AoPS Academy
[/quote]
, do not answer with "
is a solution" only. Either you post any kind of proof or at least something unexpected (like "
is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
are such that 
, find the minimum value of expression 
be a sequence of strictly positive real numbers. For each nonzero positive integer 
, define![\[
s_n = a_1 + a_2 + \cdots + a_n \quad \text{and} \quad
\sigma_n = \frac{a_1}{1 + a_1} + \frac{a_2}{1 + a_2} + \cdots + \frac{a_n}{1 + a_n}.
\]](http://latex.artofproblemsolving.com/1/9/1/191a2ebd41e3d236c90e9022ed65323461083815.png)
Show that if the sequence 
is unbounded, then the sequence 
is also unbounded.
+1 w
be positive real numbers such that 
. Find the minimum value of the following sum :
knowing that the denominators are positive real numbers.
![\( P(x) \in \mathbb{Z}[x] \)](http://latex.artofproblemsolving.com/2/1/5/215e7ef672c0effe96e63ac9254fbd7356e17637.png)
for which there exists an integer 
such that 
for integers 
.
+1 w
with 
vertices may have, so that after deleting an arbitrary cycle, is not connected anymore.
be nonzero vectors in 
-dimensional space. 
are real numbers. Show that there exists 
such that![\[\left|\left\langle\sum_{i=1}^t\varepsilon_iv_i,\frac{v_k}{|v_k|}\right\rangle-m_k\right|\ge |v_k|\]](http://latex.artofproblemsolving.com/6/c/6/6c6f82687e0fbf781f18caed5ba6fdbe9bc1125b.png)
holds for all 
be positive real numbers such that 
. Prove that 
be a convex quadrilateral with ![\[\angle ADC = 90^\circ, \ \ \angle BCD = \angle ABC > 90^\circ, \mbox{ and } AB = 2CD.\]](http://latex.artofproblemsolving.com/b/4/1/b416d5c0b29c3a69a3e87cd22b6000aa8e70d456.png)
The line through 
, parallel to 
, intersects the external angle bisector of 
at point 
. Prove that the angles 
, 
, 
, 
, 
can be divided into two groups, so that the angles in each group have a sum of 
.
for which the following statement holds: there exists at least one triple 
of integers such that
and all triples of real numbers, satisfying the equations, are such that 
are integers.
that satisfies 
for all real 
be an acute triangle with 
, and let 
be the center of its circumcircle. Let 
be the reflection of 
with respect to 
. The line through parallel to intersects 
at 
, and the tangent at to the circle 
intersects the line through parallel to at point 
. Let 
be a point on the ray 
, starting at , such that 
. lies on the circle with diameter 
.
and let 
be positive integers such that 
. Prove that 
and determine when equality holds.
on the plane. For every 
segment for which 
lies on the ray 
, and 
lies on the ray 
, consider a point 
on the segment such that 
. Find the geometric locus of such points .
on the plane. For every segment for which lies on the ray , and lies on the ray , consider a point on the segment such that . Find the geometric locus of such points .
