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 the set of integers. Determine all functions 
such that, for all integers 
and 
, 
Proposed by Liam Baker, South Africa
denote the set of positive integers. A function 
is said to be bonza if![\[
f(a)~~\text{divides}~~b^a-f(b)^{f(a)}
\]](http://latex.artofproblemsolving.com/7/b/b/7bb6b2b4fcde900ee6f7ef90b906525d61312796.png)
for all positive integers and .
such that 
for all bonza functions 
and all positive integers 
.
are primes such that 
. 
are integers and 
. Prove that: the number of positive integers 
satisfying 
is less than 
.
positive numbers 
such that 
Prove ![\[ \left( 1+a_{1}\right) \left( 1+a_{2}\right) ...\left(1+a_{n}\right) \geq 2^{n}.\]](http://latex.artofproblemsolving.com/7/7/d/77d2ff03cddbac88cf46e1d5167a6e3e92fa6766.png)
be non-negative integers such that:
. find the minimum value of 
if 
)
, then 
"
and center 
, and a line intersecting the semicircle at 
and 
and the line at 
(
, 
). The circumcircles of the triangles 
and 
meet again at 
. Prove that 
is right. be a positive integer. Find the minimum possible value of![\[
S = 2^0 x_0^2 + 2^1 x_1^2 + \dots + 2^n x_n^2,
\]](http://latex.artofproblemsolving.com/7/b/8/7b818fccfea8f1dcb9f80bde51dba224f91c3713.png)
where 
are nonnegative integers such that 
.
, Arándano and Banana play a game. Initially, numbers 
are written on the blackboard. Alternatingly and starting with Arándano, the players erase numbers from the board one at a time, until exactly three numbers remain on the board. Banana wins the game if the last three numbers on the board are the sides of a nondegenerate triangle, and Arándano wins otherwise., who has a winning strategy.
such that ![\[ f(x^2 + yf(z)) = xf(x) + zf(y) , \]](http://latex.artofproblemsolving.com/2/b/a/2baee1d8464992fc38a16561b48b18f4b35807bd.png)
for all 
.
so that 
is a quadratic number. The amount of sets of 
that satisfy, are?
and 
, 
.
is a perfect square.
points on the plane, all pairwise distances between which are different. Is there always a polyline with vertices at these points, passing through each point once, in which the link lengths increase monotonously?
points on the plane, all pairwise distances between which are different. Is there always a polyline with vertices at these points, passing through each point once, in which the link lengths increase monotonously?
Prove that
