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In the arithmetic puzzle different letters denote different digits and the same letters denote the same digit. \[P.Z + T.C + D.R + O.B + E.Y\] It turned out that all five terms are not integers, but the sum itself is an integer. Find the sum of the expression. For each possible answer, write one example with these five terms. Explain why other numbers cannot be obtained.

Peter came to the Museum of Modern Art and saw a square painting in a frame of an unusual shape, consisting of \(21\) equal triangles. Peter was interested in what the angles of these triangles were equal to. Help him find them.

Red, blue and green chameleons live on the island, one day \(35\) chameleons stood in a circle. A minute later, they all changed color at the same time, each changed into the color of one of their neighbours. A minute later, everyone again changed the colors at the same time into the color of one of their neighbours. Could it turn out that each chameleon turned red, blue, and green at some point?

Is it possible to paint \(15\) segments in the picture below in three colours in such a way, that no three segments of the same colour have a common end?

In the \(n\times n\) table, the two opposite corner squares are black and the rest are white. Find the smallest number of white cells that is enough to be repainted black in order to make all the cells of the table black with only there transformations: repaint all the cells of one column, or all the cells of one row into the opposite colour.

The monkey becomes happy when they eat three different fruits. What is the largest number of monkeys that can get happy with \(20\) pears, \(30\) bananas, \(40\) peaches, and \(50\) tangerines?

A useful common problem-solving strategy is to divide a problem into cases. We can divide the problem into familiar and unfamiliar cases; easy and difficult cases; typical and extreme cases etc. The division is sometimes suggested by the problem, but oftentimes requires a bit of work first.

If you are stuck on a problem or you are not sure where to begin, gathering data by trying out easy or typical cases first might help you with the following (this list is not exhaustive):

Gaining intuition of the problem

Isolating the difficulties

Quantifying progress on the problem

Setting up or completing inductive arguments

Let us take a look at this strategy in action.

For any real number \(x\), the absolute value of \(x\), written \(\left| x \right|\), is define to be \(x\) if \(x>0\) and \(-x\) if \(x \leq 0\). What is \(\left| 3 \right|\), \(\left| -4.3 \right|\) and \(\left| 0 \right|\)?

Prove that for any real number \(x\), \(x \leq \left| x \right|\) and \(0 \leq \left| x \right|\). Then prove that for any real numbers \(x,y\), the triangle inequality holds: \(\left| x+y \right| \leq \left| x \right|+\left| y \right|\).