be a sequence of mutually distinct nonempty subsets of a set 
.
and 
are disjoint and their union is not the whole set 
and 
,
.
,
.
![\[ \left( a + 2b + \frac{2}{a + 1} \right) \left( b + 2a + \frac{2}{b + 1} \right) \geq 16 \]](http://latex.artofproblemsolving.com/c/f/4/cf4c56bfe5d059b15747969fb279181cc4e6b0fa.png)
and 
such that 
.
has the property that the sum of its first 
terms is divisible by 
.
.
.
has side length 
.
lies on segment 
such that line 
is perpendicular to line 
.
,
,
and 
.
,
,
.
is equal to 
,
be a triangle. Point 
lies on side 
,
lies on side 
,
lies on side 
such that 
,
,
.
be the foot of the altitude from 
to 
,
,![$[AQPR] = \tfrac{3}{7} \cdot [ABC]$](http://latex.artofproblemsolving.com/e/f/1/ef1f89d7ee2b45faac7265d8e2777ba3918ae16e.png)
,
can be expressed as 
and tells Bob that the three resulting remainders are 
,
,
satisfying the system of equations 
The value of 
can be expressed as 
,
and 
are positive integers such that 
.
and 
inclusive. We say a sequence 
in 
,
that are at least 
is 
.
,
denote the value of the binary number obtained by reading the binary representation of 
such that the equation 
has at least ten positive integer solutions 
such that 
and 
for 
.
.
such that 
and 
is a multiple of 
.
be positive real numbers such that 
.![\[ \frac{(x^2 + y)(x + y^2)}{x + y}. \]](http://latex.artofproblemsolving.com/8/f/8/8f81ce62be260aaac6858fdca6522a20f4986885.png)
be the number of five-element subsets that can be chosen from the set of the first 
natural numbers so that at least two of the five numbers are consecutive. Find the remainder when 
.
Given that 
is a prime, find the number of nonnegative integers 
such that 
where 
are positive integers, 
and 
,
.
is a prime number for every 
integer.
Y by
Hi everyone,
Recently I;ve started doing a lot of nice combo/algebra Olympiad problems(JMO, PAGMO, CMO, etc.) and I’ve got to say, it’s been pretty fun(I’m enjoying it!). I was wondering if doing Olympiad problems also helps increase computational abilities slightly. Currently I am doing 75% computational, 25% oly but if anyone has any expreience I want to switch it to 25% computational and 75% Olympiad, though I still want to have computational skills for ARML, AIME, SMT, BMT, HMMT, etc.
If anyone has any experience, please let me know!
Thank you so much in advance!
Recently I;ve started doing a lot of nice combo/algebra Olympiad problems(JMO, PAGMO, CMO, etc.) and I’ve got to say, it’s been pretty fun(I’m enjoying it!). I was wondering if doing Olympiad problems also helps increase computational abilities slightly. Currently I am doing 75% computational, 25% oly but if anyone has any expreience I want to switch it to 25% computational and 75% Olympiad, though I still want to have computational skills for ARML, AIME, SMT, BMT, HMMT, etc.
If anyone has any experience, please let me know!
Thank you so much in advance!
