Problems

A natural number \(p\) is called prime if the only natural divisors of \(p\) are \(1\) and \(p\). Prime numbers are building blocks of all the natural numbers in the sense of the The Fundamental Theorem of Arithmetic: for a positive integer \(n\) there exists a unique prime factorization (or prime decomposition) \[n = p_1^{a_1}p_2^{a_2}...p_r^{a_r}.\] Today we will explore how unusual prime numbers are.
Essentially there is only one way to write an integer number as a product of prime numbers, where some of the prime numbers in the product can appear multiple times.

A graph is a finite set of points, some of which are connected with line segments. The points of a graph are called vertices. The line segments are called edges. In this problem set we only consider graphs in which every pair of vertices is connected with one or zero edges.

In a mathematical problem, one may use vertices of a graph to represent objects in the problem, i.e. people, cities, airports, and edges of the graph represent relations between the objects such as mutual friendship, railways between cities, plane routes. As you will see in the examples below, representing the initial problem as a graph can considerably simplify the solution.

A graph is called Bipartite if it is possible to split all its vertices into two groups in such a way that there are no edges connecting vertices from the same group. Find out whic of the following graphs are bipartite and which are not:

Imagine you are a manager of a very special hotel, a hotel with an infinite number of rooms, where each room has a natural number on the door \(1,2,3,4,...\). Only one guest can stay in each room and in most cases the hotel will be initially full with no vacant rooms left.
You will have to deal with unusual situations that may occur.

Today we will solve some geometric problems using the triangle inequality. This is an inequality between the lengths of the sides of any triangle, or between the distances of any three points.

The shortest path between any two points \(A\) and \(B\) is a straight segment - every other path is longer. In particular, a path through another point, \(C\), is equal or longer. \[AC + BC \ge AB\] The triangle inequality says that the sum of lengths of any two sides of a triangle is always larger than the length of the third side. The inequality only becomes an equality if \(ABC\) is not actually a triangle and the point \(C\) lies on the segment from \(A\) to \(B\).

Even though it is a simple idea, it can be a really helpful tool in problem solving.

A set is a collection of objects of any specified kind, the objects are called elements or members, the objects in one set cannot repeat, namely \(\{1,2,3\}\) and \(\{1,2,2,2,3\}\) are identical sets. We denote a set by a capital letter \(A\), or \(B\) and write \(x\in A\) if \(x\) is an element of \(A\), and \(x\notin A\) if it is not. The notation \(A=\{a,b,c,...\}\) means that the set \(A\) consists of the elements \(a,b,c,...\). The empty or void set, \(\emptyset\), has no elements. If all elements of \(A\) are also in \(B\), then we call \(A\) a subset of \(B\) and we write \(A\subseteq B\). It is an axiom that the sets \(A\) and \(B\) are equal \(A=B\) if they have the same elements. Namely, \(A\) is a subset of \(B\) and \(B\) is a subset of \(A\) at the same time.
For any sets \(A\) and \(B\), we define their union \(A\cup B\), intersection \(A\cap B\), and the difference \(A-B\) as follows:

• the union \(A\cup B\) is the set of all elements that belong to \(A\) or \(B\);

• the intersection \(A\cap B\) is the set of elements that belong to both \(A\) and \(B\);

• the difference \(A-B\) consists of those \(x \in A\) that are do not belong to \(B\).

Sometimes it is useful to draw sets as Venn diagrams, on the diagram below the pink circle represents the set \(A\), the yellow circle represents the set \(B\), the orange part is the intersection \(A\cap B\), the pink part is \(A-B\), the yellow part is \(B-A\), and the whole picture is the union \(A\cup B\).

A circle is inscribed into the triangle \(ABC\) with sides \(BC=6, AC=10\) and \(AB= 12\). A line tangent to the circle intersects two longer sides of the triangle \(AB\) and \(AC\) at the points \(F\) and \(G\) respectively. Find the perimeter of the triangle \(AFG\).

Liam saw an unusual clock in the museum: the clock had no digits, and it’s not clear how the clock should be rotated. That is, we know that \(1\) is the next digit clockwise from \(12\), \(2\) is the next digit clockwise from \(1\), and so on. Moreover all the arrows (hour, minute, and second) have the same length, so it’s not clear which is which. What time does the clock show?

Two circles are tangent to each other and the smaller circle with the center \(A\) is located inside the larger circle with the center \(C\). The radii \(CD\) and \(CE\) are tangent to the smaller circle and the angle \(\angle DCE = 60^{\circ}\). Find the ratio of the radii of the circles.

For positive real numbers \(a,b,c\) prove the inequality: \[(a^2b + b^2c + c^2a)(ab^2 + bc^2 + ca^2)\geq 9a^2b^2c^2.\]