AoPS Academy
[/quote]
, do not answer with "
is a solution" only. Either you post any kind of proof or at least something unexpected (like "
is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
of positive integers for which there exist positive integers 
and 
such that
holds for all integers 
(Note that 
denotes the greatest common divisor of integers 
and 
)
such that 
for all 
?
be a triangle with incentre 
and circumcircle 
. Let 
be the midpoints of the arcs 
of not containing 
respectively. Let 
be the point of diametrically opposite to 
. Show that 
and the midpoint 
of 
lie on a line.
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)
be integers such that, for each 
, there exists a subset 
with![\[
\sum_{j \in S} a_j \equiv i \pmod{2048}.
\]](http://latex.artofproblemsolving.com/8/b/e/8be39aaf83795f611b1c3fa0d5971905bd6ebdfe.png)
Show that for each 
, there is exactly one 
such that 
is divisible by 
but not by 
.
is the summation notation, for instance, 
, while for the empty set 
, one defines 
.
, 
. Construct a circle with center 
and radius 
, intersecting the extensions of 
and 
at 
and 
respectively. Construct a circle with center and radius 
, intersecting the extensions of 
and 
at and . Let 
be the intersection of 
and 
, 
be the intersection of 
and 
, 
(
) be the intersection of and the circle with center , and 
(
) be the intersection of 
and the circle with center . Show that , , and are concyclic.
be a natural number. At a mathematical olympiad training camp the same courses are organised every day. Each student takes exactly one of the courses each day. At the end of the camp, every student has takes each course exactly once, and any two students took the same course on at least one day, but took different courses on at least one other day. What is, in terms of , the largest possible number of students at the camp?
be natural numbers such that![\[
\frac{a^n - 1}{(a - 1)^n + 1}
\]](http://latex.artofproblemsolving.com/6/2/7/627f1d5f55e7594f2e6dcf5d541e796b7a5bfae2.png)
is a natural number.
is odd.
be a prime divisor of .![\[
a^{(q - 1)/2} \equiv 1 \pmod{q}.
\]](http://latex.artofproblemsolving.com/b/2/1/b2135d8c5526e577722718e6d02bcc52559b0555.png)
3. Prove that if a is prime and 
, then![\[
2^{(a - 1)/2} \equiv 1 \pmod{a}.
\]](http://latex.artofproblemsolving.com/2/d/c/2dc6cc1fe27cc72e72d63b13cf07939e19051245.png)
be a regular heptagon with its center 
. 
is the orthocenter of triangle 
, is the incenter of triangle 
. Let be the midpoint of 
and 
be the intersection point of 
and 
. Assume is the circumcenter of triangle 
. Prove that 
concur at a single point. be a cyclic quadrilateral such that is a diameter of it's circumcircle. Suppose that and intersect at , and 
at 
, 
and 
at 
, and let be a point on . Show that 
is perpendicular to 
if and only if is the midpoint of .
be an inscribed quadrilateral. Circles of diameters 
and 
intersect at points 
and 
, and circles of diameters 
and 
intersect at points 
and 
. The circles of diameters 
and 
intersect in two points 
and 
. Prove that the lines 
and 
are concurrent.
![$ [B B_{1} G C_{1}]$](http://latex.artofproblemsolving.com/1/4/b/14b014bf7155572341817f6208dd4039713ff284.png)
+ ![$ [C C_{1} G B_{1}]$](http://latex.artofproblemsolving.com/4/7/2/472398326f306f6a6f88f300b56f75154ff35b6d.png)
![$ \frac {4}{9} [A B C]$](http://latex.artofproblemsolving.com/0/6/4/06488950740eb239e677191e7dab2e1a1ba66351.png)
= ![$ 2 [G B_{1} C_{1}]$](http://latex.artofproblemsolving.com/1/2/7/1276bf046f65a9729220462de49ed2f19c370568.png)
+ ![$ [B B_{1} C_{1}] + [C C_{1} B_{1}]$](http://latex.artofproblemsolving.com/c/e/5/ce55fe99caa231eed73ebae9d90c6e2c3805c440.png)
= ![$ 2[G B_{1} C_{1}]$](http://latex.artofproblemsolving.com/2/6/8/268612037a6f90b8ba21e8ac0092a46c36246d3f.png)
+ ![$ 2[B_{1} M C_{1}]$](http://latex.artofproblemsolving.com/e/6/d/e6df88c731243d7713a36f5b24bd6b37bbef5692.png)
= ![$ 2 [G B_{1} M C_{1}]$](http://latex.artofproblemsolving.com/c/5/8/c587eaf84c4df6a1e172f4a31b663bad63d6f90e.png)
= ![$ \frac {2}{3}[A B_{1} M C_{1}]$](http://latex.artofproblemsolving.com/5/8/b/58bdd667ad850fb684de7085c5c9342554f34159.png)
= ![$ [B B_{1}M]$](http://latex.artofproblemsolving.com/d/0/6/d0645e30f29310e066e81be27e5dc63cba51a883.png)
+ ![$ [C C_{1} M]$](http://latex.artofproblemsolving.com/6/d/9/6d9d8828cedcdce4848d85329c3372f794956c4b.png)
M is the middle of 
![$ \frac {2}{3}([A B C] - \delta)$](http://latex.artofproblemsolving.com/2/b/8/2b81d2fd84beacbf43a96f88ecf9eff5762ae159.png)
![$ \frac {4}{9}[A B C]$](http://latex.artofproblemsolving.com/6/b/c/6bc883eb6b66caf7689383e6766993c91b87a7ec.png)
![$ [ABC] \geq 3\delta$](http://latex.artofproblemsolving.com/8/8/d/88df5f13bcd5e86568997bd97ee13e035a61dcef.png)
to be 
computing we will get 
+ 
after this we replace 
with 
where 
and 
are the middle of 
and his projection on the only thing left to prove is that 
on 
similar not dificult,have in mind here that 
passes under 
, will always be under the paralel to through 
and 
Prove that
be a line that intersects 
and 
,
and ![$[BB_{1}GC_{1}]+[CC_{1}GB_{1}] \geq \frac{4}{9}[ABC]$](http://latex.artofproblemsolving.com/0/0/d/00d43c29d9f2fb6d004c10c344e9e6696a0452fe.png)
.
