AoPS Academy
, 
, or 
, such that no two consecutive stars have the same letter.
, where 
and 
are positive integers satisfying:
be positive real numbers such that 
. Prove that:
independent and uniformly random letters from A to Z. Bob naturally tries to see if any subsequence of the said string of random letters spells "ILOVEYOU", and we call him delulu if he manages to find at least one such subsequence. What is the least value of such that the probability that Bob is delulu is at least 
? (Note: a subsequence of a sequence is a sequence formed by a subset of the terms of the original sequence, preserving the original order of the terms.)
, where 
where the curve 
is the boundary of the portion of plane 
in the first octant, traversed in the counterclockwise direction as viewed from above, using the definition of line integral and the Stokes's theorem.
![\[
A =
\begin{bmatrix}
1 & 2 & 3 & \cdots & n-1 & n \\
2 & 3 & 4 & \cdots & n & 1 \\
3 & 4 & 5 & \cdots & 1 & 2 \\
\vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\
n-1 & n & 1 & \cdots & n-3 & n-2 \\
n & 1 & 2 & \cdots & n-2 & n-1
\end{bmatrix}
\]](http://latex.artofproblemsolving.com/7/e/5/7e564602cc0cc14e1501f6654b00a4dfeac50f70.png)
satisfy the functional equation![\[[f(x)]^{2}=1+x f(x+1)\]](http://latex.artofproblemsolving.com/3/4/d/34de57b7e476314c43e0dc4cbae1e329511cbdb8.png)
![\[\frac{x+1}{2}\leq f(x) \leq 2(x+1)\]](http://latex.artofproblemsolving.com/5/5/8/55858a3de3307af5aac073230338d67840c1a45b.png)
. Prove that 
.
is twice continuously differentiable on 
, and satisfies the conditions:![\[
\lim_{x \to +\infty} x f(x) = 0 \quad \text{and} \quad \lim_{x \to +\infty} x f''(x) = 0.
\]](http://latex.artofproblemsolving.com/3/e/5/3e536d8b5357b75e68d24c12b5cb0202edd52c8c.png)
Prove that:![\[
\lim_{x \to +\infty} x f'(x) = 0.
\]](http://latex.artofproblemsolving.com/4/d/3/4d3487c0be2069b8e1a44e7fe88b8d3b4456557d.png)
such that:
. Prove that 
is invertible
be a nonzero square matrix of order 
satisfying![\[
a_{ik} a_{jk} = a_{kk} a_{ij}, \quad \text{for all } i, j, k.
\]](http://latex.artofproblemsolving.com/7/5/2/752b16a277d9fd18bfda6c3be35d23d42d75ebaa.png)
Denote by 
the trace of 
, which is the sum of the diagonal elements of .
. in terms of .
.
where 
and 
where 
is