[/quote]
,
be an acute triangle. Let 
be the midpoint of side 
,
and 
be the feet of the altitudes from 
and 
,
and 
intersect at a point 
,
is perpendicular to line 
and 
be prime numbers. Prove that if 
is a perfect square, then 
is also a perfect square.
.
and 
with conditions,
is a prime number.
is a perfect square.
is also perfect square.
be a sequence of positive real numbers such that for every 
,![\[
a_n = \max_{1 \leq i \leq 2025} a_{n-i} - \min_{1 \leq i \leq 2025} a_{n-i}
\]](http://latex.artofproblemsolving.com/e/3/0/e30e9e533eb9a41cf847484e676ac4522edda665.png)
Prove that there exists a natural number 
such that for all 
,![\[
a_n < \frac{1}{1404}
\]](http://latex.artofproblemsolving.com/0/1/f/01f1a81eed2230332d51a15501ef2a6ad3a7ec82.png)
such that:
for all 
for all 
is a natural number. A function 
is called \textit
-
with 
,
has a solution in natural numbers 
such that ![\( \frac{y_0}{x_0} \in \left[\frac{1}{100}, 100\right] \)](http://latex.artofproblemsolving.com/d/a/6/da67f464a934f890626d188b84940fc82c4f4ff9.png)
,
is a solution. Suppose 
,
is a polynomial with real coefficients, negative leading coefficients, and total degree greater than 2, and for every real number 
,
as ![\( \frac{y}{x} \in \left[\frac{1}{100}, 100\right] \)](http://latex.artofproblemsolving.com/6/5/d/65d4154dd5fb612f92f24f98406bc9bcfc713806.png)
.
is not 
such that for all 
be defined by![\[
x_1 = 2,\quad x_{n+1} = x_n + \frac{n}{x_n},\quad \text{for all } n \geq 1.
\]](http://latex.artofproblemsolving.com/f/2/8/f28193d8ea3fa6da8bbdb51f9a7e2e9d8a7d3c9f.png)
Prove that the sequences 
and 
have finite limits, and find those limits.
.
!
candy bars. Let N & M be positive integers, what is N+M?
Y by
I was wondering what I could use to practice for Mathcount Target rounds because my sprint is normally 23-25+ (Depending on the level: This is for Chapter, but some times for states) but my target is normally 8-10 (Depending on the level: This is normally for states and some time Nats) and I want that to go up to 12-14 So I was wondering what are some good resources to build that problem solving knowledge for target, and some resources that will help me learn the complex problems on there mostly for Geometry and Algebra.
This post has been edited 1 time. Last edited by sadas123, Mar 29, 2025, 8:17 PM
