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Example 5. How many different permutations of letters can be made from the words: zamok, rotor, topor, kolokol? | Solution. In the word "замок" (lock), all letters are different, and there are five of them. According to formula (1.3.1), we get
$$
P_{5}=5!=1 \cdot 2 \cdot 3 \cdot 4 \cdot 5=120
$$
In the word "ротор" (rotor), consisting of five letters, the letters $p$ and о are repeated twice. To count the different permutations,... | 210 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
Example 5. When shooting at a target, the frequency of hits $W=0.75$. Find the number of hits in 40 shots. | Solution. From formula (1.4.1), it follows that $m=W n$. Since $W=0.75, n=40$, then $m=0.75 \cdot 40=30$. Therefore, 30 hits were obtained. | 30 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
Example 6. The frequency of normal seed germination $W=0.97$. Out of the sown seeds, 970 germinated. How many seeds were sown? | Solution. From formula (1.4.1), it follows that $n=\frac{m}{W}$. Since $m=970, \quad W=0.97$, then $n=970 / 0.97=1000$. Therefore, 1000 seeds were sown. | 1000 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
Example 22. How many times do you need to roll two dice so that the probability of rolling at least one double six is greater than $1 / 2$? (This problem was first posed by the French mathematician and writer de Mere ( $1610-1684$ ), hence the problem is named after him). | Solution. Let the event $A_{i}$ be "rolling two sixes on the $i$-th throw". Since any of the six faces of the first die can match any of the six faces of the second die, there are $6 \cdot 6=36$ equally likely and mutually exclusive events. Only one of these - rolling a six on both the first and the second die - is fav... | 25 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
Example 5. The distribution law of a discrete random variable is given by the following table:
| $X$ | 0 | 1 | 2 | 3 |
| :---: | :---: | :---: | :---: | :---: |
| $P$ | 0.2 | 0.4 | 0.3 | 0.1 |
Find the distribution function of this random variable. | Solution. To construct the distribution function $F(x)$ of a discrete random variable $X$, we use formula (2.2.11).
1. For $x \leq 0 \quad F(x)=\sum_{x_{k}3 \quad F(x)=P(X=0)+P(X=1)+P(X=2)+P(X=3)=$
$$
=0.2+0.4+0.3+0.1=1
$$
The graph of the function $F(x)$ is shown in Fig. 2.7. | 1 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
Example 4. The mathematical expectations of two independent random variables $X$ and $Y$ are known: $M(X)=4, M(Y)=5$.
Find the mathematical expectation of their product. | Solution. Applying formula (2.4.15), we find
$$
M(X \cdot Y)=M(X) \cdot M(Y)=4 \cdot 5=20
$$ | 20 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
Example 5. Find the mathematical expectation of the random variable $Y=2X+7$, given that $M(X)=4$. | Solution. Using formulas (2.4.10), (2.4.11), (2.4.12), we find
$$
\begin{aligned}
& M(Y)=M(2 X+7)=M(2 X)+M(7)= \\
& =2 M(X)+7=2 \cdot 4+7=15
\end{aligned}
$$ | 15 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
Example 7. Two dice are rolled. The discrete random variable $X$ is the sum of the points that appear on both dice. Find the mathematical expectation of this random variable. | Solution. This random variable takes all integer values from 2 to 12. The distribution law can be given by the following table:
| $X$ | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| $P$ | $\frac{1}{36}$ | $\frac{2}{36}... | 7 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
Example 13. Find the mathematical expectation of a random variable $X$, the distribution function of which is given by
$$
F(x)= \begin{cases}0 & \text { if } x \leq -a \\ \frac{(a+x)^{2}}{2 a^{2}} & \text { if } -a < x \leq a \\ 1 & \text { if } x > a\end{cases}
$$ | Solution. First, let's find the probability density function of this random variable. Since \( p(x) = F'(x) \), we have:
\[
p(x)= \begin{cases}0 & \text{if } x \leq -a \\ \frac{1}{a}\left(1+\frac{x}{a}\right) & \text{if } -a < x \leq 0 \\ \frac{1}{a}\left(1-\frac{x}{a}\right) & \text{if } 0 < x \leq a \\ 0 & \text{if ... | 0 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
Example 8. The average germination rate of seeds is $80 \%$. Find the most probable number of germinated seeds among nine seeds. | Solution. The number $k_{0}$ will be determined using inequalities (3.1.5). Since $n=9, p=0.8$, and $q=0.2$, then $9 \cdot 0.8 - 0.2 \leq k_{0} \leq 9 \cdot 0.8 + 0.8 = 8$. An integer is obtained; hence, there are two most probable numbers of germinated seeds: 8 and 7. Their probabilities are the highest and equal to e... | 78 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
Example 2. Prove that
$$
\sum_{k=0}^{\infty} p_{k}=\sum_{k=0}^{\infty} \frac{a^{k} e^{-a}}{k!}=1
$$ | Solution. Taking into account the power series expansion of the function $f(x)=e^{x}$
$$
e^{x}=1+\frac{x}{1!}+\frac{x^{2}}{2!}+\frac{x^{3}}{3!}+\ldots+\frac{x^{k}}{k!}+\ldots
$$
and the resulting equality
$$
e^{a}=1+\frac{a}{1!}+\frac{a^{2}}{2!}+\frac{a^{3}}{3!}+\ldots+\frac{a^{k}}{k!}+\ldots
$$
we obtain
$$
\sum_... | 1 | Algebra | proof | Yes | Yes | olympiads | false |
Example 12. A factory sent 5000 good-quality items to a base. The probability that an item will be damaged during transportation is 0.0002. What is the probability that 3 defective items will arrive at the base?
$\mathrm{Pe} \mathrm{sh} \mathbf{e n i e . ~ F r o m ~ t h e ~ c o n d i t i o n ~ i t ~ f o l l o w s ~ t ... | Solution. Here it is required to find the probabilities: 1) $P_{1000}(2)$;
2) $P_{1000}(k \geq 2)$. According to the condition, $n=1000, p=0.001, a=n p=1000 \cdot 0.001=1$.
The probability of exactly two elements failing:
$$
P_{1000}(2)=\frac{a^{2}}{2!} e^{-a}=\frac{1}{2 e} \approx 0.1831
$$
The probability of at l... | 0 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
Example 9. Find the mathematical expectation of a random variable $X$,
uniformly distributed on the interval $[2,8]$. | Solution. The mathematical expectation of a random variable $X$, uniformly distributed on the interval $[\alpha, \beta]$, is defined by formula (3.4.7). Since in this case $\alpha=2, \beta=8$, then
$$
M(X)=\frac{8+2}{2}=5
$$ | 5 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
Example 1. Prove that the function (3.5.1), defining the density of the normal distribution, satisfies condition (2.3.6), i.e.
$$
\int_{-\infty}^{+\infty} p(x) d x=1
$$ | Solution. In the integral
$$
\int_{-\infty}^{+\infty} p(x) d x=\frac{1}{\sigma \sqrt{2 \pi}} \int_{-\infty}^{+\infty} e^{-(x-a)^{2} / 2 \sigma^{2}} d x
$$
we will transition to a new variable \( t \) using the formula
$$
t=\frac{x-a}{\sigma}
$$
Then \( x=a+\sigma t, d x=\sigma d t \). Since the new limits of integr... | 1 | Calculus | proof | Yes | Yes | olympiads | false |
Example 14. A machine manufactures bearings, which are considered suitable if the deviation $X$ from the design size in absolute value does not exceed 0.77 mm. What is the most probable number of suitable bearings out of 100, if the random variable $X$ is normally distributed with the parameter $\sigma=0.4$ mm? | Solution. First, we find the probability of deviation using formula (3.5.4) with $\delta=0.77$ and $\sigma=0.4$:
$$
P(|X-a|<0.77)=2 \Phi\left(\frac{0.77}{0.4}\right) \approx 2 \Phi(1.93)=2 \cdot 0.473197=0.946394
$$
Approximating $p=0.95$ and $q=0.05$, according to formula (3.1.5), i.e., $n p-q \leq k_{0} \leq n p+p$... | 95 | Other | math-word-problem | Yes | Yes | olympiads | false |
Example 4. Prove that the function (3.6.2) satisfies condition (2.3.6), i.e. $\int_{-\infty}^{+\infty} p(x) d x=1$. | Solution. Indeed,
$$
\int_{-\infty}^{+\infty} p(x) d x=\int_{-\infty}^{0} p(x) d x+\int_{0}^{+\infty} p(x) d x=\int_{-\infty}^{0} 0 \cdot d x+\int_{0}^{+\infty} \alpha e^{-\alpha x} d x=
$$
$$
=-\int_{0}^{+\infty} d\left(e^{-\alpha x}\right)=-\left.e^{-\alpha x}\right|_{0} ^{+\infty}=-\left(e^{-\infty}-e^{0}\right)=1... | 1 | Calculus | proof | Yes | Yes | olympiads | false |
Example 11. Find the mathematical expectation of the random variable
$X$, the density of which is determined by the function $p(x)=0.2 e^{-0.2 x}$ for $x \geq 0$. | Solution. Since in this case $\alpha=0.2$ and $M(X)=\frac{1}{\alpha}$, then
$$
M(X)=\frac{1}{2 / 10}=\frac{10}{2}=5, \quad M(X)=5
$$ | 5 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
Example 3. For what number of independent trials will the probability of the inequality $\left|\frac{m}{n}-p\right|<0.2$ being satisfied exceed 0.96, if the probability of the event occurring in a single trial $p=0.7$? | Solution. According to the problem, we have: $\varepsilon=0.2, p=0.7$, so $q=0.3$; it is required to determine $n$ using inequality (4.2.5). The condition $P>0.96$ is equivalent to the inequality
$$
\frac{p q}{n \varepsilon^{2}} < 0.04
$$
or
$$
n > \frac{p q}{\varepsilon^{2} \cdot 0.04}
$$
Substituting the values $... | 132 | Inequalities | math-word-problem | Yes | Yes | olympiads | false |
Example 6. Determine how many measurements of the cross-sectional diameter of trees need to be made on a large plot so that the average diameter of the trees differs from the true value $a$ by no more than 2 cm with a probability of at least 0.95. The standard deviation of the cross-sectional diameter of the trees does... | Solution. We will consider the selection of trees for measurements such that the measurement results can be regarded as independent random variables. Let $X_{i}$ denote the measurement result of the cross-section of the $i$-th tree. According to the problem, $\sigma\left(X_{i}\right)=\sqrt{D\left(X_{i}\right)} \leq 10$... | 500 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
Example 11. On average, $85 \%$ of the items coming off the conveyor are of the first grade. How many items need to be taken so that with a probability of 0.997, the deviation of the frequency of first-grade items from the probability $p=0.85$ in absolute value does not exceed $0.01?$ | Solution. We will use the formula
$$
P\left(\left|\frac{m}{n}-p\right| \leq \varepsilon\right)=2 \Phi\left(\varepsilon \sqrt{\frac{n}{p q}}\right)
$$
From the condition, it follows that $p=0.85, q=1-0.85=0.15, \varepsilon=0.01$, $P=0.997$. We need to determine the value of $n$. Since
$$
2 \Phi\left(\varepsilon \sqrt... | 11171 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
Example 1. Find the zeros of the function $f(z)=e^{z}-1-z$ and determine their order. | Solution.
1. Find the zeros of the function $f(z)$ by solving the equation $e^{z}-1-z=0$. We get $z=0$.
2. Determine the order of the obtained zero $z=0$. For this, we use the Taylor series expansion of the function $f(z)$ in powers of $z$:
$$
e^{z}-1-z=\left(1+z+\frac{z^{2}}{2!}+\frac{z^{3}}{3!}+\ldots\right)-1-z=\f... | 0 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
Example 2. How to find out how many fish are in the pond?
We catch $n_{a}=20$ fish, mark them, and release them back into the pond. After some time, sufficient for the marked fish to disperse throughout the pond, we catch $m=50$ fish. Suppose that among them, $k_{1}=7$ are marked. Determine the number of fish in the p... | ## Solution.
1. The random variable $\xi$ - the number of marked fish among $m$ caught - is determined by the probabilities
$$
\mathrm{P}(\xi=k)=p(k ; N)=\frac{C_{n_{a}}^{k} C_{N-n_{a}}^{m-k}}{C_{N}^{m}}=\frac{C_{20}^{k} C_{N-20}^{50-k}}{C_{N}^{50}}(k=0,1, \ldots, 50)
$$
2. We define the likelihood function
$$
L(N)... | 142 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
1.7. Calculate the determinant
$$
\Delta=\left|\begin{array}{cccc}
3 & 1 & -1 & 2 \\
-3 & 1 & 4 & -5 \\
2 & 0 & 1 & -1 \\
3 & -5 & 4 & -4
\end{array}\right|
$$ | S o l u t i o n. Notice that the second column of the determinant already contains one zero element. Add to the elements of the second row the elements of the first row multiplied by -1, and to the elements of the fourth row - the elements of the first row multiplied by 5. We get:
$$
\Delta=\left|\begin{array}{cccc}
3... | 40 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
1.9. Using the property that the determinant of the matrix $C=A \cdot B$, which represents the product of square matrices $A$ and $B$ of the same order, is equal to the product of the determinants of matrices $A$ and $B$, i.e., $\operatorname{det} C=\operatorname{det} A \cdot \operatorname{det} B$, calculate the determ... | Solution. We will find the determinants of matrices $A$ and $B$ and then multiply them. We have
$$
\operatorname{det} A=\left|\begin{array}{lll}
3 & 2 & 5 \\
0 & 2 & 8 \\
4 & 1 & 7
\end{array}\right|=\left|\begin{array}{ccc}
3 & 2 & -3 \\
0 & 2 & 0 \\
4 & 1 & 3
\end{array}\right|=2 \cdot(-1)^{2+2}\left|\begin{array}{c... | 2142 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
1.13. Using the method of bordering minors, find the rank of the matrix
$$
A=\left(\begin{array}{ccccc}
2 & 1 & 2 & 1 & 2 \\
1 & 1 & 5 & -2 & 3 \\
-1 & 0 & -4 & 4 & 1 \\
3 & 3 & 8 & 1 & 9
\end{array}\right)
$$ | Solution. The second-order minor
$$
M_{2}=\left|\begin{array}{ll}
2 & 1 \\
1 & 1
\end{array}\right|=2 \cdot 1-1 \cdot 1=1 \neq 0
$$
is non-zero and located in the upper left corner of matrix $A$. The third-order minor
$$
M_{3}=\left|\begin{array}{ccc}
2 & 1 & 2 \\
1 & 1 & 5 \\
-1 & 0 & 4
\end{array}\right|=\left|\be... | 3 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
1.14. Using elementary transformations, find the rank of the matrix
$$
A=\left(\begin{array}{ccccc}
5 & 7 & 12 & 48 & -14 \\
9 & 16 & 24 & 98 & -31 \\
14 & 24 & 25 & 146 & -45 \\
11 & 12 & 24 & 94 & -25
\end{array}\right)
$$ | Solution. Let's denote the $i$-th row of matrix $A$ by the symbol $\alpha_{i}$. In the first stage, we perform elementary transformations $\alpha_{2}^{\prime}=\alpha_{2}-\alpha_{3}+\alpha_{1}, \alpha_{3}^{\prime}=\alpha_{3}-\alpha_{2}-\alpha_{1} ; \alpha_{4}^{\prime}=\alpha_{4}-\alpha_{3}+\alpha_{1}$.
In the second st... | 3 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
1.27. Given a linear transformation using a system of equations
$$
\left\{\begin{array}{l}
g_{1}=-x_{1}+0 \cdot x_{2}+0 \cdot x_{3} \\
g_{2}=0 \cdot x_{1}+1 \cdot x_{2}+0 \cdot x_{3} \\
g_{3}=0 \cdot x_{1}+0 \cdot x_{2}+1 \cdot x_{3}
\end{array}\right.
$$
Find the transformation matrix, will it be singular? | Solution. With the matrix representation of the transformation, we get:
$$
\left(\begin{array}{l}
g_{1} \\
g_{2} \\
g_{3}
\end{array}\right)=\left(\begin{array}{ccc}
-1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{array}\right) \cdot\left(\begin{array}{l}
x_{1} \\
x_{2} \\
x_{3}
\end{array}\right)
$$
Here, the transformati... | -1 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
2.13. Are the vectors coplanar
$$
\vec{a}, \vec{b}, \vec{c}: \vec{a}=\{1,-2,1\}, \vec{b}=\{3,1,-2\}, \vec{c}=\{7,14,-13\} ?
$$ | Solution. To answer the given question, it is necessary to calculate the mixed product of these vectors, and if it turns out to be zero, this will indicate that the vectors $\ddot{a}, \vec{b}, \vec{c}$ are coplanar. We find the mixed product $\vec{a} \cdot \vec{b} \cdot \vec{c}$ using formula (2.40).
$$
\begin{aligned... | 0 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
3.3. Obtain the equation of the line passing through two given points $P_{0}(1 ; 2)$ and $P_{1}(3 ; 2)$. | Solution.
1st step. Make a schematic drawing (Fig. 3.3).
2nd step. Write down the coordinates of the vectors defining the given geometric object - the line. This is the current vector lying on the line $\overrightarrow{P_{0} M}=\{x-1 ; y-2\}$ and the vector formed by two points on the line
$$
\overrightarrow{P_{0} P... | 2 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
3.23. Derive the polar equation of the ellipse $x^{2} / a^{2}+$ $+y^{2} / b^{2}=1$ under the condition that the direction of the polar axis coincides with the positive direction of the x-axis, and the pole is at the center of the ellipse. | Solution. Substitute into the ellipse equation the expressions for $x$ and $y$ in terms of the polar radius $\rho$ and the angle $\varphi$.
We have
$$
\frac{\rho^{2} \cos ^{2} \varphi}{a^{2}}+\frac{\rho^{2} \sin ^{2} \varphi}{b^{2}}=1, \rho^{2} \frac{b^{2} \cos ^{2} \varphi+a^{2} \sin ^{2} \varphi}{a^{2} b^{2}}=1
$$
... | 5 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
4.2. Two planes are given by their general equations:
$$
\begin{aligned}
& 5 x-4 y+z-1=0 \\
& 10 x+2 y+4 z-7=0
\end{aligned}
$$
Determine the magnitude of the dihedral angle between these planes. | Solution. The dihedral angle between planes is measured, as is known, by a linear angle, and the latter is equal to the angle between vectors perpendicular to the planes. The vector perpendicular to the first plane has coordinates $\vec{n}_{1}=(5,-4,1)$, to the second plane $\ddot{n}_{2}=(10,2,4)$. Let's use the formul... | 52 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
5.2. Find the limit of the sequence
$$
\lim _{n \rightarrow \infty} \frac{\sqrt{2 n-1}-\sqrt[3]{n^{3}+10}}{\sqrt{n^{2}+1}-\sqrt[4]{n+1}}
$$ | Solution. Divide the numerator and the denominator of the fraction by $n$. After transformations and discarding infinitely small quantities, we obtain the required result:
$$
\begin{gathered}
\lim _{n \rightarrow \infty} \frac{\sqrt{2 n-1}-\sqrt[3]{n^{3}+10}}{\sqrt{n^{2}+1}-\sqrt[4]{n+1}}=\lim _{n \rightarrow \infty} ... | -1 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
5.9. Calculate the limit
$$
\lim _{x \rightarrow \pm \infty}\left(\sqrt{x^{2}+1}-\sqrt{x^{2}-1}\right)
$$ | Solution. In this case, we have an indeterminate form of type «$\infty - \infty$». To resolve it, we multiply and divide the given expression by its conjugate. We get
$$
\begin{gathered}
\lim _{x \rightarrow \pm \infty}\left(\sqrt{x^{2}+1}-\sqrt{x^{2}-1}\right)=\lim _{x \rightarrow \pm \infty} \frac{x^{2}+1-x^{2}+1}{\... | 0 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
5.10. Calculate the limit
$$
\lim _{x \rightarrow 0} \frac{\tan 6 x}{\sin 3 x}
$$ | Solution. To calculate this limit, which represents an indeterminate form of the type "0/0", it is best to use the theorem on the replacement of infinitesimal functions with equivalent quantities.
According to this theorem, as $x \rightarrow 0, \operatorname{tg} 6 x \sim 6 x$, $\sin 3 x \sim 3 x$. Taking this into acc... | 2 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
### 5.11. Compute the limit
$$
\lim _{x \rightarrow 0} \frac{1+\sin 2 x-\cos 2 x}{1-\sin 2 x-\cos 2 x}
$$ | Solution. Under the limit sign, we have an indeterminate form of type "0/0". To resolve this, we will use the known trigonometric formulas:
$$
\sin 2 x=2 \sin x \cos x, \quad 1-\cos 2 x=2 \sin ^{2} x
$$
Taking this into account, we get
$$
\begin{aligned}
& \lim _{x \rightarrow 0} \frac{1+\sin 2 x-\cos 2 x}{1-\sin 2 ... | -1 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
### 5.13. Calculate
$$
\lim _{x \rightarrow \pi}(\pi-x) \cot x
$$ | Solution. The given limit represents an indeterminate form of type «0 $\cdot \infty »$. To resolve this, we introduce a new variable $\pi-x=t$. Then
$\lim _{x \rightarrow \pi}(\pi-x) \operatorname{ctg} x=\lim _{t \rightarrow 0} t \operatorname{ctg}(\pi-t)=-\lim _{t \rightarrow 0} t \operatorname{ctg} t=-\lim _{t \righ... | -1 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
5.18. Calculate the limit
$$
\lim _{x \rightarrow \infty}\left(\frac{4 x^{2}-x+1}{2 x^{2}+x+1}\right)^{\frac{x^{3}}{2-x}}
$$ | Solution. This limit is not an indeterminate form, since when
$$
x \rightarrow \infty \frac{4 x^{2}-x+1}{2 x^{2}+x+1} \rightarrow 2, \text { and } \frac{x^{3}}{2-x} \sim -x^{2} \rightarrow -\infty
$$
Therefore:
$$
\begin{gathered}
\lim _{x \rightarrow \infty}\left(\frac{4 x^{2}-x+1}{2 x^{2}+x+1}\right)^{\frac{x^{3}}... | 0 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
5.22. Let
$$
f(x)=\left\{\begin{array}{l}
e^{3 x}, \text { if } x<0, \\
a+5 x, \text { if } x \geqslant 0
\end{array}\right.
$$
For which choice of the number $a$ will the function $f(x)$ be continuous? | The problem is solved. Let's write down the condition for the continuity of the function at the point \( x=0 \), where it is currently not continuous due to the arbitrariness of the number \( a \). We have
\[
\lim _{x \rightarrow 0-0} f(x)=\lim _{x \rightarrow 0+0} f(x)=f(0)
\]
In this case,
\[
\begin{aligned}
& \li... | 1 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
### 5.24. Investigate the function
$$
f(x)=\frac{5^{1 / x}}{1+5^{1 / x}}
$$
for continuity and determine the nature of the points of discontinuity. | Solution. The function is defined everywhere except at the point $x=0$. Let's investigate the behavior of the function in the neighborhood of the point $x=0$. We will find the one-sided limits
$$
\lim _{x \rightarrow 0-0} f(x)=\lim _{x \rightarrow 0-0} \frac{5^{1 / x}}{1+5^{1 / x}}=0
$$
since as $x \rightarrow 0-0$, ... | 0 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
6.14. Find the derivative of the function
$$
f(x)=\left\{\begin{array}{l}
\operatorname{arctg}\left(x^{2} \cdot \sin (1 /(5 x))\right), \quad x \neq 0, \\
0, x=0
\end{array}\right.
$$
and compute its value at the point $x_{0}=0$. | Solution. Let's find the derivatives for $x \neq 0$. We have
$$
\begin{gathered}
f^{\prime}(x)=\frac{1}{1+\left(x^{2} \cdot \sin (1 / 5 x)\right)^{2}} \cdot\left(x^{2} \cdot \sin (1 /(5 x))\right)^{\prime}= \\
=\frac{1}{1+\left(x^{2} \cdot \sin (1 /(5 x))\right)^{2}} \cdot\left(\left(x^{2}\right)^{\prime} \cdot \sin (... | 0 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
6.28. Determine the angle at which the graph of the curve $f(x)=e^{x}-x$ intersects the y-axis. | Solution. $f_{x}^{\prime}(x)=e^{x}-1, f_{x}^{\prime}(0)=e^{0}-1=0$, hence the angle of intersection is $\alpha=0^{\circ}$. | 0 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
6.31. Find the first differential of the function $y(x)=$ $=e^{3 x} \ln \left(1+x^{2}\right)$ and calculate its value at $x=0, d x=$ $=\Delta x=0.1$. | Solution. The first method is based on the direct application of the formula $d y(x)=y^{\prime}(x) d x$.
We have
$$
y^{\prime}(x)=3 e^{3 x} \cdot \ln \left(1+x^{2}\right)+\frac{e^{3 x} \cdot 2 x}{1+x^{2}}
$$
hence
$$
d y(x)=\left(3 e^{3 x} \cdot \ln \left(1+x^{2}\right)+\frac{e^{3 x} \cdot 2 x}{1+x^{2}}\right) \cdo... | 0 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
6.50. $\lim _{x \rightarrow 1} \frac{x^{4}-1}{\ln x}$. | Solution. Let's check the conditions for applying L'Hôpital's rule: 1) there is an indeterminate form
$$
\left.\frac{x^{4}-1}{\ln x}\right|_{x=1}=\frac{0}{0}
$$
2) the functions $x^{4}-1$ and $\ln x$ are differentiable in a neighborhood of the point $\left.x=1 ; 3)(\ln x)_{x=1}^{\prime} \neq 0 ; 4\right)$ the limit e... | 4 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
6.52. $\lim _{x \rightarrow+\infty} \frac{\ln x}{x}$. | Solution.
$$
\lim _{x \rightarrow+\infty} \frac{\ln x}{x}=(\infty / \infty)=\lim _{x \rightarrow+\infty} \frac{(\ln x)^{\prime}}{(x)^{\prime}}=\lim _{x \rightarrow+\infty} \frac{1 / x}{1}=\lim _{x \rightarrow+\infty} \frac{1}{x}=0
$$ | 0 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
6.53. $\lim _{x \rightarrow \infty} \frac{x}{e^{x}}$. | Solution.
$$
\lim _{x \rightarrow \infty} \frac{x}{e^{x}}=(\infty / \infty)=\lim _{x \rightarrow \infty} \frac{(x)^{\prime}}{\left(e^{x}\right)^{\prime}}=\lim _{x \rightarrow \infty} \frac{1}{e^{x}}=0
$$
## INDETERMINACY OF THE FORM «0 $\cdot$ | 0 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
6.54. $\lim _{x \rightarrow \pi / 2}(x-\pi / 2) \cdot \tan x$. | Solution.
$$
\begin{aligned}
& \lim _{x \rightarrow \pi / 2}(x-\pi / 2) \cdot \tan x=(0 \cdot \infty)=\lim _{x \rightarrow \pi / 2} \frac{x-\pi / 2}{\cot x}= \\
& =(0 / 0)=\lim _{x \rightarrow \pi / 2} \frac{(x-\pi / 2)'}{(\cot x)'}=\lim _{x \rightarrow \pi / 2} \frac{1}{-\frac{1}{\sin ^{2} x}}=-1
\end{aligned}
$$ | -1 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
6.55. $\lim _{x \rightarrow 1+0}(\ln x \cdot \ln (x-1))$.
6.55. $\lim _{x \rightarrow 1+0}(\ln x \cdot \ln (x-1))$.
(No change needed as the text is already in English and contains mathematical notation which is universal.) | Solution.
$$
\begin{aligned}
& \lim _{x \rightarrow 1+0}(\ln x \cdot \ln (x-1))=(0 \cdot(-\infty))=\lim _{x \rightarrow 1+0} \frac{\ln (x-1)}{\frac{1}{\ln x}}=(-\infty /-\infty)= \\
& \quad=\lim _{x \rightarrow 1+0} \frac{(\ln (x-1))^{\prime}}{\left(\frac{1}{\ln x}\right)^{\prime}}=\lim _{x \rightarrow 1+0} \frac{\fra... | 0 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
6.58. $\lim _{x \rightarrow+0} x^{x}$. | Solution.
$\lim _{x \rightarrow+0} x^{x}=e^{\lim _{x \rightarrow 0} x \ln x}=e^{\lim _{x \rightarrow 0 \rightarrow 0(1)}\left(\frac{(\ln x)'}{(1 / x)'}\right)}=e^{\lim _{x \rightarrow-0} \frac{1 / x}{-1 / x^{2}}}=e^{\lim _{x \rightarrow+0}(-x)}=e^{0}=1$. | 1 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
6.59. $\lim _{x \rightarrow 0}(1-\cos x)^{x}$. | Solution.
$$
\lim _{x \rightarrow 0}(1-\cos x)^{x}=\left(0^{0}\right)=e^{\lim _{x \rightarrow 0} x \ln (1-\cos x)}
$$
## Find separately
$$
\begin{aligned}
& \lim _{x \rightarrow 0} x \cdot \ln (1-\cos x)=(0 \cdot(-\infty))=\lim _{x \rightarrow 0} \frac{(\ln (1-\cos x))^{\prime}}{(1 / x)^{\prime}}=\lim _{x \rightarr... | 1 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
13. Let $X, Y$ be random variables, $\mathbf{E}|X|, \mathrm{E}|Y|<\infty$ and $\mathrm{E}(Y \mid X)=0$ a.s. Show that from the condition $\mathrm{E}(Y \mid X+Y)=0$ a.s. it follows that $Y=0$ with probability one. | Solution. Let $f(x)=|x|-\operatorname{arctg}|x|, x \in \mathbb{R}$. The function $f=f(x)$ is increasing on $\mathbb{R}_{+}$ and is an even, strictly convex function on $\mathbb{R}$, and $0 \leqslant f(x) \leqslant|x|$ for all $x \in \mathbb{R}$. By Jensen's inequality for conditional expectations, we have
$$
\begin{ga... | 0 | Algebra | proof | Yes | Yes | olympiads | false |
47. Find the limit of the function:
1) $f(x)=x^{3}-5 x^{2}+2 x+4$ as $x \rightarrow-3$;
2) $\varphi(t)=t \sqrt{t^{2}-20}-\lg \left(t+\sqrt{t^{2}-20}\right)$ as $t \rightarrow 6$. | Solution. The given function is elementary, it is defined at the limit point, so we find the limit of the function as its particular value at the limit point:
1) $\lim _{x \rightarrow-3} f(x)=f(-3)=(-3)^{3}-5 \cdot(-3)^{2}+2 \cdot(-3)+4=-74$;
2) $\lim \varphi(t)=\varphi(6)=6 \sqrt{6^{2}-20}-\lg \left(6+\sqrt{6^{2}-20}... | 23 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
232.
$$
\text { 1) }\left\{\begin{array}{l}
x=k \sin t+\sin k t \\
y=k \cos t+\cos k t ;\left(\frac{d y}{d x}\right)_{t=0} ?
\end{array}\right.
$$
What is the geometric meaning of the result?
2) $\left\{\begin{array}{l}x=\alpha^{2}+2 \alpha \\ y=\ln (\alpha+1) ; \frac{d^{2} y}{d x^{2}} ?\end{array}\right.$
3) $\left... | Solution. 1) We find the derivatives of $x$ and $y$ with respect to the parameter $t$:
$$
\frac{d x}{d t}=k \cos t+k \cos k t ; \quad \frac{d y}{d t}=-k \sin t-k \sin k t
$$
The desired derivative of $y$ with respect to $x$ is found as the ratio of the derivatives of $y$ and $x$ with respect to $t$:
$\frac{d y}{d x}... | 0 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
315. Find the limits:
1) $\lim x \operatorname{ctg} 2 x$
2) $\lim _{x \rightarrow+0} \sqrt[3]{x} \ln x$
3) $\lim (\operatorname{tg} \varphi-\sec \varphi)$; $\varphi \rightarrow \frac{\pi}{2}$
4) $\lim _{x \rightarrow 1}\left(\frac{1}{\ln x}-\frac{x}{x-1}\right)$;
5) $\lim _{t \rightarrow 0}\left(\frac{1}{\sin t}-\frac{... | Solution. By establishing that the case is $0 \cdot \infty$ or $\infty - \infty$, we transform the function into a fraction where both the numerator and the denominator simultaneously tend to zero or infinity, then apply L'Hôpital's rule:
1) $\lim _{x \rightarrow 0} x \operatorname{ctg} 2 x=\lim \frac{x}{\operatorname... | 0 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
411. Find the curvature of the curve: 1) $x=t^{2}, y=2 t^{3}$ at the point where $t=1$; 2) $y=\cos 2 x$ at the point where $x=\frac{\pi}{2}$. | Solution. 1) We find the derivatives $\dot{x}=2 t, \ddot{x}=2, \dot{y}=6 t^{2}$, $\ddot{y}=12 t$, and compute their values at the point where $t=1$:
$$
\dot{x}=2, \ddot{x}=2, \dot{y}=6, \ddot{y}=12
$$
and, substituting into formula (1), we get
$$
K=\frac{|\ddot{x} \ddot{y}-\ddot{y} x|}{\left(\dot{x}^{2}+\dot{y}^{2}\... | 4 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
665. A rectangular reservoir with a horizontal cross-sectional area of $S=6 \mathrm{~m}^{2}$ is filled with water to a height of $H=5$ m. Determine the time it takes for all the water to flow out of the reservoir through a small hole in its bottom with an area of $s=0.01 \mu^{2}$, assuming that the velocity of the wate... | Solution. According to the general scheme (1), divide the desired time $T$ into a large number $n$ of small intervals $\Delta t_{1}, \Delta t_{2}, \ldots, \Delta t_{n}$, and let the water level in the reservoir decrease by an amount $\Delta x=\frac{H}{n}$ during each such interval (Fig. 124).[^20]
If we assume that du... | 1010 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
701. Calculate the integral $\int_{1}^{9} \sqrt{6 x-5} d x$ using the Newton-Leibniz formula and approximate formulas for rectangles, trapezoids, and Simpson's method, dividing the integration interval into 8 equal parts. Then estimate the percentage error of the results obtained using the approximate formulas. | Solution. By the Newton-Leibniz formula
$$
I=\int_{1}^{9} \sqrt{6 x-5} d x=\frac{1}{6} \int_{1}^{9}(6 x-5)^{\frac{1}{2}} d(6 x-5)=\left.\frac{1}{9}(6 x-5)^{\frac{3}{2}}\right|_{1} ^{9}=38
$$
Next, we divide the integration interval $[1 ; 9]$ into 8 equal parts, find the length of one part $h=1$, the division points $... | 38 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
702. Using Simpson's formula, calculate the approximate value of the integral $\int_{0} \cos x d x$ with an accuracy of 0.00001. | Solution. First, we determine the number $n$ of parts into which the integration interval $\left[0, \frac{\pi}{2}\right]$ should be divided to achieve the required accuracy of the computation.
Assuming the error $\delta(n)$ of Simpson's formula is less than $10^{-5}$, we have
$$
\frac{(b-a)^{5}}{180 n^{4}} y_{H L}^{(... | 1 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
714. Find the limits:
1) $\lim _{\substack{x \rightarrow 3 \\ y \rightarrow 0}} \frac{\tan(x y)}{y}$
2) $\lim _{\substack{x \rightarrow 0 \\ y \rightarrow 0}} \frac{x}{x+y}$. | Solution. Having made sure that the function is not defined at the limit point, we perform transformations, guided by the instructions in § 7, Chapter I:
1) $\lim _{\substack{x \rightarrow 3 \\ y \rightarrow 0}} \frac{\tan(x y)}{y}=\lim x \cdot \lim \frac{\tan(x y)}{x y}=3 \cdot 1=3$, since $\lim _{a \rightarrow 0} \f... | 3 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
874. Given points $A(3, -6, 0)$ and $B(-2, 4, 5)$. Compute the line integral $I=\int_{C} x y^{2} d x + y z^{2} d y - z x^{2} d z$:
1) along the straight line segment $O B$ and
2) along the arc $A B$ of the circle defined by the equations $x^{2} + y^{2} + z^{2} = 45, 2 x + y = 0$. | Solution. 1) First, we write the equation of the line passing through two points $\frac{x-x_{1}}{x_{2}-x_{1}}=\frac{y-y_{1}}{y_{2}-y_{1}}=\frac{z-z_{1}}{z_{2}-z_{1}}$, obtaining $\frac{x}{-2}=\frac{y}{4}=\frac{z}{5}$. By equating these equal ratios to the parameter $t$, we transform the canonical equations of the line ... | 91 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
875. Compute the line integrals:
1) $\oint_{-i} 2 x d x-(x+2 y) d y \quad$ and
2) $\oint_{+l} y \cos x d x+\sin x d y$
along the perimeter of the triangle with vertices $A(-1 ; 0), B(0 ; 2)$ and $C(2 ; 0)$. | Solution. 1) Here (Fig. 186) the integration line (closed) consists of three segments lying on different lines (with different equations). Accordingly, the line integral along the broken line $A B C A$ is calculated as the sum of integrals taken over the segments $A B, B C$, and $C A$.
By formulating the equation of t... | 3 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
1167. A locomotive is moving along a horizontal section of the track c) at a speed of 72 km/hour. In what time and at what distance will it be stopped by the brake, if the resistance to motion after the start of braking is equal to 0.2 of its weight. | Solution. According to Newton's second law in mechanics, the differential equation of motion for the locomotive will be
$$
m \frac{d^{2} s}{d t^{2}}=-0.2 m g
$$
where $s$ is the distance traveled in time $t$, $m$ is the mass of the locomotive, and $g$ is the acceleration due to gravity.
Multiplying both sides of thi... | 102 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
Example 4. Find the singular solutions of the differential equation
$$
\left(y^{\prime}\right)^{2}=4 x^{2}
$$ | Solution. Differentiating (23) with respect to $y_{1}$:
$$
2 y^{\prime}=0
$$
Excluding $y^{\prime}$ from (23) and (24), we get $x^{2}=0$. The discriminant curve is the y-axis. It is not an integral curve of equation (23), but according to scheme (16) it can be the geometric locus of points of tangency of integral cur... | 0 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
Example 5. Find the singular solutions of the differential equation
$$
\left(y^{\prime}\right)^{2}(2-3 y)^{2}=4(1-y)
$$ | Solution. Let's find the PDK. Excluding $y^{\prime}$ from the system of equations
$$
\left\{\begin{aligned}
\left(y^{\prime}\right)^{2}(2-3 y)^{2}-4(1-y) & =0 \\
y^{\prime}(2-3 y)^{2} & =0
\end{aligned}\right.
$$
we obtain
$$
(2-3 y)^{2}(1-y)=0 .
$$
Transforming equation (25) to the form
$$
\frac{d x}{d y}= \pm \f... | 1 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
Example 8. Find the Wronskian determinant for the functions: $y_{1}(x)=\sin x$,
$$
y_{2}(x)=\sin \left(x+\frac{\pi}{8}\right), y_{3}(x)=\sin \left(x-\frac{\pi}{8}\right)
$$ | Solution. We have
$$
W\left[y_{1}, y_{2}, y_{3}\right]=\left|\begin{array}{rrr}
\sin x & \sin \left(x+\frac{\pi}{8}\right) & \sin \left(x-\frac{\pi}{8}\right) \\
\cos x & \cos \left(x+\frac{\pi}{8}\right) & \cos \left(x-\frac{\pi}{8}\right) \\
-\sin x & -\sin \left(x+\frac{\pi}{8}\right) & -\sin \left(x-\frac{\pi}{8}\... | 0 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
Example 2. Investigate the stability of the solution of the degenerate equation for the equation
$$
\varepsilon \frac{d x}{d t}=x\left(e^{x}-2\right)
$$ | Solution. The degenerate equation $x\left(e^{x}-2\right)=0$ has two solutions
$$
\text { 1) } x=0, \quad 2) x=\ln 2 \text {. }
$$
We have
$$
\left.\frac{\partial f(t, x)}{\partial x}\right|_{x=0}=\left.\left(e^{x}-2+x e^{x}\right)\right|_{x=0}=-1
$$
so the solution $x=0$ is stable;
$$
\left.\frac{\partial f(t, x)}... | 0 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
88. Calculate the sum:
$$
S=\frac{1}{(a-b)(a-c)}+\frac{1}{(b-a)(b-c)}+\frac{1}{(c-a)(c-b)}
$$ | $$
\begin{aligned}
& S=\frac{1}{(a-b)(b-c)(c-a)}(-(b-c)-(c-a)-(a-b))= \\
& =\frac{-1}{(a-b)(b-c)(c-a)}(b-c+c-a+a-b)=0
\end{aligned}
$$
Answer: 0. | 0 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
114. Calculate the sum:
$$
x=\sqrt[3]{9+4 \sqrt{5}}+\sqrt[3]{9-4 \sqrt{5}}
$$ | $$
\triangle$ Let's raise the equality to the third power, using the formula
$$
(a+b)^{3}=a^{3}+b^{3}+3 a b(a+b)
$$
We get:
$$
\begin{aligned}
& x^{3}=(9+4 \sqrt{5})+(9-4 \sqrt{5})+3 \sqrt[3]{81-80} \cdot(\sqrt[3]{9+4 \sqrt{5}}+\sqrt[3]{9-4 \sqrt{5}}) \\
& x^{3}=18+3 x, \quad x^{3}-3 x-18=0
\end{aligned}
$$
The las... | 3 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
128. Knowing that $x+\frac{1}{x}=4$, calculate $x^{2}+\frac{1}{x^{2}}$. | $\triangle$ It seems that the equation $x+\frac{1}{x}=4$ should be transformed into a quadratic equation, find $x$ from the quadratic equation, and substitute the obtained value of $x$ into the sum $x^{2}+\frac{1}{x^{2}}$. However, this path is long, especially since the quadratic equation has two roots.
It is simpler... | 14 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
172. Prove that the numbers formed by the last two digits of the sequence $a_{n}=6^{n}$ form a periodic sequence. Find the period of such a sequence. | $\triangle$ That such a sequence is periodic is almost obvious: after all, the numbers formed by the last two digits of the power $6^{n}$ do not exceed 10: from 06 to 96. To find the period of the sequence, we will compute its first few terms:
$$
06,36,16,96,76,56,36,16
$$
It turned out that $a_{7}=a_{2}$, which mean... | 5 | Number Theory | proof | Yes | Yes | olympiads | false |
253. The digits of a three-digit number form a geometric progression with different terms. If this number is decreased by 200, the result is a three-digit number whose digits form an arithmetic progression. Find the original three-digit number. | $\triangle$ The number of three-digit numbers with distinct digits forming a geometric progression is small, and all of them can be easily found by enumeration:
$124,421,139,931,248,842,469,964$.
We will discard the numbers 124 and 139, as they are less than 200. Subtract 200 from the remaining numbers and find the n... | 842 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
390. Solve the equation:
$$
\sqrt[3]{x-1}+\sqrt[3]{2 x-1}=1
$$ | $\triangle$ Let's raise both sides of the equation to the third power, using the formula
$$
(a+b)^{3}=a^{3}+b^{3}+3 a b(a+b)
$$
We will have:
$x-1+2 x-1+3 \sqrt[3]{(x-1)(2 x-1)} \cdot(\sqrt[3]{x-1}+\sqrt[3]{2 x-1})=1$,
$\sqrt[3]{(x-1)(2 x-1)} \cdot(\sqrt[3]{x-1}+\sqrt[3]{2 x-1})=1-x$.
But what now? Now let's use t... | 1 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
394. Solve the equation:
$$
\sqrt[3]{x}+\sqrt[3]{x+19}=5
$$ | $\triangle$ Equations of this type have already been encountered in section 11.1, with a different method of solution (see the solution to problem 390). Let's introduce two new variables:
$$
\sqrt[3]{x}=y, \quad \sqrt[3]{x+19}=z
$$
We obtain a system of rational equations:
$$
\left\{\begin{array}{l}
y+z=5 \\
y^{3}=x... | 8 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
403*. Solve the equation:
$$
\sqrt[4]{1-x^{2}}+\sqrt[4]{1-x}+\sqrt[4]{1+x}=3
$$ | $\triangle$ The domain of the equation is the interval $[-1 ; 1]$. In this domain, we can apply the inequality between the geometric mean and the arithmetic mean of two non-negative numbers to each of the radicals in the left-hand side:
$\sqrt[4]{1-x^{2}}=\sqrt{\sqrt{1+x} \cdot \sqrt{1-x}} \leq \frac{\sqrt{1+x}+\sqrt{... | 0 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
447. A motorcyclist left point A at a speed of 45 km/h. After 40 minutes, a car left A in the same direction at a speed of 60 km/h. How much time after the car's departure will the distance between it and the motorcyclist be 36 km? | $\triangle$ Important question: at the moment when the car is 36 km away from the motorcycle, will it be ahead or behind the motorcycle?
In 40 minutes, the motorcycle will travel a distance of $45 \cdot \frac{2}{3}$ km $=30$ km, which is less than 36 km. Therefore, at the moment the car departs, it is 30 km behind the... | 4 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
448. A cyclist set off from point A to point B, and 15 minutes later, a car set off after him. Halfway from A to B, the car caught up with the cyclist. When the car arrived at B, the cyclist still had to cover another third of the entire distance. How long will it take the cyclist to travel the distance from A to B? | $\triangle$ Let's take the path AB as a unit. (This can be done in cases where there is no distance given in linear units in the problem data.) Let's denote the speed of the cyclist by $x$ (in fractions of the path per hour).
During the time it took the car to travel the second half of the path, the cyclist traveled $... | 45 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
455*. From point A to point B, which is 40 km away from A, two tourists set off simultaneously: the first on foot at a speed of 6 km/h, and the second on a bicycle. When the second tourist overtook the first by 5 km, the first tourist got into a passing car traveling at a speed of 24 km/h. Two hours after leaving A, th... | $\triangle$ Let's make a drawing (Fig. 2). $\qquad$
Fig. 2
Let $\mathrm{A}_{1}$ and $\mathrm{B}_{1}$ be the points where the first and second tourists are, respectively, at the moment when the second tourist overtakes the first by $5 \mathrm{km}$, and let $\mathrm{K}$ be the point where the first tourist catches up wi... | 9 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
457. A motorcyclist and a cyclist set off towards each other from points A and B simultaneously and met 4 km from B. At the moment when the motorcyclist arrived at B, the cyclist was 15 km from A. Find the distance AB. | $\triangle$ Let the distance AB be denoted as $x$ km.
By the time they meet, the motorcyclist and the cyclist have traveled $(x-4)$ km and $x$ km, respectively, and by the time the motorcyclist arrives at $\mathrm{B}$, the distances are $x$ km and $(x-4)$ km, respectively.
Thus, the ratio of the speeds of the motorcy... | 20 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
463. From two ports A and B, two steamships set off simultaneously towards each other across the sea. The speed of each is constant. The first steamship arrived at B 16 hours after the meeting, while the second arrived at A 25 hours after the meeting. How long does it take for each steamship to travel the entire distan... | $\triangle$ Let the path AB be a unit. Denote the speeds of the steamships as $v_{1}$ and $v_{2}$ (in fractions of a unit per hour).
Suppose the steamships met at point C. Then $\frac{A C}{C B}=\frac{v_{1}}{v_{2}}$. From this,
$$
\frac{A C}{C B}+1=\frac{v_{1}}{v_{2}}+1, \quad \frac{1}{C B}=\frac{v_{1}+v_{2}}{v_{2}}, ... | 36 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
489*. Three cyclists set off simultaneously: the first and second from point A, and the third towards them from point B. After 1.5 hours, the first cyclist was at an equal distance from the other two, and after 2 hours from the start, the third cyclist was at an equal distance from the first and second. How many hours ... | $\triangle$ Let's take the path AB as a unit. Denote the speeds of the first, second, and third cyclists as $v_{1}, v_{2}$, and $v_{3}$ (in fractions of this path per hour).
We will denote the positions of the first, second, and third cyclists in each of the three specified situations by the letters $\mathbf{M}_{1}, \... | 3 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
493. Four pumps of the same capacity, working together, filled the first tanker and a third of the second tanker (of a different volume) in 11 hours. If three pumps had filled the first tanker and then one of them filled a quarter of the second tanker, the work would have taken 18 hours. How many hours would it take fo... | $\triangle$ Let one pump fill the first tanker in $x$ hours, and the second tanker in $y$ hours. Then four pumps, working together, will fill the first tanker in $\frac{x}{4}$ hours, and the second in $\frac{y}{4}$ hours.
We have the system of equations:
$$
\left\{\begin{array}{l}
\frac{x}{4}+\frac{y}{4 \cdot 3}=11 \... | 8 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
498. Three automatic lines produce the same product, but have different productivity. The combined productivity of all three lines working simultaneously is 1.5 times the productivity of the first and second lines working simultaneously. A shift assignment for the first line can be completed by the second and third lin... | $\triangle$ Let's accept the shift task for the first line as a unit.
Let the first line complete its shift task in $x$ hours, and the third line the same task in $y$ hours. Then the second line completes the task of the first in $(x-2)$ hours.
Therefore, the productivity of the first, second, and third lines are res... | 8 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
503. Three workers need to make 80 identical parts. Together, they make 20 parts per hour. The first worker started the job alone. He made 20 parts, spending more than 3 hours on their production. The remaining work was done by the second and third workers together. The entire job took 8 hours. How many hours would it ... | Let's denote the productivity of the first, second, and third workers as \( x, y, \) and \( z \) parts per hour, respectively. Based on the conditions of the problem, we obtain a system of two equations with three unknowns:
\[
\left\{\begin{array}{l}
x+y+z=20 \\
\frac{20}{x}+\frac{60}{y+z}=8
\end{array}\right.
\]
To ... | 16 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
507. Fresh mushrooms contain $90\%$ water, while dried ones contain $12\%$ water. How many kilograms of dried mushrooms can be obtained from 44 kg of fresh mushrooms? | $\triangle$ According to the condition, 44 kg of fresh mushrooms contain $44 \cdot 0.9 = 39.6$ kg of water, which means there is $44 - 39.6 = 4.4$ kg of dry matter.
Let's denote the mass of dried mushrooms that can be obtained from 44 kg of fresh mushrooms by $x$ kg. These $x$ kg consist of $0.12 x$ kg of water and $0... | 5 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
7. A mowing team had to mow two meadows, one twice as large as the other. For half a day, the team mowed the larger meadow. After that, they split in half: one half stayed on the large meadow and finished it by evening, while the other half mowed the smaller meadow but did not finish it. How many mowers were in the tea... | 7. If the whole brigade mowed a large meadow for half a day and half of the brigade mowed for another half a day, it is clear that the brigade mowed $\frac{2}{3}$ of the meadow in half a day, and half of the brigade mowed $\frac{1}{3}$ of the meadow in half a day. Since the second meadow is half the size of the first, ... | 8 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
14. A box with a square base is required to be made for laying boxes that are 9 cm wide and 21 cm long. What should be the minimum length of the side of the square base so that the boxes fit snugly in the box? | 14. The side of the base of the box is 63 cm (you need to find the LCM of the numbers 9 and 21). | 63 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
15. When adding several numbers, the following errors were made: in one of the numbers, the digit 3 in the tens place was taken for 8, and the digit 7 in the hundreds place was taken for 4; in another, the digit 2 in the thousands place was taken for 9. The sum obtained was 52000. Find the correct sum. | 15. The obtained sum is greater than the true one by 7 thousand and 5 tens, but less than it by three hundred. To get the true sum, you need to subtract 7050 from 52000 and add 300 to the result. Answer: 42250. | 42250 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
16. Columbus discovered America in the 15th century. In which year did this discovery take place, if the following is known:
a) the sum of the digits representing this year is 16,
b) if the tens digit is divided by the units digit, the quotient is 4 and the remainder is 1. | 16. Let $x$ be the number of tens, $y$ be the number of units of the year of discovery. Then: $\left\{\begin{array}{l}x+y=11 \\ x=4 y+1 .\end{array}\right.$ Answer: 1492. | 1492 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
23. A pond is overgrown with duckweed in 64 days. In how many days will a quarter of the pond be overgrown if the amount of duckweed doubles every day? | 23. 62 days (the problem should be solved from the end). | 62 | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false |
25. In the wallet, there are 71 kopecks in coins of 5, 2, and 1 kopeck. The number of 1 and 2 kopeck coins is the same. How many coins of each denomination are in the wallet if there are 31 of them in total? | 25. 1 kopeck - 12 coins, 2 kopecks - 12 coins, and 5 kopecks - 7 coins. | 1 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
12. Find the remainder of the division of $67^{283}$ by 13. | 12. $65=13 \cdot 5$, then $67^{283}=(65+2)^{283}=65^{283}+283 \cdot 65^{282} \cdot 2+\ldots+2^{283}$. Each term, except the last one, contains a factor of 65, and therefore, is divisible by 13.
$$
2^{283}=16^{70} \cdot 2^{3}=(13+3)^{70} \cdot 2^{3}=\left(13^{70}+70 \cdot 13^{69} \cdot 3+\ldots+3^{70}\right) \cdot 2^{3... | 11 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
14. What digit does the number $777^{777}$ end with? | 14. Let's determine the last digits of the powers of 777. \(777^1\) ends in 7. \(777^2\) ends in 9; \(777^3\) ends in 3; \(777^4\) ends in 1; \(777^5\) ends in 7; \(777^6\) ends in 9; \(777^7\) ends in 3; \(777^8\) ends in 1, and so on. We establish that 777776 ends in 1, therefore, \(777^{777}\) ends in 7. | 7 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
18. How many digits does the number $2^{100}$ have? | 18. The number of digits in the representation of a number is always one more than the characteristic of the decimal logarithm of the number. In our case
$$
\lg 2^{100}=100 \lg 2 \approx 100 \cdot 0.3010=30.10
$$
Therefore, the number $2^{100}$ contains 31 digits. | 31 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
19. Find the last two digits of the number $99^{99}-51^{51}$. | 19. Let's determine the last digits of the powers $99^{99}$ and $5151.99^{1}-99$, $99^{2}-01,99^{3}-99,99^{4}-01$ and so on. $99^{99} - 99$, i.e., $99^{99}=100 M+99$.
Similarly, we establish that $51^{51}=100 N+51$, then $99^{99}-5151=$ $=100(M-N)+99-51=100(M-N)+48$.
Answer: the last two digits are 4 and 8. | 48 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
39. Find a two-digit number that has the following properties:
a) if the desired number is multiplied by 2 and 1 is added to the product, the result is a perfect square;
b) if the desired number is multiplied by 3 and 1 is added to the product, the result is a perfect square. | 39. The condition of the problem can be written as: $2 a+1=N^{2}, 3 a+1=M^{2}$. The squares of numbers can end with the digits $0,1,4,5,6,9$. Then $2 a$ can end with $0,4,8$ (since $2 a$ is an even number) and $a$ can end with 5,2, 4, and 0. $10 \leqslant a \leqslant 99$, i.e., $21<2 a+1<199,31<3 a+1<298$ or $21<N^{2}<... | 40 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
40. Find a five-digit number that has the following property: when multiplied by 9, the result is a number composed of the same digits but in reverse order, i.e., the number is reversed. | 40. Multiply the desired number by 10 and subtract the desired number from the product, which is equivalent to multiplying by 9, then $\overline{a b c d e o}-\overline{a b c d e}=\overline{e d c b a}$. It is easy to establish that $a=1$, then $e=9, b=0, d=8, c=9$. The desired number is 10989. | 10989 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
43. Find a four-digit number that is a perfect square, where the digit in the thousands place is the same as the digit in the tens place, and the digit in the hundreds place is 1 more than the digit in the units place. | 43. The desired number is $1000a + 100(b+1) + 10a + b = 1010a + 101b + 100$. $1010a + 101b + 100 = x^2$ or $101(10a + b) = (x-10)(x+10)$.
The left side has a factor of 101, so the right side must also have such a factor. Since $x$ is a two-digit number, $x-10 < 101$. Therefore, $x+10 = 101$, from which $x = 91$.
The ... | 8281 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
56. If the thought number is multiplied by 3, 2 is appended to the right, the resulting number is divided by 19, and 7 is added to the quotient, the result is three times the thought number. What is this number? | 56. Let $x$ be the thought-of number. Then the condition of the problem can be written as: $(3 x \cdot 10 + 2) : 9 + 7 = 3 x$.
From which $x = 5$.
## $\S 3$ | 5 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
2. The square root of a two-digit number is expressed as an infinite decimal fraction, the first four digits of which (including the integer part) are the same. Find this number without using tables. | 2. Let the desired number be $N_{\bullet}$ Then $10 \leqslant N \leqslant 100$. Therefore, $3, \ldots \leqslant$ $10,89$, i.e., $N=11 ; 12 ; \ldots$
But $\sqrt{11}=3.31662$. Therefore, there does not exist a two-digit number, the first four digits of the square root of which are written only with threes. By performing... | 79 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
3. Find a six-digit number that, when multiplied by 2, 3, 4, 5, and 6, gives other six-digit numbers composed of the same digits but in a different order. | 3. Let the desired number be denoted as: \( N = \overline{a_{6} a_{5} a_{4} a_{3} a_{2} a_{1}} \), with \( a_{6} \neq 0 \), because otherwise \( N \) would not be a six-digit number. \( a_{6} \neq 2 \), because \( 5N \) would then be a seven-digit number. Therefore, \( a_{6} = 1 \).
It is clear that the first digits o... | 142857 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
4. Find the smallest four-digit number that is equal to the square of the sum of the numbers formed by its first two digits and its last two digits. | 4. Let the desired number be $N=100 a+b$, where $a$ is the number written by the first two digits of the desired number, and $b$ is the last two digits. According to the problem, $100 a+b=(a+b)^{2}$. From this equation, we express $a+b$ in terms of $a$. By trial and error, we establish that $a+b$ will be an integer whe... | 2025 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
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