Problems

Prove that the tangent to the graph of the function $f(x)$, constructed at coordinates $(x_0,f(x_0))$ intersects the $Ox$ axis at the coordinate: $x_0-\frac{f(x_0)}{f^{\prime }(x_0)}$.

Prove that $\sqrt{\frac{a^2+b^2}{2}}\ge \frac{a+b}{2}$.

Prove that the equation $\frac{x}{y}+\frac{y}{z}+\frac{z}{x}=1$ is unsolvable using positive integers.

Prove that if the function $f(x)$ is convex upwards on the line $[a,b]$, then for any distinct points $x_1,x_2$ in $[a;b]$ and for any positive ${\alpha }_1,{\alpha }_2$ such that ${\alpha }_1+{\alpha }_2=1$ the following inequality holds: $f({\alpha }_1{x}_1+{\alpha }_2{x}_2)>{\alpha }_{1}f(x_1)+{\alpha }_{2}f(x_2)$.

Let the sequences of numbers $\{a_n\}$ and $\{b_n\}$, that are associated with the relation $\mathrm{\Delta }{b}_n=a_n$ ($n=1,2,\dots$), be given. How are the partial sums $S_n$ of the sequence $\{a_n\}$ $S_n=a_1+a_2+\cdots +a_n$ linked to the sequence $\{b_n\}$?

Definition. Let the function $f(x,y)$ be valid at all points of a plane with integer coordinates. We call a function $f(x,y)$ harmonic if its value at each point is equal to the arithmetic mean of the values of the function at four neighbouring points, that is: $$f(x,y)=1/4(f(x+1,y)+f(x-1,y)+f(x,y+1)+f(x,y-1)).$$ Let $f(x,y)$ and $g(x,y)$ be harmonic functions. Prove that for any $a$ and $b$ the function $af(x,y)+bg(x,y)$ is also harmonic.

Let $f(x,y)$ be a harmonic function. Prove that the functions ${\mathrm{\Delta }}_{x}f(x,y)=f(x+1,y)-f(x,y)$ and ${\mathrm{\Delta }}_{y}f(x,y)=f(x,y+1)-f(x,y)$ will also be harmonic.

Definition. The sequence of numbers $a_0,a_1,\dots ,a_n,\dots$, which, with the given $p$ and $q$, satisfies the relation $a_{n+2}=p{a}_{n+1}+q{a}_n$ ($n=0,1,2,\dots$) is called a linear recurrent sequence of the second order.

The equation $$x^2-px-q=0$$ is called a characteristic equation of the sequence $\{a_n\}$.

Prove that, if the numbers $a_0$, $a_1$ are fixed, then all of the other terms of the sequence $\{a_n\}$ are uniquely determined.

The frog jumps over the vertices of the hexagon $ABCDEF$, each time moving to one of the neighbouring vertices.

a) How many ways can it get from $A$ to $C$ in $n$ jumps?

b) The same question, but on condition that it cannot jump to $D$?

c) Let the frog’s path begin at the vertex $A$, and at the vertex $D$ there is a mine. Every second it makes another jump. What is the probability that it will still be alive in $n$ seconds?

d)* What is the average life expectancy of such frogs?

Prove that the 13th day of the month is more likely to occur on a Friday than on other days of the week. It is assumed that we live in the Gregorian style calendar.