[/quote]
,
with an area of 
,
.
,
.
such that for all 
we have:
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)
Let 
be positive reals such that 
.
pieces of equal weights. Can Vova prevent Pasha's win?
be positive real numbers satisfying![\[ xy + yz + zx = 3xyz. \]](http://latex.artofproblemsolving.com/9/2/0/9200874fd6bf9a32ff0c4e82382a450be68cc444.png)
Prove that![\[
\sqrt{\frac{x}{3y^2z^2 + xyz}} + \sqrt{\frac{y}{3x^2z^2 + xyz}} + \sqrt{\frac{z}{3x^2y^2 + xyz}} \le \frac{3}{2}.
\]](http://latex.artofproblemsolving.com/f/5/f/f5fb85d81e93935601ceb5f30c25aea7aa3f23b2.png)
cups labeled 
,
-
Prove that from any configuration where each cup contains an integer amount of water, it is possible to reach a configuration in which each cup contains exactly 1 liter of water in at most 
steps.
Prove that in at most 
steps, one can go from any configuration with integer water amounts to any other configuration with the same property.
Prove that
Let 
Prove that
Prove that
be vertices of regular pentagon inscribed in a circle whose radius is 
and center is at 
.
![\[
\frac{|a|}{a}+\frac{|b|}{b}+\frac{|c|}{c}
\]](http://latex.artofproblemsolving.com/8/4/a/84a1a990f8bbde7670e9154dd9a46bb18c14d547.png)
For all non zero integers 
Y by
1. If x squared minus (x squared times pi)/4 equals 3+ 3 root 3 -2pi, solve for x.
2. You're playing a game where you need to roll at least N to win. You can either roll 2 6-sided fair dice or a 12-sided fair die. For what value of n between 1 and 12, inclusive, is the probability of winning equal no matter which dice you choose?
3. Define the polynomial P(x)=(x^3)+1. Find the sum of the coefficients of P(P(P(x))).
4. Jerry chooses 2 positive integers x and y that yield a product of 6^5, or 7776. He then computes the value of n, which is the GCD of (a,b). How many possible values of n can Jerry get?
Provide solutions to answers
Also how hard are these problems: AMC8, AMC10, MathCounts, or AMC12?
2. You're playing a game where you need to roll at least N to win. You can either roll 2 6-sided fair dice or a 12-sided fair die. For what value of n between 1 and 12, inclusive, is the probability of winning equal no matter which dice you choose?
3. Define the polynomial P(x)=(x^3)+1. Find the sum of the coefficients of P(P(P(x))).
4. Jerry chooses 2 positive integers x and y that yield a product of 6^5, or 7776. He then computes the value of n, which is the GCD of (a,b). How many possible values of n can Jerry get?
Provide solutions to answers
Also how hard are these problems: AMC8, AMC10, MathCounts, or AMC12?
This post has been edited 1 time. Last edited by hashbrown2009, Apr 6, 2025, 11:07 PM
,
.
.
,
.
,
,
to win. You can either roll two 
-
-
between 
and 
inclusive, is the probability of winning equal, no matter which dice you choose?
and 
that yield a product of 
He then computes the value of 
which is 
How many possible values of 
by factoring and dividing
.
Recall that the sum of the coefficients of a polynomial 
is 
.
.
.
or 
.
possibilities of the GCD.
