AoPS Academy
+5 w
[/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.
coins numbered from 
to 
. These coins are placed around a circle, not necesarily in order.
, we will jump to the one places from it, always in a clockwise order, beginning with coin number 1. For an example, see the figure below. for which there exists an arrangement of the coins in which every coin will be visited.
.
people and 
colors, where each person has one ball of each color. For each person, their balls have positive mass summing to one. Find the smallest constant 
such that regardless of the mass distribution, each person can choose one ball such that the sum of the chosen balls of each color does not exceed .
and 
. Prove that
Equality holds when 
Equality holds when 
be an integer. Prove that there exists a set 
of 
positive integers satisfying the following property: For every 
the set can be partitioned into two subsets with equal sums of elements, with one of subsets of cardinality 
.
, 
, 
be an infinite sequence of positive integers. Suppose that there is an integer 
such that, for each 
, the number
is an integer. Prove that there is a positive integer 
such that 
for all 
. be a positive integer. On the table, we have 
ornaments in different colours, not necessarily of each colour. Prove that we can hang the ornaments on Christmas trees in such a way that there are exactly ornaments on each tree and the ornaments on every tree are of at most 
different colours.
be a fixed positive integer and consider an 
grid of real numbers. Determine the greatest possible number of cells 
in the grid such that the entry in is both strictly greater than the average of 's column and strictly less than the average of 's row.
has side lengths 
, 
, and . Select a point 
inside , and construct the incenters of 
, 
, and 
and denote them as 
, 
, 
. What is the maximum area of the triangle 
?
. Let 
be a point on side 
, and 
be a point on the extension of 
such that 
. Let 
be the circumcenter of 
, and be a point on the side extension of 
satisfying 
. Line BP intersects AC at point Q. Prove that 
be a prime and suppose 
is prime. Prove that 
is prime
This is a special case of Pocklington's theorem. A proof of this special case is required.
be the circumcenter and 
the orthocenter of an acute triangle 
. Show that there exist points 
, 
, and on sides , 
, and 
respectively such that ![\[ OD + DH = OE + EH = OF + FH\]](http://latex.artofproblemsolving.com/a/8/7/a874af4cb802b9b341b055140a08e096aaf2112e.png)
and the lines 
, 
, and 
are concurrent.
as 
. Then we may show that 
through angle chasing, redefine as 
. Then let 
, note that 
is cyclic and tangent to 
, likewise we may angle chase that 
is cyclic so 
and thus 
is tangent to 
, thus 
, and thus 
. Finally, the result is another painful angle chase.
lies on 
, where 
is the radius of by trig bash
and then angle/arc bash the required angle equality.
,
and 
.