[/quote]
,
such that every collection of 
.
for which there exist infinitely many positive integers 
such that ![\[ \frac{a^m+a-1}{a^n+a^2-1} \]](http://latex.artofproblemsolving.com/2/8/a/28a68e409451bbdcaee7c1286341f6aeea9ffdad.png)
is itself an integer.
for which there exists a sequence of 
pairwise distinct points 
in the plane satisfying the following four conditions: [list=i] [*]no three of the 
for all 
,
;
for all 
,
;
for all 
.
be a positive integer. Find the largest real number 
as a function of 
such that the inequality![\[\max\left(|a_1+a_2|, |a_2+a_3|, \dots ,|a_{n-1}+a_{n}| , |a_n+a_1|\right) \ge t_n \cdot \max(|a_1|,|a_2|, \dots ,|a_n|)\]](http://latex.artofproblemsolving.com/2/8/2/2828c2c73c3d6d7094d50bfcc2b114f492bb4f14.png)
holds for all real numbers 
.
;
with 
you can choose a subset of the blocks whose total weight is at least 
.
and 
Prove that
where 
is some prime divisor of 
was chosen as a prime number 
is right angled (for clarity, 
)
as a point on line 
such that 
is parallel to 
.
,
meaning 
,
,
.
,
meaning 
.
,
or 
and we are done.
it can draw the line which forms the interior angle bisector of 
.
in the plane . Prove that Euclid can use the splitter several times to draw the centre of circle passing through 
and 
be a circle, and 
are two fixed points on 
and 
denote the feet of the altitudes from 
.
such that 
is tangent to 
.
the midpoint of arc 
.
meets 
and intersects 
or 
is tangent to 
,
,
are
with 
,
are all coloured blue.
for which the value of the expression![\[x^2+6x+33\]](http://latex.artofproblemsolving.com/3/d/e/3de5bc2a9b022192315975731d48af7fda9fe228.png)
is a perfect square.
denote the set of all real numbers. Find all functions 
such that
for all real numbers 
.
Y by Adventure10, Mango247
Circles 
and 
with centers 
and 
, respectively, are externally tangent at point 
. A circle 
with center 
touches at 
and at 
so that the centers , lie inside 
. The common tangent to and at intersects the circle at 
and 
. If 
is the midpoint of the segment 
, show that 
.
Greece
and 
with centers 
and 
, respectively, are externally tangent at point 
. A circle 
with center 
touches at 
and at 
so that the centers , lie inside 
. The common tangent to and at intersects the circle at 
and 
. If 
is the midpoint of the segment 
, show that 
.Greece
and 
don't need to be tangent.
i use some lemma that dont need to proof but if you need say 
a
a
a
(
)and harmonic(my friend solution 
)
(
and 
i
p
a
,
,
.
is the circumcenter of 
is the diameter of the circle containing the points 
is perpendicular to 
,
is cyclic. The rest is obvious.
,
from above.
and 
,
is cyclic. Let 
,
is cyclic, and similarly, 
is cyclic too. Therefore, 
.
