,
,
be the roots of 
Find the numerical value of the expression:![\[\frac{\alpha + 2}{\alpha - 2} + \frac{\beta + 2}{\beta - 2} + \frac{\gamma + 2}{\gamma - 2}\]](http://latex.artofproblemsolving.com/b/5/9/b59f4515409422333e10857ea775a36e58490b5d.png)
and 
be the roots of the polynomial 
.
such that![\[
\frac{11}{\log_2 x} + \frac{1}{2 \log_{25} x} - \frac{3}{\log_8 x} = \frac{1}{\log_b x} \quad \text{for all } x > 1.
\]](http://latex.artofproblemsolving.com/2/d/7/2d76319b3d4fcd1f193ea0f9f37cb4bf513376f3.png)
are the roots of the polynomial 
.
be the unique monic polynomial whose roots are 
.
.
![\[ x^{\log{x}} = 10^{2 - 3\log{x} + 2(\log{x})^2}\]](http://latex.artofproblemsolving.com/b/f/b/bfb8185992867a8926b5c62198a799c935771d53.png)
where 
is the logarithm of 
to the base 
.
Prove that
that satisfies![\[
a^3+b^3=1957
\]](http://latex.artofproblemsolving.com/5/d/8/5d8d2a40734ec6b0a9149f9a4ae3a5828d5055f3.png)
![\[
(a+b)(a+1)(b+1)=2014
\]](http://latex.artofproblemsolving.com/3/0/0/300f2d16e669a1d8d9c54b357a6eaf7a8eb5d826.png)
Hence, find the value of 
and n a positive integer 
,
cause there are 10 ways to choose 1 element that will be repeated. Similarly for 2 same elements it would be 
the answer would be ![$(2^{10}-1)-([A_1+A_3+A_5+A_7+A_9]-[A_2+A_4+A_6+A_8+A_{10}].$](http://latex.artofproblemsolving.com/7/3/1/73130cab8af6ffb5c2638548fdf6af3ecc70a6ed.png)
But this number turned out to be 
But I realized that 
So I was really confused of why i got the right answer somehow in my calculations.
written on the board. We apply the following rule: At each step, we will add all the numbers that are the sum of the two numbers on the board so that the sum does not appear on the board. For example, if the two initial numbers are 
;
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;
,
;
,
Y by Kizaruno
A positive even integer 
is called stylish if the set 
can be partitioned into 
pairs such that the sum of the elements in each pair is a power of 
. For example, 
is stylish because the set 
can be partitioned as 
, with sums , 
, and respectively. Determine the number of stylish numbers less than 
.
is called stylish if the set 
can be partitioned into 
pairs such that the sum of the elements in each pair is a power of 
. For example, 
is stylish because the set 
can be partitioned as 
, with sums , 
, and respectively. Determine the number of stylish numbers less than 
.This post has been edited 1 time. Last edited by pedronis, Apr 13, 2025, 5:40 PM
