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Problem: In how many ways can the numbers \( 1,2,3, \ldots, 1024 \) be coloured in two colours (blue and red) so that any 2 numbers, the sum of which is a power of two, are coloured in different colours.

12.4.9 Various Olympiad problems

3.58 $

Problem: Find such smallest \( k \) that for any 15-colour colouring of a \( 30 \times k \) table, there are two rows and columns, at the intersection of which there are 4 cells of the same colour.

12.4.10 Various Olympiad problems

7.17 $

Problem: How many integer solutions does the equation \( |a|+|b|+|c|+|d|+|e|=100 \) have?

12.4.11 Various Olympiad problems

3.58 $

Problem: What is the sum of all numbers obtained from the number \( k \) by permuting the digits (including \( k \) ), if \( k=5307447 \) ? Representing 0 at the beginning of the number is not allowed. As an answer, write down the remainder of dividing the obtained sum by 10000 . If the question of the problem allows several answers, then indicate them all in the form of a set.

12.4.12 Various Olympiad problems

4.78 $

Problem: Numbers from 1 to 50 are written on cards. Is it possible to arrange these cards into 11 bags (so that there is at least one card in each bag) so that the product of the numbers on the cards in each bag is divisible by 9 ?

12.4.13 Various Olympiad problems

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Problem: Several not necessarily different natural numbers are written on the board in one line from left to right. It is known that each next number, except for the first one, is either greater than the previous one by 1 , or two times smaller than the previous one. a) It is possible that the first number is 12 , and the seventh one is 2 ? b) It is possible that the first number is equal to 1200 , and the \( 25^{\text {th }} \) one is equal to 63 ? c) What is the smallest amount of numbers that can be written on the blackboard if the first number is 1200 and the last number is 5 ?

12.4.14 Various Olympiad problems

7.17 $

Problem: How many ways are there to colour \( n \) balls in 3 colours (different variants are those in which the number of balls of a certain colour differs).

12.4.15 Various Olympiad problems

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Problem: 4399 optionally different positive numbers \( v_{1}, v_{2}, \ldots, v_{4399} \) are written around the circle. For any 4 numbers \( h, k, z \) and \( p \), in a row in the indicated order clockwise, the inequality holds: \[ 1,6(h+k) \geq \frac{1}{z}+\frac{1}{p} \text {. } \] What is the smallest possible value of the arithmetic mean of these numbers? If the question of the problem allows several answers, then indicate all of them in the form of a set.

12.3.1.6 Algebraic

5.97 $

Problem: Find the sum of natural values of the parameter \( a \), for each of which the system has at least one solution. \[ \left\{\begin{array}{c} \sqrt{(x+a)^{2}+(y+3)^{2}}+\sqrt{(x-4)^{2}+(y-a)^{2}}= \\ =\sqrt{2 a^{2}+14 a+25} \\ y=a^{2}-5 a-7 \end{array}\right. \]

12.3.1.7 Algebraic

2.87 $

Problem: Find the sum of natural values of the parameter \( a \), for each of which the system has at least one solution \[ \left\{\begin{array}{c} \sqrt{(x-a)^{2}+(y+3)^{2}}+\sqrt{(x+2)^{2}+(y-a)^{2}}= \\ =\sqrt{2 a^{2}+10 a+13} \\ y=a^{2}-4 a-6 \end{array}\right. \]

12.3.1.8 Algebraic

2.87 $

Problem: Consider the inequality \[ \left|\log _{2} x\right| \cdot \frac{x^{2}-2 x-a}{(10 x-3 a-16)^{2}} \ldots 0 \] Which inequality \( \operatorname{sign}(<,> \), \( \leq \) or \( \geq) \) must be there instead of the ellipses, so that for at least one value of the parameter \( a \) the solution of the inequality with respect to \( x \) there was an interval (a nonempty open connected bounded set on the real line)? Find the sum of all integer values of the parameter \( a \), for which the solution of this inequality with the chosen sign is an interval.

12.3.1.9 Algebraic

2.39 $

Problem: Find the derivatives of the functions: a) \( y=3 \sqrt[3]{\frac{x+1}{(x-1)^{2}}} \), b) \( y=x-3 \ln \left[\left(1+e^{\frac{x}{6}}\right) \sqrt{1+e^{\frac{x}{3}}}\right. \), c) \( y=\frac{1}{2} \sqrt{\frac{1}{x^{2}}-1}-\frac{\arccos x}{2 x^{2}} \), d) \( y=\ln \left(\arccos \frac{1}{\sqrt{x}}\right) \), e) \( y=\frac{6^{x}(\sin 4 x \ln 6-4 \cos 4 x)}{16+\ln ^{2} 6} \), f) \( x=\left(1+\cos ^{2} t\right)^{2}, y=\frac{\cos t}{\sin ^{2} t} \).

2.2.23 Derivatives and differentials

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Problem: Find the derivatives of the functions: a) \( y=\frac{1+x^{2}}{2 \sqrt{1+2 x^{2}}} \), b) \( y=e^{a x}\left[\frac{1}{2 a}+\frac{a \cos 2 b x+2 b \sin 2 b x}{2\left(a^{2}+4 b^{2}\right)}\right] \), c) \( y=\ln \sin \frac{2 x+4}{x+1} \) d) \( y=\frac{4+x^{4}}{x^{3}} \arctan \frac{x^{2}}{2}+\frac{4}{x} \), e) \( y=\left(x^{3}+4\right)^{\tan x} \), f) \( x=\arcsin \sqrt{1-t^{2}}, \quad y=(\arccos t)^{2} \).

2.2.24 Derivatives and differentials

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Problem: Find the limits of the functions without using L'Hopital's rule. a) \( \lim _{x \rightarrow 3} \frac{x^{2}-9}{x^{2}+2 x-15} \), b) \( \lim _{x \rightarrow+\infty} \frac{\sqrt{x}+1}{\sqrt{x}} \), c) \( \lim _{x \rightarrow \infty}\left(\sqrt{x^{2}+1}-\sqrt{x^{2}-1}\right) \), d) \( \lim _{x \rightarrow 0} \frac{5 x^{2}}{\arcsin 3 x} \), e) \( \lim _{x \rightarrow \infty}\left(\frac{x-8}{x}\right)^{\frac{x}{2}} \), f) \( \lim _{x \rightarrow a} \frac{x^{m}-a^{m}}{x^{n}-a^{n}} \).

2.1.27 Calculation of limits

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Problem: Find the limits of the function without using L'Hopital's rule. a) \( \lim _{x \rightarrow 1} \frac{2-x-x^{2}}{1-x} \), b) \( \lim _{x \rightarrow \infty} \frac{x^{3}+x^{2}+x+1}{2 x^{3}+4} \), c) \( \lim _{x \rightarrow 4} \frac{\sqrt{x}-2}{x-4} \), d) \( \lim _{x \rightarrow 0} \frac{\tan x-\sin x}{x^{3}} \), e) \( \lim _{x \rightarrow 0}(1+2 x)^{\frac{3}{x}} \) f) \( \lim _{x \rightarrow 0} \frac{\ln x}{\cot x} \).

2.1.28 Calculation of limits

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