MathDB
Problems
Contests
National and Regional Contests
Netherlands Contests
Dutch Mathematical Olympiad
2003 Dutch Mathematical Olympiad
5
5
Part of
2003 Dutch Mathematical Olympiad
Problems
(1)
card game, replace a and b cards with (a+b+ab) card
Source: Dutch NMO 2003 p5
9/21/2019
There are a number of cards on a table. A number is written on each card. The "pick and replace" operation involves the following: two random cards are taken from the table and replaced by one new card. If the numbers
a
a
a
and
b
b
b
appear on the two packed cards, the number
a
+
b
+
a
b
a + b + ab
a
+
b
+
ab
is set on the new card. If we start with ten cards with the numbers
1
,
2
,
3
,
4
,
5
,
6
,
7
,
8
,
9
1, 2, 3, 4, 5, 6, 7, 8, 9
1
,
2
,
3
,
4
,
5
,
6
,
7
,
8
,
9
and
10
10
10
respectively, what value(s) can the number have that "grab and replace" nine times is on the only card still on the table? Prove your answer
combinatorics