PROBLEM 4. We say that a number of the form $\overline{0, a b c d e}$ has property $(P)$ if the digits $a, b, c, d, e$ belong to the set $\{4,6\}$.

a) Show that the solution to the equation $x+0,46646=1,1111$ has property $(P)$;

b) Determine how many different numbers of the form $\overline{0, a b c d e}$ have property $(P)$;

c) Show that from any 17 different numbers of the form $\overline{0, a b c d e}$ that have property $(P)$, two can be chosen whose sum is 1,1111 .[^0]

## NATIONAL MATHEMATICS OLYMPIAD  Local stage - 5.03. 2016  GRADING GUIDE - 6th Grade
PROBLEM 4. We say that a number of the form $\overline{0, a b c d e}$ has property $(P)$ if the digits $a, b, c, d, e$ belong to the set $\{4,6\}$.

a) Show that the solution to the equation $x+0,46646=1,1111$ has property $(P)$;

b) Determine how many different numbers of the form $\overline{0, a b c d e}$ have property $(P)$;

c) Show that from any 17 different numbers of the form $\overline{0, a b c d e}$ that have property $(P)$, two can be chosen whose sum is 1,1111.

## Grading Rubric.

(2p) a) The solution to the equation is $x=1,1111-0,46646=0,64464$, which has property $(P)$;

$(2 p)$ b) For each digit $a, b, c, d, e$ there are two possibilities for selection; with 5 digits, we have $2^{5}=32$ numbers with property $(P)$;

$(3 p) c$ ) Numbers with property $(P)$ can be grouped into pairs of the form $\overline{0, a_{1} a_{2} a_{3} a_{4} a_{5}}$ and $\overline{0, b_{1} b_{2} b_{3} b_{4} b_{5}}$ such that

$$
a_{1}+b_{1}=a_{2}+b_{2}=\ldots=a_{5}+b_{5}=10
$$

The sum of two such numbers from the 16 pairs is 1,1111; thus, with 17 numbers, there exist two numbers that form such a pair.[^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.