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.
be a fixed integer. There are 
beads on a circular necklace. You wish to paint the beads using 
colors, such that among any 
consecutive beads every color appears at least once. Find the largest value of 
for which this task is 
possible.
, let 
, 
, and 
denote the feet of the altitudes from 
, 
, and 
, respectively, and let 
denote the circumcenter of 
. Points 
and 
denote the projections of and , respectively, onto 
, and 
. There exists a point 
such that 
and 
. If 
and 
lies on 
such that 
, prove that line 
bisects 
.
aluminum coins and bronze coins arranged in a row in some arbitrary initial order. A chain is any subsequence of consecutive coins of the same type. Given a fixed positive integer 
, Gilberty repeatedly performs the following operation: he identifies the longest chain containing the 
coin from the left and moves all coins in that chain to the left end of the row. For example, if 
and 
, the process starting from the ordering 
would be 
with 
such that for every initial ordering, at some moment during the process, the leftmost coins will all be of the same type.
and 
Prove that
Let and 
Prove that
Let and 
Prove that
prove that
such that for all naturals 
the function satisfies:![\[a + b \mid a^{f(a)} + b^{f(b)} \]](http://latex.artofproblemsolving.com/e/b/b/ebb44041cedd29de516192d983a334d46e7fe7d6.png)
Bob wants to find all possible functions Alice could have. Help Bob and find all functions that Alice could have.
be the medial triangle of an acute-angle triangle . Suppose the line through perpendicular to 
meet 
at 
. Define 
analogously. Let 
. Similarly define 
and 
. Suppose the circle with diameter 
meet the -altitude at 
, where lies inside . Define 
and 
similarly. Let 
be the circumcenter of , and let 
be the circle with diameter 
, which meets 
at 
. Similarly define 
and 
.
are collinear.
lie on a circle centered at .
are coaxial. is perpendicular to the radical axis of .
denote the set of all real numbers. Find all functions 
such that
for all real numbers 
, 
.![\[x^2-cx+1 = \dfrac{f(x)}{g(x)},\]](http://latex.artofproblemsolving.com/1/3/3/1339acdaf1999e06e1a4b7dec7d1217389e2afef.png)
where 
and 
are polynomials with nonnegative real coefficients. For each 
, determine the minimum possible degree of , or show that no such 
exist.
such that every collection of consecutive positive integers contains an integer whose sum of digits is divisible by 
. for which he can do this, or prove that none exist., consider points on line such that 
are in that order and 
and 
. Similarly consider points 
on line 
, and 
on line . Prove that orthocenters of triangles 
and 
are equidistant from the circumcenter of .
be six positive real numbers. Prove that![\[a^2+b^2+c^2+\frac{4(ax+by+cz)\sqrt{ab+bc+ca}}{x+y+z} \geqslant 2(ab+bc+ca).\]](http://latex.artofproblemsolving.com/f/8/0/f808f7857c2150960f2a430837286487666797d8.png)
be a cyclic quadrilateral with circumcentre and circumcircle 
. Let be the intersection of tangents at and to . Let 
be the circumcircle of triangle 
and let 
, 
denote the second intersections of 
with respectively. Let 
and intersect at and 
be the circumcircle of triangle 
. If intersects again at 
, prove that the line tangent to at is also tangent to . be an acute triangle with orthocenter 
. Let 
be the midpoint of , and 
be the intersection of the tangents from and to the circumcircle of . Denote by the circle centered at and tangent to line 
.
intersects at two distinct points , .
and 
meet at 
, while lines 
and 
meet at . Prove that either or lies on .
with 
non-negative integers, a square of side 
with center at the point is constructed. Compute the area of the union of all these squares.
