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Problem: On a \( 100 \times 100 \) board there are 2500 kings, not hitting each other. Let's prove that the number of these arrangements is \( \leq 51^{100} \).

12.4.5 Various Olympiad problems

10.45 $

Problem: On the side \( C D \) of the parallelogram \( A B C D \) points \( E_{1} \) and \( E_{2} \) are chosen so, that \( A B=B E_{1}=B E_{2} \). On the beam \( A E_{1} \) the point \( F_{1} \) is chosen so, that \( B E_{1} \| C F_{1} \), and on the beam \( A E_{2} \) the point \( F_{2} \) is chosen so, that \( B E_{2} \| C F_{2} \). Prove that \( D F_{1}=D F_{2} \).

12.1.5 Olympic geometry

3.92 $

Problem: Let \( n \) be a positive even integer and \( a, b \) are positive coprime integers. Find \( a, b \) if \( a+b \) divides \( a^{n}+b^{n} \).

12.2.5 Number theory

2.61 $

Problem: Find all positive integers \( n \), for which the number, obtained by erasing the last digit of \( n \) is a divisor of \( n \).

12.2.6 Number theory

2.61 $

Problem: Determine all integers \( x \), for which \( \frac{x^{3}-3 x+2}{2 x+1} \) is an integer.

12.2.7 Number theory

3.92 $

Problem: Let \( (a, b)=1 \). Prove that \( (a+b, a-b)=1 \) or 2 .

12.2.8 Number theory

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Problem: At the vertices of a 178-gon numbers are written, and the sum of numbers at any successive \( k \) vertices is equal. What is the smallest \( k \) for which all numbers are equal, given the condition \( k>1 \) ?

12.4.6 Various Olympiad problems

5.22 $

Problem: Find the general solution (the first integrals) of the system of differential equations: \[ \frac{d x}{x\left(y^{2}-z^{2}\right)}=\frac{d y}{-y\left(x^{2}+z^{2}\right)}=\frac{d z}{z\left(x^{2}+y^{2}\right)} \text {. } \]

8.3.6 Systems of ordinary differential equations

2.61 $

Problem: The sum of two positive numbers is equal to 5432 , and their least common multiple is 223020 . Find these numbers.

12.2.9 Number theory

3.92 $

Problem: Let \( (a, b)=10 \). Find all possible values of \( \left(a^{3}, b^{4}\right) \).

12.2.10 Number theory

3.13 $

Problem: Let's prove that \( n !+1 \) and \( (n+1) !+1 \) are coprime.

12.2.11 Number theory

3.92 $

Problem: Find the general solution of the second order inhomogeneous differential equation: \[ y^{\prime \prime}+2 i y^{\prime}-y=-20 e^{-i \cdot x} \text {. } \]

8.1.2.10 Second order differential equations

1.83 $

Problem: Find a particular solution to the first order inhomogeneous differential equation: \[ y^{\prime}-\frac{y}{x+1}=e^{x} \cdot(x+1), y(0)=1 . \]

8.1.1.11 First order differential equations

1.31 $

Problem: Find the particular solution of the differential equation: \[ \tan x \cdot\left(x y^{\prime \prime}+y^{\prime}\right)=x y^{\prime}-1, x=\sqrt{e}, y=\frac{1}{2}, \quad y^{\prime}=\frac{1}{\sqrt{e}} \text {. } \] clip2net.com

8.1.2.12 Second order differential equations

2.61 $

Problem: Find the general solution of the linear homogeneous differential equation with constant coefficients: \[ y^{\prime \prime \prime}+5 y^{\prime \prime}-6 y^{\prime}=0 . \] clip2net.com

8.1.3.13 Higher order differential equations

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