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 
is a perfect square.
have altitudes 
and 
which meet at 
. The reflection of 
over 
is 
. Let 
meet 
at 
and 
at 
. Let meet 
at 
. Prove that 
.
from the set of positive integers to the set of positive integers such that, for all positive integers 
and 
, there exists a non-degenerate triangle with sides of lengths![\[ a, f(b) \text{ and } f(b + f(a) - 1).\]](http://latex.artofproblemsolving.com/4/a/b/4ab97da27e05d0f9e7762f74d0d0bd2dc8b9842f.png)
the following inequality takes place![\[ \frac{1}{b(a+b)}+ \frac{1}{c(b+c)}+ \frac{1}{a(c+a)} \geq \frac{27}{2(a+b+c)^2} . \]](http://latex.artofproblemsolving.com/6/5/3/6533ef78f4a0b3bd90624c0722e4eaeb0ffbf0a6.png)
be a point interior to triangle (with 
). The lines 
, 
and 
meet again its circumcircle 
at 
, 
, respectively 
. The tangent line at 
to meets the line 
at 
. Show that from 
follows 
. with 
prove that 
, we can post new and original problems, my idea is to make an competition like 
, 6 problems for 2 rounds 
are non-negative reals such that 
.
.
so that![\[f(f(x)+x+y) = f(x+y) + y f(y)\]](http://latex.artofproblemsolving.com/c/4/6/c466d41cd26543767d6ccc2bab6be70bc1f8f36b.png)
for all real numbers 
.
has points 
,
,
on segment in that order, where is between 
and , and 
and 
trisect angle 
. If 
, 
, and 
, find 
.
has the property that the sum of its first 
terms is divisible by for every positive integer 
. Let be the number of such sequences satisfying 
. Compute the remainder when is divided by 
.
has side length 
. Point 
lies on segment 
such that line 
is perpendicular to line 
. Given that 
, the area of can be expressed as 
, where 
and are relatively prime positive integers. Compute 
.
and 
satisfy 
, 
, and 
. If the number of possible ordered pairs 
is equal to 
, compute the remainder when is divided by . be a triangle. Point lies on side , point lies on side , and point lies on side 
such that 
, 
, and 
. Let be the foot of the altitude from to . Given that 
, 
, and ![$[AQPR] = \tfrac{3}{7} \cdot [ABC]$](http://latex.artofproblemsolving.com/e/f/1/ef1f89d7ee2b45faac7265d8e2777ba3918ae16e.png)
, the value of 
can be expressed as , where and are relatively prime positive integers. Compute .
and tells Bob that the three resulting remainders are 
, 
, and , in some order. For how many values of is it possible for Bob to uniquely determine ?
satisfying the system of equations 
The value of 
can be expressed as 
, where 
and 
are positive integers such that is square-free. Compute 
. be the set of all monotonically increasing six-term sequences whose terms are all integers between 
and 
inclusive. We say a sequence 
in is symmetric if for every integer 
, the number of terms of 
that are at least 
is 
. The probability that a randomly chosen element of is symmetric is 
, where and are relatively prime positive integers. Compute ., let 
denote the value of the binary number obtained by reading the binary representation of from right to left. Find the smallest positive integer 
such that the equation 
has at least ten positive integer solutions . be a quadratic polynomial with a positive leading coefficient. There exists a positive real number 
such that 
and 
for 
. Compute 
.
such that 
and 
is a multiple of 
.
can be defined as a sequence such that 
, and 
, such that 
. The function 
is such that 
. How many numbers within the interval 
for the function 
are perfect squares?
can be defined as a sequence such that , and , such that . The function is such that . How many numbers within the interval for the function are perfect squares? and 
are related?
is defined on the positive integers where 
for all positive integers . Also, the sequence is defined in two different ways that contradict each other if is allowed to be any positive integer. Maybe it is intended that we can only take 
?
for all positive integers such that , which is probably a crucial idea for whatever this problem is meant to be. there are such that and 
is a perfect square. Everything else is just confusing.
be distinct positive integers. Prove that![\[
\operatorname{lcm}\bigl\{\,a_i + a_j : 1 \le i < j \le n\bigr\}
\;\ge\;
\sum_{i=1}^{n} a_i.
\]](http://latex.artofproblemsolving.com/6/8/c/68cb27e6a5e8218299ce3015b09d34b84bfe5b7c.png)
,
to be a perfect square. There are 
perfect squares between 
and 
inclusive, so our answer is