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 for all positive real numbers 
,![\[
f^2(x) \geq f(x+y)\left(f(x) + y\right).
\]](http://latex.artofproblemsolving.com/7/1/f/71fb0d51492ee4c3b2bf4049e8224882fec16020.png)
is called a good prime if there exists a positive integer 
such that![\[
n^2 + 1 \text{ is divisible by } p^{2025}.
\]](http://latex.artofproblemsolving.com/3/3/4/334cd7b5e4a4bfb6b0dd3a4607cdd793ecadd66c.png)
Prove that there are infinitely many good primes.
. We call a distribution the act of distributing the integers from 
represented by tokens to players 
to 
so that they all have the same number of tokens in their urns.
beats 
when, when and each draw a token from their urn, has a strictly greater chance of drawing a larger number than . We then denote 
. A distribution is said to be chicken-fox-viper when 
What is 
Let there be a variable positive integer whose last two digits are 
. Prove that this number is divisible by a prime greater than 
.
, determine the smallest integer 
, such that it is possible to place numbers 
around a circle so that the sum of every consecutive numbers takes one of at most values.
is inscribed in circle 
. The tangent lines to at 
and 
meet at 
. Point 
lies on ray 
such that 
. Points 
and 
lie on ray 
(with in between and ) such that 
. Prove that triangles and 
are similar to each other.
be a positive integer such that divides the sum![\[
1 + \sum_{i=1}^{n-1} i^{n-1}.
\]](http://latex.artofproblemsolving.com/e/2/d/e2daf7800da3f984cd761ed1fecf10083444db25.png)
Prove that is square-free.
, all the roots added will give 
. I also learned that if every single root, multiplies every single root, it will give 
. I also learned that if all the roots are multiplied, it will give 
. Is this right? And is there any purpose for these equations?
and 
Then the number of elements in the set ${C⊆A:C∩B=ϕ}$ is
and 
If each person remains there for 
minutes, what's the probability that all three people meet each other?
Find the maximum positive integer 
such that for every subset 
with n elements, there always exist two elements a, b in T such that:
with 
.![\[
\frac{a^5}{b^2 + 2c^3} + \frac{2b^5}{3c + a^6} + \frac{c^7}{a + b^4} \geq 2.
\]](http://latex.artofproblemsolving.com/f/f/5/ff5043705025592aeefc55a764a8b7170e13ff4b.png)
and 
Prove that
. Prove that
and solve for the area of a hexagon in that 
square.
?