AoPS Academy
be a positive integer. Find all functions 
such that 
and![\[
f(af(x) - f(y) + y) \leq x + f(y) - y, \quad \forall x, y \in \mathbb{R}^+.
\]](http://latex.artofproblemsolving.com/e/2/9/e296b71ad8b8bd977e2bb2a3630a5bc020b8e03a.png)
. Find all functions 
such that for all 
,![\[
f(xf(x+y)+z) = f(z) + f(x)y + f(xf(x)).
\]](http://latex.artofproblemsolving.com/f/9/3/f93b7526ff4338eea89a1c59bc009aa001ffc15d.png)
be an acute triangle with 
and orthocenter 
. Let 
and 
be the feet of the altitudes from 
and 
onto sides 
and 
, respectively. Let 
be the midpoint of 
, and 
be the midpoint of 
. Let the perpendicular line through to 
meet at 
and at 
. The line 
meets at 
and at 
. Let 
be the second intersection point (different from ) of the circumcircles of triangles 
and 
. Prove that the points , , , , and lie on the same circle.
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)
with 
. Show that: 
.
, we define a sequence by 
, 
, and 
for 
. Prove that if 
for some 
, then the sequence is periodic.
contains only one set of three distinct points, 
and 
which are the vertices of an equilateral triangle.
be the set of all triples 
of positive integers for which there exist triangles with side lengths 
Express ![\[\sum_{(a,b,c)\in T}\frac{2^a}{3^b5^c}\]](http://latex.artofproblemsolving.com/9/f/6/9f6e0cee87e5b6958e143ff23b887bcd876f6cba.png)
as a rational number in lowest terms.
is a finite set, let denote the number of elements in . Call an ordered pair 
of subsets of 
if 
for each 
, and 
for each 
. How many admissible ordered pairs of subsets 
are there? Prove your answer.
of nonnegative integers, let 
denote the number of ordered pairs 
such that 
, 
, 
, and 
. Is it possible to partition the nonnegative integers into two sets 
and in such a way that 
for all ?
be real numbers. Consider the following curves on 
plane.![\[ y=e^{|x|}\ \ \ \ \ \ \ \ \ \ [1]\]](http://latex.artofproblemsolving.com/2/6/9/269e966da23aef955a98054744956624fce0f0ca.png)
![\[ \ \ \ \ \ \ y=ax+b\ \ \ \ \ \ \ [2]\]](http://latex.artofproblemsolving.com/1/4/a/14ab8d9a27c07c7b155d586708553b16d4365f01.png)
interms of 
, sketch the graph on 
plane.
the graph of (1). For a constant 
, find the maximum value of 
and the value of giving the maximum value.
. Find the shortest distance between its incenter and its centroid. Denote this shortest distance by 
. When a varies in the interval 
, find the maximum value of .
![\[ \int_{-1}^{1} e^x \sin \sqrt{1-x^2} \, \mathrm dx .\]](http://latex.artofproblemsolving.com/0/f/0/0f01e4820da5328db8b1b55947de2148cbdd3e83.png)
