.
intersects 
at 
and 
.
is the midpoint of 
.
be the altitude in 
,
intersects 
.
be the midpoint of 
be the reflection of 
with respect to 
and 
are similar.
be the set of all positive integers. Find, with proof, all functions 
with the property that 
for all positive integers 
.
and 
.
be a positive integer and 
be a polynomial with integer coefficients such that 
.
is divisible by 
.
and 
.
be a prime number. Define the set![\[
M_q = \left\{ x \in \mathbb{Z}^* \,\middle|\, \sqrt{x^2 + 2q^{2025} x} \in \mathbb{Q} \right\}.
\]](http://latex.artofproblemsolving.com/9/8/b/98b266a869cf5686cea86be51152ef234486993e.png)
.
and 
be points on a plane such that 
Let 
be a point on that plane such that 
Prove that ![$$AP \in \left[\frac{5-\sqrt{5}}{10}, \frac{-1+\sqrt{5}}{2}\right] \cup \left[\frac{5+\sqrt{5}}{10}, \frac{1+\sqrt{5}}{2}\right].$$](http://latex.artofproblemsolving.com/7/8/e/78e428cbc99ce4e9448f5c4cec86f50567123812.png)
is bounded.
.
.
![$\color[rgb]{0.35,0.35,0.35}\displaystyle\sum_{n=0}^\infty\frac{\sin (nx)}{3^n}$](http://latex.artofproblemsolving.com/a/f/3/af33799c6924b61378db323645ea3e1b9c4e8c9d.png)
if ![$\color[rgb]{0.35,0.35,0.35}\sin x=1/3$](http://latex.artofproblemsolving.com/c/e/a/cea5c46445caa512d94d655350170fb19518a2c8.png)
and ![$\color[rgb]{0.35,0.35,0.35} 0\le x\le \pi/2$](http://latex.artofproblemsolving.com/9/8/b/98b7d75c944002b8b9c11dbe8cca0d402e6fbb67.png)
.
red marbles and 
blue marbles. The rules of the game are as follows:
Y by
Given an acute triangle 
, in which 
. On the sides 
and 
take the points 
and 
, respectively, such that 
. Prove that the center of the inscribed circle of triangle 
lies on the side 
.
(Dmitry Shvetsov)
, in which 
. On the sides 
and 
take the points 
and 
, respectively, such that 
. Prove that the center of the inscribed circle of triangle 
lies on the side 
.(Dmitry Shvetsov)
This post has been edited 1 time. Last edited by parmenides51, Feb 25, 2024, 5:18 PM
