A \(7 \times 7\) square was tiled using \(1 \times 3\) rectangular blocks in such a way that one of the squares has not been covered. Find all the squares that could be left without being covered.
On a \(9\times 9\) board \(65\) bugs are placed in the centers of some of the squares. The bugs start moving at the same time and speed to a square that shares a side with the one they were in. When they reach the center of that square, they make a \(90\) degrees turn and keep walking (without leaving the board). Prove that at some moment of time there are two bugs in the same square. Note: When they turn it can be either to the right or to the left.
In an \(n \times n\) board the squares are painted black or white in some way. Three of the squares in the corners are white and one is black. Show that there is a \(2\times 2\) square with an odd number of white unit squares.
On an \(8\times 8\) board there is a lamp in every square. Initially every lamp is turned off. In a move we choose a lamp and a direction (it can be the vertical direction or the horizontal one) and change the state of that lamp and all its neighbours in that direction. After a certain number of moves, there is exactly one lamp turned on. Find all the possible positions of that lamp.
Given a natural number \(n\) you are allowed to perform two operations: "double up", namely get \(2n\) from \(n\), and "increase by \(1\)", i.e. to get \(n+1\) from \(n\). Find the smallest amount of operations one needs to perform to get the number \(n\) from \(1\).
A set includes weights weighing \(1\) gram, \(2\) grams, \(4\) grams, ... (all powers of the number \(2\)), and in the set some of the weights might be the same. Weights were placed on two cups of the scales so that the scales are in balance. It is known that on the left cup, all weights are different. Prove that there are as many weights on the right cup as there are on the left.
\(CD\) is a chord of a circle with centre \(A\). The line \(CD\) is parallel to the tangent to the circle at the point \(B\). Prove that the triangle \(BCD\) is isosceles.
Four lines, intersecting at the point \(D\), are tangent to two circles with a common center \(A\) at the points \(C,F\) and \(B,E\). Prove that there exists a circle passing through all the points \(A,B,C,D,E,F\).
A circle with center \(A\) is inscribed into the triangle \(CDE\), so that all the sides of the triangle are tangent to the circle. We know the lengths of the segments \(ED=c, CD=a, EC=b\). The line \(CD\) is tangent to the circle at the point \(B\) - find the lengths of segments \(BD\) and \(BC\).
A circle with center \(A\) is tangent to all the sides of the quadrilateral \(FGHI\) at the points \(B,C,D,E\). Prove that \(FG+HI = GH+FI\).