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3) Problem: Calculate the definite integral: \[ \int_{1}^{e^{\pi / 2}} \cos \ln x d x \]

9.3.3.6 Definite Integrals

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Problem: Circle \( P \) is inscribed in the triangle TCS. Find the perimeter of the trapezoid, formed by the side \( T C \) and the line, passing through the centre \( P \) and the parallel \( T C \), if the bases of the trapezoid are equal to 3 and 5 .

12.1.4 Olympic geometry

3.76 $

Problem: Geppetto and Pinocchio play according to the following rules: Geppetto writes on the board six different numbers in a row, and Pinocchio comes up with his own four numbers for them \( x_{1}, x_{2}, x_{3}, x_{4} \) and under each number Geppetto writes respectively any of the sums \( x_{1}+x_{2}, x_{1}+x_{3}, x_{1}+ \) \( x_{4}, x_{2}+x_{3}, x_{2}+x_{4}, x_{3}+x_{4} \) (each only once), after which for each sum, equal to the number under it, Pinocchio gets 3 apples, and for a larger one 1 apple. What is the maximum number of apples Pinocchio can be guaranteed to get?

12.4.1 Various Olympiad problems

5.01 $

Problem: There are \( n \) people gathered, each tow of which either know each other, or have exactly one mutual acquaintance. At the same time, there is no one, who is familiar with everyone. Prove that \( n-1 \) is the square of an integer.

12.4.2 Various Olympiad problems

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Problem: In a class of 21 pupils, each three pupils have done either Mathematics or English homework together exactly once. Can we claim that there are four pupils in this class, any three of which have done homework together in the same subject?

12.4.3 Various Olympiad problems

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Problem: John, leaving the house, looked at his watch and saw that the line, dividing the angle between the hour and minute hands in half, passes through number 12. When he came to school, the line, dividing the angle between the hour and minute hands in half, passed through the mark, corresponding to 13 minutes. How long did John walk from home to school if it is known that he left after 8:00, and arrived at school before 9:00?

12.4.4 Various Olympiad problems

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Problem: Solve the inequality: \[ \frac{\sqrt{x}}{x-2} \leq 3 \sqrt{x} \]

12.3.1.1 Algebraic

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Problem: What is the maximum value the expression can take: \[ \left(1-a^{2}\right) \sqrt{1-b^{2}}-\left(1-b^{2}\right) \sqrt{1-a^{2}} . \]

12.3.1.2 Algebraic

2.51 $

Problem: One stone was put in each of the points \( n \) with the cooridinates \( x=1, x=2, \ldots, x=n \) in ascending order of their weight. The weight of the lightest stone is \( 3 \mathrm{~kg} \). The weight of each next stone is \( 1 \mathrm{~kg} \) less, than the twice weight of the previous one. Find the total weight of the first 10 stones. For what \( n(9

12.2.4 Number theory

7.52 $

Problem: Investigate the stability of the system at zeroresting point. \[ \left\{\begin{array}{c} \frac{d y}{d t}=\tan ^{-1} \frac{A x}{\sqrt{1-(A x)^{2}}}+B \sin \left(\frac{\pi}{2}-y\right)-(B-y) e^{-D y} \\ \frac{d x}{d t}=C \cdot \cosh y+D \cdot \sinh (A x)-C \end{array}\right. \] \begin{tabular}{|l|l|l|l|} \hline\( A \) & \( B \) & \( C \) & \( D \) \\ \hline- & + & - & + \\ \hline \end{tabular}

8.4.1.2 Stability of the systems of equations

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Problem: Investigate the stability of the system at the zeroresting point. \[ \left\{\begin{array}{c} \frac{d y}{d t}=\tan ^{-1} \frac{A x}{\sqrt{1-(A x)^{2}}}+B \sin \left(\frac{\pi}{2}-y\right)-(B-y) e^{-D y} \\ \frac{d x}{d t}=C \cdot \cosh y+D \cdot \sinh (A x)-C \end{array}\right. \] \begin{tabular}{|l|l|l|l|} \hline- & - & + & + \\ \hline \end{tabular}

8.4.1.4 Stability of the systems of equations

6.26 $

Problem: Find the resting points (equilibrium) of the system and investigate the stability of the system at these points. \[ \left\{\begin{array}{c} \frac{d y}{d t}=A \sin y+C x e^{-\frac{B}{D} x} \\ \frac{d x}{d t}=B \sinh (A x)+D \tan (C y) \end{array}\right. \] \begin{tabular}{|c|c|c|c|} \hline\( A \) & \( B \) & \( C \) & \( D \) \\ \hline+ & + & - & - \\ \hline \end{tabular}

8.4.1.3 Stability of the systems of equations

6.26 $

Problem: Find the general solution of the second order inhomogeneous differential equation with variable coefficients: \[ \frac{1}{49}(7 x-8)^{2} y^{\prime \prime}-\frac{1}{7}(7 x-8) y^{\prime}-3 y=-5|7 x-8|^{-6} \text {. } \] clip2net.com

8.1.2.7 Second order differential equations

2.51 $

Problem: Find the general solution of the second order inhomogeneous differential equation: \[ y^{\prime \prime}+y=-\frac{5}{\sin x} \text {. } \]

8.1.2.8 Second order differential equations

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Problem: Find the homogeneous linear differential equation with constant real coefficients, possibly of a lower order, a) which has the solution \( y_{1}=e^{4 x} \sin 2 x \), b) which has the solution \( y_{1}=x e^{-4 x} \sin 2 x \), c) which has the solutions \( y_{1}=x^{3}, y_{2}=e^{4 x} \sin 2 x \). clip2net.com

8.1.3.9 Higher order differential equations

3.76 $

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