PROBLEM 4. On a board, 20 white circles, 21 red circles, and 22 green circles are drawn. Two circles of different colors are erased and a circle of the third color is drawn in their place. This operation is repeated until only one circle remains on the board. Specify the color of the remaining circle. Justify your answer.[^0]

## NATIONAL MATHEMATICS OLYMPIAD  Local stage - 5.03.2016  GRADING SCALE - 5th Grade
PROBLEM 4. On a board, 20 white circles, 21 red circles, and 22 green circles are drawn. Two circles of different colors are erased and a circle of the third color is drawn in their place. This operation is repeated until only one circle remains on the board. Specify the color of the remaining circle. Justify your answer.

Grading rubric. We will illustrate the changes that occur with each operation, as follows:

$(2 p)$

Initially: even odd even

(2p) After the first operation: odd even odd

After the second operation: even odd even

e.t.c.

(3p) To have one circle left on the board, it is necessary to have $0,0,1$ (not necessarily in this order), i.e., two even numbers and one odd number. In conclusion, the remaining circle is a red one.[^1]


[^0]:    ${ }^{1}$ The actual working time is 2 hours;

    ${ }^{2}$ All problems are mandatory;

    ${ }^{3}$ Each problem is graded from 0 to 7 .

[^1]:    ${ }^{1}$ Each grader awards an integer number of points;

    ${ }^{2}$ Any other correct solution is graded accordingly.