Problems

In the triangle \(ABC\) the points \(D,E,F\) are chosen on the sides \(AB, BC, AC\) in such a way that \(\angle ADF = \angle BDE\), \(\angle AFD = \angle CFE\), \(\angle CEF = \angle BED\). Prove that the segments \(AE, BF, CD\) are the heights of the triangle \(ABC\).

There are real numbers written on each field of a \(m \times n\) chessboard. Some of them are negative, some are positive. In one move we can multiply all the numbers in one column or row by \(-1\). Is that always possible to obtain a chessboard where sums of numbers in each row and column are non-negative?

Tom found a large, old clock face and put \(12\) sweets on the number \(12\). Then he started to play a game: in each move he moves one sweet to the next number clockwise, and some other to the next number anticlockwise. Is it possible that after finite number of steps there is exactly \(1\) of the sweets on each number?

There are numbers \(1,2,3,4,5,6,7,8,9\) and \(10\) written on a board. Each time you make a "move" you can erase three of the remaining numbers, \(a,b\) and \(c\), and replace them with the numbers \(2a+b,2b+c\) and \(2c+a\). Is it possible to make all the \(10\) numbers left on the board equal?

On a certain island there are \(17\) grey, \(15\) brown and \(13\) crimson chameleons. If two chameleons of different colours meet, both of them change to the third colour. No other colour changes are allowed. Is it possible that after a few such colour transitions all the chameleons have the same colour?

Sixteen lightbulbs are arranged in a \(4 \times 4\) grid. Some of them are on, the other ones are off. You are allowed to change the state of all the bulbs in a column, in a row, or along any diagonal (note: there are \(14\) diagonals in total!). Is it possible to go from the arrangement on the left to the one on the right by repeating this operation?

The numbers \(1\) to \(2025\) are written on a board. In one move we can erase two numbers and replace them with the absolute value of their difference. Can we achieve a sequence consisting of only \(0\) after some number of moves?

In the quadrilateral \(ABCD\) the diagonals \(AC\) and \(BD\) intersect at the point \(E\). It is known that the perimeter of the triangle \(ABC\) is equal to the perimeter of the triangle \(ABD\), and the perimeter of the triangle \(ACD\) equals the perimeter of the triangle \(BCD\).

Prove that \(AE=BE\).

A magic square is a square filled with numbers, one in each cell, in such a way that the sums of the numbers in each row, each column and along each of the two main diagonals are the same. The value of this sum is known as the magic constant of the square. Show that in any \(4 \times 4\) magic square (which contains \(16\) numbers) the sum of all the numbers in the \(4\) central squares is also equal to the magic constant of the square.

Show that for any natural number \(n\geq 1\), the number \[n^4+4n^2+9\] is not a perfect square.