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
, from 
outside the circle, draw tangents 
(
are tangential points) and secant 
(
lie on circle 
, 
is between and 
). 
intersects 
at 
; 
is midpoint of 
. The line crossing , paralleling with 
, intersects at 
. 
intersects 
at 
. Let 
lie inside the triangle 
such that: 
=
. From draw chords 
, 
, 
on circle . Prove that: 
is a isosceles triangle
, how many 
-tuples 
of positive integers with 
satisfy 
mod ?
be an acute triangle with orthocenter , and let 
be a point on the side , lying strictly between and . The points 
and are the feet of the altitudes from and , respectively. Denote by 
is the circumcircle of 
, and let 
be the point on such that 
is a diameter of . Analogously, denote by 
the circumcircle of triangle 
, and let 
be the point such that 
is a diameter of . Prove that 
and are collinear.
and 
Prove that
Let 
and 
Prove that
Let 
and 
Prove that![$$a + b \geq \frac{1+\sqrt[3]{28-3\sqrt {87}}+\sqrt[3]{28+3\sqrt {87}}}{3}$$](http://latex.artofproblemsolving.com/4/f/9/4f9529f5dec193ec87a99134d9628ae59078eed7.png)
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 
, 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?
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.
because the main diagonals 
diagonals
