Problems

Four lines, intersecting at the point $D$, are tangent to two circles with a common center $A$ at the points $C,F$ and $B,E$. Prove that there exists a circle passing through all the points $A,B,C,D,E,F$.

A circle with center $A$ is inscribed into the triangle $CDE$, so that all the sides of the triangle are tangent to the circle. We know the lengths of the segments $ED=c,CD=a,EC=b$. The line $CD$ is tangent to the circle at the point $B$ - find the lengths of segments $BD$ and $BC$.

A circle with center $A$ is tangent to all the sides of the quadrilateral $FGHI$ at the points $B,C,D,E$. Prove that $FG+HI=GH+FI$.

Two circles with centres $A$ and $C$ are tangent to each other at the point $B$. Both circles are tangent to the sides of an angle with vertex $D$. It is known that the angle $\mathrm{\angle }EDF={60}^{\circ }$ and the radius of the smaller circle $AF=5$. Find the radius of the large circle.

Two circles with centres $A$ and $C$ are tangent to each other at the point $B$. Two points $D$ and $E$ are chosen on the circles in such a way that a segment $DE$ passes through the point $B$. Prove that the tangent line to one circle at the point $D$ is parallel to the tangent line to the other circle at the point $E$.

Is it true that if $a$ is a positive number, then $a^2\ge a$? What about $a^2+1\ge a$?

Show for positive $a$ and $b$ that $a^2+b^2\ge 2ab$.

Consider the following sum: $$\frac{1}{1\times 2}+\frac{1}{2\times 3}+\frac{1}{3\times 4}+\dots$$ Show that no matter how many terms it has, the sum will never be larger than $1$.

Is it true that if $b$ is a positive number, then $b^3+b^2\ge b$? What about $b^3+1\ge b$?

Show that if $a$ is positive, then $1+a\ge 2\sqrt{a}$.