Problems

Prove that $\mathrm{\angle }ABC>{90}^{\circ }$ if and only if the point $B$ lies inside a circle with diameter $AC$.

The radii of two circles are $R$ and $r$, and the distance between their centres is equal $d$. Prove that these circles intersect if and only if $|R-r|<d<R+r$.

Prove that $(a+b-c)/2<m_c<(a+b)/2$, where $a$, $b$ and $c$ are the lengths of the sides of an arbitrary triangle and $m_c$ is the median to side $c$.

$a$, $b$ and $c$ are the lengths of the sides of an arbitrary triangle. Prove that $a=y+z$, $b=x+z$ and $c=x+y$, where $x$, $y$ and $z$ are positive numbers.

a, b and c are the lengths of the sides of an arbitrary triangle. Prove that $a^2+b^2+c^2<2(ab+bc+ca)$.

In a triangle, the lengths of two of the sides are 3.14 and 0.67. Find the length of the third side if it is known that it is an integer.

A triangle of area 1 with sides $a\le b\le c$ is given. Prove that $b\ge \sqrt{2}$.

In the quadrilateral $ABCD$, the angles $A$ and $B$ are equal, and $\mathrm{\angle }D>\mathrm{\angle }C$. Prove that $AD<BC$.

In the trapezoid $ABCD$, the angles at the base $AD$ satisfy the inequalities $\mathrm{\angle }A<\mathrm{\angle }D<{90}^{\circ }$. Prove that $AC>BD$.

Prove that if two opposite angles of a quadrilateral are obtuse, then the diagonal connecting the vertices of these angles is shorter than the other diagonal.