Roman numerals are a numeral system that originated in ancient Rome and remained the usual way of writing numbers throughout Europe well into the Late Middle Ages. Numbers are written as combinations of letters from the Latin alphabet, each letter with a fixed integer value:
I&V&X&L&C&D&M
1&5&10&50&100&500&1000
For example the first \(12\) numbers in Roman Numerals are written as: \(I,\,II,\, III,\, IV,\, V,\, VI,\, VII,\, VIII,\, IX,\, X,\, XI,\, XII\), where the notations \(IV\) and \(IX\) can be read as "one less than five" and "one less than ten" correspondingly. A number containing two or more decimal digits is built by appending the Roman numeral equivalent for each digit, from highest to lowest, as in the following examples: the current year \(2024\) as \(MMXXIV\), number \(17\) as \(XVII\) and number \(42\) as \(XLII\) or \(XXXXII\). Let’s see how to multiply Roman numerals by multiplying \(17\) and \(42\).
Write down in Roman numerals the numbers \(14\) and \(61\) and multiply them as Roman numerals.
Detective Nero Wolf investigates a crime. He’s got \(80\) people involved in the case, among whom one is a criminal and another is a witness to the crime (but it is not known who either of them are). Each day the detective may invite one or more of these \(80\) people, and if there is a witness among those invited, but not the perpetrator, the witness will report who the perpetrator is. Can the detective solve a case in \(12\) 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.
Generally, when a line intersects a circle, it creates two different points of intersection. However, sometimes there is only one point. In such case we say the line is tangent to the circle. For example on the picture below the line \(CD\) intersects the circle at two points \(D\) and \(E\) and the line \(CB\) is tangent to the circle. To solve the problems today we will need the following theorem.
Theorem: The radius \(AB\) is perpendicular to the tangent line \(BC\).
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 \(CD\).