $AC_{1}$, $M$ and $N$ are the midpoints of $D_{1}C_{1}$ and $AB$ respectively. Determine the distance from $C$ to the plane $MB_{1}ND$.
|
\frac{\sqrt{6}}{3}
| 0.75 |
Find the positive integers $n$ such that $n^{2}+1$ divides $n+1$.
|
1
| 0.75 |
The pages in a book are numbered as follows: the first sheet contains two pages (numbered 1 and 2), the second sheet contains the next two pages (numbered 3 and 4), and so on. A mischievous boy named Petya tore out several consecutive sheets: the first torn-out page has the number 185, and the number of the last torn-out page consists of the same digits but in a different order. How many sheets did Petya tear out?
|
167
| 0.75 |
Given the function \( f_{1}(x) = \frac{2x - 1}{x + 1} \), for a positive integer \( n \), define \( f_{n+1}(x) = f_{1}\left[ f_{n}(x) \right] \). Find the explicit formula for \( f_{1234}(x) \).
|
\frac{1}{1-x}
| 0.125 |
Find all integers $n$ such that $2^{n} + 3$ is a perfect square. Same question with $2^{n} + 1$.
|
3
| 0.25 |
If \( 3y - 2x = 4 \), determine the value of \( \frac{16^{x+1}}{8^{2y-1}} \).
|
\frac{1}{2}
| 0.875 |
The dwarves painted cubic blocks with green and white paint so that each face was entirely painted with one of these two colors. After a while, they noticed that some painted blocks looked exactly the same after suitable rotations and began sorting them into groups based on this criterion (blocks that look the same are in the same group).
What is the maximum number of such groups they could get?
Hint: In what relationships can pairs of cube faces be?
|
10
| 0.125 |
The game takes place on a $9 \times 9$ grid of squared paper. Two players take turns; the player who starts the game places crosses in free cells while their partner places noughts. When all the cells are filled, the number of rows and columns $K$ in which crosses outnumber noughts, and the number of rows and columns $H$ in which noughts outnumber crosses, is counted. The difference $B=K-H$ is considered the winning score of the player who starts. Find the value of $B$ such that:
1) The first player can ensure a win of at least $B$, regardless of how the second player plays;
2) The second player can always ensure that the first player’s winning score is no more than $B$, regardless of how the first player plays.
|
B = 2
| 0.25 |
In triangle \(ABC\), let \(M\) be the midpoint of \(BC\), \(H\) be the orthocenter, and \(O\) be the circumcenter. Let \(N\) be the reflection of \(M\) over \(H\). Suppose that \(OA = ON = 11\) and \(OH = 7\). Compute \(BC^2\).
|
288
| 0.375 |
Points \( Q \) and \( R \) are located on the sides \( MN \) and \( MP \) of triangle \( MNP \), respectively, where \( MQ = 3 \) and \( MR = 4 \). Find the area of triangle \( MQR \), given \( MN = 4 \), \( MP = 5 \), and \( NP = 6 \).
|
\frac{9\sqrt{7}}{4}
| 0.875 |
How many six-digit multiples of 27 have only 3, 6, or 9 as their digits?
|
51
| 0.375 |
If the numbers \(a, b, c\) are pairwise coprime, then
$$
(a b + b c + a c, a b c) = 1
$$
|
1
| 0.875 |
Choose one vertex of a cube and consider the rays leading from it to the other vertices. How many different angles do we get when these rays are paired in all possible ways?
|
5
| 0.5 |
A sphere is the set of points at a fixed positive distance \( r \) from its center. Let \(\mathcal{S}\) be a set of 2010-dimensional spheres. Suppose that the number of points lying on every element of \(\mathcal{S}\) is a finite number \( n \). Find the maximum possible value of \( n \).
|
2
| 0.75 |
Given that \(\alpha\) and \(\beta\) satisfy \(\tan \left(\alpha + \frac{\pi}{3}\right) = -3\) and \(\tan \left(\beta - \frac{\pi}{6}\right) = 5\), find the value of \(\tan (\alpha - \beta)\).
|
-\frac{7}{4}
| 0.75 |
Simplify the expression: \(\frac{a^{7 / 3}-2 a^{5 / 3} b^{2 / 3}+a b^{4 / 3}}{a^{5 / 3}-a^{4 / 3} b^{1 / 3}-a b^{2 / 3}+a^{2 / 3} b}\) divided by \(a^{1 / 3}\).
|
a^{1/3} + b^{1/3}
| 0.125 |
Find the sum of the digits of the number $\underbrace{44 \ldots 4}_{2012 \text { times }} \cdot \underbrace{99 \ldots 9}_{2012 \text { times }}$.
|
18108
| 0.625 |
In how many ways can we paint 16 seats in a row, each red or green, in such a way that the number of consecutive seats painted in the same colour is always odd?
|
1974
| 0.125 |
Five guys join five girls for a night of bridge. Bridge games are always played by a team of two guys against a team of two girls. The guys and girls want to make sure that every guy and girl play against each other an equal number of times. Given that at least one game is played, what is the least number of games necessary to accomplish this?
|
25
| 0.875 |
If \( a > b > c > d \) is an arithmetic progression of positive reals and \( a > h > k > d \) is a geometric progression of positive reals, show that \( bc > hk \).
|
bc > hk
| 0.875 |
Let \( C \) be the complex numbers. \( f : C \rightarrow C \) satisfies \( f(z) + z f(1 - z) = 1 + z \) for all \( z \). Find \( f \).
|
f(w) = 1
| 0.875 |
In triangle \(ABC\), the angle bisector \(AL\) (where \(L \in BC\)) is drawn. Points \(M\) and \(N\) lie on the other two angle bisectors (or their extensions) such that \(MA = ML\) and \(NA = NL\). Given that \(\angle BAC = 50^\circ\).
Find the measure of \(\angle MAN\) in degrees.
|
65^\circ
| 0.125 |
Let \( G \) be the group \( \{ (m, n) : m, n \text{ are integers} \} \) with the operation \( (a, b) + (c, d) = (a + c, b + d) \). Let \( H \) be the smallest subgroup containing \( (3, 8) \), \( (4, -1) \) and \( (5, 4) \). Let \( H_{ab} \) be the smallest subgroup containing \( (0, a) \) and \( (1, b) \). Find \( a > 0 \) such that \( H_{ab} = H \).
|
7
| 0.25 |
For each positive integer \( n \), let \( \varphi(n) \) be the number of positive integers from 1 to \( n \) that are relatively prime to \( n \). Evaluate:
$$
\sum_{n=1}^{\infty} \frac{\varphi(n) 4^{n}}{7^{n}-4^{n}}
$$
|
\frac{28}{9}
| 0.75 |
Different numbers \(a\), \(b\), and \(c\) are such that the equations \(x^{2}+a x+1=0\) and \(x^{2}+b x+c=0\) have a common real root. In addition, the equations \(x^{2}+x+a=0\) and \(x^{2}+c x+b=0\) also have a common real root. Find the sum \(a+b+c\).
|
-3
| 0.875 |
What is the minimum number of cells that need to be colored on a $6 \times 6$ board so that, no matter how a 4-cell L-shaped figure is placed on the board (it can be rotated or flipped), there is at least one colored cell in that figure?
|
12
| 0.125 |
Given an odd prime number \( p \), if there exists a positive integer \( k \) such that \( \sqrt{k^2 - pk} \) is also a positive integer, find the value of \( k \).
|
\frac{(p+1)^2}{4}
| 0.5 |
The 3rd term of an arithmetic sequence is 14, and the 18th term is 23. Determine which term(s) among the first 2010 terms of the sequence are integers.
|
402
| 0.75 |
Determine the number of 7-combinations of the multiset $S = \{4 \cdot a, 4 \cdot b, 3 \cdot c, 3 \cdot d\}$.
|
60
| 0.25 |
Calculate (without using a calculator) \(\sqrt[3]{9+4 \sqrt{5}}+\sqrt[3]{9-4 \sqrt{5}}\), given that this number is an integer.
|
3
| 0.875 |
If the set \( S = \{1, 2, 3, \cdots, 16\} \) is arbitrarily divided into \( n \) subsets, there must exist some subset that contains elements \( a, b, \) and \( c \) (which can be the same) such that \( a + b = c \). Find the maximum value of \( n \).
**Note**: If the subsets \( A_1, A_2, \cdots, A_n \) of set \( S \) satisfy the following conditions:
1. \( A_i \neq \varnothing \) for \( i = 1, 2, \cdots, n \);
2. \( A_i \cap A_j = \varnothing \);
3. \( \bigcup_{i=1}^{n} A_i = S \),
then \( A_1, A_2, \cdots, A_n \) are called a partition of set \( S \).
|
3
| 0.375 |
Given $n$ rays in space such that any two rays form an obtuse angle, what is the maximum value of $n$?
|
4
| 0.5 |
Let \( x, y, z \) be strictly positive real numbers. Show that the following inequality is true:
\[ \frac{x}{x+2y+3z} + \frac{y}{y+2z+3x} + \frac{z}{z+2x+3y} \geq \frac{1}{2} \]
|
\frac{1}{2}
| 0.25 |
Evaluate the sum
$$
\cos \left(\frac{2\pi}{18}\right) + \cos \left(\frac{4\pi}{18}\right) + \cdots + \cos \left(\frac{34\pi}{18}\right).
$$
|
-1
| 0.625 |
In an isosceles triangle \(ABC\) where \(AB = AC\) and \(\angle BAD = 30^{\circ}\), \(D\) is an interior point on side \(BC\). Furthermore, triangle \(ADE\) is also isosceles, where \(E\) is an interior point on side \(AC\). What is the measure of \(\angle EDC\)?
|
15^\circ
| 0.75 |
Two fair octahedral dice, each with the numbers 1 through 8 on their faces, are rolled. Let \( N \) be the remainder when the product of the numbers showing on the two dice is divided by 8. Find the expected value of \( N \).
|
\frac{11}{4}
| 0.125 |
Let's call a natural number interesting if all its digits are distinct and the sum of any two adjacent digits is a square of a natural number. Find the largest interesting number.
|
6310972
| 0.25 |
Four points are independently chosen uniformly at random from the interior of a regular dodecahedron. What is the probability that they form a tetrahedron whose interior contains the dodecahedron's center?
|
\frac{1}{8}
| 0.875 |
Compute \( t(0)-t\left(\frac{\pi}{5}\right)+t\left(\frac{2\pi}{5}\right)-t\left(\frac{3\pi}{5}\right)+\ldots+t\left(\frac{8\pi}{5}\right)-t\left(\frac{9\pi}{5}\right) \), where \( t(x) = \cos 5x + * \cos 4x + * \cos 3x + * \cos 2x + *^2 \cos x + * \).
The coefficients indicated by * are missing. A math student claimed that he could compute the sum without knowing the values of the missing coefficients. Is he right?
|
10
| 0.625 |
ABC is a triangle. The points P and Q trisect BC, so BP = PQ = QC. A line parallel to AC meets the lines AB, AP, AQ at X, Y, and Z respectively. Show that YZ = 3 XY.
|
YZ = 3XY
| 0.625 |
Calculate the volume of the solid bounded by the surfaces.
$$
z=2 x^{2}+8 y^{2}, z=4
$$
|
2\pi
| 0.375 |
It is known that in 3 out of 250 cases, twins are born, and in one out of three of these cases, the twins are identical (monozygotic) twins. What is the a priori probability that a particular pregnant woman will give birth to twins - a boy and a girl?
|
\frac{1}{250}
| 0.625 |
Let \( A = \{a_1, a_2, a_3\} \) and \( B = \{b_1, b_2, b_3, b_4\} \).
1. Write a function \( f: A \rightarrow B \) such that \( f \) is injective, and find the number of injective functions from \( A \) to \( B \).
2. Write a function \( f: A \rightarrow B \) such that \( f \) is not injective, and find the number of such functions.
3. Can a function from \( A \) to \( B \) be surjective?
|
\text{No}
| 0.375 |
In a $3 \times 3$ table, numbers are placed such that each number is 4 times smaller than the number in the adjacent cell to the right and 3 times smaller than the number in the adjacent cell above. The sum of all the numbers in the table is 546. Find the number in the central cell.
|
24
| 0.75 |
Let \( f: \mathbb{N} \rightarrow \mathbb{Q} \) be a function, where \( \mathbb{N} \) denotes the set of natural numbers, and \( \mathbb{Q} \) denotes the set of rational numbers. Suppose that \( f(1) = \frac{3}{2} \), and
\[ f(x+y) = \left(1 + \frac{y}{x+1}\right) f(x) + \left(1 + \frac{x}{y+1}\right) f(y) + x^2 y + xy + xy^2 \]
for all natural numbers \( x, y \). Find the value of \( f(20) \).
|
4305
| 0.875 |
Find the derivative of the implicit function $y$ given by the equations:
1) $x^{2} + y^{2} + 2x - 6y + 2 = 0$
2) $x^{y} = y^{x}$
and calculate its value at $x = 1$.
|
1
| 0.75 |
In the company, there are elves, fairies, and dwarves. Each elf is friends with all fairies except for three of them, and each fairy is friends with twice as many elves. Each elf is friends with exactly three dwarves, and each fairy is friends with all the dwarves. Each dwarf is friends with exactly half of the total number of elves and fairies. How many dwarves are there in the company?
|
12
| 0.5 |
Find the increments of the argument and the function for \( y = 2x^2 + 1 \) when the argument \( x \) changes from 1 to 1.02.
|
0.0808
| 0.125 |
Find the greatest natural number that cannot be expressed as the sum of two composite numbers.
|
11
| 0.625 |
Is \(65^{1000} - 8^{2001}\) greater than, less than, or equal to 0? (Fill in the blank with ">", "<", or "=").
|
>
| 0.75 |
Find all natural numbers \(a\) and \(b\) such that for all natural numbers \(n\), the number \((a n + 1)^{6} + b\) is divisible by \(n^{2} + n + 1\).
|
a = 2, b = 27
| 0.375 |
Maria Ivanovna is a strict algebra teacher. She only gives grades of 2, 3, and 4, and never gives two consecutive 2s to the same student. It is known that she gave Vovochka 6 grades in a quarter. In how many different ways could she have done this?
|
448
| 0.125 |
The number \( N = 3^{16} - 1 \) has a divisor of 193. It also has some divisors between 75 and 85 inclusive. What is the sum of these divisors?
|
247
| 0.375 |
A group with 7 young men and 7 young women was divided into pairs randomly. Find the probability that at least one pair consists of two women. Round the answer to two decimal places.
|
0.96
| 0.625 |
Using the digits 1, 2, 3, 4 only once to form a 4-digit number, how many of them are divisible by 11?
|
8
| 0.625 |
Find the largest positive integer \( n \) such that \( n^{3}+100 \) is divisible by \( n+10 \).
|
890
| 0.75 |
A certain sports team has members with distinct numbers chosen from the positive integers 1 to 100. If the number of any member is neither the sum of the numbers of two other members nor twice the number of another member, what is the maximum number of members this sports team can have?
|
50
| 0.75 |
Given \( x, y \in \mathbf{R} \), find the minimum value of the function \( f(x, y)=\sqrt{x^{2}+y^{2}}+\sqrt{(x-1)^{2}+(y-1)^{2}}+\sqrt{(x+2)^{2}+(y+2)^{2}} \).
|
3\sqrt{2}
| 0.875 |
From an $m \times n$ chessboard, how many ways are there to select an L-shaped piece consisting of three squares?
|
4(m-1)(n-1)
| 0.125 |
Let \( S = \{1, 2, \ldots, 1963\} \). What is the maximum number of elements that can be chosen from \( S \) such that the sum of any two chosen numbers is not divisible by their difference?
|
655
| 0.375 |
On a board, the numbers $1, 2, 3, \ldots, 235$ were written. Petya erased several of them. It turned out that among the remaining numbers, no number is divisible by the difference of any two others. What is the maximum number of numbers that could remain on the board?
|
118
| 0.375 |
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