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is 
, then the value of
is 
.
and this sort of becomes an equation for a distance from a point but i am not able to move ahead )
, 
, 
be real numbers such that:![\[
\begin{cases}
x+y+z&=1\\
x^2+y^2+z^2&=12 \\
x^3+y^3+z^3&=18
\end{cases}
\]](http://latex.artofproblemsolving.com/1/c/6/1c6ebcfbb8e8a60d6bec1ecd8971740cd81b247c.png)
Find the value of 
.
and 
such that the function 
, which has a second derivative for all 
, satisfies the following conditions:![\[
(\gamma) \quad [f(x)]^5 + [f(x)]^3 + ax + b = \ln \left( x^2 + x + \frac{5}{2} \right)
\]](http://latex.artofproblemsolving.com/5/8/7/587911fe6b85ec528d64a5f5884902419c8a56b9.png)
![\[
(\delta) \quad f(-3) f(3) < 0, \quad f'(2) > 0
\]](http://latex.artofproblemsolving.com/0/d/f/0dfdf33c2e20aab8e660ea0403a129367f0bc02b.png)
-inch-wide rectangle of some length 
, and a bunch of x
dominos. how many different ways can you completely cover the rectangle with dominos? you can place the dominos horizontally or vertically - for example, for a -by-
rectangle, a valid arrangement of dominos is vertical domino on the left and horizontal dominos on the right.
be a sequence of integers defined by 
, 
, and 
for all 
. Find the remainder when 
is divided by 
.
are either prime numbers or the negatives of prime numbers?
, 
, and 
for 
. Define![\[A_n=\sum_{i=0}^n s_{i}\]](http://latex.artofproblemsolving.com/3/d/4/3d416f32205acdec15a910d056d4a6e9ad4f85d8.png)
Find 
. Express your answer in the form 
, where 
, 
, 
, and 
are positive integers.
, satisfies the recurrence relation![\[ \frac{x_{n+1}}{x_n} = \frac{(x_{n+1})^2 + 27}{x_n^2 + 27} \]](http://latex.artofproblemsolving.com/3/0/4/304d56991f9955dc93c16413478b604dd7042ebf.png)
for all positive integers 
. Suppose that 
. Let 
be the maximum value of![\[ \sum_{n=1}^{2025} x_n. \]](http://latex.artofproblemsolving.com/c/4/d/c4da21c0341fdffb56b9a110dd6df874b6e0e8b2.png)
What is 
?
has the property that the sum of its first terms is divisible by for every positive integer 
. Let 
be the number of such sequences satisfying 
. Compute the remainder when is divided by 
.
has side length . Point 
lies on segment 
such that line 
is perpendicular to line 
. Given that 
, the area of can be expressed as 
, where 
and are relatively prime positive integers. Compute 
. and satisfy 
, 
, and 
. If the number of possible ordered pairs 
is equal to 
, compute the remainder when is divided by .
be a triangle. Point 
lies on side 
, point 
lies on side 
, and point 
lies on side 
such that 
, 
, and 
. Let 
be the foot of the altitude from 
to . Given that 
, 
, and ![$[AQPR] = \tfrac{3}{7} \cdot [ABC]$](http://latex.artofproblemsolving.com/e/f/1/ef1f89d7ee2b45faac7265d8e2777ba3918ae16e.png)
, the value of 
can be expressed as , where and are relatively prime positive integers. Compute .
and tells Bob that the three resulting remainders are 
, 
, and , in some order. For how many values of is it possible for Bob to uniquely determine ? satisfying the system of equations 
The value of 
can be expressed as 
, where 
and 
are positive integers such that is square-free. Compute 
. be the set of all monotonically increasing six-term sequences whose terms are all integers between 
and 
inclusive. We say a sequence 
in is symmetric if for every integer 
, the number of terms of 
that are at least 
is 
. The probability that a randomly chosen element of is symmetric is 
, where and are relatively prime positive integers. Compute ., let 
denote the value of the binary number obtained by reading the binary representation of from right to left. Find the smallest positive integer such that the equation 
has at least ten positive integer solutions . be a quadratic polynomial with a positive leading coefficient. There exists a positive real number 
such that 
and 
for 
. Compute 
.
such that 
and 
is a multiple of 
.
are such that 
and ![\[\frac{x}{26-x^2}=\frac{y}{26-y^2}=\frac{xy}{26-(xy)^2}.\]](http://latex.artofproblemsolving.com/d/7/4/d744d40b3d9e719d30777cc47035fe9e9f6fc44d.png)
What is 
?
. [/center]
, and 
, 
. This is proven in solution 1a-1. We will refer to this as Trick 1.
be a polynomial with integer coefficients such that 
and 
for some integer . Prove that there exists no integer such that 
.
. Let 
such that 
for 
. This yields:
an irrational number. Thus, no such 
exists under these conditions.
an irrational number. Thus, no such exists under these conditions.
just because 
In fact, this clearly wrong when 
and 
isn't irrational when 
, yet such 
and exist as alexheinis pointed out. The rest of your solution is neat, its just this one step that makes an assumption. In fact, I believe the statement is false for all 
except constant and quadratic polynomials. Here is a proof: has degree 
, then take
where 
has degree 
with integer coefficients. Then 
and 
(This type of construction is common when you have one polynomial behaving like a smaller degree polynomial, such as 
, for a set of .) For the quadratic case, take 
. Then and 
give, respectively,
and
The first equation tells us that 
from which we get
and
We can do similar analysis for the second equation:
and
and produce a nonzero, integral value for (nonzero since we are looking for quadratics), and hence no such quadratics exist.
is an integer for integers and 
so 
or 
so 
so 
works.
where 
![\[
\frac{1}{6}
\]](http://latex.artofproblemsolving.com/6/b/2/6b2edc8ad71f66480061d0fdc2d64c8d2177084e.png)
and 
.