Difference between revisions of "2020 IMO Problems/Problem 5"
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Contradiction: Let us assume this is not true and for a certain n, there are k distinct positive integers which can be written in ascending order as follows : | Contradiction: Let us assume this is not true and for a certain n, there are k distinct positive integers which can be written in ascending order as follows : | ||
| − | z<sub>k</sub> > z<sub>k-1</sub> | + | z<sub>k</sub> > z<sub>k-1</sub> > z<sub>k-2</sub> > … > > z<sub>1</sub> |
| + | |||
| + | Since z<sub>k</sub> is the largest of the numbers, it has to be greater than 1. This implies that there will be a prime p<sub>1</sub> that divides z<sub>k</sub>. Now we know that the arithmetic mean of z<sub>k</sub> and > z<sub>k-1</sub> is more than > z<sub>k-1</sub>, thus the geometric mean which it is equivalent to must include the term > z<sub>k</sub>. | ||
== Video solution == | == Video solution == | ||
Revision as of 23:10, 12 August 2025
Contents
Problem
A deck of 
cards is given. A positive integer is written on each card. The deck has the property that the arithmetic mean of the numbers on each pair of cards is also the geometric mean of the numbers on some collection of one or more cards.
For which 
does it follow that the numbers on the cards are all equal?
Solution 1
Claim : For all n > 1, all numbers must be equal
Contradiction: Let us assume this is not true and for a certain n, there are k distinct positive integers which can be written in ascending order as follows :
zk > zk-1 > zk-2 > … > > z1
Since zk is the largest of the numbers, it has to be greater than 1. This implies that there will be a prime p1 that divides zk. Now we know that the arithmetic mean of zk and > zk-1 is more than > zk-1, thus the geometric mean which it is equivalent to must include the term > zk.
Video solution
https://www.youtube.com/watch?v=dTqwOoSfaAA [video covers all day 2 problems]
See Also
| 2020 IMO (Problems) • Resources | ||
| Preceded by Problem 4 |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 6 |
| All IMO Problems and Solutions | ||
