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 an equilateral triangle, and 
is a point on its incircle. Let 
be the circle tangent to 
passing through and 
. Similarly, let 
be the circle tangent to 
passing through and 
, and 
be the circle tangent to 
passing through and 
., , has a common tangent line.
with degree 
, such that for all positive reals 
with product 
, then; 
Suppose there exist positive reals 
with product , such that; 
Is it possible that the values are all distinct?
satisfying 
Prove that 
Proposed by Phan Ngoc Chau
is called \textit{good}, if 
are non-negative integers, and 
is either 
or for all 
.
, and Ivan has a whiteboard with some ones written on it. In each step, he may erase any good sequence that appears on the whiteboard, and then he writes the number 
such that 
Suppose Ivan starts with the least possible number of ones to obtain 
after some steps, determine the minimum number of steps he will need in order to do so. and 
, and 
positive (not neccesarily distinct) real numbers 
, 
. An equation is written on a whiteboard: 
where 
is a fixed positive real number, with exactly asterisks.
, while Rubi fills each asterisk with a number from 
, so that the equation on the whiteboard is correct. Suppose for every positive real number , the number of ways for Ebi and Rubi to do so are equal. and are permutations of each other.
and 
are considered different ways to fill the asterisks, and the chosen terms need not be distinct, for example 
.)
be a triangle, with incenter 
and -excenter 
. The lines 
, 
, 
and 
intersect the circumcircle of at , 
, 
and 
respectively. Let 
, 
be diameters in the circumcircles of triangles 
and 
respectively.
.
be the set of real numbers. Find all functions 
where there exist a real constant 
such that 
holds for all reals 
, 
, 
that satisfy 
.
be a positive integer. Megavan and Minivan are playing a game, where Megavan secretly chooses a real number in ![$[0, 1]$](http://latex.artofproblemsolving.com/a/b/1/ab178d831a786b92cb4c9ddc2d33578223036f98.png)
. At the start of the game, the only information Minivan has about is in . He needs to now learn about based on the following protocols: at each turn of his, Minivan chooses a number and submits to Megavan, where Megavan replies immediately with one of 
, 
, or 
, subject to two rules:
The answers in the form of and must be truthful; Define the score of a round, known only to Megavan, as follows: if the answer is in the form and , and 
if in the form . Then for every positive integer and every consecutive rounds, at least one round has score no more than 
.
such that 
and 
. Let 
be the minimum number of queries that Minivan needs in order to guarantee success, regardless of Megavan's strategy. Prove that 
, where 
and 
are prime numbers and 
, such that 
such that for any two infinite sequences of positive integers 
and 
, such that 
for all 
, there always exists a real number 
such that 
for all 
.
and 
respectively be the incentre, circumcentre and circumcircle of triangle . Points 
are chosen on , such that 
, and point 
is the foot of the altitude from to 
. If 
are similarly defined, prove that lines 
and 
concurr on 
. for which all positive divisors of can be put into the cells of a rectangular table under the following constraints:
, such that 
, 
and 
.
satisfying:
and let 
and 
be tangency points of the incircle of 
and 
.
,
and 
,
.
and 
be feet of perpendiculars from 
and 
respectively. Prove that 
,
and 
are concurrent.
