problem stringlengths 2 5.64k | solution stringlengths 2 13.5k | answer stringlengths 1 43 | problem_type stringclasses 8
values | question_type stringclasses 4
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102. Find a four-digit number that, when divided by 131, gives a remainder of 112, and when divided by 132, gives a remainder of 98. | 102. Let $N$ be the desired number. According to the problem, we have:
$$
N=131 k+112=132 l+98
$$
where $k$ and $l$ are positive integers. Since $N$ is a four-digit number, it is clear that:
$$
l=\frac{N-98}{132}<\frac{10000-98}{132} \leqslant 75
$$
Further, we have:
$$
131 k+112=132 l+98 ; \quad 131(k-l)=l-14
$$
... | 1946 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
103. a) Prove that the sum of all n-digit numbers $(n>2)$ is $494 \underbrace{99 \ldots 9}_{(n-3) \text { times }} 5 \underbrace{00 \ldots 0}_{(n-2) \text { times }}$ (thus, the sum of all three-digit numbers is 494550, and the sum of all six-digit numbers is 494999550000).
b) Find the sum of all four-digit even numbe... | 103. a) The $2n$-digit number given in the condition of the problem can be expressed as follows:
$$
\begin{gathered}
4 \cdot 10^{2n-1} + 9 \cdot 10^{2n-2} + 4 \cdot 10^{2n-3} + 9(10^{2n-4} + 10^{2n-5} + \cdots \\
\ldots + 10^n) + 5 \cdot 10^{n-1} + 5 \cdot 10^{n-2} = 4 \cdot 10^{2n-1} + 9 \cdot 10^{2n-2} + \\
+ 4 \cdo... | 1769580 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
104. How many and which digits will be needed to write down all integers from 1 to 100000000 inclusive? | 104. Let's first consider all integers from 0 to 99999999; for those numbers that have fewer than eight digits, we will pad them with leading zeros to make them eight-digit numbers. We will thus have 100000000 eight-digit numbers, which will require, obviously, 800000000 digits for their representation. In this case, e... | 68888897 | 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 |
110. Find all ten-digit numbers such that the 1st digit of the number equals the number of zeros in its decimal representation, the 2nd digit - the number of ones, and so on up to the 10th digit, which equals the number of nines in the number's representation. | 110. Let the desired number $X=\overline{a_{0} a_{1} a_{2} \ldots a_{9}}$ (where $a_{0}, a_{1}, \ldots, a_{9}$ are the digits of the number); in this case, $a_{0}$ is the number of zeros among the digits of $X$, $a_{1}$ is the number of ones, $a_{2}$ is the number of twos, and so on. Therefore, the sum of all the digit... | 6210001000 | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false |
116. The distance from point $A$ to point $B$ is 899 km. Along the road connecting $A$ and $B$, there are kilometer markers indicating the distances to $A$ and to $B$ - the inscriptions on them look like this:
$$
0|999,1| 998,2|997, \ldots, 999| 0
$$
How many of these markers have only two different digits? | 116. It is clear that if the numbers on the pole are $\overline{\lambda y z} \mid \overline{x_{1} y_{1} z_{i}}$ (where $x$, $\boldsymbol{y}, \ldots$ are digits), then $\overline{x_{1} y_{1} z_{1}}=999-x y z$ and, therefore, $z_{1}=9-z, y_{1}=$ $=9-y, x_{1}=9-x$. (If $x=9$ or $x=y=9$, then the digits $x_{1}=$ $=0$, or $... | 40 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
119. What is the greatest possible value of the ratio of a three-digit number to the sum of its digits? | 119. Let \( N = \overline{a b} c \), where \( a, b, c \) are the digits of the number; it is clear that for "round" numbers \( N = 100, 200, \ldots, 900 \) we have \( \frac{N}{a+b+c} = 100 \). Furthermore, if the number \( N \) is not "round," then \( b+c > 0 \) and \( a+b+c \geqslant a+1 \), and since the leading digi... | 100 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
150. a) Find a four-digit number that is a perfect square and such that its first two digits are equal to each other and the last two digits are also equal to each other.
b) A two-digit number, when added to the number formed by the same digits in reverse order, results in a perfect square. Find all such numbers. | 150. a) Let $a$ be the first and $b$ the last digit of the number $N$ we are looking for. Then this number is equal to $1000 a + 100 a + 10 b + b = 1100 a + 11 b = 11(100 a + b)$. Since the number $N$ is a perfect square, it follows from its divisibility by 11 that it is also divisible by 121, i.e., $\frac{N}{11} = 100... | 7744 | 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 |
188. Among the first 100000001 Fibonacci numbers (see problem 185), is there a number that ends with four zeros? | 188. Let's leave in each member of the Fibonacci series, written with five or more digits, only the last four digits. We will get a sequence of numbers, each of which is less than $10^{4}$. Let's denote by $a_{k}$ the member of this sequence standing at the $k$-th place. Note that if we know $a_{k+1}$ and $a_{k}$, then... | 7501 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
195. Let's consider the following sequence of sets of (natural) numbers. The first set $I_{0}$ consists of
two ones 1,1. Then we insert their sum $1+1=2$ between these numbers; we get the set $I_{1}: 1,2,1$. Next, between every two numbers in the set $I_{1}$, we insert the sum of these numbers; we get the set $I_{2}: 1... | 195. First of all, note that each of the sets $I_{0}, I_{1}$, $I_{2}, \ldots$ is obtained from the preceding one by adding a certain number of new numbers; at the same time, all the numbers that were previously present are retained in the new set. It is easy to see, furthermore, that all the new numbers appearing in th... | 1972 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
200. Let $t$ be an arbitrary positive number; the number of irreducible fractions $\frac{p}{q}$, where the numerator $p$ and the denominator $q$ do not exceed $t$, is denoted by $d(t)$,
## What is the sum
$-S=d\left(\frac{100}{1}\right)+d\left(\frac{100}{2}\right)+d\left(\frac{100}{3}\right)+\ldots+d\left(\frac{100}{... | 200. To each irreducible fraction $\frac{p}{q}$, where
$$
0<p \leqslant 100, \quad 0<q \leqslant 100
$$
we associate a point $M$ with coordinates $p$, $q$; the inequalities ( ${ }^{*}$ ) indicate that the point $M$ is located inside the shaded square $\mathscr{K}=O A C B$ bounded by the coordinate axes and the lines ... | 10000 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
211. How many digits does the number $2^{100}$ have? | 211. It is easy to check that $2^{10}=1024$; thus, $2^{100}=$ $=1024^{10}$. Since $1000^{10}=10^{30}$ represents a number consisting of a one followed by 30 zeros, and $1024^{10}>1000^{10}$, the number $2^{100}=$ $=1024^{10}$ cannot have fewer than 31 digits. On the other hand,
$$
\begin{aligned}
\frac{1024^{10}}{1000... | 31 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
229*. Find the integer part of the number
$$
\frac{1}{\sqrt[3]{4}}+\frac{1}{\sqrt[3]{5}}+\frac{1}{\sqrt[3]{6}}+\ldots+\frac{1}{\sqrt[3]{1000000}}
$$ | 229. First, note that from the comparison of two expressions
$$
\left(1+\frac{1}{n}\right)^{2}=1+2 \frac{1}{n}+\frac{1}{n^{2}}
$$
and
$$
\left(1+\frac{2}{3} \frac{1}{n}\right)^{3}=1+2 \frac{1}{n}+\frac{4}{3} \frac{1}{n^{2}}+\frac{8}{27} \frac{1}{n^{3}}
$$
it follows that for each positive integer $n$
$$
\left(1+\f... | 14996 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
232. Prove that if in the sum
$$
1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\ldots+\frac{1}{n}
$$
all terms where the denominator contains the digit 9 are removed, then the sum of the remaining terms for any $n$ will be less than 80. | 232. Let $n_{k}$ denote the number of unstruck terms between $\frac{1}{10^{k}}$ and $\frac{1}{10^{k+1}}$, including $\frac{1}{10^{k}}$ but not $\frac{1}{10^{k+1}}$. If the term $-\frac{1}{q}$, located between $\frac{1}{10^{h-1}}$ and $\frac{1}{10^{h}}$, is not struck out, then among the terms $\frac{1}{10 q}, \frac{1}{... | 80 | Number Theory | proof | 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:
$$
x_{1} x_{2} x_{3} \ldots x_{n}=a_{1} a_{3} a_{5} \ldots a_{n-1} \text { and } x_{1} x_{2} x_{3} \ldots x_{n}=a_{2} a_{4} a_{6} \ldots a_{n}... | 0 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
257. a) How many roots does the equation
$$
\sin x=\frac{x}{100} ?
$$
b) How many roots does the equation
$$
\sin x=\lg x ?
$$ | 257. a) First of all, note that if $x_{0}$ is a root of the equation, then $-x_{0}$ is also a root. Therefore, the number of negative roots is the same as the number of positive roots. Furthermore, the number 0 is a root of the equation. Thus, it is sufficient to find the number of positive roots. Now, let's note that ... | 63 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
299. When dividing the polynomial $x^{1051}-1$ by $x^{4}+x^{3}+2 x^{2}+x+1$, a quotient and a remainder are obtained. Find the coefficient of $x^{14}$ in the quotient. | 299. The polynomial $x^{4}+x^{8}+2 x^{2}+x+1$ can be factored; it equals $\left(x^{2}+1\right)\left(x^{2}+x+1\right)$. From this, it is easy to see that this polynomial is a divisor of the polynomial
$$
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 |
7. In a bus without a conductor, there were 20 people. Although they only had coins worth 10, 15, and 20 kopecks, each of them paid for the fare and received the change they were due. How could this have happened? Prove that they had no fewer than 25 coins. (One ticket costs 5 kopecks.) | 7. Let's first prove that the passengers had no less than 25 coins. Indeed, each passenger should receive change, i.e., no fewer than twenty coins should remain in the hands of the passengers. One ruble was dropped into the cash box, so there must be no fewer than five coins (since the largest coin available is 20 kope... | 25 | Logic and Puzzles | 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 |
18. Prince Gvidon had three sons. Among his descendants, 93 each had two sons and no daughters, while all the others died childless. How many descendants did Prince Gvidon have in total? | 18. Each of the descendants who have children adds two more descendants to the three descendants of the first generation. Since 93 descendants had children, this added $2 \cdot 93 = 186$ descendants, making a total of $3 + 186 = 189$. This problem ad-
 | 25. Each time we break one piece, we get two smaller pieces, i.e., the total number of pieces increases by one. Initially, there was one piece. Therefore, we will have to break the chocolate bar 39 times.
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。 | 27. Let's put three matches on the table and arrange the coins in a row so that the coins of the first schoolboy lie before the first match, between the first and second - the coins of the second, between the second and third - the coins of the third, and finally, after the third match - the coins of the fourth schoolb... | 3276 | Number Theory | 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 |
32. On a $10 \times 10$ board for playing "Battleship," a four-cell "ship" $\square \square$ ( $\square$ is located. What is the minimum number of "shots" needed to hit the ship? (Indicate the method of delivering this number of shots and prove that with fewer shots, the ship can always be placed in such a way that it ... | 32. Let's color the fields of the board as shown in Fig. 31 (instead of colors, we will write letters: k - red, s - blue, 3 - green, y - yellow). A direct count shows that the board has 24 yellow, 26 blue, 25 green, and 25 red fields. - It is clear that each four-cell ship occupies exactly one cell of each color. There... | 24 | 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 |
74. 30 students from five courses came up with 40 problems for the olympiad, with students from the same course coming up with the same number of problems, and students from different courses coming up with a different number of problems. How many students came up with exactly one problem? | 74. Let's choose 5 students, one from each year. Each of them came up with a different number of problems. Therefore, the total number of problems proposed by them is no less than $1+2+3+4+5=15$. The remaining 25 students came up with no more than $40-15=25$ problems. It is clear that each of them came up with one prob... | 26 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
85. Bacteria have the following development rule: each one lives for 1 hour and every half hour it produces one new one (a total of two during its life). What will be the offspring of one bacterium 6 hours after its birth? | 85. Let's call bacteria "young" if their age does not exceed half an hour, and "old" if they have lived for more than half an hour. Let's take half an hour as the unit of time. Denote the number of bacteria at moment $n$ by $u_{n}$. At moment $n-1$, $u_{n-1}$ bacteria are born, so by moment $n$ there are $u_{n-1}$ youn... | 377 | Combinatorics | 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 |
99. There is a certain number of cards. Each card has a natural number written on it. For each $n \leqslant 1968$, there are exactly $n$ cards on which the numbers are divisors of $n$. Prove that the number $2^{10}=$ $=1024$ is written on at least one card. | 99. We will prove by induction the following statement: the number $2^{n}$ (where $n \leqslant 10)$ is written on exactly $2^{n-1}$ cards. For $n=1$, the statement is trivial. Consider the number $2^{n}$. Its divisors are written on $2^{n}$ cards. But the divisors of the number $2^{n}$ are it itself and the divisors of... | 1024 | Number Theory | proof | 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 |
Example 6.21. On average, $85 \%$ of the items coming off the conveyor are of the first grade. How many items need to be taken so that with a probability of 0.997, the deviation of the frequency of first-grade items from 0.85 in absolute value does not exceed 0.01? | Solution. Here $p=0.85, q=1-0.85=0.15, \varepsilon=0.01, P=0.997$, $n-$? Since in the equation $P\left|\frac{m}{n}-p\right|<\varepsilon \approx \Phi \varepsilon \sqrt{\frac{n}{p q}}$ the probability $P$ on the left is known, we first solve the equation $\Phi(t)=P$. Let $t_{P}$ be the root of this equation. Then $\varep... | 11475 | Other | math-word-problem | Yes | Yes | olympiads | false |
### 7.1.1. Optimal Production Task
For the production of two types of products I and II, three types of raw materials are used.
To produce one unit of product I, it is required to spend 13 kg of the first type of raw material, 32 kg of the second type of raw material, and 58 kg of the third type of raw material.
To ... | Solution. Let's consider the mathematical model of the problem. If we take $x_{1}$ as the number of items I planned for production and $x_{2}$ as the number of items II, we obtain a linear programming problem.
Subject to the constraints
$$
\begin{gathered}
13 x_{1}+24 x_{2} \leq 312 \\
32 x_{1}+32 x_{2} \leq 480 \\
5... | 54 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
6. Find the distance $d$ between points $A$ and $B$ in each of the following cases:
1) $A(2), B(3) ; 2) A(-4), B(-8)$. | Solution. Applying formula (2), we get:
1) $d=|3-2|=1 ; 2) d=|-8-(-4)|=4$. | 4 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
75. Write the equation of a line parallel to the $O x$ axis and cutting off a segment equal to -2 from the $O y$ axis. Check if the points $A(2; -2)$ and $B(3; 2)$ lie on this line. | Solution. According to the problem, the slope $k=0$, the initial ordinate $b=-2$, therefore, the equation of the line we are looking for is $y=-2$. It is easy to see that the coordinates of any point $M(x; -2)$ satisfy this equation.
Point $A$ lies on the line we are looking for, since its coordinates satisfy the equa... | -2 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
312. Find the scalar square of the vector $\bar{a}=2 \bar{i}-\bar{j}-2 \bar{k}$ and its length. | Solution. We will use formulas (10), (13), and (14):
$$
(\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
$$ | -6 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
314. Find the angle $\varphi$ between two vectors
$$
\bar{a}=\bar{i}+\bar{j}-4 \bar{k} \text { and } \bar{b}=\bar{i}-2 \bar{j}+2 \bar{k} .
$$ | Solution. We will use formula (16):
$$
\cos \varphi=\frac{\bar{a} \bar{b}}{a b}=\frac{1-2-8}{\sqrt{1+1+16} \sqrt{1+4+4}}=-\frac{\sqrt{2}}{2},
$$
therefore, $\varphi=135^{\circ}$. | 135 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
626. Find $f^{\prime}(0)$, if $f(x)=e^{x} \arcsin x+\operatorname{arctg} x$. | Solution. $\quad f^{\prime}(x)=\left(e^{x} \arcsin x\right)^{\prime}+(\operatorname{arctg} x)^{\prime}=\left(e^{x}\right)^{\prime} \arcsin x+$ $+e^{x}(\arcsin x)^{\prime}+(\operatorname{arctg} x)^{\prime}=e^{x} \arcsin x+\frac{e^{x}}{\sqrt{1-\lambda^{2}}}+$ $+\frac{1}{x^{2}+1}=e^{x} \frac{1+\sqrt{1-x^{2}} \arcsin x}{\s... | 2 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
976. Find the following limits:
1) $\lim _{\substack{x \rightarrow 2 \\ y \rightarrow 0}} \frac{\tan x y}{y}$
2) $\lim _{\substack{x \rightarrow 0 \\ y \rightarrow 0}} \frac{x+y}{x}$. | Solution. 1) $\lim _{\substack{x \rightarrow 2 \\ y \rightarrow 0}} \frac{\operatorname{tg} x y}{y}=\lim _{\substack{x \rightarrow 2 \\ y \rightarrow 0}} x \frac{\operatorname{tg} x y}{x y}=2 \cdot 1=2$,
since
$$
\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.
$$
from which $x=0, y=2$.
The function at the critical point $M_{0}(0 ... | 0 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
1287. Calculate the area of the figure bounded by the parabola $y=-x^{2}+$ $+6 x-5$ and the coordinate axes. | Solution. If the figure is located on opposite sides of the $O x$ axis (Fig. $82, a$), then the area $S$ should be calculated using the formula
$$
S=\int_{a}^{c} f(x) d x+\left|\int_{c}^{b} f(x) d x\right|
$$
 d \bar{x}$ using the Newton-Leibniz formula and approximate formulas of rectangles and trapezoids, dividing the interval of integration into 6 equal parts $(n=6)$. Find the absolute and relative errors of the results obtained by the approximate formulas. | Solution. By the Newton-Leibniz formula
$$
I=\int_{0}^{6}\left(x^{2}+3\right) d x=\left.\left(\frac{x^{3}}{3}+3 x\right)\right|_{0} ^{6}=90
$$
The interval $[0,6]$ is divided into 6 equal parts, and we construct a table of values of the integrand function $y=x^{2}+3$ :
$$
\begin{array}{lllr}
x_{0}=0 & y_{0}=3 & x_{4... | 90 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
1429. Check that the series $\sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{\sqrt{n}}$ converges. How many terms of this series need to be taken to compute its sum with an accuracy of 0.01? | Solution. For this series, all conditions of the Leibniz criterion are satisfied, so the series converges. We can write the sum of the series in the form: $S=S_{n}+r_{n},\left|r_{n}\right|<\left|a_{n+1}\right|$. We need to find such an $n$ that $\left|a_{n+1}\right|=\frac{1}{\sqrt{n+1}} \leqslant 0.01$. Solving the equ... | 9999 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
1586. The share of premium quality items in this enterprise is $31 \%$. What is the most probable number of premium quality items in a randomly selected batch of 75 items? | Solution. It is known that $p=0.31, n=75$, and $q=1-p=$ $=0.69$. We form the double inequality (1):
$$
\begin{gathered}
75 \cdot 0.31 - 0.69 \leqslant m_{0} \leqslant 75 \cdot 0.31 + 0.31 ; \\
22.56 \leqslant m_{0} \leqslant 23.56 .
\end{gathered}
$$
From this, it follows that $m_{0}=23$. | 23 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
1615. On average, $85 \%$ of the items coming off the conveyor are of the first grade. How many items need to be taken so that with a probability of 0.997, the deviation of the frequency of first-grade items from 0.85 in absolute value does not exceed 0.01? | Solution. Here $p=0.85, q=1-0.85=0.15, \varepsilon=0.01$, $P=0.997, n$-? Since in the equality $P\left(\left|\frac{m}{n}-p\right|<\varepsilon\right) \approx$ $\approx \Phi\left(\varepsilon \sqrt{\frac{n}{p q}}\right)$ the probability $P$ on the left is known, we first solve the equation $\Phi(t)=P$. Let $t_{P}$ be the ... | 11475 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
1702. How many times does one need to measure a given quantity, the true value of which is equal to $a$, to be able to state with a probability of at least 0.95 that the arithmetic mean of these measurements differs from $a$ in absolute value by less than 2, if the root mean square deviation of each measurement is less... | Solution. Let $X_{i}$ be the result of the $i$-th measurement. From the condition of the problem, it follows that $E X_{i}=a, D X_{i}<10^{2}=100$. Therefore,
$$
\frac{E X_{1}+E X_{2}+\ldots+E X_{n}}{n}=\frac{a+a+\ldots+a}{n}=a
$$
We need to find $n$ such that
$$
P\left(\left|\frac{X_{1}+X_{2}+\ldots+X_{n}}{n}-a\righ... | 500 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
1737. Let the dairy farm of a collective farm have data on the milk yield of cows for the lactation period:
| Milk yield, kg | Number of cows | Milk yield, kg | Number of cows |
| :---: | :---: | :---: | :---: |
| $400-600$ | 1 | $1600-1800$ | 14 |
| $600-800$ | 3 | $1800-2000$ | 12 |
| $800-1000$ | 6 | $2000-2200$ | ... | Solution. The arithmetic mean will be found using the simplified formula
$$
\bar{X}=k \bar{U}+x_{0} \quad\left(U=\frac{X-x_{0}}{k}\right)
$$
by setting $x_{0}=1500$ and $k=200$. All necessary calculations are presented in the following table:
| Interval $X_{i}^{\prime}-X_{i}^{\prime \prime}$ | Frequency $m_{i}$ | M... | 1560 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
1776. What should be the sample size to ensure that when determining the percentage of non-standard items, the maximum error $\varepsilon=5\%$ is achieved with a confidence probability $P=0.99896$. | Solution. Given the confidence probability $P=$ $=0.99896$, we find the argument $t$ of the Laplace function $\Phi(t)$, for which $\Phi(t)=0.99896 ; t=3.28$. Therefore, the sample size, at which with a probability of 0.99896 it can be stated that the sample proportion differs from the population proportion in absolute ... | 1076 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
1777. It is planned to conduct a selective survey of 5000 light bulbs to determine the average duration of their burning. What should be the volume of the non-repeated sample to ensure with a probability of 0.9802 that the general average differs from the sample average by less than 15 hours in absolute value, if the g... | Solution. According to the table (Appendix 2), we find that the argument $t=2.33$ corresponds to the given confidence probability of 0.9802. For sampling without replacement, the sample size is determined by the formula
$$
n \approx \frac{N t^{2} \sigma_{0}^{2}}{N \varepsilon^{2}+t^{2} \sigma_{0}^{2}}
$$
In our case ... | 230 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
1.7. Convert to polar coordinates and evaluate the double integrals.
a) $\iint_{\Omega} x y^{2} d x d y$, where the region $\Omega$ is bounded by the circles $x^{2}+(y-1)^{2}=1$ and $x^{2}+y^{2}=4 y$
b) $\iint_{\Omega} e^{-x^{2}-y^{2}} d x d y$, where $\Omega-$ is the circle $x^{2}+y^{2} \leqslant R^{2}$. | Solution.
a) First, let's depict the region $\Omega$ in the Cartesian coordinate system (Fig. 1.32).
The equations of the boundary circles in polar coordinates are obtained after the substitution $x=\rho \cos \varphi, y=\rho \sin \varphi$:
$$
\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:
$$
\begin{gathered}
S=\left|\int_{0}^{1}... | 1 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
Example 2.58. Compute the length of the arc of the astroid $x=\cos ^{3} t$, $y=\sin ^{3} t, 0 \leqslant t \leqslant 2 \pi$. | Solution. From Fig. 2.9, it is clear that due to symmetry, it is sufficient to compute only a quarter of the astroid for
Fig. 2.9.
$0 \leqslant t \leqslant \pi / 2$. We will use formula (2... | 12 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
Example 2.59. Calculate the length of the arc of the cardioid $\rho=$ $=2(1+\cos \varphi), 0 \leqslant \varphi \leqslant 2 \pi$ (see Fig. 2.7). | Solution. We will use formula (2.28):
$$
\begin{aligned}
L & =\int_{0}^{2 \pi} \sqrt{(2(1+\cos \varphi))^{2}+(-2 \sin \varphi)^{2}} d \varphi=\int_{0}^{2 \pi} \sqrt{16 \cos ^{2}(\varphi / 2)} d \varphi= \\
& =4 \int_{0}^{2 \pi}|\cos (\varphi / 2)| d \varphi=4 \cdot 2 \int_{0}^{\pi} \cos (\varphi / 2) d \varphi=\left.1... | 16 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
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