On a plane \(n\) randomly placed lines are given. What is the number of triangles formed by them?
On two parallel lines \(a\) and \(b\), the points \(A_1, A_2, \dots , A_m\) and \(B_1, B_2, \dots , B_n\) are chosen, respectively, and all of the segments of the form \(A_iB_j\), where \(1 \leq i \leq m\), \(1 \leq j \leq n\). How many intersection points will there be if it is known that no three of these segments intersect at one point?
Find \(m\) and \(n\) knowing the relation \(\binom{n+1}{m+1}: \binom{n+1}{m}:\binom{n+1}{m-1} = 5:5:3\).
Tickets cost 50 cents, and \(2n\) buyers stand in line at a cash register. Half of them have one dollar, the rest – 50 cents. The cashier starts selling tickets without having any money. How many different orders of people can there be in the queue, such that the cashier can always give change?
Prove that the Catalan numbers satisfy the recurrence relationship \(C_n = C_0C_{n-1} + C_1C_{n-2} + \dots + C_{n-1}C_0\). The definition of the Catalan numbers \(C_n\) is given in the handbook.
A square grid on the plane and a triangle with vertices at the nodes of the grid are given. Prove that the tangent of any angle in the triangle is a rational number.
Prove that there is at most one point of an integer lattice on a circle with centre at \((\sqrt 2, \sqrt 3)\).
\(N\) points are given, no three of which lie on one line. Each two of these points are connected by a segment, and each segment is coloured in one of the \(k\) colours. Prove that if \(N > \lfloor k!e\rfloor\), then among these points one can choose three such that all sides of the triangle formed by them will be colored in one colour.
There are 4 weights and scales. How many loads that are different by weight can be accurately weighed using these weights, if
a) weights can be placed only on one side of the scales;
b) weights can be placed on both sides of the scales?
Peter marks several cells on a \(5 \times 5\) board. His friend, Richard, will win if he can cover all of these cells with non-overlapping corners of three squares, that do not overlap with the border of the square (you can only place the corners on the squares). What is the smallest number of cells that Peter should mark so that Richard cannot win?