x(x + y) + y(x + y) = 90 + 54

(x + y)(x + y) = 144

(x + y)² = 144 = ( ± 12 )²

=> x + y = ± 12

Mà |x + y| ≥ 0 => |x + y| = 12

quá cảm ơn

2^x = 8^y+1 <=> 2^x = ( 2³ )^y+1 = 2^3y+3

=> x = 3y + 3 (1)

9^y = 3^x-9 <=> 3^2.y = 3^x-9 

=> 2y = x - 9 => x = 2y + 9 (2)

Từ (1); (2) => 3y + 3 = 2y + 9

<=> 3y - 2y = 9 - 3=> y = 6

=> 2.6 = x - 9 <=> 12 = x - 9 => x = 21

=> x + y = 21 + 6 = 27

\(\Leftrightarrow\left(\frac{x-1}{2009}-1\right)+\left(\frac{x-2}{2008}-1\right)=\left(\frac{x-3}{2007}-1\right)+\left(\frac{x-4}{2006}-1\right)\)

\(\Leftrightarrow\frac{x-2010}{2009}+\frac{x-2010}{2008}=\frac{x-2010}{2007}+\frac{x-2010}{2006}\)

\(\Leftrightarrow\frac{x-2010}{2009}+\frac{x-2010}{2008}-\frac{x-2010}{2007}-\frac{x-2010}{2006}=0\)

\(\Leftrightarrow\left(x-2010\right)\left(\frac{1}{2009}+\frac{1}{2008}+\frac{1}{2007}+\frac{1}{2006}\right)=0\)

\(\Rightarrow x-2010=0\Rightarrow x=2010\)

A=1+2+2^2+2^3+....+2^9

2A=2+2^2+2^3+....+2^10

2A-A=2^10-1

A=2^10-1/2

B=5.2^8=(2^2+1).2^8=2^10+2^8

=>B>A

2A = 2(1 + 2 + 2² + .... + 2⁹ )

= 2 + 2² + 2³ + ..... + 2¹⁰

2A - A = (2 + 2² + 2³ + ..... + 2¹⁰) - (1 + 2 + 2² + .... + 2⁹ )

A = 2¹⁰ - 1  

B = 5.2⁸ = (2² + 1).2⁸ = 2¹⁰ + 2⁸

2¹⁰ - 1 < 2¹⁰ + 2⁸

=> A < B

1 )Vì \(\left(x+2\right)^2\ge0;\left(y-3\right)^2\ge0\)

\(\Rightarrow\left(x+2\right)^2+\left(y-3\right)^2\ge0\)

\(\Rightarrow\left(x+2\right)^2+\left(y-3\right)^2+1\ge1\)

Dấu "=: xảy ra <=> \(\orbr{\begin{cases}\left(x+2\right)^2=0\\\left(y-3\right)^2=0\end{cases}\Rightarrow\orbr{\begin{cases}x=-2\\y=3\end{cases}}}\)

Vậy ........

2 ) \(\frac{1}{\left(x-2\right)^2+2}\ge\frac{1}{2}\)

Dấu "=" xảy ra <=> x = 2

Vậy ..........

Đặt A = 1 + 3 + 3² + 3³ + .... + 3²⁰⁰⁰

3A = 3(1 + 3 + 3² + 3³ + .... + 3²⁰⁰⁰)

= 3 + 3² + 3³ + 3⁴ + .... + 3²⁰⁰¹ 

3A - A = (3 + 3² + 3³ + 3⁴ + .... + 3²⁰⁰¹ ) - (1 + 3 + 3² + 3³ + .... + 3²⁰⁰⁰)

2A = 3²⁰⁰¹ - 1

\(\Rightarrow A=\frac{3^{2001}-1}{2}\)

ĐẶT A LÀM BIỂU THỨC 1+3+3^2+3^3+..+3^2000

=>3A=3+3^2+3^3+.....+3^2001

=>3A-A=(1+3+3^2+3^3+..+3^2000)-(3+3^2+3^3+......+3^2001)

=>A=3^2001-1/3