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In the \(4 \times 4\) square, the cells in the left half are painted black, and the rest – in white. In one go, it is allowed to repaint all cells inside any rectangle in the opposite colour. How, in three goes, can one repaint the cells to get the board to look like a chessboard?

The product of two natural numbers, each of which is not divisible by 10, is equal to 1000. Find the sum of these two numbers.

An old analogue clock speeds up by 9 minutes after 24 hours. If you went to sleep at 22:00 and set the correct time on the clock, then for what time should the alarm be set if you want it to go off at exactly 6:00? Explain your answer.

A digital clock shows the time in hours and minutes (for example, 16:15). While practising his counting, Pinocchio finds the sum of all the numbers on this clock (for example, \(1+6+1+5=13\)). Find the time at which the sum of these numbers will be at its maximum.

Prove that if the numbers \(x, y, z\) satisfy the following system of equations for some values of \(p\) and \(q\): \[\begin{aligned} y &= x^2 + px + q,\\ z &= y^2 + py + q,\\ x &= z^2 + pz + q, \end{aligned}\] then the inequality \(x^2y + y^2z + z^2x \geq x^2z + y^2x + z^2y\) is satisfied.

On the \(xy\)-plane shown below is the graph of the function \(y=ax^2 +c\). At which points does the graph of the function \(y=cx+a\) intersect the \(x\) and \(y\) axes?

Does there exist a natural number which, when divided by the sum of its digits, gives a quotient and remainder both equal to the number 2011?

Along the path between Fiona’s and Jane’s house there is a row of flowers: 15 peonies and 15 tulips in a random order. Before visiting Fiona’s house, Jane started watering all of the plants from the beginning of the row. After the 10th tulip the water finished and 10 flowers were left unwatered. The next day, before visiting Jane’s house, Fiona started picking flowers for a bouquet starting from the end of the row. After picking the 6th tulip, she decided that the bouquet was big enough. How many flowers were left growing beside the path?