Example 5. Let's find all values of $a$ for which the system

$$
\left\{\begin{array}{l}
2^{b x}+(a+1) b y^{2}=a^{2} \\
(a-1) x^{3}+y^{3}=1
\end{array}\right.
$$

has at least one solution for any value of $b,(a, b, x, y \in \mathbf{R})$.
S o l u t i o n. Suppose there exists some value $a$ for which the system has at least one solution, for example, for $b=0$.

In this case, the system will take the form:

$$
\left\{\begin{array}{l}
1=a^{2} \\
(a-1) x^{3}+y^{3}=1
\end{array}\right.
$$

It is clear that this system is consistent only if $a=1$ or $a=-1$. Therefore, if $b=0$, then for these values of $a$ the original system is consistent. We will now prove that the original system will be consistent for any value of $b$.

32

Let $a=1$. Then the original system will take the form:

$$
\left\{\begin{array}{l}
2^{b x}+2 b y^{2}=1 \\
y^{3}=1
\end{array}\right.
$$

From the second equation, we find: $y=1$ (by the condition $y \in \mathbf{R}$). Substituting $y=1$ into the first equation, we get:

$$
2^{b x}=1-2 b
$$

It is easy to see that this equation has no solutions for $b \geqslant \frac{1}{2}$. Therefore, when $a=1$, the system has solutions, but not for any value of $b$, i.e., the value $a=1$ does not satisfy the condition of the problem.

Now let's check $a=-1$. Then the original system will take the form:

$$
\left\{\begin{array}{l}
2^{b x}=1 \\
-2 x^{3}+y^{3}=1
\end{array}\right.
$$

It is completely obvious that the first equation has a solution $x=0$ for any $b$. Then $y=1$. Therefore, the condition of the problem is satisfied only by $a=-1$.

## EXERCISES

57. Prove that the equation

$$
x^{2}-x \sin x y+1=0
$$

has no solutions.

58. Find all pairs of real values of $x$ and $y$ that satisfy the equations:

a) $\sin 2 x+\cos (x+y)=2$

b) $4^{\sin x}-2^{1+\sin x} \cos x y+2^{|y|}=0$;

c) $\operatorname{tg}^{2} x+2 \operatorname{tg} x(\sin x+\cos y)+2=0$;

d) $\left(\sin ^{2} x+\frac{1}{\sin ^{2} x}\right)^{2}+\left(\cos ^{2} x+\frac{1}{\cos ^{2} y}\right)^{2}=12+\frac{1}{2} \sin x y$;

e) $\operatorname{tg}^{4} x+\operatorname{tg}^{4} y+2 \operatorname{ctg}^{2} x \operatorname{ctg}^{2} y=3+\sin ^{2}(x+y)$.

59. Solve the inequalities:
a) $\cos x \geqslant y^{2}+\sqrt{y-x^{2}-1}$

b) $y \geqslant\left|\sec ^{2} x\right|+\sqrt{1-y-x^{2}}$

c) $x-|y| \geqslant 1+\sqrt{x^{2}+y^{2}-1}$.

60. Find the general form of the solution of the system:

$$
\left\{\begin{array}{l}
\operatorname{tg} x=\operatorname{tg}(y-z), \\
\operatorname{tg} y=\operatorname{tg}(z-x) .
\end{array}\right.
$$

61. Find the real solutions of the systems:
a) $\left\{\begin{array}{l}x^{2}+4 y^{2}+5=4 z \\ x-y \geqslant z\end{array}\right.$

2 Order 454
b) $\left\{\begin{array}{l}x^{2}+y^{2}+20=z \\ 8 x+4 y \geqslant z .\end{array}\right.$

62. Find the real solutions of the system:

$$
\left\{\begin{array}{l}
x+y+z=2 \\
2 x y-z^{2}=4
\end{array}\right.
$$

63. Prove that if three numbers $x, y, z$ satisfy the system of equations:

$$
\left\{\begin{array}{l}
x+y+z=a \\
\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\frac{1}{a}
\end{array}\right.
$$

then at least one of these numbers is equal to $a$.

64. Find all values of $a$ for which the system

$$
\left\{\begin{array}{l}
2^{|x|}+|x|=y+x^{2}+a \\
x^{2}+y^{2}=1
\end{array}\right.
$$

has only one solution.

65. Find all values of $a$ and $b$ for which the system

$$
\left\{\begin{array}{l}
\left|\frac{x^{y}-1}{x^{y}+1}\right|=a \\
x^{2}+y^{2}=b
\end{array}\right.
$$

has only one solution ( $a, b, x, y \in \mathbf{R}, x>0, x \neq 1$ ).

66. Find all values of $a$ for which the system

$$
\left\{\begin{array}{l}
\left(x^{2}+1\right)^{a}+\left(b^{2}+1\right)^{y}=2 \\
a+b x y+x^{2} y=1
\end{array}\right.
$$

has at least one solution for any $b(a, b, x, y \in \mathbf{R})$.

67. Determine for which values of the parameter $a$ the system

$$
\left\{\begin{array}{l}
a x^{2}+a-1=y-|\sin x| \\
\operatorname{tg}^{2} x+y^{2}=1
\end{array}\right.
$$

has a unique real solution.

68. Given the system:

$$
\left\{\begin{array}{l}
x^{y}=a, x>0 \\
\operatorname{arctg} x=\frac{\pi}{4}+y
\end{array}\right.
$$

For which values of $a$ does the system have a unique solution?

69. Find all values of $a$ for which the system

$$
\left\{\begin{array}{l}
x^{3}-a y^{3}=\frac{1}{2}(a+1)^{2} \\
x^{3}+a x^{2} y+x y^{2}=1
\end{array}\right.
$$

has at least one solution and any solution satisfies the equation $x+y=0 \quad(a, x, y \in \mathbf{R})$.

34

70. Given the system of equations:

$$
\left\{\begin{array}{l}
\frac{x}{y}+\sin x=a \\
\frac{y}{x}+\sin y=a
\end{array}\right.
$$

For which values of $a$ does the system have a unique solution satisfying the conditions: $0 \leqslant x \leqslant 2 \pi, 0 \leqslant y \leqslant 2 \pi$ ?

## Chapter II

## POWER MEAN AND ITS APPLICATIONS

The expression of the form

$$
C_{\alpha}(a)=\left(\frac{a_{1}^{\alpha}+a_{2}^{\alpha}+\ldots+a_{n}^{\alpha}}{n}\right)^{\frac{1}{\alpha}}
$$

is called the power mean of order $\alpha$ of positive numbers $a_{1}, a_{2}, \ldots, a_{n}$.

In particular, for $\alpha=-1$ we get the harmonic mean

$$
C_{-1}(a)=\frac{n}{\frac{1}{a_{1}}+\frac{1}{a_{2}}+\ldots+\frac{1}{a_{n}}}
$$

for $\alpha=1$ we get the arithmetic mean

$$
C_{1}(a)=\frac{a_{1}+a_{2}+\ldots+a_{n}}{n}
$$

for $\alpha=2$ we get the quadratic mean

$$
C_{2}(a)=\sqrt{\frac{a_{1}^{2}+a_{2}^{2}+\ldots+a_{n}^{2}}{n}}
$$

Let us note some properties of the power mean:

$1^{0}$. The following equality holds:

$$
\lim _{\alpha \rightarrow 0} C_{\alpha}(a)=G
$$

where $G=\sqrt[n]{a_{1} a_{2} \ldots a_{n}}$ is the geometric mean. Therefore, we adopt

$$
C_{0}(a)=\sqrt[n]{a_{1} a_{2} \ldots a_{n}}
$$

$2^{0}$. For any real $\alpha$ and $\beta$, such that $\alpha \leqslant \beta$, the following inequality holds (monotonicity property):

$$
C_{\alpha}(a) \leqslant C_{\beta}(a)
$$

$3^{0}$. Let $\min a$ be the smallest among the numbers $a_{1}, a_{2}, \ldots a_{n}$, and $\max a$ be the largest among these same numbers. Then

$$
\lim _{\alpha \rightarrow-\infty} C_{\alpha}(a)=\min a, \lim _{\alpha \rightarrow+\infty} C_{\alpha}(a)=\max a
$$

2*

From this, the following chain of inequalities follows:

$\min a \leqslant \ldots \leqslant C_{-2} \leqslant C_{-1} \leqslant C_{0} \leqslant C_{1} \leqslant C_{2} \leqslant \ldots \leqslant \max a$.

Equality occurs only when $a_{1}=a_{2}=\ldots=a_{n}$.

## § 1. Proof of Inequalities