Problems

Geometry reminder

We call two polygons congruent if all their corresponding sides and angles are equal. Triangles are the easiest sort of polygons to deal with. Assume we are given two triangles $ABC$ and $A_1{B}_1{C}_1$ and we need to check whether they are congruent or not, some rules that help are:

• If all three corresponding sides of the triangles are equal, then the triangles are congruent.

• If, in the given triangles $ABC$ and $A_1{B}_1{C}_1$, two corresponding sides $AB=A_1{B}_1$, $AC=A_1{C}_1$ and the angles between them $\mathrm{\angle }BAC=\mathrm{\angle }{B}_1{A}_1{C}_1$ are equal, then the triangles are congruent.

• If the sides $AB=A_1{B}_1$ and pairs of the corresponding angles next to them $\mathrm{\angle }CAB=\mathrm{\angle }{C}_1{A}_1{B}_1$ and $\mathrm{\angle }CBA=\mathrm{\angle }{C}_1{B}_1{A}_1$ are equal, then the triangles are congruent.

The basic principles about parallel lines and general triangles are:
1. The supplementary angles (angles "hugging" a straight line) add up to ${180}^{\circ }$.
2. The sum of all internal angles of a triangle is also ${180}^{\circ }$.
3. A line cutting two parallel lines cuts them at the same angles (these are called corresponding angles).
4. In an isosceles triangle (which has two sides of equal lengths), two angles touching the third side are equal.

Today’s topic is inequalities, expressions like $a\ge b$, or $a>b$. There are certain rules for operating inequalities: one can subtract the same number from both sides of the inequality, namely if $a\ge b$, then $a-b\ge 0$. If $a\ge b$ and $b\ge c$, then $a\ge c$. If a number $c\ge 0$, then from $a\ge b$ it follows that $ac\ge bc$. However, in case of multiplication by a negative number $c\le 0$, the inequality sign reverses: from $a\ge b$ it follows that $ac\le bc$. One should also remember that the square of any real number is non-negative.

Coloring is a very neat technique in problems involving boards since it allows us to simplify the problem a great deal. The important part is focusing on an adequate subset of the squares, however doing it with colors is a lot easier.
The kinds of colorings can be very different and there is no general rule for determining which one is going to solve the problem. There are some colorings (such as a chessboard coloring) that are frequently used, but the only way to learn how to use this technique is by solving several problems of this style.
When the problem is related to pieces covering a certain figure, the “good colorings” are those that yield an invariant associated with the pieces. This can be the number of squares of one color they cover, the number of colors they may use, some parity argument, etc. Coloring is basically an illustrative way to describe invariants.

Today we will solve some problems about finding areas of geometric figures. All you need to know in order to solve every problem in this set is: to calculate the area of a triangle we multiply the length of a side by the length of a height to that side and divide by $2$, namely:$\frac{1}{2}AB\times CD$, as for rectangle we just multiply two adjacent sides ($EF\times GF$), and when we have a circle we calculate the area by $\pi r^2$, where $r$ is the radius of the circle.

Sometimes proof of a statement requires elaborate reasoning, but sometimes it enough to provide an example when the described construction works. Often enough the problem is asking whether an event is possible, or if an object exists under certain conditions making the existence seemingly unlikely, in such cases all you need to do is to provide an example to solve the problem. Today we will see how to construct such examples.

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.

One can hardly imagine modern life without numbers, but have you wondered when and how the numbers were invented? It turns out people started using numbers about $42000$ years BCE supposedly to mark the dates in calendar. But how do we represent the numbers in writing? Well, there are two ways: examples of the first abstract numeral systems are generally tallying systems, the ones where the value or contribution of a digit does not depend on its position, a good example is the famous Roman numeral system: $IVXLCDM$, here a digit has only one value: $I$ means one, $X$ means ten and $C$ a hundred. However, one might struggle to express large numbers in Roman system.
Majority of ancient civilisations, Sumerian, Egyptian, Babylonian, Chinese, Japanese, Indian used what is called positional numeral systems, where the contribution of a digit to the value of a number is the value of the digit multiplied by a factor determined by the position of the digit. All these systems, even when invented independently, have something in common, they are what is called "base-$10$".
Try to guess why do we use the decimal numeral system, which has exactly $10$ digits in our everyday use. Because it does not actually have to be $10$ digits, it could easily be $3,8,16$, the binary system (with only digits $0$ and $1$) is used in all electronic devices, since it is enough to represent any bit of information we might possibly know.

There exist various ways to prove mathematical statements, one of the possible methods, which might come handy in certain situations is called Proof by contradiction. To prove a statement we first assume that the statement is false and then deduce something that contradicts either the condition, or the assumption itself, or just common sense. Thereby concluding that the first assumption must have been wrong, so the statement is actually true.

In a lot of geometric problems the main idea is to find congruent figures. We call two polygons congruent if all their corresponding sides and angles are equal. Triangles are the easiest sort of polygons to deal with. Assume we are given two triangles $ABC$ and $A_1{B}_1{C}_1$ and we need to check whether they are congruent or not, some rules that help are:

• If all three corresponding sides of the triangles are equal, then the triangles are congruent.

• If, in the given triangles $ABC$ and $A_1{B}_1{C}_1$, two corresponding sides $AB=A_1{B}_1$, $AC=A_1{C}_1$ and the angles between them $\mathrm{\angle }BAC=\mathrm{\angle }{B}_1{A}_1{C}_1$ are equal, then the triangles are congruent.

• If the sides $AB=A_1{B}_1$ and pairs of the corresponding angles next to them $\mathrm{\angle }CAB=\mathrm{\angle }{C}_1{A}_1{B}_1$ and $\mathrm{\angle }CBA=\mathrm{\angle }{C}_1{B}_1{A}_1$ are equal, then the triangles are congruent.

At a previous geometry lesson we have derived these rules from the axioms of Euclidean geometry, so now we can just use them.

Today we will be solving problems using the pigeonhole principle. What is it? Simply put, we are asked to place pigeons in pigeonholes, but the number of pigeons is larger than the number of pigeonholes. No matter how we try to do that, at least one pigeonhole will have to contain at least 2 pigeons. By ”pigeonholes” we can mean any containers and by ”pigeons” we mean any items, which are placed in these containers. This is a simple observation, but it is helpful in solving some very difficult problems. Some of these problems might seem obvious or intuitively true. Pigeonhole principle is a useful way of formalising things that seem intuitive but can be difficult to describe mathematically.

There is also a more general version of the pigeonhole principle, where the number of pigeons is more than $k$ times larger than the number of pigeonholes. Then, by the same logic, there will be one pigeonhole containing $k+1$ pigeons or more.

A formal way to prove the pigeonhole principle is by contradiction - imagine what would happen if each pigeonhole contained only one pigeon? Well, the total number of pigeons could not be larger than the number of pigeonholes! What if each pigeonhole had $k$ pigeons or fewer? The total number of pigeons could be $k$ times larger than the number of pigeonholes, but not greater than that.