be a trapezoid. By 
we denote the angle bisector of angle 
. Let 
and 
.
is cyclic.
such that
Proposed by Ngo Van Trang, Vietnam
and 
Prove that
Let 
Prove that
such that 
.
.
,![\[ \dfrac {1}{2ne} < \dfrac {1}{e} - \left( 1 - \dfrac {1}{n} \right)^n < \dfrac {1}{ne}. \]](http://latex.artofproblemsolving.com/1/3/e/13e2e3859e16ca0e4453330eeac03465d2eb8429.png)
let the numbers 
be determined by the rules 
and 
Find the value of ![\[\sum_{n=1}^{2013}c(n)c(n+2).\]](http://latex.artofproblemsolving.com/3/f/a/3faf9da9b035e02298650c79251370d0c8b61c1f.png)
be expressed as a polynomial in 
with real coefficients?
be a twice differentiable function such that![\[
f(0) = 2, \quad f'(0) = -2, \quad \text{and} \quad f(1) = 1.
\]](http://latex.artofproblemsolving.com/1/e/1/1e178debf8ae0b6b76d2e60ffca95f5a4c4ad4e5.png)
Prove that there exists a point 
such that![\[
f(\xi) \cdot f'(\xi) + f''(\xi) = 0.
\]](http://latex.artofproblemsolving.com/4/6/b/46b8d103981b838ae0aa5afdee104edfae4e4b8f.png)
positive such that 
,
positive such that 
converges and 
diverges ?
is continuous, positive with 
,
continuous and positive exist on 
such that 
converges and 
diverges ?
is in the 
class, increasing : I chose 
.
but it is not clear that it is ok, even if 
,
diverges ?
converges ?
does not converge to 
,
such that for every polynomial 
with integer coefficients and for every integer 
the integer![\[p^{(j)}(k)=\left. \frac{d^j}{dx^j}p(x) \right|_{x=k}\]](http://latex.artofproblemsolving.com/a/2/7/a2793d7f46345bcba4ccaa3082cc47afcd35074b.png)
(the 
)
order, and if the constituents of its first, second, ..., 
figurate numbers of the first, second, ..., 
holds inequality 
Y by
Let 
be a prime number. Find the number of elements of the set
![\[
M_p = \left\{ x \in \mathbb{Z}^* \,\middle|\, \sqrt{x^2 + 2p^{n} x} \in \mathbb{Q} \right\}.
\]](//latex.artofproblemsolving.com/5/4/1/54152836e072da8d4ec5984fcb7fea428cda8e48.png)
Edit: I edited with a variation of the same problem because I intend to give this one to an exam for my pupils.
be a prime number. Find the number of elements of the set![\[
M_p = \left\{ x \in \mathbb{Z}^* \,\middle|\, \sqrt{x^2 + 2p^{n} x} \in \mathbb{Q} \right\}.
\]](http://latex.artofproblemsolving.com/5/4/1/54152836e072da8d4ec5984fcb7fea428cda8e48.png)
Edit: I edited with a variation of the same problem because I intend to give this one to an exam for my pupils.
This post has been edited 1 time. Last edited by td12345, Apr 14, 2025, 6:17 PM
Reason: variation of the problem
Reason: variation of the problem
,
comes from but I think when 
we get 2 extra: 
.
not the set of the square root expression no? I got a different number from yours and I don't know where I went wrong, I will lay down the 
:
where 
. This leads to ![\( [\frac{2025+1}{2}]\)](http://latex.artofproblemsolving.com/d/9/3/d93cfced933adf97b0b51a17da9558e88d0838fc.png)
values of 
where 
.
extra values for 
:
where 
. This leads to 
distinct non-zero values of 
where 
distinct non-zero values of 
?
it leads to 
for the very first family and 
we shall have 
.
,
extra values for 
,
.
.
is natural numbers (positive integers) and 
is natural numbers without 
?
