Problems

Write down in Roman numerals the numbers \(14\) and \(61\) and multiply them as Roman numerals.

Detective Nero Wolf is investigating a crime. There are \(80\) people involved in the case. Among them, one is the criminal and another is the only witness to the crime (but the detective does not know who they are).

Each day, the detective may invite any group of these \(80\) people for questioning. If the group contains the witness but does not contain the criminal, then the witness will reveal who the criminal is. Otherwise, nothing happens.

Can the detective guarantee that he solves the case within \(12\) days?

A set includes weights weighing \(1\) gram, \(2\) grams, \(4\) grams, ... (all powers of the number \(2\)), and in the set some of the weights might be the same. Weights were placed on two cups of the scales so that the scales are in balance. It is known that on the left cup, all weights are different. Prove that there are as many weights on the right cup as there are on the left.

Two lines \(CD\) and \(CB\) are tangent to a circle with the center \(A\) and radius \(R\), see the picture. The angle \(\angle BCD\) equals \(120^{\circ}\). Find the length of \(BD\) in terms of \(R\).

Given two circles, one has centre \(A\) and radius \(r\), another has centre \(C\) and radius \(R\). Both circles are tangent to a line at the points \(B\) and \(D\) respectively and the angles \(\angle CED = \angle AEB = 30^{\circ}\). Find the length of \(AC\) in terms of \(r\) and \(R\).

Consider a triangle \(CDE\). The lines \(CD\), \(DE\), and \(CE\) are tangent to a circle with centre \(A\) at the points \(F,G\), and \(B\) respectively. We also have that the angle \(\angle DCE = 120^{\circ}\). Prove that the length of the segment \(AC\) equals the perimeter of the triangle \(CDE\).

A circle with center \(A\) is tangent to the lines \(CB\) and \(CD\), see picture. Find the angles of the triangle \(BCD\) if \(BD=BC\).

Take two circles with a common centre \(A\). A chord \(CD\) of the bigger circle is tangent to the smaller one at the point \(B\). Prove that \(B\) is the midpoint of \(CD\).

Prove that the lines tangent to a circle in two opposite points of a diameter are parallel.

\(CD\) is a chord of a circle with centre \(A\). The line \(CD\) is parallel to the tangent to the circle at the point \(B\). Prove that the triangle \(BCD\) is isosceles.