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be an integer and 
is a sequence of real numbers for which 
and:
Prove that all elements of the sequence are positive.
is made up of 64 small unit cubes. Exactly 16 of these small cubes must be colored red, subject to the following condition:
, 
, and 
cubes, there must be exactly one red cube.
from the set of positive integers to the set of positive integers such that, for all positive integers 
and 
, there exists a non-degenerate triangle with sides of lengths![\[ a, f(b) \text{ and } f(b + f(a) - 1).\]](http://latex.artofproblemsolving.com/4/a/b/4ab97da27e05d0f9e7762f74d0d0bd2dc8b9842f.png)
,such that the inequality 
holds for all positive reals 
and for every positive real 
, there exist positive reals 
, such that the equality holds.
satisfy 
for all 
. Prove that
be a fixed circle, and let 
be a point outside such that 
. A variable line through intersects the circle at two points 
and 
, such that the quadrilateral 
is cyclic, where 
are fixed points on the circle.
,
,
is the tangent from to the circle , and 
is the point of tangency.
, 
, and are collinear.
is perpendicular to chord 
. varies, the point traces a fixed straight line, which is parallel to the isogonal conjugate axis (the so-called *isotropic line*) of the centers 
and .
lie on a circle, their cross-ratio is harmonic:![\[
(ANMB) = -1.
\]](http://latex.artofproblemsolving.com/b/d/5/bd5301ba76f493aec0821c454dffc46dea93c0f2.png)
- The intersection points , and form a well-known harmonic setup along the diagonals of the quadrilateral., , :** is tangent to , and due to harmonicity and projective duality, the polar of passes through ., , and must lie on a common line. is tangent at and is a chord, the angle between and chord is right. is perpendicular to .:** varies, the point moves. lie on a fixed line, which is perpendicular to 
, and is parallel to the isogonal (or isotropic) line determined by the centers and ., or of line , as varies?![\[y''+by'+cy=g(t)\]](http://latex.artofproblemsolving.com/7/8/8/7882dc954ab6dba2445b68d370d630ce1c66a6db.png)
with 
, where 
are constants has the form ![\[y(t)=\int^{t}_{t_0}K(t-s)g(s)ds\,\]](http://latex.artofproblemsolving.com/6/3/c/63c34df1c0013bf659bc76506a93873a5f879070.png)
![$f:[1,8]\to R$](http://latex.artofproblemsolving.com/c/e/c/cec97ea0372e37977e8a77e8e77518a2344c1c43.png)
that:
that exist function 
such 
continuous on 
and discontinuous on 
be a continuous function defined on ![$[0,1]$](http://latex.artofproblemsolving.com/e/8/6/e861e10e1c19918756b9c8b7717684593c63aeb8.png)
with 
and 
. We define the point 
to be the projection of point on the x-axis. Prove that there exist points 
such that 
is a square.
is a finite group of 
matrices with respect to multiplication. Prove that:
that 
be an infinite set of translated copies (i.e., obtained by parallel translation) of a given ellipse 
in the plane, and let 
be a fixed straight line. It is known that every straight line parallel to intersects at least one ellipse from . Prove that there exist infinitely many triples of ellipses from such that there exists a straight line intersecting all three ellipses in the triple.
![\[ \frac12 = \sin(3 \theta) - \sin \theta \]](http://latex.artofproblemsolving.com/6/d/6/6d60415cf2c7ab69e8690a7f79125906fac2e4dc.png)
where 
. Note that 
is sufficient to find all the needed angles. Now the above equation is easily solved by using the addition theorems of Sines and Cosines, as that gives us a polynomial equation in degree 
and it's easy to guess one answer.
and use variation of parameters in each case to find the solution. Is there any short way to calculate solution?
be points on a circle with radius 
and 
.
.
or 
