Mathematical and Physical Journal
for High Schools
Issued by the MATFUND Foundation
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Problem K. 198. (January 2009)

K. 198. The inhabitants of a certain planet use not four but five basic arithmetic operations. The operations of addition, multiplication, subtraction and division are the same as ours, but they also have a special operation denoted by the operation sign @. We do not know exactly how this operation works, but we have found out that the following properties are valid for all x and y:

(a)  x \mathrel{@} 0 = x,

(b)  x \mathrel{@} y = y \mathrel{@} x,

(c)  (x+1) \mathrel{@} y = (x \mathrel{@} y) + y + 1.

What is the value of 12
\mathrel{@} 5 on the planet in question?

(6 pont)

Deadline expired on February 10, 2009.


Sorry, the solution is available only in Hungarian. Google translation

Megoldás. A művelet tulajdonságait felhasználva:

12@5=5@12=(4@12)+12+1=3@12+2.(12+1)=2@12+3.(12+1)=1@12+4.(12+1)=0@12+5.(12+1)=12@0+5.(12+1)=12+5.(12+1)=77.


Statistics:

136 students sent a solution.
6 points:92 students.
5 points:11 students.
4 points:2 students.
3 points:3 students.
1 point:5 students.
0 point:15 students.
Unfair, not evaluated:8 solutions.

Problems in Mathematics of KöMaL, January 2009