Problems

The statement of the problem 9 can be generalised for higher dimensions, in particular for the \(3\)-dimensional space. Prove that in order to colour the entire \(3\)-dimensional space in such a way that no two points distance \(1\) apart are of the same colour Alex needs at least \(5\) colours.

Among all natural numbers we can distinguish prime and composite numbers.

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.

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: \[630=2\times3\times3\times5\times7=2\times3^2\times5\times7.\]

Modulo operation: Given any two natural numbers \(a\) and \(b\), called the dividend and the divisor respectively, we can divide \(a\) by \(b\) with the remainder. That is to find such non-negative integer numbers \(c\) and \(d\) (\(d<b\)), called the quotient and the remainder respectively, that \(a=c\times b+d\). For example \(41=2\times15+11\) is the division of 41 by 15 with the remainder 11, and \(5=0\times7+5\) is the division of 5 by 7 with the remainder 5.
If \(a\) is divided by \(b\) with zero remainder (without a remainder) we say that "\(a\) is divisible by \(b\)"\(\;\)or "\(b\) divides \(a\)". From the definition of modulo operation for \(a\) the property to be divisible by \(b\) is equivalent to the existence of non-negative integer \(c\) such that \(a=c\times b\). We denote it by \(b|a\) for "\(b\) divides \(a\)". For example \(7 \mid 105\) and \(9|111111111\) because \(105=15\times7\) and \(111111111=12345679\times9\).
We immediately 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.

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.

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.