Problems

Vincent would like to colour the entire \(3\)-dimensional space in such a way that no two points distance \(1\) apart are of the same colour. Prove that he needs at least \(5\) colours.

Just like atoms are the building blocks of molecules, prime numbers are the building blocks of integers, or whole numbers. What do we mean by this?

Well, every molecule is composed of atoms, and each atom is an atom of a particular element, like carbon or nitrogen. Similarly, every positive integer (except \(1\)) can be broken down into prime numbers. We can say this formally as follows:

The Fundamental Theorem of Arithmetic says that any natural number greater than \(1\) can be uniquely expressed as a product of prime numbers in non-decreasing order. For example: \[630=2\times3\times3\times5\times7=2\times3^2\times5\times7.\]

Recall that a number is composite if it is a product of two smaller, natural numbers. For example, \(6 = 2\times3\). Otherwise, and if the number is not equal to \(1\), it is called prime. The number \(1\) is neither prime nor composite.

Modulo operation: We look at division. For example \(41=2\times15+11\) is the division of \(41\) (the dividend) by \(15\) (the divisor) with remainder \(11\), and \(5=0\times7+5\) is the division of \(5\) by \(7\) with the remainder \(5\). More generally, when we divide \(a\) by \(b\), we’re looking for non-negative integers \(c\) and \(d\) (\(d<b\)) such that \(a=c\times b+d\).
In the case of \(45\) divided by \(15\), we get \(3\) with remainder \(0\) - in which case we say “\(15\) divides \(45\)", or “\(45\) is divisible by \(15\)". We can write this as \(15|45\).

We can deduce from the Fundamental Theorem of Arithmetic that if a product of two natural numbers is divisible by a prime number, then one of these numbers is divisible by this prime number. For example, \(7|7007=13\times539\) tells us that \(7\) divides \(13\) or \(539\). Clearly \(7\nmid13\), so we know \(7|539\).

It is often the case in geometric situations that figures look very similar, but not quite equal. Two polygons on a plane are called similar, if and only if ALL their corresponding angles are equal AND the ratio between ALL the corresponding sides is the same.
The relation between the corresponding sides, in our case it is \(\frac{AB}{IH}\) is called the similarity coefficient between the figures. It is common practice to write vertices of similar figures in the order that respects the similarity.

We will derive some criterions for determining similarity and apply them to solve problems. This means the statement of some examples and problems may be helpful for problems coming after them!

Let \(ABC\) and \(DEF\) be such triangles that angles \(\angle ABC = \angle DEF\), \(\angle ACB = \angle DFE\). Prove that the triangles \(ABC\) and \(DEF\) are similar.

The medians \(AD\) and \(BE\) of the triangle \(ABC\) intersect at the point \(F\). Prove that the triangles \(AFB\) and \(DFE\) are similar. What is their similarity coefficient?

In a triangle \(\triangle ABC\), the angle \(\angle B = 90^{\circ}\) . The altitude from point \(B\) intersects \(AC\) at \(D\). We know the lengths \(AD = 9\) and \(CD = 25\). What is the length \(BD\)?

Let \(ABC\) and \(DEF\) be two triangles such that \(\angle ACB = \angle DFE\) and \(\frac{DF}{AC} = \frac{EF}{BC}\). Prove that triangles \(ABC\) and \(DEF\) are similar.

Let \(AA_1\) and \(BB_1\) be the medians of the triangle \(ABC\). Prove that triangles \(A_1B_1C\) and \(BAC\) are similar. What is the similarity coefficient?

Let \(AD\) and \(BE\) be the heights of the triangle \(ABC\), which intersect at the point \(F\). Prove that the triangles \(AFE\) and \(BFD\) are similar.

Let \(AD\) and \(BE\) be the heights of the triangle \(ABC\). Prove that triangles \(DEC\) and \(ABC\) are similar.