Logarithmic Function Graph Intersection or Geometric Analysis
The question involves the intersection of logarithmic curves with lines or other curves, or asks about geometric properties (e.g., coordinates of intersection points, lattice point counting, or graphical reasoning).
For a natural number $n ( n \geqq 2 )$, let the $x$-coordinates of the two distinct points where the line $y = - x + n$ and the curve $y = \left| \log _ { 2 } x \right|$ meet be $a _ { n }$ and $b _ { n }$ respectively ($a _ { n } < b _ { n }$). Which of the following statements in are correct? [4 points] ㄱ. $a _ { 2 } < \frac { 1 } { 4 }$ ㄴ. $0 < \frac { a _ { n + 1 } } { a _ { n } } < 1$ ㄷ. $1 - \frac { \log _ { 2 } n } { n } < \frac { b _ { n } } { n } < 1$ (1) ㄱ (2) ㄴ (3) ㄷ (4) ㄴ, ㄷ (5) ㄱ, ㄴ, ㄷ
For a natural number $n ( n \geqq 2 )$, let $a _ { n }$ and $b _ { n } \left( a _ { n } < b _ { n } \right)$ be the $x$-coordinates of the two distinct points where the line $y = - x + n$ and the curve $y = \left| \log _ { 2 } x \right|$ meet. Which of the following statements in are correct? [4 points] Remarks ㄱ. $a _ { 2 } < \frac { 1 } { 4 }$ ㄴ. $0 < \frac { a _ { n + 1 } } { a _ { n } } < 1$ ㄷ. $1 - \frac { \log _ { 2 } n } { n } < \frac { b _ { n } } { n } < 1$ (1) ᄀ (2) ᄂ (3) ᄃ (4) ᄂ, ᄃ (5) ᄀ, ᄂ, ᄃ
On the coordinate plane, for natural numbers $n$, consider the region $$\left\{ (x, y) \mid 2^x - n \leq y \leq \log_2(x + n) \right\}$$ Let $a_n$ be the number of points in this region satisfying the following conditions. (가) The $x$-coordinate and $y$-coordinate are equal. (나) Both the $x$-coordinate and $y$-coordinate are integers. For example, $a_1 = 2, a_2 = 4$. Find the value of $\sum_{n=1}^{30} a_n$. [4 points]
In the coordinate plane, for a natural number $n$, let $a _ { n }$ be the number of points in the region $$\left\{ ( x , y ) \mid 2 ^ { x } - n \leq y \leq \log _ { 2 } ( x + n ) \right\}$$ that satisfy the following conditions. (a) The $x$-coordinate and $y$-coordinate are equal. (b) The $x$-coordinate and $y$-coordinate are both integers. For example, $a _ { 1 } = 2$ and $a _ { 2 } = 4$. Find the value of $\sum _ { n = 1 } ^ { 30 } a _ { n }$. [4 points]
For a natural number $n$, let $a _ { n }$ be the smallest natural number $m$ satisfying the following conditions. What is the value of $\sum _ { n = 1 } ^ { 10 } a _ { n }$? [4 points] (가) The coordinates of point A are $\left( 2 ^ { n } , 0 \right)$. (나) Let D be the point on the line passing through two points $\mathrm { B } ( 1,0 )$ and $\mathrm { C } \left( 2 ^ { m } , m \right)$ whose $x$-coordinate is $2 ^ { n }$. The area of triangle ABD is less than or equal to $\frac { m } { 2 }$. (1) 109 (2) 111 (3) 113 (4) 115 (5) 117
For a real number $x \geq \frac { 1 } { 100 }$, let $f ( x )$ be the mantissa of $\log x$. Let $R$ be the region representing the ordered pairs $( a , b )$ of two real numbers satisfying the following conditions on the coordinate plane. (가) $a < 0$ and $b > 10$. (나) The graph of the function $y = 9 f ( x )$ and the line $y = a x + b$ meet at exactly one point. For a point $( a , b )$ in region $R$, the minimum value of $( a + 20 ) ^ { 2 } + b ^ { 2 }$ is $100 \times \frac { q } { p }$. Find the value of $p + q$. (where $p$ and $q$ are coprime natural numbers.) [4 points]
For a real number $a$ with $\frac { 1 } { 4 } < a < 1$, let A and B be the points where the line $y = 1$ meets the curves $y = \log _ { a } x$ and $y = \log _ { 4 a } x$ respectively, and let C and D be the points where the line $y = - 1$ meets the curves $y = \log _ { a } x$ and $y = \log _ { 4 a } x$ respectively. Which of the following statements in the given options are correct? [4 points] $\langle$Given Options$\rangle$ ㄱ. The coordinates of the point that divides segment AB externally in the ratio $1 : 4$ are $( 0,1 )$. ㄴ. If quadrilateral ABCD is a rectangle, then $a = \frac { 1 } { 2 }$. ㄷ. If $\overline { \mathrm { AB } } < \overline { \mathrm { CD } }$, then $\frac { 1 } { 2 } < a < 1$. (1) ㄱ (2) ㄷ (3) ㄱ, ㄴ (4) ㄴ, ㄷ (5) ㄱ, ㄴ, ㄷ
For a real number $a$ with $\frac { 1 } { 4 } < a < 1$, let A and B be the points where the line $y = 1$ meets the curves $y = \log _ { a } x$ and $y = \log _ { 4 a } x$ respectively, and let C and D be the points where the line $y = - 1$ meets the curves $y = \log _ { a } x$ and $y = \log _ { 4 a } x$ respectively. Choose all correct statements from the following. [3 points]
$\langle$Statements$\rangle$
ㄱ. The point that divides segment AB externally in the ratio $1 : 4$ has coordinates $( 0,1 )$. ㄴ. If quadrilateral ABCD is a rectangle, then $a = \frac { 1 } { 2 }$. ㄷ. If $\overline { \mathrm { AB } } < \overline { \mathrm { CD } }$, then $\frac { 1 } { 2 } < a < 1$. (1) ㄱ (2) ㄷ (3) ㄱ, ㄴ (4) ㄴ, ㄷ (5) ㄱ, ㄴ, ㄷ
Consider the following equation where $a$ is a real number and $a > 1$ : $$( * ) \quad a ^ { x } = x$$ Which of the following equations must have the same number of real solutions as $( * )$ ? I $\quad \log _ { a } x = x$ II $\quad a ^ { 2 x } = x ^ { 2 }$ III $a ^ { 2 x } = 2 x$ A none of them B I only C II only D III only E I and II only F I and III only G II and III only H I, II and III
In the Cartesian coordinate plane, the graphs of functions $f(x) = \log_{2} x$ and $g(x) = \log_{\frac{1}{4}} x$ are given in the figure. Points $A$ and $D$ are on the graph of function $f$, and points $B$ and $C$ are on the graph of function $g$. In the figure, the line segment $[AB]$ passing through the point $(4, 0)$ and the line segment $[CD]$ are both perpendicular to the x-axis, and the area of triangle $ABC$ is 6 square units. Accordingly, what is the area of triangle $ACD$ in square units? A) 12 B) 11 C) 10 D) 9 E) 8