differents subjects, each consist of 
students. Given that for any 
different subjects, there exists a student compete in both subjects. Prove that there exists a student who compete in at least 
different subjects.
,
and 
,
and 
denote the product and sum of the digits of 
for which the following statement holds: there exists at least one triple 
of integers such that
and all triples 
are integers.
students and a teacher with 
marbles. The teacher then play a Marble distribution according to the following rules. At the start, the teacher distributed all her marbles to students, so that each student receives at least 
marbles from the teacher. Then, the teacher chooses a student , who has never been chosen before, such that the number of marbles that he owns in a multiple of 
.
other students. The same goes on until the teacher is not able to choose anymore student.
Y by
Ivan bought 
cats consisting of five different breeds. He records the number of cats of each breed and after multiplying these five numbers he obtains the number 
. How many cats of each breed does he have?
cats consisting of five different breeds. He records the number of cats of each breed and after multiplying these five numbers he obtains the number 
. How many cats of each breed does he have?This post has been edited 5 times. Last edited by jl_, Apr 23, 2025, 10:21 AM
Reason: Change source
Reason: Change source
each.
denote the number of cats of each breed. We know that 
and 
.![$$10=\frac{a+b+c+d+e}{5} \geq \sqrt[5]{abcde}=10.$$](http://latex.artofproblemsolving.com/5/3/c/53c2e29741af54049854cd86e84fc52424715f55.png)
This is precisely the equality case, which can only happen when 
.
