AoPS Academy
be positive integers, 
is even such that 
. Prove that there exist positive integers 
satisfying 
. [/center]
, and 
, 
. This is proven in solution 1a-1. We will refer to this as Trick 1.
does 
~![$f([x])$](http://latex.artofproblemsolving.com/d/d/6/dd69757bd098b197e75a45140479c40828f5c227.png)
hold?
. Is this a necessary condition?
then find the 
?
then find the ?
be a region and the function 
defined by 
is well-defined and analytic in 
show that 
, 
for all integers 
, show that for all integers 
,
, define 
as the smallest positive integer satisfying the following property: for any integer 
coprime with , we have 
.
, and integers 
all coprime with , prove that there exists a non-empty subset 
of 
such that![\[\prod_{i \in I} a_i \equiv 1 \pmod{n}.\]](http://latex.artofproblemsolving.com/8/f/a/8fadde1e14e28dda0cb2371b69e783fd7e572352.png)
Proposed by Zhenqian Peng from High School Affiliated to Renmin University of China
be a three times differentiable function (defined on 
and real-valued) such that has at least five distinct real zeros. Prove that 
has at least two distinct real zeros.
be a differentiable function. If 
, 
and 
for all 
. Which of the followings are necessarily true:
f is strictly increasing
![$f \colon [0,1] \to \mathbb{R} $](http://latex.artofproblemsolving.com/2/1/8/21859d37ed035abd07d0be112f49d52598fa477e.png)
be a differentiable function such that its derivative is an integrable function on ![$[0,1]$](http://latex.artofproblemsolving.com/e/8/6/e861e10e1c19918756b9c8b7717684593c63aeb8.png)
, and 
. Prove that ![\[ \int_0^1 (xf'(x))^2 dx \geq 12 \cdot \left( \int_0^1 xf(x) dx\right)^2 \]](http://latex.artofproblemsolving.com/2/3/e/23e33eed39fc1e20486bbb86261d3566e8fd2017.png)
has property 
if :![\[
\begin{cases}
\text{the set } A \text{ has at least } 4 \text{ elements} \\
\text{the element } 1+1 \text{ is invertible}\\
x+x^4=x^2+x^3 \text{ holds for all } x \in A
\end{cases}
\]](http://latex.artofproblemsolving.com/b/6/9/b6993ee76eb3792e08d441f5d5c463247c392c4a.png)
a) Prove that if a ring has property , and 
are distinct elements, such that 
and 
are units, then 
is also a unit, but 
is not a unit..
be a positive integer such that there exists exactly 26 pairs of positive integers 
to the equation 
with 
. Find the smallest possible value of .
" to "Find the smallest possible value of ". That's what I call "bad style" because it destroys the traceability of the thread.
Prove that
Prove that
such that 
for all 
Let's experiment for some small values of 
;
and thus we have the solutions 
,
solutions (ignoring distinguishability and the above condition). Thus, using the condition, there are 5 solutions that satisfy the original equation.
must have 51 divisors. Thus, the value of 
;
;
,
,
positive solutions 
without condition 
.
and symmetry we get
,
.
. 
![\[ \frac 1x + \frac 1y = \frac 1N \iff xy - xN - yN = 0 \iff (x-N)(y-N) = N^2\]](http://latex.artofproblemsolving.com/e/8/e/e8e1b14905d298f71357ee16eddf3d7d9562b0a1.png)
Note that if 
of divisors (note 
is the number of pairs 
with 
divisors 
.
.