Circle Second (Lesson1) – We Solve Problems

Circle Second (Lesson1)

Circle name: Circle Second
Lesson name: Lesson1
Starts at : 14.05.2020 09:00

Problems:

1.

Painting problems , Pigeonhole principle (other) , Proof by contradiction , Tables and tournaments (other) 13-15

The cells of a $15 \times 15$ square table are painted red, blue and green.
Prove that there are two lines which at least have the same number of cells of one colour.

2.

13-15

In the US, it is customary to record the date as follows: the number of the month, then the number of the day and then the year. In Europe, the number comes first, then the month and then the year. How many days are there in the year, the date of which can be read definitively, without knowing how it was written?

4.

Visual geometry in space 12-14

Propose a method for measuring the diagonal of a conventional brick, which is easily realied in practice $($without the Pythagorean theorem$)$.

5.

Equations of higher order (other) , Integer and fractional parts. Archimedean property 13-15

Solve the equation [$x^3$] + [$x^2$] + [x] = {x} – 1.

6.

Integer lattices (other) , Pigeonhole principle (finite number of poits, lines etc.) 13-15

Inside a square with side 1 there are several circles, the sum of the radii of which is 0.51. Prove that there is a line that is parallel to one side of the square and that intersects at least 2 circles.

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