Problems

The circles ${\sigma }_1$ and ${\sigma }_2$ intersect at points $A$ and $B$. At the point $A$ to ${\sigma }_1$ and ${\sigma }_2$, respectively, the tangents $l_1$ and $l_2$ are drawn. The points $T_1$ and $T_2$ are chosen respectively on the circles ${\sigma }_1$ and ${\sigma }_2$ so that the angular measures of the arcs $T_{1}A$ and $A{T}_2$ are equal (the arc value of the circle is considered in the clockwise direction). The tangent $t_1$ at the point $T_1$ to the circle ${\sigma }_1$ intersects $l_2$ at the point $M_1$. Similarly, the tangent $t_2$ at the point $T_2$ to the circle ${\sigma }_2$ intersects $l_1$ at the point $M_2$. Prove that the midpoints of the segments $M_1{M}_2$ are on the same line, independent of the positions of the points $T_1,T_2$.

The quadratic trinomials $f(x)$ and $g(x)$ are such that $f^{\prime }(x)g^{\prime }(x)\ge |f(x)|+|g(x)|$ for all real $x$. Prove that the product $f(x)g(x)$ is equal to the square of some trinomial.

When water is drained from a pool, the water level $h$ in it varies depending on the time $t$ according to the function $h(t)=a{t}^2+bt+c$, and at the time $t_0$ of when the draining is ending, the equalities $h(t_0)=h^{\prime }(t_0)=0$ are satisfied. For how many hours does the pool drain completely, if in the first hour the water level in it is reduced by half?

Prove that the root a of the polynomial $P(x)$ has multiplicity greater than 1 if and only if $P(a)=0$ and $P^{\prime }(a)=0$.

For a given polynomial $P(x)$ we describe a method that allows us to construct a polynomial $R(x)$ that has the same roots as $P(x)$, but all multiplicities of 1. Set $Q(x)=(P(x),P^{\prime }(x))$ and $R(x)=P(x)Q^{-1}(x)$. Prove that

a) all the roots of the polynomial $P(x)$ are the roots of $R(x)$;

b) the polynomial $R(x)$ has no multiple roots.

Construct the polynomial $R(x)$ from the problem 61019 if:

a) $P(x)=x^6-6x^4-4x^3+9x^2+12x+4$;

b)$P(x)=x^5+x^4-2x^3-2x^2+x+1$.

Prove that the following polynomial does not have any identical roots: $P(x)=1+x+x^2/2!+\cdots +x^n/n!$

For which $A$ and $B$ does the polynomial $A{x}^{n+1}+B{x}^n+1$ have the number $x=1$ at least two times as its root?

Prove that the polynomial $x^{2n}-n{x}^{n+1}+n{x}^{n-1}-1$ for $n>1$ has a triple root of $x=1$.

Prove that for $n>0$ the polynomial $n{x}^{n+1}-(n+1)x^n+1$ is divisible by $(x-1{)}^2$.