,
be the result.
,
is the Euler's totient theorem, which calculates the number of integers less than 
.
for 
.
be the remainder 
leaves when you divide it by 
.
be your predicted USA(J)MO score.![$\lfloor \frac{x + \sqrt[r]{r^2} + r \ln(r)}{r} \rfloor$](http://latex.artofproblemsolving.com/c/d/3/cd3f7d917b0ec944ec3017e1f737c538e1edd10e.png)
.
unless of your age can be written as 
.
Y by Soccerstar9, OronSH, akliu
A group of 
students from different countries meet at a mathematics competition. Each student speaks the same number of languages, and, for every pair of students 
and 
, student speaks some language that student does not speak, and student speaks some language that student does not speak. What is the least possible total number of languages spoken by all the students?
students from different countries meet at a mathematics competition. Each student speaks the same number of languages, and, for every pair of students 
and 
, student speaks some language that student does not speak, and student speaks some language that student does not speak. What is the least possible total number of languages spoken by all the students?
This post has been edited 3 times. Last edited by ccarolyn4, Dec 23, 2024, 11:31 PM
where n is the number of languages in total & k is the number of languages each student speaks, we need 
,
is the number of languages that each person speaks, then we need to have 
Note that 
so the answer is 
.
and 
are guaranteed to satisfy the problem condition as long as 
.
.
works.
at first) then I read the problem and it was 
so I just made an entire pascals triangle
