is a two times differentiable function satisfying 
and for all 
,![\[ f^{\prime \prime}(x)-5f^{\prime}(x)+6f(x)\ge 0 \]](http://latex.artofproblemsolving.com/e/1/5/e15e1397c2810c849571b3efa36d0da69bc31dde.png)
![\[ f(x)\ge 3e^{2x}-2e^{3x} \]](http://latex.artofproblemsolving.com/5/2/4/5246591711bdedc2dd68e7064795a059aa110d2d.png)
be a twice continuously differentiable function, and suppose that 
and 
.
and find all such functions for which equality holds.
is continuous on [0, 1] and diffrentiable on (0, 1). Given that 
and 
.
is twice differentiable and satisfies 
.![$\xi \in ]0,1[$](http://latex.artofproblemsolving.com/f/a/c/fac71dc4df50908a85c3133787a84ccb2ae3d79b.png)
for which we have 
.
and an operator defined as 
.
such 
now ?
,
be the number of pairs of integers 
such that the number 
is a perfect square. Prove that the limit
exists and find its value.
be a positive integer. Consider the following random process which produces 
.
is chosen randomly with 
for every positive integer 
.
.
,
,
randomly with 
for every positive integer 
.![$$
\mathbb{E}\left[Y_n\right]=\sum_{i=1}^n \frac{2^i}{2^i-1}
$$](http://latex.artofproblemsolving.com/4/e/3/4e38d740482044146ce1f6be2601ef217d63661b.png)
where ![$\mathbb{E}\left[Y_n\right]$](http://latex.artofproblemsolving.com/1/e/1/1e106e23a601c28219befc3d4f2c0f79114f6dda.png)
is the expected value of 
.
be real numbers. Define 
.![\[
\sum_{1 \leq i, j \leq n} 2^{\|x_i - x_j\|} \leq \sum_{1 \leq i, j \leq n} 2^{\|x_i - x_j + \frac{1}{2}\|}.
\]](http://latex.artofproblemsolving.com/1/a/3/1a3c78a907f136fa4f8f9bead7306918775597eb.png)
Proposed by Site Mu
Y by GA34-261
Recently, I have noticed that the AoPS Wiki takes rather long to load. Normally, this would not be an issue, as some webpages are just a little sluggish on the server side. However, I recall the Wiki being no slower than any other page(forums, profile, etc.) when loading. Now, however, any Wiki page(problem archives, LaTeX tutorials, etc.) takes about 15 seconds to load compared to the regular site, which almost always loads instantly. I have attempted accessing the Wiki page using multiple different browsers such as Chrome and Safari, and have also tried switching operating systems and computers to no avail. Not an internet issue, either, as other sites load fine. Not sure if this is an issue other people experience? It seems as if the load times have changed in the last few months. I can recall when the Wiki loaded instantly just like the rest of the site.
