Problems
A circle with center \(A\) is tangent to all the sides of the quadrilateral \(FGHI\) at the points \(B,C,D,E\). Prove that \(FG+HI = GH+FI\).
Is it true that if \(a\) is a positive number, then \(a^2 \ge a\)? What about \(a^2 +1 \ge a\)?
The numbers \(a\), \(b\) and \(c\) are positive. By completing the square, show that \[\frac{a^2}4 + b^2 + c^2 \ge ab-ac+2bc.\]
Let \(m\) and \(n\) be natural numbers such that \(m>n\). Show that: \[\frac1{n^2} + \frac1{(n+1)^2} + \frac1{(n+2)^2} + \dots + \frac1{m^2} > \frac1{n} - \frac1{m}.\]
The numbers \(a,b,c\) are positive. Show that: \[\frac{ab}{c} + \frac{bc}{a} + \frac{ac}{b} \ge a +b+c.\]
Recall that a line is tangent to a circle if they have only one point of intersection, a circle is called inscribed in a polygon if it is tangent to every side as a segment of that polygon.
In the triangle \(CDE\) the angle \(\angle CDE = 90^{\circ}\) and the line \(DH\) is the median. A circle with center \(A\) is inscribed in the triangle \(CDH\) and is tangent to the segment \(DH\) in its middle, let’s denote it as \(G\), so \(GH=DG\). Find the angles of the triangle \(CDE\).
Recall that a line is tangent to a circle if they have only one point of intersection, a circle is called inscribed in a polygon if it is tangent to every side as a segment of that polygon.
In the triangle \(EFG\) the line \(EH\) is the median. Two circles with centres \(A\) and \(C\) are inscribed into triangles \(EFH\) and \(EGH\) respectively, they are tangent to the median \(EH\) at the points \(B\) and \(D\). Find the length of \(BD\) if \(EF-EG=2\).
Is it possible to cover a \((4n+2) \times (4n+2)\) board with the \(L\)-tetraminos without overlapping for any \(n\)? The pieces can be flipped and turned.
On the diagram each number denotes the area of a rectangle it is written into. What are the areas of the other rectangles?
