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Convert the binary number 10011 into decimal, and convert the decimal number 28 into binary. Multiply by 2 as binary numbers both 10011 and the result of conversion of 28 into binary numbers.

The ternary numeral system has only 3 digits: 0, 1 and 2. Therefore the number 3 is written in ternary as 10. Write down the numbers 23 and 156 in ternary and add them as ternary.

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.

Two lines CD and CB are tangent to a circle with the center A and radius R, see the picture. The angle BCD equals 120. 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 CED=AEB=30. 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 DCE=120. 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.