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Problem: Conduct a complete examination of the function by methods of differential calculus and plot the graph. \[ y=\frac{x^{3}-4}{x^{2}} \text {. } \]

2.4.9 Graphing functions using derivatives

2.54 $

Problem: Conduct a complete examination of the function by methods of differential calculus and plot the graph: \[ y=\frac{x^{2}-4 x+1}{x-4} . \]

2.4.10 Graphing functions using derivatives

2.54 $

Problem: Conduct a complete examination of the function by methods of differential calculus and plot the graph: \[ y=\sqrt[3]{(x-1)(x+2)^{2}} . \]

2.4.11 Graphing functions using derivatives

2.54 $

Problem: Conduct a complete examination of the function by methods of differential calculus and plot the graph: \[ y=-\frac{e^{-(x+2)}}{x+2} \]

2.4.12 Graphing functions using derivatives

2.54 $

Problem: Find the derivative of function \( \varphi \) at point \( A \) in the direction of vector \( \overrightarrow{A B} \) : \[ \varphi=\frac{y^{2}}{x}+\frac{y^{2}}{z}+\frac{z^{2}}{x^{2}}, \quad A(1 ; 0 ; 1), \quad B(2 ; 2 ; 3) . \]

2.3.21 Gradient and directional derivative

0.76 $

Problem: Calculate the derivative of function \( f(x, y)=x^{5}+x \arccos y^{2} \) at point \( A=(3 ; 0) \) in the direction of vector \( \vec{a}=(2 ; 2) \).

2.3.22 Gradient and directional derivative

0.76 $

Problem: Find the value and the direction of the gradient of field \( u=x y \) at point \( A(1 ; 1), B(1 ;-1) \).

2.3.23 Gradient and directional derivative

1.27 $

Problem: Find the integer part of the largest value of the parameter \( a \), for which the equation \[ 3 \log _{5} \frac{x^{2}+a x+4}{x}+5\left(x^{2}+(a-1) a+4\right)=0 \] has positive roots and all these roots are integers. If the desired value doesn't exist, write the word "no" as an answer.

12.5.1 Olympic algebra

2.54 $

Problem: Calculate the sum: \[ \begin{array}{l} \left(\frac{1+2}{3}+\frac{4+5}{6}+\frac{7+8}{9}+\cdots+\frac{2017+2018}{2019}\right)+ \\ +\left(1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{673}\right) . \end{array} \]

12.5.2 Olympic algebra

0 $

Problem: Simplify the sum: \[ 1-\frac{1}{5}+\frac{1}{4}-\frac{1}{25}+\frac{1}{16}-\frac{1}{125}+\frac{1}{64}-\cdots-\frac{1}{5^{n}}+\frac{1}{4^{n}} . \] Ellipses and summation signs can't be used in the answer.

12.5.3 Olympic algebra

0 $

Problem: Let \( 1,4, \ldots, \ldots \) and \( 9,16, \ldots, \ldots \) be two arithmetic progressions. \( S \) is the set, obtained by the union of the first 2004 elements of each sequence. How many different elements are there in the set \( S \) ?

12.5.4 Olympic algebra

2.03 $

Problem: To what and for what \( \alpha \in \mathbb{R} \) does the sequence converge \[ x_{n}=\frac{n 2^{n}+\alpha^{n}}{(n+1) 2^{n}+(2 n+3) \alpha^{n}} ? \]

12.6.1 Higher mathematics

2.54 $

Problem: Prove that the sequence \( n \sin (n) \) has a bounded subsequence. Hint: \( \quad \) use the assertion \( \forall \alpha \in \mathbb{R}, \alpha>0 \), \( \forall M \in \mathbb{N} \exists p, q \in \mathbb{N}, \quad q>M, \quad|\alpha-p / q| \leq 1 / q^{2} \). In other words, any real number is approximated up to \( q^{-2} \) by an infinite set of rational numbers of the form \( p / q \).

12.6.2 Higher mathematics

3.81 $

Problem: Find all functions, differentiable on \( \mathbb{R} \), \( f: \mathbb{R} \rightarrow[1 ;+\infty) \) for which \[ \int_{1}^{f(x)} e^{u^{2}} d u=\int_{0}^{x} \frac{u d u}{f(u)}, \quad \forall x \in \mathbb{R} . \]

12.6.3 Higher mathematics

5.09 $

Problem: Find all such \( f \in C^{2}[0 ; 1] \) functions that 1) \( f(0)=f^{\prime}(0)=1 \), 2) \( f^{\prime \prime}(x) \geq 0, \forall x \in(0 ; 1) \), 3) \( \int_{0}^{1} f(x) d x=\frac{3}{2} \).

12.6.4 Higher mathematics

5.09 $

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