Problems

Prove that the product of five consecutive integers is divisible (a) by \(30\); (b) by \(120\).

Prove that if \(n\) is a composite number, then \(n\) is divisible by some natural number \(x\) such that \(1 < x\leq \sqrt{n}\).

The natural numbers \(a,b,c,d\) are such that \(ab=cd\). Prove that the number \(a^{2023} + b^{2023} + c^{2023} + d^{2023}\) is composite.

Prove that for an arbitrary odd \(n = 2m - 1\) the sum \(S = 1^n + 2^n + ... + n^n\) is divisible by \(1 + 2 + ... + n = nm\).

Certain geometric objects nicely blend when they happen to be together in a problem, one possible example of such a pair of objects is a circle and an inscribed angle.
We will be using the following statements in the examples and problems:
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. Two triangles are pronounced "congruent" if ALL their corresponding sides and angles are equal.
We recommend solving the problems in this sheet in the order of appearance, as some problems use statements from previous problems as a step in the solution. Specifically, problem 6.2’s statement is required to solve every other problem that comes after it.

Let \(ABC\) and \(A_1B_1C_1\) be two triangles with the following properties: \(AB = A_1B_1\), \(AC = A_1C_1\), and angles \(\angle BAC = \angle B_1A_1C_1\). Then the triangles \(ABC\) and \(A_1B_1C_1\) are congruent.

In the triangle \(\triangle ABC\) the sides \(AC\) and \(BC\) are equal. Prove that the angles \(\angle CAB\) and \(\angle CBA\) are equal.

Point \(A\) is the centre of a circle and points \(B,C,D\) lie on that circle. The segment \(BD\) is a diameter of the circle. Show that \(\angle CAD = 2 \angle CBD\).

Let \(ABC\) be a triangle with given angles \(\angle BAC\) and \(\angle ABC\). What is the value of the angle \(\angle BCD\) in terms of \(\angle BAC\) and \(\angle ABC\)?

Point \(A\) is the centre of a circle and points \(B,C,D\) lie on that circle. Show that \(\angle CAD = 2 \angle CBD\).