[/quote]
,
and 
be different points on a circle 
such that 
is not a diameter. Let 
be the tangent line to 
is such that 
.
is chosen on the shorter arc 
of triangle 
intersects 
be the common point of 
meets 
.
is tangent to 
such that
![\[
\int_{0}^{1} \frac{dx}{(\ln x + i\pi)^2\,(1 - x + e^{i\pi})}
\]](http://latex.artofproblemsolving.com/2/7/d/27d389a41148543ea478644e4bc6e0a1258f0d04.png)
table. It is permissible to reverse the signs of all the numbers in any row or column. Prove that after a number of these operations, we can make the sum of the numbers along each line (row or column) nonnegative.
,
is the circumcenter of 
,
are points on segments 
and 
respectively with 
.
be a point such that 
and 
.
intersect 
and 
intersect 
,
defined by 
and![\[
u_{n+1} = \frac{1}{2}u_n^2 - 4 \quad \text{for all } n \in \mathbb{N}.
\]](http://latex.artofproblemsolving.com/9/9/4/994aa754cc1288ce4f28a95a0276e64282fb5f66.png)
a) Prove that there exist infinitely many positive integers 
such that 
.![\[
\lim_{n \to \infty} \frac{2u_{n+1}}{u_0u_1\cdots u_n}.
\]](http://latex.artofproblemsolving.com/f/f/1/ff174e7431cbfbc17c650d109651241286756a1a.png)
and 
be altitudes in an acute triangle 
which meet at 
.
meets the circumcircle of 
and 
such that 
and 
.
and 
meet at 
.
and 
and the line 
concur.
aand 
Prove that 
and 
for 
Compute ![\[\prod_{k=0}^{\infty}\left(1-\frac1{a_k}\right)\]](http://latex.artofproblemsolving.com/1/9/7/197b63d6a2a31e18dae30ef70595b27318667338.png)
in closed form.
,
.
satisfies the quadratic equation 
and deduce that the value of 
.
.
,
is 
.
Y by Euler_Gauss
Let 
be a 
-dimensional inner product space of column vectors, where for 
and 
, the inner product of 
and 
is defined as ![\[\langle v, w \rangle = \sum_{i=1}^{10} v_i w_i.\]](//latex.artofproblemsolving.com/9/f/9/9f9b7ff2fb19fee7e935c1b18dea3ec80547b9f9.png)
For 
, define a linear transformation 
on as follows:
![\[ P_u : V \to V, \quad x \mapsto x - \frac{2\langle x, u \rangle u}{\langle u, u \rangle} \]](//latex.artofproblemsolving.com/9/6/4/9646290e2b5e540b7f85d0d71e4d8992ccd8f542.png)
Given 
satisfying
![\[ 0 < \langle v, w \rangle < \sqrt{\langle v, v \rangle \langle w, w \rangle} \]](//latex.artofproblemsolving.com/d/8/0/d800fe48958cfee9ac01aa8bf11fe431ef45f8ef.png)
let 
. Then the dimension of the linear space formed by all linear transformations 
satisfying 
is 
be a 
-dimensional inner product space of column vectors, where for 
and 
, the inner product of 
and 
is defined as ![\[\langle v, w \rangle = \sum_{i=1}^{10} v_i w_i.\]](http://latex.artofproblemsolving.com/9/f/9/9f9b7ff2fb19fee7e935c1b18dea3ec80547b9f9.png)
For 
, define a linear transformation 
on as follows:![\[ P_u : V \to V, \quad x \mapsto x - \frac{2\langle x, u \rangle u}{\langle u, u \rangle} \]](http://latex.artofproblemsolving.com/9/6/4/9646290e2b5e540b7f85d0d71e4d8992ccd8f542.png)
Given 
satisfying![\[ 0 < \langle v, w \rangle < \sqrt{\langle v, v \rangle \langle w, w \rangle} \]](http://latex.artofproblemsolving.com/d/8/0/d800fe48958cfee9ac01aa8bf11fe431ef45f8ef.png)
let 
. Then the dimension of the linear space formed by all linear transformations 
satisfying 
is 
is the orthogonal symmetry with respect to 
.
;
and 
is a rotation with angle 
.
,
.
.
