problem stringlengths 15 4.7k | solution stringlengths 2 11.9k | answer stringclasses 51
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105. Is it true that there exists a number $C$ such that for all integers $k$ the inequality
$$
\left|\frac{k^{8}-2 k+1}{k^{4}-3}\right|<C ?
$$ | 105. Let's see how the expression $\left|\frac{k^{3}-2 k+1}{k^{4}-3}\right|$ behaves for large (in absolute value) values of $k$. Clearly, in the numerator, the term $k^{3}$ plays the main role, and in the denominator, $k^{4}$. Therefore, we can expect that for large values of $k$, our expression is approximately equal... | 2 | Inequalities | math-word-problem | Yes | Yes | olympiads | false |
148. To compute the square root of a positive number $a$, one can use the following method of successive approximations. Take any number $x_{0}$ and construct a sequence according to the following rule:
$$
x_{n+1}=\frac{1}{2}\left(x_{n}+\frac{a}{x_{n}}\right)
$$
Prove that if $x_{0}>0$, then $\lim _{n \rightarrow \in... | 148. First, let's prove that if the limit of $\{x_{n}\}$ exists, then it equals $\pm \sqrt{a}$. Indeed, let $\lim _{n \rightarrow \infty} x_{n}=b$. Then $\lim _{n \rightarrow \infty} \frac{1}{2}\left(x_{n}+\frac{a}{x_{n}}\right)=\frac{1}{2}\left(b+\frac{a}{b}\right)$. We obtain the equation $b=\frac{1}{2}\left(b+\frac{... | 2 | Algebra | proof | Yes | Yes | olympiads | false |
228. Given an angle MAN and a point $O$ not lying on the side of the angle. Draw a line through $O$ intersecting the sides of the angle at points $X$ and $Y$, such that the product $O X \cdot O Y$ has a given value $k$. | 228. Suppose the problem is solved. From OX. $O Y=k$, it follows that $X$ is obtained from point $Y$ by the inversion with center $O$ and power $k$; therefore, $X$ lies on the circle $S$, which is obtained from the line $A N$ by the inversion with center $O$ and power $k$ (i.e., $k$), i.e., $X$ is the intersection poin... | 4 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
7.1. (GDR, 74). What is greater: $\sqrt{4+\sqrt{7}}-\sqrt{4-\sqrt{7}}-\sqrt{2}$ or 0? | 7.1. Using Theorem 4, we get
$$
\begin{aligned}
& \sqrt{4+\sqrt{7}}=\sqrt{4-\sqrt{7}}-\sqrt{\overline{2}}= \\
& =\left(\sqrt{\frac{4+\sqrt{16-7}}{2}}+\sqrt{\frac{4-\sqrt{16-7}}{2}}\right)- \\
& -\left(\sqrt{\frac{4+\sqrt{16-7}}{2}}-\sqrt{\frac{4-\sqrt{16-7}}{2}}\right)-\sqrt{2}= \\
& =\sqrt{\frac{7}{2}}+\sqrt{\frac{1}... | 0 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
8.2. (England, 75). Solve the equation
$$
[\sqrt[3]{1}]+[\sqrt[3]{2}]+\ldots+\left[\sqrt[3]{x^{3}-1}\right]=400
$$
in natural numbers. | 8.2. Note that the relation $[\sqrt[3]{m}]=k$, where $m, k \in \mathrm{N}$, is equivalent to the inequality $k^{3} \leqslant m \leqslant(k+1)^{3}-1$. The number of natural numbers $m$ satisfying this condition (for a fixed $k$) is $(k+1)^{3}-k^{3}=3 k^{2}+3 k+1$. Therefore, the left side of the equation is $\sum_{k=1}^... | 5 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
11.16. (England, 66). Find the number of sides of a regular polygon if for four of its consecutive vertices \( A, B, C, D \) the equality
\[
\frac{1}{A B}=\frac{1}{A C}+\frac{1}{A D}
\]
is satisfied. | 11.16. Let a circle with center 0 and radius $R$ be circumscribed around a polygon (Fig. 58). Denote $\alpha=\angle A O B$, then $0<\alpha<$ $<120^{\circ}$ and
$A B=2 R \sin (\alpha / 2), A C=2 R \sin \alpha$,
$$
A D=2 R \sin (3 \alpha / 2)
$$
from which we have
$$
\frac{1}{\sin (\alpha / 2)}=\frac{1}{\sin \alpha}+... | 7 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
12.10. (SFRY, 76). Find all values of $n \in \mathbf{N}$, greater than 2, for which it is possible to select $n$ points on a plane such that any two of them are vertices of an equilateral triangle, the third vertex of which is also one of the selected points. | 12.10. We will prove that the condition of the problem is satisfied only by the value $n=3$ (in which case the points can be placed at the vertices of an equilateral triangle). Indeed, suppose that it is possible to arrange $n \geqslant 4$ points in the manner specified in the problem. We select two points $A$ and $B$,... | 3 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
12.11. (CSSR, 80). The set $M$ is obtained from the plane by removing three distinct points $A, B$, and $C$. Find the smallest number of convex sets whose union is the set $M$. | 12.11. Let points $A, B$, and $C$ initially lie on the same line. Then these points divide the line into 4 intervals, and no points from different intervals can lie in the same convex set. Therefore, the number of required sets cannot be less than 4. The number 4 is achieved if the set $M$ is divided into parts as show... | 4 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
12.12. (Jury, SRP, 79). Prove that for any value of $n \in \mathbf{N}$, greater than some number $n_{0}$, the entire plane can be divided into $n$ parts by drawing several lines, among which there must be intersecting ones. Find the smallest such value of $n_{0}$. | 12.12. Among the lines drawn according to the condition, there are necessarily intersecting lines that already divide the plane into 4 parts. If another line is drawn, then, as a simple case analysis shows, the number of parts will increase by at least 2. Therefore, it is impossible to get exactly 5 parts, from which i... | 5 | Geometry | proof | Yes | Yes | olympiads | false |
15.7. (NBR, 68). Inside the tetrahedron $A B C D$ is a point $O$ such that the lines $A O, B O, C O, D O$ intersect the faces $B C D, A C D, A B D, A B C$ of the tetrahedron at points $A_{1}, B_{1}$, $C_{1}, D_{1}$ respectively, and the ratios
$$
\frac{A O}{A_{1} O}, \frac{B O}{B_{1} O}, \frac{C O}{C_{1} O}, \frac{D O... | 15.7. Let \( V \) be the volume of the tetrahedron \( ABCD \), and \( k \) be the desired number. Then we have
\[
\frac{V}{V_{OBCD}} = \frac{AA_1}{OA_1} = \frac{AO}{A_1O} + \frac{OA_1}{OA_1} = k + 1
\]
\[
\frac{V}{V_{OACD}} = \frac{V}{V_{OABD}} = \frac{V}{V_{OABC}} = k + 1,
\]
from which
\[
k + 1 = \frac{4V}{V_{OBC... | 3 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
16.15. (England, 68). Find the maximum number of points that can be placed on a sphere of radius 1 so that the distance between any two of them is: a) not less than $\sqrt{2} ;$ b) greater than $\sqrt{2}$. | 16.15. a) Let's prove that if points $A_{1}, A_{2}, \ldots, A_{n}$ are located on a sphere with center $O$ and radius 1 such that the distance between any points $A_{i}, A_{j} (i \neq j)$ is at least $\sqrt{2}$, then $n \leqslant 6$. Indeed, let $n>6$. By the cosine theorem, we have
$$
A_{i} A_{j}^{2}=2-2 \cos \angle ... | 4 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
17.3. (New York, 74). Let
$$
a_{n}=\frac{1 \cdot 3 \cdot 5 \ldots(2 n-1)}{2 \cdot 4 \cdot 6 \ldots 2 n}, n \in \mathrm{N}
$$
Find $\lim a_{n}$.
$$
n \rightarrow \infty
$$ | 17.3. Since
$$
\begin{aligned}
a_{n}^{2}=\frac{1^{2} \cdot 3^{2} \cdot \ldots \cdot(2 n-1)^{2}}{2^{2} \cdot 4^{2} \cdot \ldots \cdot(2 n)^{2}} & = \\
& =\frac{1 \cdot 3}{2^{2}} \cdot \frac{3 \cdot 5}{4^{2}} \cdots \frac{(2 n-1)(2 n+1)}{(2 n)^{2}} \cdot \frac{1}{2 n+1}<\frac{1}{2 n+1}
\end{aligned}
$$
for any $n \in \... | 0 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
21.5. (Austria, 83). Find all values of $a$ for which the roots $x_{1}, x_{2}, x_{3}$ of the polynomial $x^{3}-6 x^{2}+a x+a$ satisfy the equation
$$
\left(x_{1}-3\right)^{3}+\left(x_{2}-3\right)^{3}+\left(x_{3}-3\right)^{3}=0
$$ | 21.5. Let's make the substitution $y=x-3$, then the numbers $y_{1}=x_{1}-3, y_{2}=$ $=x_{2}-3$ and $y_{3}=x_{3}-3$ are the roots of the polynomial
$$
(y+3)^{3}-6(y+3)^{2}+a(y+3)+a=y^{3}+3 y^{2}+(a-9) y+4 a-27
$$
By Vieta's theorem, we have the equalities
$$
\begin{aligned}
& y_{1}+y_{2}+y_{3}=-3 \\
& y_{1} y_{2}+y_{... | -9 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
23.13*. (CSSR, 74). Let $M$ be the set of all polynomials of the form
$$
P(x)=a x^{3}+b x^{2}+c x+d \quad(a, b, c, d \in \mathbf{R})
$$
satisfying the inequality $|P(x)| \leqslant 1$ for $x \in[-1 ; 1]$. Prove that some number $k$ provides the estimate $|a| \leqslant k$ for all polynomials $P(x) \in M$. Find the smal... | 23.13. The polynomial $P_{0}(x)=4 x^{3}-3 x$ belongs to the set $M$, since $P_{0}(-1)=-1, P_{0}(1)=1$, and at its extremum points we have $P_{0}(-1 / 2)=1, P_{0}(1 / 2)=-1$. Let's prove that for any polynomial $P(x) \in M$ the estimate $|a| \leqslant 4$ holds. Suppose, on the contrary, that there exists a polynomial $P... | 4 | Algebra | proof | Yes | Yes | olympiads | false |
25.6. (USSR, 81; USA, 81). In a certain country, any two cities are directly connected by one of the following means of transportation: bus, train, or airplane. It is known that there is no city provided with all three types of transportation, and at the same time, there do not exist three cities such that any two of t... | 25.6. Suppose there are five cities connected in the manner indicated in the problem. First, let's prove that no city has three lines of the same type of transport leading out of it. Let city \( A \) be connected to cities \( B, C \), and \( D \), for example, by airplane. Then, according to the condition, no pair of c... | 4 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
27.5. (Jury, Brazil, 82; Australia, 83). An urn contains $n$ white and $m$ black balls, and next to the urn is a box with a sufficiently large number of black balls. The following operation is performed: a pair of balls is randomly drawn from the urn; if they are of the same color, a black ball from the box is moved to... | 27.5. Since the parity of the number of white balls contained in the urn does not change after each operation, the last ball will be white if and only if the number $n$ is odd. Therefore, the desired probability is either 1 (if $n$ is odd), or 0 (if $n$ is even). | 1 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
27.11. (Belgium, 77). Three shooters $A, B, C$ decided to duel simultaneously. They positioned themselves at the vertices of an equilateral triangle and agreed on the following: the first shot is taken by $A$, the second by $B$, the third by $C$, and so on in a circle; if one of the shooters is eliminated, the duel con... | 27.11. Let's consider three events that may occur after the first shot of shooter $A$.
1) $C$ is hit. Then with probability 1, shooter $A$ will be hit by the first shot of $B$.
2) $B$ is hit. Then: either with probability 0.5, shooter $C$ will hit $A$ with his first shot, or with probability $0.5 \cdot 0.3$, shooter $... | 3 | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false |
14. Some of the 20 metal cubes, identical in size and appearance, are aluminum, the rest are duralumin (heavier). How can you determine the number of duralumin cubes using no more than 11 weighings on a balance scale without weights?
Note. The problem assumes that all cubes can be aluminum, but they cannot all be dura... | 14. Let's place one cube on each pan of the scales (first weighing). In this case, two different scenarios may occur.
$1^{\circ}$. During the first weighing, one pan of the scales tips. In this case, one of the two weighed cubes is definitely aluminum, and the other is duralumin. Next, we place these two cubes on one ... | 11 | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false |
16. a) One day, a guest came to the hotel owner, K, without money but with a silver chain consisting of seven links. The owner agreed to keep the guest for a week on the condition that the guest would give him one of the chain links as payment each day. What is the minimum number of links that need to be cut so that th... | 16. a) It is sufficient to saw off one third link; in this case, the chain will break into two parts, containing 2 and 4 links respectively, and one separate (sawed) link. On the first day, the guest will give this link; on the second day, he will take it back and give in exchange the part of the chain consisting of tw... | 7 | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false |
28. Two brothers sold a flock of sheep that belonged to both of them, taking as many rubles for each sheep as there were sheep in the flock. The money received was divided as follows: first, the elder brother took ten rubles from the total amount, then the younger brother took ten rubles, after that the elder brother t... | 28. Let the number of sheep in the flock be denoted by $n$; in this case, the brothers received $n$ rubles for each sheep, and thus the total amount they received is $N=n \cdot n=n^{2}$ rubles. Let $d$ be the number of whole tens in the number $n$, and $e$ be the number of units; then $n=10 d+e$ and
$$
N=(10 d+e)^{2}=... | 2 | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false |
51. What remainders can the hundredth power of an integer give when divided by 125? | 51. Any integer either is divisible by 5 or can be represented in one of the following four forms: $5k+1, 5k+2, 5k-2$, or $5k-1$. If a number is divisible by 5, then its hundredth power is clearly divisible by $5^3 = 125$. Further, using the binomial theorem, we get:
$$
(5k \pm 1)^{100} = (5k)^{100} \pm \ldots + \frac... | 1 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
54*. Let $N$ be an even number not divisible by 10. What will be the tens digit of the number $N^{20}$? What will be the hundreds digit of the number $N^{200}$? | 54. Let's find the last two digits of the number \(N^{20}\). The number \(N^{20}\) is divisible by 4 (since \(N\) is even). Further, the number \(N\) is not divisible by 5 (otherwise it would be divisible by 10) and, therefore, can be represented in the form \(5k \pm 1\) or in the form \(5k \pm 2\) (see the solution to... | 7 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
57. The number $123456789(10)(11)(12)(13)(14)$ is written in the base-15 numeral system, i.e., this number is equal to
(14) $+(13) \cdot 15+(12) \cdot 15^{2}+(11) \cdot 15^{3}+\ldots+2 \cdot 15^{12}+15^{13}$. What remainder does it give when divided by 7? | 57. The number 15 gives a remainder of 1 when divided by 7. Therefore, it follows that
$$
15^{2}=(7 \cdot 2+1) \cdot(7 \cdot 2+1)=7 n_{1}+1
$$
gives a remainder of 1 when divided by 7,
$$
15^{3}=15^{2} \cdot 15=\left(7 n_{1}+1\right) \cdot(7 \cdot 2+1)=7 n_{2}+1
$$
gives a remainder of 1 when divided by 7, and gene... | 0 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
65. Find the remainder when the number
$$
10^{10}+10^{\left(10^{0}\right)}+\ldots+10^{\left(10^{10}\right)}
$$
is divided by 7. | 65. First of all, note that \(10^{6}-1=999999\) is divisible by 7 (since \(999999=7 \cdot 142857\)). From this, it easily follows that \(10^{\mathrm{N}}\), where \(N\) is any integer, gives the same remainder when divided by 7 as \(10^{r}\), where \(r\) is the remainder from dividing \(N\) by 6. Indeed, if \(N=6k+r\), ... | 5 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
68. a) What is the last digit of the number
$$
\left(\ldots\left(\left(\left(7^{7}\right)^{7}\right)^{7}\right)^{\ldots 7}\right)
$$
(raising to the power of 7 is repeated 1000 times)? What are the last two digits of this number?
b) What is the last digit of the number
$$
7\left(7^{\left(.7^{\left(7^{7}\right)}\righ... | 68. a) If you multiply two numbers, one of which ends in the digit $a$, and the second in the digit $b$, then their product will end in the same digit as the product $a b$. This observation allows us to easily solve the given problem. We will sequentially raise to powers, keeping track only of the last digit of the num... | 7 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
70. For which natural numbers $n$ is the sum $5^{n}+n^{5}$ divisible by 13? What is the smallest $n$ that satisfies this condition? | 70. First, let's find the remainders of the division by 13 of the numbers \(5^n\) and \(n^5\) for the first few values of \(n = 0, 1, 2, \ldots\). It is more convenient to start with the numbers \(5^n\):
\(n\)
| 0 | 1 | 2 | 3 | 4 | . | . | . |
| ---: | ---: | ---: | ---: | ---: | ---: | ---: | ---: |
| 1 | 5 | 25 | 1... | 12 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
81. The square of an integer ends with four identical digits. Which ones? | 81. A perfect square can only end in the digits 0, 1, 4, 9, 6, and 5. Furthermore, the square of any even number is clearly divisible by 4, while the square of any odd number gives a remainder of 1 when divided by 4 \(\left((2 k)^{2}=4 k^{2},(2 k+1)^{2}=4\left(k^{2}+k\right)+1\right)\); therefore, the square of no numb... | 0 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
105. All integers are written in a row, starting from one. Determine which digit stands at the $206788-\mathrm{th}$ place.
将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果。 | 105. There are nine single-digit numbers, $99-9=90$ two-digit numbers, $999-99=900$ three-digit numbers, and generally $9 \cdot 10^{n-1}$ $n$-digit numbers.
Single-digit numbers will occupy nine places in the sequence we have written, two-digit numbers $90 \cdot 2=180$ places, three-digit numbers $900 \cdot 3=2700$ pl... | 7 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
107. Each of the integers from one to a billion inclusive is replaced by the sum of the digits of the number; single-digit numbers, of course, do not change, while the others decrease. Then each of the newly obtained numbers is again replaced by the sum of its digits, and so on until a billion single-digit numbers are ... | 107. It is a good fact to know that every positive integer $N$ gives the same remainder when divided by 9 as the sum of its digits. (This follows from the fact that the digit $a_{k}$ in the $(k+1)$-th position from the end in the decimal representation of $N$ represents the term $a_{k} \cdot 10^{k}$ in the expansion of... | 1 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
153. a) Find all three-digit numbers that are equal to the sum of the factorials of their digits.
b) Find all integers that are equal to the sum of the squares of their digits. | 153. a) Let the hundreds, tens, and units digits of the desired number \( N \) be denoted by \( x, y \), and \( z \), respectively, so that \( N = 100x + 10y + z \). In this case, the condition of the problem gives
\[
100x + 10y + z = x! + y! + z!
\]
Note that \( 7! = 5040 \) is a four-digit number; therefore, no dig... | 1 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
169. In a chess tournament, two 7th-grade students and a certain number of 8th-grade students participated. Each participant played one game with every other participant. The two 7th-graders together scored 8 points, and all the 8th-graders scored the same number of points (in the tournament, each participant earns 1 p... | 169. Let $n$ be the number of eighth-graders, and $m$ be the number of points earned by each of them. In this case, the number of points scored by all participants in the tournament is $m n + 8$. This number is equal to the number of games played. Since the number of participants in the tournament is $n + 2$ and each p... | 7 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
256. Let all numbers $a_{1}, a_{2}, \ldots, a_{n}$ (where $n \geqslant 2$) be positive; how many real solutions does the system of equations have:
$$
x_{1} x_{2}=a_{1}, x_{2} x_{2}=a_{2}, \ldots, x_{n-1} x_{n}=a_{n-1}, x_{n} x_{1}=a_{n} ?
$$ | 256. Let's consider two cases separately.
$1^{\circ} . n$ is even. By multiplying the "odd" (i.e., $1$-st, $3$-rd, ..., $(n-1)$-th) and "even" equations of our system, we get:
$$
x_{1} x_{2} x_{3} \ldots x_{n}=a_{1} a_{3} a_{5} \ldots a_{n-1} \text { and } x_{1} x_{2} x_{3} \ldots x_{n}=a_{2} a_{4} a_{6} \ldots a_{n}... | 0 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
299. When dividing the polynomial $x^{1051}-1$ by $x^{4}+x^{3}+2 x^{2}+x+1$, a quotient and a remainder are obtained. Find the coefficient of $x^{14}$ in the quotient. | 299. The polynomial $x^{4}+x^{8}+2 x^{2}+x+1$ can be factored; it equals $\left(x^{2}+1\right)\left(x^{2}+x+1\right)$. From this, it is easy to see that this polynomial is a divisor of the polynomial
$$
x^{12}-1=\left(x^{6}-1\right)\left(x^{6}+1\right)=\left(x^{3}-1\right)\left(x^{3}+1\right)\left(x^{2}+1\right)\left(... | -1 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
53. Prove that for any convex polyhedron with a sufficiently large number of faces, at least one face has no fewer than six neighbors. Find the minimum number $N$ such that for any convex polyhedron with a number of faces not less than $N$, this statement is true. | 53. Let a certain convex polyhedron $W$ have the number of neighbors of each face less than six, i.e., not exceeding five.
It is easy to see that in each vertex of such a polyhedron, no more than five edges (and thus faces) can meet. To show this, consider any face of the polyhedral angle. This face has at least three... | 12 | Geometry | proof | Yes | Yes | olympiads | false |
85. The diagonals of a convex 17-gon, drawn from one vertex, divide it into 15 triangles. Can a convex 17-gon be cut into 14 triangles?
What about a non-convex 17-gon? What is the smallest number of triangles into which a 17-gon can be cut? | 85. Each of the angles of a convex polygon is less than $2 d$, so the vertices cannot lie on the sides of the triangles of the partition. When a convex polygon is divided into triangles (not necessarily by diagonals), all the angles of the polygon will be divided into parts, which will
. From these parts, we form a new
Fig. 165.
par... | 12 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
71. A segment $AB$ of unit length, which is a chord of a sphere with radius 1, is positioned at an angle of $\pi / 3$ to the diameter $CD$ of this sphere. The distance from the end $C$ of the diameter to the nearest end $A$ of the chord $AB$ is $\sqrt{2}$. Determine the length of the segment $BD$. | 71. Draw a line through $C$ parallel to $A B$, and take a point $E$ on it such that $|C E|=|A B|$, making $A B E C$ a parallelogram. If $O$ is the center of the sphere, then since $\widehat{O C E}=\pi / 3$ and $|C E|=1$ (as follows from the condition), $\triangle O C E$ is equilateral. Therefore, point $O$ is equidista... | 1 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
104. The centers of three spheres, with radii of 3, 4, and 6, are located at the vertices of an equilateral triangle with a side length of 11. How many planes exist that are tangent to all three spheres simultaneously? | 104. Any tangent plane divides space into two parts, and either all three spheres are located on one side, or two are on one side and one on the other. It is obvious that if a certain plane is tangent to the spheres, then the plane symmetric to it relative to the plane passing through the centers of the spheres is also... | 6 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
108. The volume of the tetrahedron \(ABCD\) is 5. A plane is drawn through the midpoints of the edges \(AD\) and \(BC\), intersecting the edge \(CD\) at point \(M\). The ratio of the length of segment \(DM\) to the length of segment \(CM\) is \(2/3\). Calculate the area of the section of the tetrahedron by the specifie... | 108. Let $K$ and $L$ be the midpoints of edges $AD$ and $BC$, and let $N$ and $P$ be the points of intersection of the plane with lines $AB$ and $AC$ (Fig. 16). We need to find the ratios $\frac{|PA|}{|PC|}$ and $\frac{|PK|}{|PM|}$. Draw $KQ$ and $AR$ parallel to $DC$, with $Q$ being the midpoint of $AC$.
$$
|AR|=|DM|... | 3 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
156. Given a cube $A B C D A_{1} B_{1} C_{1} D_{1}$. Points $M$ and $N$ are taken on segments $A A_{1}$ and $B C_{1}$ such that the line $M N$ intersects the line $B_{1} D$. Find
$$
\frac{\left|B C_{1}\right|}{|B N|}-\frac{|A M|}{\left|A A_{1}\right|}
$$ | 156. Projecting a cube onto a plane perpendicular to $B_{i} D_{\text {, }}$ we obtain a regular hexagon $A B C C_{1} D_{1} A_{1}$ (Fig. 35) with side
Fig. $35_{s}$ equal to $\sqrt{\frac{2}... | 1 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
195. Given a tetrahedron $A B C D$. In the planes defining its faces, points $A_{i}, B_{i}, C_{i}, D_{i}$ are taken such that the lines $A A_{i}, B B_{i}, C C_{i}, D D_{i}$ are parallel to each other. Find the ratio of the volumes of the tetrahedrons $A B C D$ and $A_{1} B_{1} C_{1} D_{\mathrm{i}}$. | 195. Let $M$ be the point of intersection of the lines $C B_{1}$ and $C_{1} B$. The vertex $A$ lies on $D M$. We draw a plane through the points $D, D_{1}$, and $A$. Denote by $K$ and $L$ its points of intersection with $C_{1} B_{i}$ and $C B$, and by $A_{2}$ the point of intersection of the line $A A_{\hat{1}}$ with $... | 3 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
212. Given a parallelepiped $A B C D A_{1} B_{1} C_{1} D_{i}$, the diagonal $A C_{1}$ of which is equal to $d$, and the volume $V$. Prove that from the segments equal to the distances from the vertices $A_{i}, B$ and $D$ to the diagonal $A C_{1}$, a triangle can be constructed and that if $s$ is the area of this triang... | 212. Let $M$ be the point of intersection of the diagonal $A C_{i}$ with the plane $A_{1} B D$. Then $M$ is the point of intersection of the medians of triangle $A_{1} B D$, and, moreover, $M$ divides the diagonal $A C_{1}$ in the ratio $1: 2$, i.e., $|A M|=\frac{1}{3}d$.
Consider the pyramid $A B A_{1} D$ (Fig. 45). ... | 2 | Geometry | proof | Yes | Yes | olympiads | false |
306. The sum of the planar angles of a trihedral angle is $180^{\circ}$. Find the sum of the cosines of the dihedral angles of this trihedral angle. | 306. Consider a tetrahedron, all faces of which are equal triangles, the angles of which are respectively equal to the plane angles of a given trihedral angle. (Prove that such a tetrahedron exists.) All trihedral angles of this tetrahedron are equal to the given trihedral angle. The sum of the cosines of the dihedral ... | 1 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
335. What is the greatest number of balls of radius 7 that can simultaneously touch, without intersecting, a ball of radius 3?
Transition to space. | 335. We will prove that there cannot be more than six such spheres. Suppose there are seven. We connect the centers of all seven spheres with the center of the given sphere and denote by $O_{1}, O_{2}, \ldots, O_{7}$ the points of intersection of these segments with the surface of the given sphere. For each point $O_{i... | 6 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
10. The bus network in the city of Lisse is arranged in such a way that: a) each route has three stops; b) any two routes either have no common stops at all or have only one common stop. What is the maximum number of routes that can exist in this city, given that there are only nine different stops? | 10. Let's consider some stop $A$. Define how many routes can pass through it. Besides $A$, there are eight other stops in the city. On each route passing through $A_{\text {r }}$, there are two more stops. Since no two of these routes can have common stops other than $A$, a total of no more than $8: 2=4$ routes can pas... | 12 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
14**. Let the game be played until three signs in a row are achieved. What is the minimum number of cells the board should contain so that the first player can win, regardless of how his opponent plays? [Draw a board (of arbitrary shape) with the minimum number of cells and prove that on any board with fewer cells, the... | 14. On a board of seven cells, shown in Fig. $16, a$, the first player can win regardless of how the opponent plays: first, he places a cross at the intersection of rows, and then on one of the middle cells in the row that does not have a zero.
No matter what move the opponent makes, the first player can win on the th... | 7 | Logic and Puzzles | proof | Yes | Yes | olympiads | false |
30. There are 111 lamps, and each lamp has its own switch. It is allowed to simultaneously switch 13 of them. At the initial moment, some lamps are on, and some are off.
a) Is it possible to turn off all the lamps?
b) How many switches will be required for this if all the lamps were initially on? | 30. Let's show that we can always turn off all the lamps. We can assume that initially more than 13 lamps were on (otherwise, we would have turned on 13 lamps from the extinguished ones). Moreover, for the same reason, we can assume that the number of extinguished lamps is more than 6. We will select 13 lamps such that... | 9 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
58. The numbers $1,2,3, \ldots, 1000000000$ are written down. Then each number is replaced by the sum of its digits, and so on, until only single-digit numbers remain in the sequence. Which digit appears more frequently in this sequence: ones or fives? | 58. From the divisibility rule for 9, which we have used several times, it follows that the numbers in the last row give the same remainders when divided by 9 as the numbers in the first row above them. From this, it easily follows that in the first 999999999 positions of the last row, the digits $1,2,3,4,5,6$, $7,8,9$... | 1 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
71. Kolya, Lena, and Misha pooled their money and bought a football. The amount of money each of them contributed does not exceed half of the total amount contributed by the other two. How much money did Misha contribute if the ball cost 6 rubles? | 71. According to the condition, the doubled amount of money invested by each boy does not exceed the sum invested by the other two. If one of the boys had given more than two rubles, then the other two would have given less than four, i.e., less than the doubled amount of money of the first. Therefore, each gave no mor... | 2 | Inequalities | math-word-problem | Yes | Yes | olympiads | false |
86. At some point on a straight line, there is a particle. In the first second, it splits in half, and the halves move in opposite directions to a distance I from the previous position. In the next second, the resulting particles again split in half, and the halves move in opposite directions to a distance 1 from their... | 86. The number and position of particles at moments $t=0$, $1,2,3$ and 4 sec are shown in Fig. 44. We will prove that after
Fig. 44.
$2^{n}-1$ sec there will be $2^{n}$ particles, arranged i... | 4 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
Example 1 (to item $1^{\circ}$). Given the matrix
$$
A=\left(\begin{array}{rrrrr}
1 & 2 & 3 & 5 & 8 \\
0 & 1 & 4 & 6 & 9 \\
0 & 0 & 1 & 7 & 10
\end{array}\right)
$$
Determine its rank. | Solution. We have
$$
M_{1}=|1| \neq 0, \quad M_{2}=\left|\begin{array}{ll}
1 & 2 \\
0 & 1
\end{array}\right| \neq 0, \quad M_{3}=\left|\begin{array}{lll}
1 & 2 & 3 \\
0 & 1 & 4 \\
0 & 0 & 1
\end{array}\right| \neq 0
$$
Minors of higher orders cannot be formed.
Answer: $\operatorname{rank} A=3$. | 3 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
Example 2 (to item $2^{\circ}$). Find the rank of the matrix
$$
A=\left(\begin{array}{rrrrr}
3 & -1 & 1 & 2 & -8 \\
7 & -1 & 2 & 1 & -12 \\
11 & -1 & 3 & 0 & -16 \\
10 & -2 & 3 & 3 & -20
\end{array}\right)
$$ | Solution. After subtracting the first row from all the others (from the last one with a factor of 2), we obtain the equivalent matrix
$$
A \sim\left(\begin{array}{rrrrr}
3 & -1 & 1 & 2 & -8 \\
4 & 0 & 1 & -1 & -4 \\
8 & 0 & 2 & -2 & -8 \\
4 & 0 & 1 & -1 & -4
\end{array}\right) \sim\left(\begin{array}{rrrrr}
3 & -1 & 1... | 2 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
Example 1. Find the volume of the parallelepiped constructed on the vectors $\vec{a}\{1,2,3\}, \vec{b}\{0,1,1\}, \vec{c}\{2,1,-1\}$. | Solution. The desired volume $V=|\vec{a} \cdot \vec{b} \cdot \vec{c}|$. Since
$$
\left|\begin{array}{rrr}
1 & 2 & 3 \\
0 & 1 & 1 \\
2 & 1 & -1
\end{array}\right|=-4
$$
then $V=4$ | 4 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
Example 4. Determine that the planes with equations $2 x+3 y-$ $-4 z+1=0$ and $5 x-2 y+z+6=0$ are perpendicular. | Solution. Let's write down the normal vectors of the given planes: $\vec{N}_{1}=\{2,3,-4\}$ and $\vec{N}_{2}=\{5,-2,1\}$. The planes are perpendicular if and only if the scalar product $\vec{N}_{1} \vec{N}_{2}=0$. We have $2 \cdot 5+3 \cdot(-2)+(-4) \cdot 1=0$ (see point $\left.8^{\circ}\right)$ | 0 | Geometry | proof | Yes | Yes | olympiads | false |
Example 1. Given the vertices of the tetrahedron \( A(2,3,1), B(4,1,-2) \), \( C(6,3,7), D(-5,-4,8) \). Find:
Fig. 4.11
1) the length of the edge \( A B \)
2) the angle between the edges \(... | Solution. The condition of the problem is satisfied by the constructed drawing (Fig. 4.11).
1) $AB$ is calculated by the formula
$$
\begin{aligned}
d & =\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}+\left(z_{2}-z_{1}\right)^{2}} \\
AB & =\sqrt{(4-2)^{2}+(1-3)^{2}+(-2-1)^{2}}=\sqrt{17}
\end{aligned}
... | 11 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
Example 3. Find the differential of the function $y=e^{x}\left(x^{2}+3\right)$. Calculate the value of the differential at the point $x=0$. | Solution. We have:
$$
\begin{gathered}
d y=y^{\prime} d x=\left(e^{x}\left(x^{2}+3\right)+2 x e^{x}\right) d x=\left(x^{2}+2 x+3\right) e^{x} d x \\
d y(0)=\left.\left(x^{2}+2 x+3\right) e^{x}\right|_{x=0} d x=3 d x
\end{gathered}
$$
## Exercises
Find the differentials of the functions.
1. $y=\ln \left(\frac{1+x}{1... | 3 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
Example 1. Check if Rolle's theorem is valid for the function $f(x)=x^{2}+2 x+7$ on the interval $[-6 ; 4]$, and if so, find the corresponding value of c. | Solution. $f(x)=x^{2}+2 x+7$ is continuous and differentiable on any interval, in particular, on $[-6 ; 4]$, and $f(6)=f(4)=31$. Therefore, $f^{\prime}(x)=2 x+2=0$ for some $x \in(-6 ; 4)$. We have $x=c=-1$. | -1 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
Example 1. Calculate the limits:
a) $\lim _{\substack{x \rightarrow 0 \\ y \rightarrow 0}} \frac{\tan\left(x^{2}+y^{2}\right)}{x^{2}+y^{2}}$, b) $\lim _{\substack{x \rightarrow 0 \\ y \rightarrow 0}} \frac{x^{2}-y^{2}}{x^{2}+y^{2}}$, c) $\lim _{\substack{x \rightarrow 0 \\ y \rightarrow 0}} \frac{x y}{\sqrt{4-x y}-2}$... | Solution. Note that the functions $\frac{\operatorname{tg}\left(x^{2}+y^{2}\right)}{x^{2}+y^{2}}$ and $\frac{x^{2}-y^{2}}{x^{2}+y^{2}}$ are undefined only at the point $(0,0)$, while $\frac{x y}{\sqrt{4-x y}-2}$ is undefined on the coordinate axes $x=0$ and $y=0$.
a) Transition to polar coordinates $x=r \cos \varphi, ... | 1 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
Example 1. Find the derivative of the function $z=x^{2}+y^{2}$ at the point $M(3,1)$ in the direction of the point $M_{1}(0,5)$. | Solution. We have: $\overrightarrow{M M}_{1}=\{0-3,5-1\}=\{-3,4\},\left|\overrightarrow{M M}_{\mathrm{I}}\right|=$ $=\sqrt{(-3)^{2}+4^{2}}=5, \cos \alpha=-\frac{3}{5}, \sin \alpha=\frac{4}{5}$.
Let $\vec{l}=\left\{-\frac{3}{5}, \frac{4}{5}\right\}$ and find $\frac{\partial z}{\partial l}$.
We have: $z_{x}^{\prime}=2 ... | -2 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
Example 2. Find the direction of the maximum increase of the function $z=3 x^{2}-2 y^{2}$ at the point $M(1,2)$. Also, find the greatest value of the derivatives in different directions at the point $M$. | Solution. Let's find the gradient of the function $z$ at the given point $(1,2)$. We have $z_{x}^{\prime}=6 x, z_{x}^{\prime}(1,2)=6, z_{y}^{\prime}=-4 y, z_{y}^{\prime}(1,2)=-8$. The gradient of the field at point $M(1,2)$ is $\overrightarrow{\operatorname{grad}} z=\{6,-8\}$. This vector indicates the direction of the... | 10 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
Example 1.1. Calculate
$$
\left|\begin{array}{rrr}
2 & -1 & 1 \\
3 & 2 & 2 \\
1 & -2 & 1
\end{array}\right|
$$ | Solution:
$$
\begin{gathered}
\left|\begin{array}{rrr}
2 & -1 & 1 \\
3 & 2 & 2 \\
1 & -2 & 1
\end{array}\right|=2 \cdot\left|\begin{array}{rr}
2 & 2 \\
-2 & 1
\end{array}\right|-(-1) \cdot\left|\begin{array}{ll}
3 & 2 \\
1 & 1
\end{array}\right|+1 \cdot\left|\begin{array}{rr}
3 & 2 \\
1 & -2
\end{array}\right|= \\
=2 ... | 5 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
Example 1.17. Form the equation of the line $l$ passing through the point $A(2; -4)$ and being at a distance of 2 units from the origin. | Solution. Let the equation of the desired line have the form:
$$
y-y_{A}=k\left(x-x_{A}\right),
$$
or
$$
y+4=k(x-2)
$$
or
$$
k x-y-(4+2 k)=0 .
$$
To determine the slope $k$ of this line, we will use the fact that it is 2 units away from the origin. We will find this distance directly. The equation of the perpendi... | 2 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
Example 5.2. Using the definition directly, show that the series converges, and find its sum.
\[
\sum_{n=1}^{\infty} \frac{1}{n(n+1)}=\frac{1}{1 \cdot 2}+\frac{1}{2 \cdot 3}+\frac{1}{3 \cdot 4}+\ldots+\frac{1}{n(n+1)}+\ldots
\] | Solution. By the definition of the partial sum of a series, we have:
$$
\begin{aligned}
& S_{1}=a_{1}=\frac{1}{2} \\
& S_{2}=a_{1}+a_{2}=\frac{1}{2}+\frac{1}{6}=\frac{2}{3} \\
& S_{3}=a_{1}+a_{2}+a_{3}=\frac{2}{3}+\frac{1}{12}=\frac{3}{4} \\
& S_{4}=a_{1}+a_{2}+a_{3}+a_{4}=\frac{3}{4}+\frac{1}{20}=\frac{4}{5}
\end{ali... | 1 | Algebra | proof | Yes | Yes | olympiads | false |
Example 5.9. Based on the D'Alembert's criterion, investigate the convergence of the series
$$
3+\frac{3^{2}}{2^{2}}+\frac{3^{3}}{3^{3}}+\frac{3^{4}}{4^{4}}+\ldots+\frac{3^{n}}{n^{n}}+\ldots
$$ | Solution. Knowing the $n$-th term of the series $a_{n}=\frac{3^{n}}{n^{n}}$, we write the $(n+1)$-th term: $a_{n+1}=\frac{3^{n+1}}{(n+1)^{n+1}} \cdot$ From this,
\[
\begin{gathered}
\lim _{n \rightarrow \infty} \frac{a_{n+1}}{a_{n}}=\lim _{n \rightarrow \infty} \frac{3^{n+1}}{(n+1)^{n+1}}: \frac{3^{n}}{n^{n}}=\lim _{n... | 0 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
Example 5.18. Find the region of convergence of the power series
$$
x+2!x^{2}+3!x^{3}+\ldots+n!x^{n}+\ldots
$$ | Solution. Here $u_{n}=n!|x|^{n}, u_{n+1}=(n+1)!|x|^{n+1}$.
From this,
$$
l=\lim _{n \rightarrow \infty} \frac{u_{n+1}}{u_{n}}=\lim _{n \rightarrow \infty} \frac{(n+1)!|x|^{n+1}}{n!|x|^{n}}=|x| \lim _{n \rightarrow \infty}(n+1)
$$
Thus,
$$
l=\begin{array}{ccc}
\infty & \text { when } & x \neq 0, \\
0 & \text { when ... | 0 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
6.2.1. Find the integral $\iiint_{V} x d x d y d z$, if the body $V$ is bounded by the planes: $x=0, y=0, x+4 y-2 z=0$ and $x+y+z-6=0$. | Solution. The body V is bounded by the coordinate planes $x \mathrm{Oz}$ and $y \mathrm{Oz}$, and from below and above by the planes: $z=\frac{1}{2}(x+4 y)$ and $z=6-x-y$. Let's find the line of intersection of these planes:
$$
\begin{aligned}
& z=\frac{1}{2}(x+4 y) \\
& z=6-x-y
\end{aligned}
$$
or
$$
\frac{1}{2}(x+... | 8 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
6. Find the distance $d$ between points $A$ and $B$ in each of the following cases:
1) $A(2), B(3) ; 2) A(-4), B(-8)$. | Solution. Applying formula (2), we get:
1) $d=|3-2|=1 ; 2) d=|-8-(-4)|=4$. | 4 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
75. Write the equation of a line parallel to the $O x$ axis and cutting off a segment equal to -2 from the $O y$ axis. Check if the points $A(2; -2)$ and $B(3; 2)$ lie on this line. | Solution. According to the problem, the slope $k=0$, the initial ordinate $b=-2$, therefore, the equation of the line we are looking for is $y=-2$. It is easy to see that the coordinates of any point $M(x; -2)$ satisfy this equation.
Point $A$ lies on the line we are looking for, since its coordinates satisfy the equa... | -2 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
312. Find the scalar square of the vector $\bar{a}=2 \bar{i}-\bar{j}-2 \bar{k}$ and its length. | Solution. We will use formulas (10), (13), and (14):
$$
(\bar{a})^{2}=a^{2}=2^{2}+(-1)^{2}+(-2)^{2}=9
$$
from which $a=3$. | 3 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
313. Find the scalar product of the vectors
$$
\bar{p}=\bar{i}-3 \bar{j}+\bar{k}, \bar{q}=\bar{i}+\bar{j}-4 \bar{k}
$$ | Solution. The scalar product of the vectors will be found using formula (12):
$$
\bar{p} \bar{q}=1 \cdot 1+(-3) \cdot 1+1 \cdot (-4)=-6
$$ | -6 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
626. Find $f^{\prime}(0)$, if $f(x)=e^{x} \arcsin x+\operatorname{arctg} x$. | Solution. $\quad f^{\prime}(x)=\left(e^{x} \arcsin x\right)^{\prime}+(\operatorname{arctg} x)^{\prime}=\left(e^{x}\right)^{\prime} \arcsin x+$ $+e^{x}(\arcsin x)^{\prime}+(\operatorname{arctg} x)^{\prime}=e^{x} \arcsin x+\frac{e^{x}}{\sqrt{1-\lambda^{2}}}+$ $+\frac{1}{x^{2}+1}=e^{x} \frac{1+\sqrt{1-x^{2}} \arcsin x}{\s... | 2 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
976. Find the following limits:
1) $\lim _{\substack{x \rightarrow 2 \\ y \rightarrow 0}} \frac{\tan x y}{y}$
2) $\lim _{\substack{x \rightarrow 0 \\ y \rightarrow 0}} \frac{x+y}{x}$. | Solution. 1) $\lim _{\substack{x \rightarrow 2 \\ y \rightarrow 0}} \frac{\operatorname{tg} x y}{y}=\lim _{\substack{x \rightarrow 2 \\ y \rightarrow 0}} x \frac{\operatorname{tg} x y}{x y}=2 \cdot 1=2$,
since
$$
\lim _{\alpha \rightarrow 0} \frac{\operatorname{tg} \alpha}{\alpha}=1
$$
2) Let's use the definition of... | 2 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
1020. Investigate the function for extremum
$$
f(x, y)=x^{2}+y^{2}-4 y+4
$$ | Solution. Let's find the partial derivatives:
$$
f_{x}^{\prime}(x, y)=2 x ; f_{y}^{\prime}(x, y)=2 y-4
$$
Set them equal to zero, we get a system of two equations with two unknowns:
$$
\left\{\begin{array}{l}
2 x=0 \\
2 y-4=0
\end{array}\right.
$$
from which $x=0, y=2$.
The function at the critical point $M_{0}(0 ... | 0 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
1.7. Convert to polar coordinates and evaluate the double integrals.
a) $\iint_{\Omega} x y^{2} d x d y$, where the region $\Omega$ is bounded by the circles $x^{2}+(y-1)^{2}=1$ and $x^{2}+y^{2}=4 y$
b) $\iint_{\Omega} e^{-x^{2}-y^{2}} d x d y$, where $\Omega-$ is the circle $x^{2}+y^{2} \leqslant R^{2}$. | Solution.
a) First, let's depict the region $\Omega$ in the Cartesian coordinate system (Fig. 1.32).
The equations of the boundary circles in polar coordinates are obtained after the substitution $x=\rho \cos \varphi, y=\rho \sin \varphi$:
$$
\begin{aligned}
x^{2}+(y-1)^{2}=1 \Rightarrow x^{2}+y^{2} & =2 y \Rightarr... | 0 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
2.5. Compute the specified line integrals of the second kind:
a) $\int_{L} 2 y \sin 2 x d x - \cos 2 x d y$, where $L$ is any piecewise-smooth curve connecting the points $A\left(\frac{\pi}{2}, 2\right)$ and $B\left(\frac{\pi}{6}, 1\right)$.
b) $\int_{L} y x e^{x} d x - (x-1) e^{x} d y$, where $L$ is any piecewise-sm... | Solution.
a) It is not difficult to understand that the integrand is a complete differential of the function $u=-y \cos 2 x$, i.e.
$$
\begin{gathered}
\int_{L} 2 y \sin 2 x d x-\cos 2 x d y=\int_{L} d u=u(B)-u(A)= \\
=-\cos \frac{\pi}{3}+2 \cos \pi=-2-\frac{1}{2}=-\frac{5}{2}
\end{gathered}
$$
b) In this problem, th... | 0 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
2.21. Calculate the circulation of the vector field:
a) $\vec{A}=x^{2} y^{2} \vec{i}+\vec{j}+z \vec{k}$ along the circle $x^{2}+y^{2}=a^{2}, z=0$;
b) $\dot{A}=(x-2 z) \dot{i}+(x+3 y+z) \dot{j}+(5 x+y) \vec{k}$ along the perimeter of the triangle $A B C$ with vertices $A(1,0,0), B(0,1,0), C(0,0,1)$. | Solution.
a) For $z=0$ we obtain the planar field $\vec{A}=x^{2} y^{2} \dot{i}+\ddot{j}$, where $P=x^{2} y^{2}, Q=1$. To compute the circulation of this field along the circle $x^{2}+y^{2}=a^{2}$, we can use Green's formula (2.62):
$$
\begin{gathered}
\oint_{x^{2}+y^{2}=a^{2}} \vec{A} \cdot d \vec{l}=\iint_{x^{2}+y^{... | 0 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
4.179. $\frac{d x}{d t}=y, \frac{d y}{d t}=-4 x-2 y$. | Solution. We form the characteristic equation
$$
\left|\begin{array}{cc}
(0-\lambda) & 1 \\
-4 & (-2-\lambda)
\end{array}\right|=2 \lambda^{2}+2 \lambda+4=0
$$
The roots of the characteristic equation are complex conjugates, the real part is negative, and the imaginary part is non-zero:
$$
\left.\lambda_{1,2}=-\frac... | 0 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
4.180. $\frac{d x}{d t}=x-y, \frac{d y}{d t}=x+y$. | Solution. We form the characteristic equation
$$
\left|\begin{array}{cc}
(1-\lambda) & -1 \\
1 & (1-\lambda)
\end{array}\right|=\lambda^{2}-2 \lambda+2=0
$$
The roots of the characteristic equation are complex conjugates, with a positive real part and a non-zero imaginary part:
$$
\lambda_{1,2}=1 \pm i (\text{case }... | 0 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
Example 2.1. $I=\int_{0}^{2} x^{3} d x$ | Solution. $I=\int_{0}^{2} x^{3} d x=\left.\frac{x^{4}}{4}\right|_{0} ^{2}=\frac{2^{4}}{4}-\frac{0^{4}}{4}=4$. | 4 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
Example 2.2. $I=\int_{0}^{2}(x-1)^{3} d x$.
| Solution.
$$
\begin{aligned}
I= & \int_{0}^{2}(x-1)^{3} d x=\int_{0}^{2}\left(x^{3}-3 x^{2}+3 x-1\right) d x= \\
& =\int_{0}^{2} x^{3} d x-3 \int_{0}^{2} x^{2} d x+3 \int_{0}^{2} x d x-\int_{0}^{2} d x= \\
& =\left.\frac{x^{4}}{4}\right|_{0} ^{2}-\left.3 \frac{x^{3}}{3}\right|_{0} ^{2}+\left.3 \frac{x^{2}}{2}\right|_{... | 0 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
Example 2.5. $I=\int_{-1}^{1} x|x| d x$.
Translate the above text into English, please keep the original text's line breaks and format, and output the translation result directly.
However, since the text provided is already in a form that is commonly used in English for mathematical expressions, the translation is e... | Solution. Since the integrand is an odd function and the limits of integration are symmetric about zero, then
$$
\int_{-1}^{1} x|x| d x=0
$$
In the following two examples, it will be demonstrated that the formal application of the Newton-Leibniz formula (without considering the integrand functions) can lead to an inc... | 0 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
Example 2.13. $\int_{-1}^{1}|x| e^{|x|} d x$. | Solution. Here the integrand is an even function, which needs to be utilized, and then apply the integration by parts formula (2.3):
$$
\begin{gathered}
\int_{-1}^{1}|x| e^{|x|} d x=2 \int_{0}^{1} x e^{x} d x=\left|\begin{array}{l}
u=x, d u=d x \\
d v=e^{x} d x, v=\int e^{x} d x=e^{x}
\end{array}\right|= \\
=2\left(\l... | 2 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
Example 2.14. $\int_{0}^{(\pi / 2)^{2}} \sin \sqrt{x} d x$. | Solution. First, let's show an incorrect solution leading to a non-integrable function. Transform the integrand as follows:
$$
\begin{gathered}
\int_{0}^{(\pi / 2)^{2}} \frac{\sin \sqrt{x} \cdot \sqrt{x} d x}{\sqrt{x}}=\left|\begin{array}{l}
u=\sqrt{x}, d u=\frac{d x}{2 \sqrt{x}} \\
d v=\frac{\sin \sqrt{x} d x}{\sqrt{... | 2 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
Example 2.21. Find the limit $\lim _{x \rightarrow 0}\left(\left(\int_{0}^{x^{2}} \cos x d x\right) / x\right)$. | Solution. We will use L'Hôpital's rule, as there is an indeterminate form of type «0/0».
$$
\lim _{x \rightarrow 0} \frac{\int_{0}^{x^{2}} \cos x d x}{x}=\lim _{x \rightarrow 0} \frac{\left(\int_{0}^{x^{2}} \cos x d x\right)^{\prime}}{x^{\prime}}=\lim _{x \rightarrow 0} \frac{\cos \left(x^{2}\right) \cdot 2 x}{1}=0
$$ | 0 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
Example 2.22. Find the limit
$$
\lim _{x \rightarrow 0}\left(\int_{0}^{\operatorname{arctg} x} e^{\sin x} d x / \int_{0}^{x} \cos \left(x^{2}\right) d x\right)
$$ | Solution. According to L'Hôpital's rule and the rules for differentiating definite integrals, we transform the limit as follows:
$$
\lim _{x \rightarrow 0}\left(\int_{0}^{\operatorname{arctg} x} e^{\sin x} d x\right)^{\prime} /\left(\int_{0}^{x} \cos \left(x^{2}\right) d x\right)^{\prime}=\lim _{x \rightarrow 0} \frac... | 1 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
Example 2.25. $\int_{-\infty}^{0} e^{x} d x$.
Translate the text above into English, keeping the original text's line breaks and format, and output the translation result directly.
Example 2.25. $\int_{-\infty}^{0} e^{x} d x$. | Solution. According to formula (2.6), we get
$$
\int_{-\infty}^{0} e^{x} d x=\lim _{a \rightarrow-\infty} \int_{a}^{0} e^{x} d x=\left.\lim _{a \rightarrow -\infty} e^{x}\right|_{a} ^{0}=1
$$ | 1 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
Example 2.36. $I=\mathrm{V}$.p. $\int_{-\infty}^{\infty} \operatorname{arctg} x d x$.
Translating the above text into English, while preserving the original text's line breaks and format, yields:
Example 2.36. $I=\mathrm{V}$.p. $\int_{-\infty}^{\infty} \arctan x d x$. | Solution.
$$
\begin{gathered}
I=\lim _{a \rightarrow \infty} \int_{-a}^{a} \operatorname{arctg} x d x=\left|\begin{array}{l}
\left.u=\operatorname{arctg} x . d u=\frac{d x}{1+x^{2}} \right\rvert\,= \\
d v=x, v=x
\end{array}\right|= \\
=\lim _{a \rightarrow \infty}\left(\left.x \operatorname{arctg} x\right|_{-a} ^{a}-\... | 0 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
Example 2.41. $I=\int_{0}^{1} \ln x d x$. | Solution.
$$
\begin{gathered}
I=\left|\begin{array}{l}
u=\ln x, d u=\frac{d x}{x} \\
d v=d x, v=x
\end{array}\right|=\lim _{a \rightarrow+0}\left(\left.x \ln x\right|_{a} ^{1}-\int_{a}^{1} x \frac{d x}{x}\right)= \\
=-\lim _{a \rightarrow+0} \frac{\ln a}{1 / a}-1=[\text{ apply L'Hôpital's rule] }= \\
=\lim _{a \righta... | -1 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
Example 2.45. $I=\mathrm{V} . \mathrm{p} . \int_{1 / e}^{e} \frac{d x}{x \ln x}$. Calculate the improper integral of the second kind in the sense of the principal value. | Solution. According to formula (2.18)
$$
\begin{gathered}
I=\lim _{a \rightarrow+0}\left(\int_{1 / e}^{1-a} \frac{d(\ln x)}{\ln x}+\int_{1+a}^{e} \frac{d(\ln x)}{\ln x}\right)= \\
=\lim _{a \rightarrow+0}\left(\ln \left|\ln x\left\|_{1 / e}^{1-a}+\ln \mid \ln x\right\|_{1+a}^{e}\right)=\right. \\
=\lim _{a \rightarrow... | 0 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
Example 2.46. Find the average value of the function $u(x)=$ $=1 / \sqrt{x}$ on the half-interval $x \in(0,1]$. | Solution.
$$
M(x)=\lim _{a \rightarrow+0}\left(\int_{a}^{1} \frac{d x / \sqrt{x}}{1-0}\right)=\lim _{a \rightarrow+0} \frac{2 \sqrt{1}-2 \sqrt{a}}{1}=2
$$
## 2.4. GEOMETRIC APPLICATIONS OF DEFINITE INTEGRALS
## Area of a Plane Curve
The area of a plane figure bounded by curves given by their equations in Cartesian ... | 2 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
Example 2.48. Calculate the area of the figure bounded by the parabola $y=-x^{2}+3 x-2$ and the coordinate axes. | Solution. Since the figure $y=-x^{2}+3 x-2$ is located in the regions with $y \geqslant 0$ and $y<0$ (Fig. 2.3), the area $S$ should be calculated separately for the part $y \geqslant 0$ and the part $y<0$, and then the absolute values of the obtained integrals should be added:
$$
\begin{gathered}
S=\left|\int_{0}^{1}... | 1 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
Example 2.58. Compute the length of the arc of the astroid $x=\cos ^{3} t$, $y=\sin ^{3} t, 0 \leqslant t \leqslant 2 \pi$. | Solution. From Fig. 2.9, it is clear that due to symmetry, it is sufficient to compute only a quarter of the astroid for
Fig. 2.9.
$0 \leqslant t \leqslant \pi / 2$. We will use formula (2... | 12 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
Example 3.3. Find the level surfaces of the function $y=$ $=\sqrt{36-x_{1}^{2}-x_{2}^{2}-x_{3}^{2}}$ and the value of the function at the point $P(1,1,3)$. | Solution. According to the definition of level surfaces, we have: $\sqrt{36-x_{1}^{2}-x_{2}^{2}-x_{3}^{2}}=C$, where $C \geqslant 0$. From this, it follows that $36-x_{1}^{2}-x_{2}^{2}-x_{3}^{2}=C^{2}$, that is, $x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=$ $=36-C^{2}$ (obviously, $0 \leqslant C \leqslant 6$). The obtained equation... | 5 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
Example 3.9. Find the limit of the function $f(x, y)=\left(x^{2}+\right.$ $\left.+y^{2}\right)^{2} x^{2} y^{2}$ as $x \rightarrow 0$ and $y \rightarrow 0$. | Solution. For the calculation of the specified limit, it is more convenient to switch to polar coordinates $x=r \cos \varphi, y=r \sin \varphi$. We obtain
$$
\begin{gathered}
\lim _{\substack{x \rightarrow 0 \\
y \rightarrow 0}}\left(x^{2}+y^{2}\right)^{2 x^{2} y^{2}}=\lim _{r \rightarrow 0}\left(r^{2}\right)^{2 r^{4}... | 1 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
Example 4.37. How many times can the trigonometric series $\sum_{k=1}^{\infty} \frac{\cos k x}{k^{4}}$ be differentiated term by term? | Solution. In this example, $a_{k}=\frac{1}{k^{4}}$ and the series with the general term $k^{s} a_{k}$ will converge for $s=1$ and $s=2$ (for $s=3$ we get the harmonic series, and for $s>3$ the general term of the series will be an infinitely large quantity). Therefore, by Theorem 4.28, this series can be term-by-term d... | 2 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
Example 5. Expand the function
$$
f(z)=\frac{z}{z^{2}-2 z-3}
$$
into a Taylor series in the neighborhood of the point $z_{0}=0$ using expansion (12), and find the radius of convergence of the series. | Solution. Let's decompose the given function into partial fractions:
$$
\frac{z}{z^{2}-2 z-3}=\frac{1}{4} \frac{1}{z+1}-\frac{3}{4} \frac{1}{z-3}
$$
Transform the right-hand side as follows:
$$
f(z)=\frac{1}{4} \frac{1}{1+z}-\frac{1}{4} \frac{1}{1-\frac{2}{3}}
$$
Using the expansion (12) of the function $\frac{1}{1... | 1 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
Example 5. Investigate the convergence of the infinite product
$$
\prod_{k=1}^{\infty}\left(1-\frac{1}{k+1}\right)=\left(1-\frac{1}{2}\right)\left(1-\frac{1}{3}\right) \ldots\left(1-\frac{1}{k+1}\right) \ldots
$$ | Solution. Here all $u_{k}=-\frac{1}{k+1}$ are negative and the series (14)
$$
\sum_{k=1}^{\infty} u_{k}=-\sum_{k=1}^{\infty} \frac{1}{k+1}=-\left(\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\ldots\right)
$$
obviously diverges.
Then, by Theorem 4, the infinite product (15) diverges.
Remark. Calculating the $n$-th partial pr... | 0 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
Example 3. Find the order of the zero $z_{0}=0$ for the function
$$
f(z)=\frac{z^{8}}{z-\sin z}
$$ | Solution. Using the Taylor series expansion of the function $\sin z$ in the neighborhood of the point $z_{0}=0$, we obtain
$$
\begin{aligned}
f(z) & =\frac{z^{8}}{z-\sin z}=\frac{z^{8}}{z-\left(z-\frac{z^{3}}{3!}+\frac{z^{5}}{5!}-\ldots\right)}= \\
& =\frac{z^{8}}{\frac{z^{3}}{3!}-\frac{z^{5}}{5!}+\ldots}=\frac{z^{5}}... | 5 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
Example 10. Determine the nature of the singular point $z=1$ of the function
$$
f(z)=\frac{\sin \pi z}{2 e^{z-1}-z^{2}-1}
$$ | Solution. Consider the function
$$
\varphi(z)=\frac{1}{f(z)}=\frac{2 e^{z-1}-z^{2}-1}{\sin \pi z}
$$
The point $z=1$ is a zero of the third order for the numerator
$$
\psi(z)=2 e^{z-1}-z^{2}-1
$$
since
$$
\begin{gathered}
\psi(1)=0 ; \quad \psi^{\prime}(1)=\left.\left(2 e^{z-1}-2 z\right)\right|_{z=1}=0 ; \\
\psi^... | 1 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
Example 4. Find the residue of the function
$$
f(z)=z^{3} \cdot \sin \frac{1}{z^{2}}
$$
at its singular point. | Solution. A singular point of the function $f(z)$ is the point $z=0$. It is an essential singular point of the function $f(z)$. Indeed, the Laurent series expansion of the function in the neighborhood of the point $z=0$ is
$$
f(z)=z^{3}\left(\frac{1}{z^{2}}-\frac{1}{3!z^{6}}+\frac{1}{5!z^{10}}-\cdots\right)=z-\frac{1}... | 0 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
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