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#### Examples and counterexamples. Constructive proofs , Pigeonhole principle (other) , Theory of algorithms (other)

The function F is given on the whole real axis, and for each x the equality holds: F $(x + 1)$ F $(x)$ + F $(x + 1)$ + 1 = 0.
Prove that the function F can not be continuous.

#### Painting problems , Pigeonhole principle (other) , Proof by contradiction , Tables and tournaments (other)

Four outwardly identical coins weigh 1, 2, 3 and 4 grams respectively.
Is it possible to find out in four weighings on a set of scales without weights, which one weighs how much?

#### Painting problems , Pigeonhole principle (other) , Proof by contradiction , Tables and tournaments (other)

Replace the letters with numbers $($ all digits must be different $)$ so that the correct equality is obtained: A/ B/ C + D/ E/ F + G/ H/ I = 1.

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