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.
, from 
outside the circle, draw tangents 
(
is tangential point) and secant 
(
lie inside 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 
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.
is such that 
for any positive integers 
. Assume there exists a positive integer such that 
for all positive integers 
. Determine all possible values of 
.
is such that for any positive integers . Assume there exists a positive integer such that for all positive integers . Determine all possible values of .
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
satisfies the condition and 
.
.
and 
,
for all 
,
for all non-negative integers 
and 
,![\[
g(a, b) + g(c, b) = g(ac, b^2).
\]](http://latex.artofproblemsolving.com/9/1/d/91dd66cfb45616ccfaa1691e942e816060decf3d.png)
denote the assumption on 
in this equation. For 
,
depends only on the value of 
for all prime numbers ![\[
h(x) = \sum_{p \text{ prime}} h(p)\nu_p(x). \tag{2}
\]](http://latex.artofproblemsolving.com/6/8/b/68b83a00528564c6a54d5e2ac13d438b4266babf.png)
or 
.![\[
h(45) = g(45, 1) = 2g(3, 1) + g(5, 1) = 0.
\]](http://latex.artofproblemsolving.com/8/e/0/8e02c396ee210441ab72d69160d7f57de8ab2805.png)
and 
follows that 
.
we get that 
.
from 
,
,
,
.
,![\[
g(n, x) = g(n, x^n) + (x^2 - 1)h(x) + (x^4 - 1)h(x) + \ldots.
\]](http://latex.artofproblemsolving.com/4/4/b/44b899e553765ac801138cdc87c7900ea3c368da.png)
for some 
,![\[
g(n, x) + (x^2 - 1)h(x) = g(n, x^2) \le g(n, (x + 1)^2) = g(n, x + 1) + (x^2 - 1)h(x + 1).
\]](http://latex.artofproblemsolving.com/9/d/0/9d07a67f85a3666f1bb39273106e7079e9f29a88.png)
,
are constants. Therefore 
for all 
from the beginning of the problem to case 
.
or 
,
for every prime 
because 
is monotonically increasing. Let 
.
,
(the case 
is analogous). Then we have![\[
h(a^k) = kh(5^k)(7) = kh(5)(7) = h(b^k).
\]](http://latex.artofproblemsolving.com/0/1/7/017694b47f0bd9755bf3e2458aaea70e3f458104.png)
.
,![\[
q + 1 = \frac{a^{q + 1} - 1}{a - 1} > \frac{b^{q + 1} - 1}{b - 1} > b^q + 1.
\]](http://latex.artofproblemsolving.com/f/5/f/f5fdfd3799129cc139d96f18b8d683e47f815745.png)
.
,
and 
.