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.
such that:![\[ f(xy + f(x))+f(y) = xf(y)+f(x+y) \]](http://latex.artofproblemsolving.com/6/0/4/604d965a38e7df743b97369340946ef8d1c56e5e.png)
for all real numbers 
Find the minimum value of 
be a non-empty set of positive integers such that for any 
, all positive divisors of 
are also in . Prove that contains an integer of the form 
, where 
are distinct prime numbers, all greater than 
.
![\[\frac{\operatorname{lcm}(x, y) + \operatorname{lcm}(y, z)}{\operatorname{lcm}(x, z)}\]](http://latex.artofproblemsolving.com/1/e/8/1e8b3e2d9a4d0917ec622449b058702700542f22.png)
for positive integers 
, 
, 
?
be a trapezoid with parallel sides 
. Points 
and 
lie on the line segments 
and 
, respectively, so that 
. Suppose that there are points 
and 
on the line segment 
satisfying ![\[\angle{APB} = \angle{BCD}\qquad\text{and}\qquad \angle{CQD} = \angle{ABC}.\]](http://latex.artofproblemsolving.com/b/1/a/b1ae208955104081bafe221832509d9dd20eeb83.png)
Prove that the points , , 
and 
are concyclic.
that satisfies: 
be an integer. Ivan writes the numbers 
each on different cards. He then shuffles these 
cards, and divides them into two piles. Prove that at least one of the piles contains two cards such that the sum of their numbers is a perfect square.
be real numbers such that
Prove
be the set of integers. Determine all functions 
such that, for all integers 
and 
, 
Proposed by Liam Baker, South Africa
be a positive integer. Determine all positive integers 
that satisfy the following condition:
of divisors of (not necessarily distinct) such that![\[
\frac{1}{d_1} + \frac{1}{d_2} + \cdots + \frac{1}{d_k} > N,
\]](http://latex.artofproblemsolving.com/2/c/2/2c20e94d7b7d5d5e9bcc32d67c375e29cdef9ae1.png)
there exists a subset of the fractions 
whose sum is exactly .
be a non-isosceles triangle with incircle (
). Denote by 
the points where () touches 
, respectively. The A-excircle of is tangent to 
at 
. The lines 
and 
meet 
at 
and 
, respectively.
, 
, and 
are concurrent at a point.
and 
be the midpoints of and , respectively, and let 
be the orthocenter of triangle 
. Show that the points 
are collinear.
such that the following inequality holds:![\[
x^k y^k z^k (x^2 + y^2 + z^2) \leq 3
\]](http://latex.artofproblemsolving.com/1/1/4/114213c088a60405d584cf25d20b8e82cc07c9ac.png)
for all positive real numbers 
satisfying the condition 
such that the following inequality holds:![\[
x^k y^k z^k (x^2 + y^2 + z^2) \leq xy+yz+zx
\]](http://latex.artofproblemsolving.com/2/1/f/21f7e38350bd3856d283f0f826819b4d96898c79.png)
for all positive real numbers satisfying the condition 
be a positive integer and let 
and 
be positive integers congruent to 1 modulo 4. Prove that there exists a positive integer 
such that at least one of the numbers 
and 
is divisible by 
confirms this.
Suppose corresponds to 
, and corresponds to 
.
implies that 
.
, one is a multiple of the other, but this is clear by looking at highest power of 
dividing 
. 
divides 
or 
for every 
. For the base case, take 
. We get that 
. Now suppose there exists 
such that 
for a sufficiently large .
just take the same .
we have that 
take 
and rewrite 
. From LTE we get that 
so we can choose 
such that 
and since 
we get that 
and we are done because this implies that 
.
. This is trivially true for 
, since 
. So assume that the hypothesis is true for some 
: Then just choose the same k, and it shall suffice for 
. Then, 
. For the construction, let 
. Pick 
such that 
. Then, note that 
(Here 
so LTE can be used). So we get 
, so 
![\[\tau\big(\varphi(n)\big) \geq \varphi\big(\tau(n)\big)\]](http://latex.artofproblemsolving.com/1/e/f/1ef41ab06470c85fd23226c2b470b1811629b559.png)
with all integer n.
Prove that
Let 
Prove that
Let 
Prove that
