such that![\[\begin{aligned}
\begin{cases}
x+y^2+z^2=a,\\
x^2+y+z^2=a,\\
x^2+y^2+z=a.
\end{cases}
\end{aligned}\]](http://latex.artofproblemsolving.com/9/8/6/986593653bf128bf7b3d497abed78e1459134c5f.png)
such that ![\[ \dfrac{a^2}{2ab^2-b^3+1} \]](http://latex.artofproblemsolving.com/0/d/2/0d2f10786d22af2ea94ebeed6b15e723f4042734.png)
is a positive integer.
?
.
s.t. 
.
,![$K=\mathbb{Q}[\sin^2(\dfrac{2}{3}\arctan(3\sqrt{3}))]$](http://latex.artofproblemsolving.com/5/b/8/5b87dd74c6659350c000071d5a9228f2a80b1466.png)
.
.
and not its conjugate transpose.
be a finite abelian group. There is a magic box 
.
contains only the group identity, 
.
be a positive real number. Divide the positive real axis into intervals 
,
,
,
,
,
satisfying the following differential equations:![\[
u''(x) + 9^2u(x) = 0, \quad \text{for } x \text{ in black intervals},
\]](http://latex.artofproblemsolving.com/2/6/8/268b7bc6150b13eed666d566b540df7e643d8344.png)
![\[
u''(x) + 63^2u(x) = 0, \quad \text{for } x \text{ in white intervals},
\]](http://latex.artofproblemsolving.com/b/4/6/b46d0d00b34cd3fb72ec310625384547b14871c5.png)
with the initial conditions:![\[
u(0) = 1, \quad u'(0) = 1,
\]](http://latex.artofproblemsolving.com/4/1/6/416754efbff23b7efbf02b32149b1e1fd5cb9905.png)
and the continuity conditions:![\[
u(x) \text{ and } u'(x) \text{ are continuous functions}.
\]](http://latex.artofproblemsolving.com/b/3/0/b309f7b8c70cb27683a96af8452ebf9e36f67a4c.png)
Show that![\[
\lim_{\omega \to 0} u\left(\frac{\pi}{45}\right) = 0.
\]](http://latex.artofproblemsolving.com/f/0/b/f0b9105523f0561e0a4d151a2695cbf6917e5cc7.png)
![$x\in [0,4]$](http://latex.artofproblemsolving.com/a/8/9/a89e045d4c66cdb788e64e613bd6c3c9b5931cc2.png)
at which the function 
takes its maximum value
be an algebraic extension of 
,
be an algebraic closure of 
is an embedding such that 
for all 
,
extends to an automorphism of 
is an embedding. Then, there exists an embedding 
that extends 
,
is an isomorphism.
and 
that extends 
is also an algebraically closed field.
,
be the field of algebraic numbers, 
.
defined by
Then, according to the exercise above, 
How should we interpret 
?
,
.
such that![\[\mathcal{E}(B \cup C) = B \cup C.\]](http://latex.artofproblemsolving.com/e/c/3/ec34115c9e6e169c19c739220065f647934a1a5f.png)
Y by
A pair of positive integers 
is good if they satisfy 
and they do not divide each-other. Given coprime positive integers 
and 
, show that there exist infinitely many 
for which there exists a positive integer 
such that 
is good.
(Here,
denotes the product of 
's prime divisors, as usual.)
is good if they satisfy 
and they do not divide each-other. Given coprime positive integers 
and 
, show that there exist infinitely many 
for which there exists a positive integer 
such that 
is good.(Here,
denotes the product of 
's prime divisors, as usual.)This post has been edited 4 times. Last edited by Iveela, Jan 20, 2025, 2:38 PM
be an arbitrary prime number. If ![\[a^n + bm \quad ; \quad a^{n + 1} - b^{n + 1} \quad ; \quad b^n + am\]](http://latex.artofproblemsolving.com/f/3/7/f379c0956d1ff755f8e371ad1923649b42f94b20.png)
and 
,![\[p \mid a(a^n + bm) - b(b^n + am) = a^{n + 1} - b^{n + 1}\]](http://latex.artofproblemsolving.com/0/f/5/0f5b0e432b7cf26880d1a19532ce8f4031170b45.png)
and ![\[p \mid a(a^n + bm) - (a^{n + 1} - b^{n + 1}) = b^{n + 1} - abm\]](http://latex.artofproblemsolving.com/3/e/9/3e929d46650b0a2d816f672dd4b1cefdeffe20a5.png)
Since 
,
which implies 
satisfy the following three conditions, then there exists 
and 
.
satisfy 
.
divides 
.
which implies 
and hence, 
.
divisible by 
.
and ![\[a(bk + 1) - b(ak + 1) = a - b\]](http://latex.artofproblemsolving.com/3/e/3/3e30cf674e0ec1e313580e9ae86b9f927003a0a1.png)
.
and 
.
.
which satisfies the conditions of Claim 3. Let 
and 
.![\[\frac{a^n- b^n}{a - b} = a^{n - 1} + a^{n - 2}b + \dots + b^{n - 2}a + b^{n - 1} > n \cdot a^{n - 1} > a^n\]](http://latex.artofproblemsolving.com/a/d/3/ad320a370dea0f044c24490c2d1f03369e2f525d.png)
We also have 
and 
.
.
by 
