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Example 1. Find $\lim _{x \rightarrow+\infty}\left(\sqrt{x^{2}+6 x+5}-x\right)$. | Solution. When $x \rightarrow+\infty$, the given function represents the difference of two infinitely large quantities taking positive values (case $\infty-\infty$). By multiplying and dividing the given function by $\left(\sqrt{x^{2}+6 x+5}+x\right)$, we get
$$
\begin{gathered}
\lim _{x \rightarrow+\infty}\left(\sqrt... | 3 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
Example 4. Find $\lim _{x \rightarrow 0} x \operatorname{ctg} \frac{x}{3}$. | Solution. When $x \rightarrow 0$, we get an indeterminate form of $0 \cdot \infty$. Rewriting the given function in another form and applying formula (1.59), we find
$$
\lim _{x \rightarrow 0} x \operatorname{ctg} \frac{x}{3}=\lim _{x \rightarrow 0} x \frac{\cos \frac{x}{3}}{\sin \frac{x}{3}}=\lim _{x \rightarrow 0} \... | 3 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
Example 6. Find $\lim _{x \rightarrow+\infty} x\left(\operatorname{arctg} x-\frac{\pi}{2}\right)$. | Solution. Let $\operatorname{arctg} x=\alpha$, then $x=\operatorname{tg} \alpha$, if $x \rightarrow+\infty$, then $\alpha \rightarrow \frac{\pi}{2}$.
Consequently,
$$
\begin{gathered}
\lim _{x \rightarrow+\infty} x\left(\operatorname{arctg} x-\frac{\pi}{2}\right)=\lim _{\alpha \rightarrow \frac{\pi}{2}} \operatorname... | -1 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
Example 6. Find $\lim _{x \rightarrow 3} \frac{\ln \left(x^{2}-5 x+7\right)}{x-3}$. | Solution. When $x \rightarrow 3$, we get an indeterminate form of $\frac{0}{0}$, since $x^{2}-5 x+7 \rightarrow 1$ and $\ln \left(x^{2}-5 x+7\right) \rightarrow 0$. The expression $x^{2}-5 x+7$ can be represented as:
$$
x^{2}-5 x+7=1+\left(x^{2}-5 x+6\right)=1+z
$$
where $\left(x^{2}-5 x+6\right)=z \rightarrow 0$ as ... | 1 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
Example 3. Calculate the value of the derivative of the implicit function $x y^{2}=4$ at the point $M(1,2)$. | Solution. First, let's find the derivative:
$$
x^{\prime} y^{2}+x 2 y y^{\prime}=0, y^{\prime}=-\frac{y}{2 x}
$$
Substituting the values \(x=1\), \(y=2\) into the right-hand side of the last equation, we get
$$
y^{\prime}=-\frac{2}{2 \cdot 1}=-1
$$ | -1 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
Example 2. Find the third-order derivative of the function $y=x^{2}+3 x+2$. | Solution. Differentiating successively, we obtain:
$$
\begin{gathered}
y^{\prime}=\left(x^{2}+3 x+2\right)^{\prime}=2 x+3 ; y^{\prime \prime}=(2 x+3)^{\prime}=2 \\
y^{\prime \prime \prime}=(2)^{\prime}=0
\end{gathered}
$$ | 0 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
Example 4. Find the derivative $y_{x}^{\prime}$ of the function $x=e^{t} \cos t$; $y=e^{t} \sin t$ at $t=0$. | Solution. The functions $x$ and $y$ have the following derivatives with respect to $t$:
$$
\begin{aligned}
& x_{i}^{\prime}=e^{t} \cos t-e^{t} \sin t=e^{t}(\cos t-\sin t) \\
& y_{t}^{\prime}=e^{t} \sin t+e^{t} \cos t=e^{t}(\sin t+\cos t)
\end{aligned}
$$
therefore
$$
y_{x}^{\prime}=\frac{\sin t+\cos t}{\cos t-\sin t... | 1 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
Example 3. Find $\lim _{x \rightarrow \frac{\pi}{2}} \frac{\tan x}{\tan 3 x}$. | Solution. Using the L'Hôpital-Bernoulli rule, we get
$$
\lim _{x \rightarrow \frac{\pi}{2}} \frac{\tan x}{\tan 3 x}=\lim _{x \rightarrow \frac{\pi}{2}} \frac{\frac{1}{\cos ^{2} x}}{\frac{3}{\cos ^{2} 3 x}}=\lim _{x \rightarrow \frac{\pi}{2}} \frac{\cos ^{2} 3 x}{3 \cos ^{2} x}
$$
The limit of the ratio of the first d... | 3 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
Example 6. Find $\lim _{x \rightarrow 0}\left(\frac{1}{x}-\frac{1}{\sin x}\right)$. | Solution. When $x \rightarrow \mathbf{0}$, we get an indeterminate form of $\infty-\infty$. We will resolve this indeterminacy by converting it to an indeterminate form of $\frac{0}{0}$ and applying L'Hôpital-Bernoulli's rule,
$$
\begin{gathered}
\lim _{x \rightarrow 0}\left(\frac{1}{x}-\frac{1}{\sin x}\right)=\lim _{... | 0 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
Example 7. Find $\lim _{x \rightarrow 0}[(x-\sin x) \ln x]$. | Solution. Here we have an indeterminate form $0 \cdot \infty$. The given function can be represented as
$$
(x-\sin x) \ln x=\frac{\ln x}{\frac{1}{x-\sin x}}
$$
The resulting indeterminate form $\frac{\infty}{\infty}$ is resolved using L'Hôpital's rule:
$$
\lim _{x \rightarrow 0} \frac{\ln x}{\frac{1}{x-\sin x}}=\lim... | 0 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
Example 8. Find $\lim _{x \rightarrow 1}(x-1)^{\ln x}$. | Solution. When $x \rightarrow 1$, we have the indeterminate form $0^{0}$. We will use the identity
$$
[f(x)]^{\varphi(x)}=e^{\varphi(x) \ln f(x)}
$$
which in this case will be
$$
(x-1)^{\ln x}=e^{\ln x \cdot \ln (x-1)}
$$
We have
$$
\lim _{x \rightarrow 1}(x-1)^{\ln x}=\lim _{x \rightarrow 1} e^{\ln x \cdot \ln (x... | 1 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
Example 9. Find $\lim _{x \rightarrow 0}\left[\frac{\sin x}{x}\right]^{\frac{1}{x}}$. | Solution. Since $\lim _{x \rightarrow 0} \frac{\sin x}{x}=1, \lim _{x \rightarrow 0} \frac{1}{x}=\infty$, we have an indeterminate form of $1^{\infty}$ here.
Taking into account identity (A) (see example 8), using L'Hôpital-Bernoulli's rule, we find
$$
\lim _{x \rightarrow 0}\left[\frac{\sin x}{x}\right]^{\frac{1}{x}... | 1 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
Example 11. Find $\lim _{x \rightarrow \infty} \frac{x-\sin x}{x+\sin x}$. | Solution. Apply L'Hôpital's rule:
$$
\lim _{x \rightarrow \infty} \frac{x-\sin x}{x+\sin x}=\lim _{x \rightarrow \infty} \frac{1-\cos x}{1+\cos x}
$$
In the right-hand side of the last equality, the limit does not exist, so L'Hôpital's rule is not applicable here.
The specified limit can be found directly:
$$
\lim ... | 1 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
Example 2. Calculate the integral $\int_{0}^{\frac{\pi}{2}} \cos x d x$. | Solution.
$$
\int_{0}^{\frac{\pi}{2}} \cos x d x=\left.\sin x\right|_{0} ^{\frac{\pi}{2}}=\sin \frac{\pi}{2}-\sin 0=1-0=1
$$ | 1 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
Example 6. Calculate $\int_{1}^{e} \ln x d x$. | Solution. We apply the integration by parts formula (5.5).
Letting $u=\ln x, d v=d x$, we determine $d u=\frac{1}{x} d x, v=x$.
Therefore,
$$
\begin{gathered}
\int_{1}^{e} \ln x d x=\left.x \ln x\right|_{1} ^{e}-\int_{1}^{e} \frac{1}{x} x d x=\left.x \ln x\right|_{1} ^{e}-\int_{1}^{e} d x=\left.x \ln x\right|_{1} ^{... | 1 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
Example 8. Determine the area bounded by the arc of the cosine curve from $x=-\frac{\pi}{2}$ to $x=\frac{\pi}{2}$ and the $O x$ axis. | Solution. Based on the geometric meaning of the definite integral, we conclude that the desired area is expressed by the integral
$$
S=\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \cos x d x
$$
Evaluating this integral, we get
$$
S=\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \cos x d x=\left.\sin x\right|_{-\frac{\pi}{2}} ^{\fra... | 2 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
Example 1. Determine the area bounded by the lines $x y=6$, $x=1, x=e, y=0$ (Fig. 5.5).
Fig. 5.5
 into formula (5.13), we find:
$$
S=\int_{1}^{e} \frac{6}{x} d x=6 \int_{1}^{e} \frac{d x}{x}=\left.6 \ln x\rig... | 6 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
Example 1. Find the length of the arc of the astroid $x^{\frac{2}{3}}+y^{\frac{2}{3}}=a^{\frac{2}{3}}$. What is the length of the astroid when $a=1, a=\frac{2}{3}?$ | Solution. Since the astroid
Fig. 5.13 is symmetric with respect to the coordinate axes (Fig. 5.13), it is sufficient to compute the length of the arc $AB$ and multiply the result by 4.
Diff... | 4 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
Example 1. Find $\lim _{\substack{x \rightarrow 2 \\ y \rightarrow 0}} \frac{\sin x y}{y}$. | Solution. At the point $M(2,0)$, the function $z=\frac{\sin x y}{y}$ is undefined.
By multiplying and dividing the given function by $x \neq 0$, we get
$$
\frac{\sin x y}{y}=\frac{x \sin x y}{x y}=x \frac{\sin x y}{x y}
$$
Taking the limit in the last equality, we obtain
$$
\lim _{\substack{x \rightarrow 2 \\ y \ri... | 2 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
Example 2. Given the function
$$
f(x, y)=x \sin \frac{1}{y}+y \sin \frac{1}{x} \quad\left(x^{2}+y^{2} \neq 0\right), \quad f(0, y)=0, \quad f(x, 0)=0
$$
find $\lim _{\substack{x \rightarrow 0 \\ y \rightarrow 0}} f(x, y)$. | Solution. Let $\varepsilon>0$, then for $|x|<\frac{\varepsilon}{2},|y|<\frac{\varepsilon}{2}$ we get $\rho=$
$$
=\sqrt{\left(\frac{\varepsilon}{2}\right)^{2}+\left(\frac{\varepsilon}{2}\right)^{2}}=\frac{\sqrt{2}}{2} \varepsilon . \text { We form the difference } f(x, y)-0 \text { and estimate it: }
$$
$$
|f(x, y)-0|... | 0 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
Example 3. Find $\lim _{\substack{x \rightarrow 0 \\ y \rightarrow 0}} \frac{x^{2} y}{x^{2}+y^{2}}$. | Solution. The given function can be represented as:
$$
f(x, y)=\frac{x^{2} y}{x^{2}+y^{2}}=\frac{x y}{x^{2}+y^{2}} x .
$$
Since
$$
\left|\frac{x y}{x^{2}+y^{2}}\right| \leq \frac{1}{2}
$$
(this can be derived from the inequality $(x-y)^{2} \geq 0$ : $x^{2}-2 x y+y^{2} \geq 0$, $\left.x^{2}+y^{2} \geq 2 x y, \frac{1... | 0 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
Example 2. Into how many parts should the interval of integration be divided to calculate $\int_{2}^{7} \frac{d x}{\sqrt{x+2}}$ with an accuracy of 0.1? | Solution. The absolute error in calculating a definite integral using the rectangle method is determined by inequality (13.4). In the problem, the condition $\left|R_{n}(f)\right| \leq \varepsilon$ is set, where $\varepsilon=0.1$. The inequality $\left|R_{n}(f)\right| \leq \varepsilon$ will be satisfied if $\frac{(b-a)... | 8 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
Example 2. Construct a table of divided differences of various orders for the following values of $x$ and $y=f(x)$:
\[
\begin{gathered}
x_{0}=-3, x_{1}=-2, x_{2}=-1, x_{3}=1, x_{4}=2 \\
y_{0}=-9, y_{1}=-16, y_{2}=-3, y_{3}=11, y_{4}=36
\end{gathered}
\] | Solution. According to the definitions, we find the divided differences of the first order:
$$
\begin{gathered}
f\left(x_{1}, x_{0}\right)=\frac{y_{1}-y_{0}}{x_{1}-x_{0}}=\frac{-16-(-9)}{-2-(-3)}=-7 ; f\left(x_{2}, x_{1}\right)=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}=\frac{-3-(-16)}{-1-(-2)}=13 \\
f\left(x_{3}, x_{2}\right)=\... | 1 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
Example 2. How many elementary outcomes favor the event "the same number of points fell on both dice" when two gaming dice are rolled? | Solution. This event is favored by 6 elementary outcomes (see Table 1.1$):(1 ; 1),(2 ; 2),(3 ; 3),(4 ; 4),(5 ; 5),(6 ; 6)$. | 6 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
Example 5. The distribution law of a discrete random variable is given by the following table:
| $X$ | 0 | 1 | 2 | 3 |
| :---: | :---: | :---: | :---: | :---: |
| $P$ | 0.2 | 0.4 | 0.3 | 0.1 |
Find the distribution function of this random variable. | Solution. To construct the distribution function $F(x)$ of a discrete random variable $X$, we use formula (2.2.11).
1. For $x \leq 0 \quad F(x)=\sum_{x_{k}3 \quad F(x)=P(X=0)+P(X=1)+P(X=2)+P(X=3)=$
$$
=0.2+0.4+0.3+0.1=1
$$
The graph of the function $F(x)$ is shown in Fig. 2.7. | 1 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
Example 7. Two dice are rolled. The discrete random variable $X$ is the sum of the points that appear on both dice. Find the mathematical expectation of this random variable. | Solution. This random variable takes all integer values from 2 to 12. The distribution law can be given by the following table:
| $X$ | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| $P$ | $\frac{1}{36}$ | $\frac{2}{36}... | 7 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
Example 13. Find the mathematical expectation of a random variable $X$, the distribution function of which is given by
$$
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 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 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
$$ | 5 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
Example 1. Find the zeros of the function $f(z)=e^{z}-1-z$ and determine their order. | Solution.
1. Find the zeros of the function $f(z)$ by solving the equation $e^{z}-1-z=0$. We get $z=0$.
2. Determine the order of the obtained zero $z=0$. For this, we use the Taylor series expansion of the function $f(z)$ in powers of $z$:
$$
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 |
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
$$
... | 5 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
5.2. Find the limit of the sequence
$$
\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
$$
\begin{ga... | 0 | Algebra | proof | Yes | Yes | olympiads | false |
232.
$$
\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 |
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
$$
\frac{(b-a)^{5}}{180 n^{4}} y_{H L}^{(... | 1 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
714. Find the limits:
1) $\lim _{\substack{x \rightarrow 3 \\ y \rightarrow 0}} \frac{\tan(x y)}{y}$
2) $\lim _{\substack{x \rightarrow 0 \\ y \rightarrow 0}} \frac{x}{x+y}$. | Solution. Having made sure that the function is not defined at the limit point, we perform transformations, guided by the instructions in § 7, Chapter I:
1) $\lim _{\substack{x \rightarrow 3 \\ y \rightarrow 0}} \frac{\tan(x y)}{y}=\lim x \cdot \lim \frac{\tan(x y)}{x y}=3 \cdot 1=3$, since $\lim _{a \rightarrow 0} \f... | 3 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
875. Compute the line integrals:
1) $\oint_{-i} 2 x d x-(x+2 y) d y \quad$ and
2) $\oint_{+l} y \cos x d x+\sin x d y$
along the perimeter of the triangle with vertices $A(-1 ; 0), B(0 ; 2)$ and $C(2 ; 0)$. | Solution. 1) Here (Fig. 186) the integration line (closed) consists of three segments lying on different lines (with different equations). Accordingly, the line integral along the broken line $A B C A$ is calculated as the sum of integrals taken over the segments $A B, B C$, and $C A$.
By formulating the equation of t... | 3 | Calculus | math-word-problem | Yes | Yes | olympiads | false |
Example 4. Find the singular solutions of the differential equation
$$
\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}
$$
The las... | 3 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
172. Prove that the numbers formed by the last two digits of the sequence $a_{n}=6^{n}$ form a periodic sequence. Find the period of such a sequence. | $\triangle$ That such a sequence is periodic is almost obvious: after all, the numbers formed by the last two digits of the power $6^{n}$ do not exceed 10: from 06 to 96. To find the period of the sequence, we will compute its first few terms:
$$
06,36,16,96,76,56,36,16
$$
It turned out that $a_{7}=a_{2}$, which mean... | 5 | Number Theory | proof | Yes | Yes | olympiads | false |
390. Solve the equation:
$$
\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 |
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 |
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 |
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 |
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 |
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 |
10. The dividing circle of the dividing head is uniformly marked with 24 holes. What regular polygons can be marked using this instrument? | 10. By connecting all 24 holes sequentially, we get a twenty-four-sided polygon, connecting every other hole - a twelve-sided polygon,
connecting every two holes - an eight-sided polygon, ... | 6 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
14. $\log _{\frac{\sqrt{3}}{3}}\left(\log _{8} \frac{\sqrt{2}}{2}-\log _{3} \frac{\sqrt{3}}{3}\right)$.
In № $15-27$ solve the equations: | 14. We should switch to the logarithm with base $\frac{\sqrt{3}}{3}$.
Then replace $\frac{1}{3}$ with $\left(\frac{\sqrt{3}}{3}\right)^{2}$. Answer: 2. | 2 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
16. $\log _{4}\left\{2 \log _{3}\left[1+\log _{2}\left(1+3 \log _{2} x\right)\right]\right\}=\frac{1}{2}$. | 16. Gradually potentiating, we arrive at the equation $1+3 \log _{2} x=4$. From which $x=2$. | 2 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
19. $\log _{5} 120+(x-3)-2 \log _{5}\left(1-5^{x-3}\right)=-\log _{5}\left(0.2-5^{x-4}\right)$. | 19. $1-5^{x-3} \neq 0, x \neq 3,0,2-5^{x-4}=\frac{1}{5}\left(1-5^{x-3}\right)$.
Raising the entire expression, we get
$$
\frac{120 \cdot 5^{x-3}}{\left(1-5^{x-3}\right)^{2}}=\frac{5}{1-5^{x-3}} ; \quad 5^{x-3}=5^{-2} . \quad \text { Hence } x=1
$$ | 1 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
14. Calculate the product $p=1 \cdot 2^{\frac{1}{2}} \cdot 4^{\frac{1}{4}} \cdot 8^{\frac{1}{8}} \cdot 16^{\frac{1}{16}} \cdot \ldots$
In № $15-20$ find the sums: | 14. Let's write all powers as powers with base 2
$$
p=1 \cdot 2^{\frac{1}{2}} \cdot 2^{\frac{2}{4}} \cdot 2^{\frac{3}{8}} \cdot 2^{\frac{4}{16}} \ldots=2^{\frac{1}{2}+\frac{2}{4}+\frac{3}{8}+\frac{4}{16}+\cdots}
$$
The problem has been reduced to finding the sum $S=\frac{1}{2}+\frac{2}{4}+\frac{3}{8}+\frac{4}{16}+\ld... | 4 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
21. Find the sum $\frac{1}{2!}+\frac{2}{3!}+\frac{3}{4!}+\ldots+\frac{n}{(n+1)!}$ and compute its limit as $n \rightarrow \infty$. | 21. By forming the sequence of partial sums, we get
$$
S_{1}=\frac{2!-1}{2!}, \quad S_{2}=\frac{3!-1}{3!}, \ldots \quad \text { Hence } S_{n}=\frac{(n+1)!-1}{(n+1)!}, \quad S=\lim _{n \rightarrow \infty} S_{n}=1
$$ | 1 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
15. $y=\frac{|x|-2}{|x|}$.
Translate the text above into English, keeping the original text's line breaks and format, and output the translation result directly.
15. $y=\frac{|x|-2}{|x|}$. | 15. (Fig. 15) The root of the submodular expression is 0.
For $x>0, \quad y=1-\frac{2}{x}$.
For $x<0, \quad y=1+\frac{2}{x}$. | 0 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
31. How many planes of symmetry do the following have: a) a cube; b) a regular tetrahedron? | 31. a) A cube has 9 planes of symmetry; b) a regular tetrahedron has 6 planes of symmetry. | 6 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
21. What is the maximum area that a triangle with sides \(a, b, c\) can have, given that the sides are within the following limits: \(0 \leqslant a \leqslant 1 \leqslant b \leqslant 2 \leqslant c \leqslant 3\) ? | 21. $S=\frac{1}{2} a b \sin \alpha$. Takes the maximum value when $a=1, b=2, \sin \alpha=1$,
$S=\frac{1}{2} \cdot 1 \cdot 2 \cdot 1=1$. Under these conditions, $c=\sqrt{1+4}=\sqrt{5}$, which satisfies the problem's condition ( $2 \leqslant c \leqslant 3$). | 1 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
33. Determine $a$ so that the sum of the squares of the roots of the equation $x^{2}+(2-a) x-a-3=0$ is the smallest. | 33. $a=1$ | 1 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
30. Several families lived in one house. In total, there are more children in these families than adults: there are more adults than boys; more boys than girls; and more girls than families. There are no childless families, and no families have the same number of children. Each girl has at least one brother and at most... | 30. Three families lived in the house. In one of these families, there is a single child, a boy. In another family - two girls and a boy. In the third - two girls and three boys. | 3 | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false |
32. During the lunch break, the members of the communist labor brigade started talking about how many newspapers each of them reads. It turned out that each member subscribes to and reads exactly two newspapers, each newspaper is read by five people, and any combination of two newspapers is read by one person. How many... | 32. 6 newspaper names, 15 members. | 6 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
51. There are three villages: $A, B$, and $C$. The inhabitants of $A$ always tell the truth, the inhabitants of $B$ always lie, and the inhabitants of $C$, depending on their mood, tell the truth or lie. Since the villages are located close to each other, the inhabitants visit each other. A tourist ended up in one of t... | 51. Four questions: 1) Am I in one of the settlements $A$ and $B$? 2) Am I in settlement C? 3) Do you live in settlement C? 4) Am I in settlement $A^{*}$? | 4 | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false |
52. My friend has thought of an integer between 10 and 19. To guess which number he has in mind, I can ask him questions that he will answer with "yes" or "no": What is the smallest number of questions and which specific questions can I ask to determine which number he thought of? | 52. The least number of questions is three. The first question: "Is the number you are thinking of among the first four numbers (11-14)?" If the answer is "Yes," then the second question can be: "Is the number you are thinking of among the numbers 11 and 12?" If the answer is "No," then the third question can be: "Is t... | 3 | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false |
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