9.095. $\log _{4}(x+7)>\log _{2}(x+1)$.
Solution.

Domain of definition: $x>-1$.

Let's switch to base 2. We have

$$
\begin{aligned}
& \frac{1}{2} \log _{2}(x+7)>\log _{2}(x+1) \Leftrightarrow \log _{2}(x+7)>2 \log _{2}(x+1) \Leftrightarrow \\
& \Leftrightarrow \log _{2}(x+7)>\log _{2}(x+1)^{2}
\end{aligned}
$$

The last inequality is equivalent to the system of inequalities, taking into account the domain of definition:

$$
\left\{\begin{array} { l } 
{ x + 7 > ( x + 1 ) ^ { 2 } } \\
{ x + 1 > 0 }
\end{array} \Leftrightarrow \left\{\begin{array} { l } 
{ x ^ { 2 } + x - 6 < 0 }
\end{array} \Leftrightarrow \left\{\begin{array}{l}
-3 < x < 2
\end{array} \Leftrightarrow -1 < x < 2\right.\right.\right.
$$

Answer: $\quad x \in(-1 ; 2)$.

## Solutions to Chapter 10

## PROBLEMS IN PLANE GEOMETRY

## BASIC FORMULAS

1. Arbitrary triangle ( $a, b, c$ - sides; $\boldsymbol{\alpha}, \boldsymbol{\beta}, \boldsymbol{\gamma}-$ opposite angles; $p$ - semiperimeter; $R$ - radius of the circumscribed circle; $r$ - radius of the inscribed circle; $S$ - area; $h_{a}$ - height to side $a$ ):

$$
\begin{gathered}
S=\frac{1}{2} a h_{a} ; \\
S=\frac{1}{2} b c \sin \alpha ; \\
S=\sqrt{p(p-a)(p-b)(p-c)} ; \\
r=\frac{S}{p} ; \\
R=\frac{a b c}{4 S} ; \\
S=\frac{a^{2} \sin B \sin C}{2 \sin A} ; \\
\frac{a}{\sin \alpha}=\frac{b}{\sin \beta}=\frac{c}{\sin \gamma}=2 R \text { (sine theorem). }
\end{gathered}
$$

2. Right triangle ( $a, b$ - legs; $c$ - hypotenuse; $a_{c}, b_{c}$ - projections of the legs on the hypotenuse):

$$
S=\frac{1}{2} a b
$$

$$
\begin{gathered}
S=\frac{1}{2} c h_{c} \\
r=\frac{a+b-c}{2} ; \\
R=\frac{c}{2} ; \\
a^{2}+b^{2}=c^{2} \text { (Pythagorean theorem) } \\
\frac{a_{c}}{h_{c}}=\frac{h_{c}}{b_{c}} \\
\frac{a_{c}}{a}=\frac{a}{c} \\
\frac{b_{c}}{b}=\frac{b}{c} ; \\
a=c \sin \alpha=c \cos \beta=b \tan \alpha=b \cot \beta
\end{gathered}
$$

3. Equilateral triangle:

$$
\begin{aligned}
& S=\frac{a^{2} \sqrt{3}}{4} \\
& r=\frac{a \sqrt{3}}{6} \\
& R=\frac{a \sqrt{3}}{3}
\end{aligned}
$$

4. Arbitrary convex quadrilateral ( $d_{1}$ and $d_{2}$ - diagonals; $\varphi$ - angle between them; $S$ - area):

$$
S=\frac{1}{2} d_{1} d_{2} \sin \varphi
$$

5. Parallelogram ( $a$ and $b$ - adjacent sides; $\alpha$ - angle between them; $h_{a}$ - height to side $a$ ):

$$
S=a h_{a}=a b \sin \alpha=\frac{1}{2} d_{1} d_{2} \sin \varphi
$$

6. Rhombus:

$$
S=a h_{a}=a^{2} \sin \alpha=\frac{1}{2} d_{1} d_{2}
$$

7. Rectangle:

$$
S=a b=\frac{1}{2} d_{1} d_{2} \sin \varphi
$$

8. Square ( $d$ - diagonal):

$$
S=a^{2}=d^{2} / 2
$$

9. Trapezoid ( $a$ and $b$ - bases; $h$ - distance between them; $l$ - midline):

$$
\begin{gathered}
l=\frac{a+b}{2} \\
S=\frac{a+b}{2} \cdot h=l h
\end{gathered}
$$

10. Circumscribed polygon ( $p$ - semiperimeter; $r$ - radius of the inscribed circle):

$$
S=p r
$$

11. Regular polygon ( $a_{n}$ - side of a regular $n$-gon; $R$ - radius of the circumscribed circle; $r$ - radius of the inscribed circle):

$$
\begin{gathered}
a_{3}=R \sqrt{3} ; a_{4}=R \sqrt{2} ; a_{6}=R ; \\
S=\frac{n a_{n} r}{2} .
\end{gathered}
$$

12. Circle, disk ( $r$ - radius; $C$ - circumference; $S$ - area of the disk):

$$
\begin{aligned}
& C=2 \pi r \\
& S=\pi r^{2}
\end{aligned}
$$

13. Sector ( $l$ - length of the arc bounding the sector; $n^{\circ}$ - degree measure of the central angle; $\alpha$ - radian measure of the central angle):

$$
\begin{gathered}
l=\frac{\pi r n^{\circ}}{180^{\circ}}=r \alpha \\
S=\frac{\pi r^{2} n^{\circ}}{360^{\circ}}=\frac{1}{2} r^{2} \alpha
\end{gathered}
$$

## ADDITIONAL RELATIONS BETWEEN ELEMENTS OF FIGURES

1. The three medians of a triangle intersect at one point, which divides each median in the ratio $2: 1$, counting from the vertex of the triangle.
2. The length of a median of a triangle is given by the formula

$$
m_{a}=\frac{1}{2} \sqrt{2\left(b^{2}+c^{2}\right)-a^{2}}
$$

where $a, b, c$ are the lengths of the sides of the triangle.

3. The length of a side of a triangle is given by the formula

$$
a=\frac{2}{3} \sqrt{2\left(m_{b}^{2}+m_{c}^{2}\right)-m_{a}^{2}},
$$

where $m_{a}, m_{b}, m_{c}$ are the lengths of the medians of the triangle.

4. The bisector divides the side of a triangle into segments proportional to the other two sides.
5. The length of the bisector of a triangle is given by the formula

$$
l_{c}=\sqrt{a b-a_{1} b_{1}},
$$

where $a$ and $b$ are the lengths of two sides of triangle $A B C ; a_{1}$ and $b_{1}$ are the segments of the third side.

6. The length of the bisector of a triangle is given by the formula

$$
l_{c}=\frac{\sqrt{a b(a+b+c)(a+b-c)}}{a+b}
$$

7. For any triangle, the relationship between its heights $h_{a}$, $h_{b}, h_{c}$ and the radius $r$ of the inscribed circle is given by the formula

$$
\frac{1}{h_{a}}+\frac{1}{h_{b}}+\frac{1}{h_{c}}=\frac{1}{r}
$$

8. The area $S$ of an isosceles trapezoid, whose diagonals are perpendicular to each other, is equal to the square of its height, i.e., $S=h^{2}$.
9. The height of an isosceles trapezoid, in which a circle can be inscribed, is the geometric mean of its bases.

Proofs of all these additional relations can be found in any edition of this problem book from recent years.

![](https://cdn.mathpix.com/cropped/2024_05_22_fa9db84b44b98ec0a5c7g-556.jpg?height=426&width=408&top_left_y=79&top_left_x=186)

Fig. 10.1

![](https://cdn.mathpix.com/cropped/2024_05_22_fa9db84b44b98ec0a5c7g-556.jpg?height=354&width=406&top_left_y=159&top_left_x=777)

Fig. 10.2