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.
be a natural number. Prove that,![\[ \left\lfloor \frac{n}{1} \right\rfloor+ \left\lfloor \frac{n}{2} \right\rfloor + \cdots + \left\lfloor \frac{n}{n} \right\rfloor + \left\lfloor \sqrt{n} \right\rfloor \]](http://latex.artofproblemsolving.com/5/6/0/560a9568b2c09512cac1d317a011931a439e6a02.png)
+1 w
such that for any 
, 
+1 w
, 
, 
, 
, where and 
are odd primes and![\[2^ap^b=(p+2)^c-1.\]](http://latex.artofproblemsolving.com/5/8/5/58518a8ebed14836b72af743ff100eae2efba5e2.png)
![\[
2^a + 3^b + 1 = 6^c
\]](http://latex.artofproblemsolving.com/1/9/e/19eb5efa2ccdf54cd277339f8d3b99f75168779d.png)
positive integers. Product of any one of them with sum of remaining numbers increased by 
is divisible with sum of all numbers. Prove that sum of squares of all numbers is divisible with sum of all numbers
be a triangle with 
, and let 
be the foot of the altitude from 
. Let 
be a point in the interior of the segment 
. Let 
be the point on the segment 
such that 
. Similarly, let 
be the point on the segment 
such that 
. Let 
be the point of intersection of 
and 
.
. be a scalene triangle with circumcircle 
. Let be the midpoint of 
. A variable point 
is selected in the line segment 
. The circumcircles of triangles 
and 
intersect again at points and 
, respectively. The lines 
and 
intersect (a second time) the circumcircles to triangles and at and 
, respectively. Prove that as varies, the circumcircle of 
passes through a fixed point 
distinct from 
.
, 
, 
, ... be an infinite sequence of real numbers satisfying the equation 
for all 
, where and are two different positive reals., , , ... be bounded?
be two positive sequences defined by 
and![\[ \begin{cases} {{x}_{n+1}}{{y}_{n+1}}-{{x}_{n}}=0 \\ x_{n+1}^{2}+{{y}_{n}}=2 \end{cases} \]](http://latex.artofproblemsolving.com/2/0/1/2012dc6ff3a41478547cea4aa9b6bccbe17a3623.png)
for all 
.
be an infinite sequence of positive integers. Prove that there exists a unique integer 
such that![\[a_n < \frac{a_0+a_1+a_2+\cdots+a_n}{n} \leq a_{n+1}.\]](http://latex.artofproblemsolving.com/9/4/f/94fc5a51b7e68588123c5b527fe75183bc4c4937.png)
with 
, 
and 
for all positive integers .
where is a positive integer and is a prime.(Don't get mad at me,I've used the search function and did not see a correct and complete solution anywhere.)
, define a sequence of triples 
by
.Show that each of the sequences 
converges to a limit and find these limits.
, define a sequence of triples by
,.Show that each of these sequences converges to a limit and find these limits.
are the three positive sidelengths or a triangle, we can rather easily show that :
are the sides of a new triangle with same area
, 
and 
are decreasing sequences
have same limits, sides of an equilateral triangle of same area
![$l=\sqrt[4]{\frac{(x_0+y_0+z_0)(x_0+y_0-z_0)(x_0-y_0+z_0)(-x_0+y_0+z_0)}3}$](http://latex.artofproblemsolving.com/7/4/9/749d0a7955dd294a78fcf172aa31ed1225df1dc7.png)
, this formula gives 
. Also, the thing under the radical suspiciously looks like the product of length of tangent from a vertex and some other thing. This motivates the introduction of a triangle with sides 
. Usually we need some kind of invariant in multivariable limits. So, we try the squared area of the triangle since it is a polynomial in , more precisely, 
. This surprisingly comes out to be invariant. Also to show the convergence, it seems better to choose the difference between squares.
