In a square which has sides of length 1 there are 100 figures, the total area of which sums to more than 99. Prove that in the square there is a point which belongs to all of these figures.
A pentagon is inscribed in a circle of radius 1. Prove that the sum of the lengths of its sides and diagonals is less than 17.
Prove that the segment connecting the vertex of an isosceles triangle to a point lying on the base is no greater than the lateral side of the triangle.
In an isosceles triangle, the sides are equal to either 3 or 7. Which side length is the base?
Prove that \(S_{ABCD} \leq (AB \times BC + AD \times DC)/2\).
Prove that \(\angle ABC > 90^{\circ}\) 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 \(a^2 + b^2 + c^2 < 2 (ab + bc + ca)\).