[/quote]
,
such that 
is an integer for every positive integer 
,
be all the positive integers smaller than 
such that![\[gcd(N, c_i + c_{i + 1}) \neq 1\]](http://latex.artofproblemsolving.com/1/0/7/1075290d442e8c1f54237aad8ad07126f455f1f1.png)
for all 
.
of the triangle 
is extended beyond 
to 
so that 
.
is extended beyond 
so that 
.
,
,
,
be positive real numbers such that 
.![$$\sqrt[3]{\frac{a^3+b^3}{2}}+\sqrt[3]{\frac{b^3+c^3}{2}}+\sqrt[3]{\frac{c^3+a^3}{2}}\leqslant a^2+b^2+c^2.$$](http://latex.artofproblemsolving.com/5/a/b/5ab64bd6c09363206f936a4a49a9ac7968218600.png)
be all the positive integers smaller than 
for all 
satisfied 
Find minimum and maximum of 
of positive integers is called central if for every positive integer 
,
terms of the sequence is equal to 
,
,
,
of positive integers such that for every central sequence 
,
,
,
.
and 
.
and 
and 
,
is acute, and let 
be its orthocentre. Points 
and 
lie on lines 
and 
,
and 
and 
and 
and 
.![\[
\frac{1}{x} + \frac{1}{y} + 2 \sqrt{\frac{2}{x^2 + y^2}} + 4 \geq 4 \left( \sqrt{\frac{2}{x^2 + 1}} + \sqrt{\frac{2}{y^2 + 1}} \right).
\]](http://latex.artofproblemsolving.com/1/b/8/1b8ab2e343a3043271f5a392d32b12aca748f1b9.png)
.
,
,
are concurrent and the point of concurrence is 
,
is written in base 
,
.
for Girls, PRISMS Online Math Meet for Girls, is happening this spring! 
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 
,
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
