<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">% !TeX encoding = UTF-8
% !TeX program = xelatex
% !TeX spellcheck = en_US


\documentclass{exam-zh}
\usepackage{siunitx}

\examsetup{
  page = {
    size            = a3paper,
    show-columnline = false
  },
  style/fullwidth-stop = catcode,
  sealline = {
    show        = true,
    scope       = mod-2,
    circle-show = false,
    line-type   = solid,
    odd-info-content = {
      {\heiti \zihao{4}å§“å} {\underline{\hspace*{8em}}},
      {\heiti \zihao{4}å‡†è€ƒè¯å·} {\examsquare{9}},
      {\heiti \zihao{4}è€ƒåœºå·} {\examsquare{2}},
      {\heiti \zihao{4}åº§ä½å·} {\examsquare{2}},
    },
    odd-info-xshift = 12mm,
    text = {æ­¤å·åªè£…è®¢ä¸å¯†å°},
    text-width = 0.98\textheight,
    text-format  = \zihao{-3}\sffamily,
    text-xshift = 20mm
  },
  square = {
    x-length = 1.8em,
    y-length = 1.6em
  }
}


% è¡Œå†…å…¬å¼ç»Ÿä¸€æŒ‰ç…§è¡Œé—´çš„æ&nbsp;·å¼
\everymath{\displaystyle}

\title{2021 å¹´æ™®é€šé«˜ç­‰å­¦æ&nbsp;¡æ‹›ç”Ÿå…¨å›½ç»Ÿä¸€è€ƒè¯•}

\subject{æ•°å­¦ï¼ˆç†ç§‘ï¼‰}


\begin{document}

\tableofcontents

\chapter{2021 å¹´æ™®é€šé«˜ç­‰å­¦æ&nbsp;¡æ‹›ç”Ÿå…¨å›½ç»Ÿä¸€è€ƒè¯•}


\title{2021 å¹´æ™®é€šé«˜ç­‰å­¦æ&nbsp;¡æ‹›ç”Ÿå…¨å›½ç»Ÿä¸€è€ƒè¯•}

\subject{æ•°å­¦ï¼ˆç†ç§‘ï¼‰}

\secret

\maketitle

æœ¬è¯•å·å…± 4 é¡µï¼Œ22 é¢˜ã€‚å…¨å·æ»¡åˆ† 150 åˆ†ã€‚è€ƒè¯•ç”¨æ—¶ 120 åˆ†é’Ÿã€‚


\begin{notice}
  \item ç­”å·å‰ï¼Œè€ƒç”ŸåŠ¡å¿…å°†è‡ªå·±çš„å§“åã€è€ƒç”Ÿå·ã€è€ƒåœºå·å’Œåº§ä½å·å¡«å†™ç­”é¢˜å¡ä¸Šï¼Œç”¨2Bé“…ç¬”å°†è¯•å·ç±»åž‹ï¼ˆBï¼‰å¡«æ¶‚åœ¨ç­”é¢˜å¡ç›¸åº”ä½ç½®ä¸Šï¼Œå°†æ¡å½¢ç&nbsp;æ¨ªè´´åœ¨ç­”é¢˜å¡å³ä¸Šè§’â€œæ¡å½¢ç&nbsp;ç²˜è´´å¤„â€ã€‚
  \item ä½œç­”é€‰æ‹©é¢˜æ—¶ï¼Œé€‰å‡ºæ¯å°é¢˜ç­”æ¡ˆåŽï¼Œç”¨2Bé“…ç¬”åœ¨ç­”é¢˜å¡ä¸Šå¯¹åº”é¢˜ç›®é€‰é¡¹çš„ç­”æ¡ˆä¿¡æ¯ç‚¹æ¶‚é»‘ï¼›å¦‚éœ€æ”¹åŠ¨ï¼Œç”¨æ©¡çš®æ“¦å¹²å‡€åŽï¼Œå†é€‰æ¶‚å…¶ä»–ç­”æ¡ˆã€‚ç­”æ¡ˆä¸èƒ½ç­”åœ¨è¯•å·ä¸Šã€‚
    å†™åœ¨è¯•å·ã€è‰ç¨¿çº¸å’Œç­”é¢˜å¡ä¸Šçš„éžç­”é¢˜åŒºåŸŸå‡æ—&nbsp;æ•ˆã€‚
  \item éžé€‰æ‹©é¢˜å¿…é¡»ç”¨é»‘è‰²å­—è¿¹çš„é’¢ç¬”æˆ–ç­¾å­—ç¬”ä½œç­”ï¼Œç­”æ¡ˆå¿…é¡»å†™åœ¨ç­”é¢˜å¡å„é¢˜ç›®æŒ‡å®šåŒºåŸŸå†…ç›¸åº”ä½ç½®ä¸Šï¼›å¦‚éœ€æ”¹åŠ¨ï¼Œå…ˆåˆ’æŽ‰åŽŸæ¥çš„ç­”æ¡ˆï¼Œç„¶åŽå†å†™ä¸Šæ–°ç­”æ¡ˆï¼›ä¸å‡†ä½¿ç”¨é“…ç¬”å’Œæ¶‚æ”¹æ¶²ã€‚ä¸æŒ‰ä»¥ä¸Šè¦æ±‚ä½œç­”æ—&nbsp;æ•ˆã€‚
  \item è€ƒç”Ÿå¿…é¡»ä¿æŒç­”é¢˜å¡çš„æ•´æ´ã€‚è€ƒè¯•ç»“æŸåŽï¼Œå°†è¯•å·å’Œç­”é¢˜å¡ä¸€å¹¶äº¤å›žã€‚
\end{notice}



\section{%
  é€‰æ‹©é¢˜ï¼šæœ¬é¢˜å…± 8 å°é¢˜ï¼Œæ¯å°é¢˜ 5 åˆ†ï¼Œå…± 40 åˆ†ã€‚
  åœ¨æ¯å°é¢˜ç»™å‡ºçš„å››ä¸ªé€‰é¡¹ä¸­ï¼Œåªæœ‰ä¸€é¡¹æ˜¯ç¬¦åˆé¢˜ç›®è¦æ±‚çš„ã€‚
}

% 1.
\begin{question}[points = 2]
  è®¾é›†åˆ $A = \{x \mid -2 &lt; x &lt; 4\}$ï¼Œ$B = \{2, 3, 4, 5\}$ï¼Œåˆ™ $A \cap B = $ \paren[]

  \begin{choices}
    \item $\{2\}$
    \item $\{2, 3\}$
    \item $\{3, 4\}$
    \item $\{2, 3, 4\}$
  \end{choices}
\end{question}

% 2.
\begin{question}
  å·²çŸ¥ $z = 2 - \iu$ï¼Œåˆ™ $z (\bar{z} + \iu) = $ \paren
  \begin{choices}
    \item $6 - 2\iu$
    \item $4 - 2\iu$
    \item $6 + 2\iu$
    \item $4 + 2\iu$
  \end{choices}
\end{question}

% 3.
\begin{question}
  å·²çŸ¥åœ†é”¥çš„åº•é¢åŠå¾„ä¸º $\sqrt{2}$ï¼Œå…¶ä¾§é¢å±•å¼€å›¾ä¸ºä¸€ä¸ªåŠåœ†ï¼Œåˆ™è¯¥åœ†é”¥çš„æ¯çº¿é•¿ä¸º \paren
  \begin{choices}
    \item $2$
    \item $2 \sqrt{2}$
    \item $4$
    \item $4 \sqrt{2}$
  \end{choices}
\end{question}

% 4.
\begin{question}
  ä¸‹åˆ—åŒºé—´ä¸­ï¼Œå‡½æ•° $f(x) = 7 \sin \left( x - \frac{\uppi}{6} \right)$ å•è°ƒé€’å¢žçš„åŒºé—´æ˜¯ \paren
  \begin{choices}
    \item $\left( 0               , \frac{\uppi}{2}  \right)$
    \item $\left( \frac{\uppi}{2} , \uppi            \right)$
    \item $\left( \uppi           , \frac{3\uppi}{2} \right)$
    \item $\left( \frac{3\uppi}{2}, 2\uppi           \right)$
  \end{choices}
\end{question}

% 5.
\begin{question}
  å·²çŸ¥ $F_1$ï¼Œ$F_2$ æ˜¯æ¤­åœ† $C \colon \frac{x^2}{9} + \frac{y^2}{4} = 1$ çš„ä¸¤ä¸ªç„¦ç‚¹ï¼Œ
  ç‚¹ $M$ åœ¨ $C$ ä¸Šï¼Œåˆ™ $|M F_1| \cdot |M F_2|$ çš„æœ€å¤§å€¼ä¸º \paren
  \begin{choices}
    \item $13$
    \item $12$
    \item $9$
    \item $6$
  \end{choices}
\end{question}

% 6.
\begin{question}
  è‹¥ $\tan\theta = -2$ï¼Œåˆ™ $\frac{\sin\theta (1 + \sin 2\theta)}{\sin\theta + \cos\theta} = $ \paren
  \begin{choices}
    \item $-\frac{6}{5}$
    \item $-\frac{2}{5}$
    \item $\frac{2}{5}$
    \item $\frac{6}{5}$
  \end{choices}
\end{question}

% 7.
\begin{question}
  è‹¥è¿‡ç‚¹ $(a, b)$ å¯ä»¥ä½œæ›²çº¿ $y = \eu^x$ çš„ä¸¤æ¡åˆ‡çº¿ï¼Œåˆ™ \paren
  \begin{choices}
    \item $\eu^b &lt; a$
    \item $\eu^a &lt; b$
    \item $0 &lt; a &lt; \eu^b$
    \item $0 &lt; b &lt; \eu^a$
  \end{choices}
\end{question}

% 8.
\begin{question}
  æœ‰ $6$ ä¸ªç›¸åŒçš„çƒï¼Œåˆ†åˆ«æ&nbsp;‡æœ‰æ•°å­— $1$ï¼Œ$2$ï¼Œ$3$ï¼Œ$4$ï¼Œ$5$ï¼Œ$6$ï¼Œä»Žä¸­æœ‰æ”¾å›žåœ°éšæœºå–ä¸¤æ¬¡ï¼Œæ¯æ¬¡å– $1$ ä¸ªçƒï¼Œ
  ç”²è¡¨ç¤ºäº‹ä»¶â€œç¬¬ä¸€æ¬¡å–å‡ºçš„çƒçš„æ•°å­—æ˜¯ $1$â€ï¼Œ
  ä¹™è¡¨ç¤ºäº‹ä»¶â€œç¬¬äºŒæ¬¡å–å‡ºçš„çƒçš„æ•°å­—æ˜¯ $2$â€ï¼Œ
  ä¸™è¡¨ç¤ºäº‹ä»¶â€œä¸¤æ¬¡å–å‡ºçš„çƒçš„æ•°å­—ä¹‹å’Œæ˜¯ $8$â€ï¼Œ
  ä¸è¡¨ç¤ºäº‹ä»¶â€œä¸¤æ¬¡å–å‡ºçš„çƒçš„æ•°å­—ä¹‹å’Œæ˜¯ $7$â€ï¼Œåˆ™ \paren
  \begin{choices}
    \item ç”²ä¸Žä¸™ç›¸äº’ç‹¬ç«‹
    \item ç”²ä¸Žä¸ç›¸äº’ç‹¬ç«‹
    \item ä¹™ä¸Žä¸™ç›¸äº’ç‹¬ç«‹
    \item ä¸™ä¸Žä¸ç›¸äº’ç‹¬ç«‹
  \end{choices}
\end{question}



\section{%
  é€‰æ‹©é¢˜ï¼šæœ¬é¢˜å…± 4 å°é¢˜ï¼Œæ¯å°é¢˜ 5 åˆ†ï¼Œå…± 20 åˆ†ã€‚
  åœ¨æ¯å°é¢˜ç»™å‡ºçš„é€‰é¡¹ä¸­ï¼Œæœ‰å¤šé¡¹ç¬¦åˆé¢˜ç›®è¦æ±‚ã€‚
  å…¨éƒ¨é€‰å¯¹çš„å¾— 5 åˆ†ï¼Œéƒ¨åˆ†é€‰å¯¹çš„å¾— 2 åˆ†ï¼Œæœ‰é€‰é”™çš„å¾— 0 åˆ†ã€‚
}

% 9.
\begin{question}
  æœ‰ä¸€ç»„æ&nbsp;·æœ¬æ•°æ® $x_1, x_2, \dots, x_n$ï¼Œç”±è¿™ç»„æ•°æ®å¾—åˆ°æ–°æ&nbsp;·æœ¬æ•°æ® $y_1, y_2, \dots, y_n$ï¼Œ
  å…¶ä¸­ $y_i = x_i + c$ï¼ˆ$i = 1, 2, \dots, n$ï¼‰ï¼Œ$c$ ä¸ºéžé›¶å¸¸æ•°ï¼Œåˆ™ \paren
  \begin{choices}
    \item ä¸¤ç»„æ&nbsp;·æœ¬æ•°æ®çš„æ&nbsp;·æœ¬å¹³å‡æ•°ç›¸åŒ
    \item ä¸¤ç»„æ&nbsp;·æœ¬æ•°æ®çš„æ&nbsp;·æœ¬ä¸­ä½æ•°ç›¸åŒ
    \item ä¸¤ç»„æ&nbsp;·æœ¬æ•°æ®çš„æ&nbsp;·æœ¬æ&nbsp;‡å‡†å·®ç›¸åŒ
    \item ä¸¤ç»„æ&nbsp;·æœ¬æ•°æ®çš„æ&nbsp;·æœ¬æžå·®ç›¸åŒ
  \end{choices}
\end{question}

% 10.
\begin{question}
  å·²çŸ¥ $O$ ä¸ºåæ&nbsp;‡åŽŸç‚¹ï¼Œç‚¹
  $P_1(\cos\alpha,  \sin\alpha)$ï¼Œ
  $P_2(\cos\beta , -\sin\beta)$ï¼Œ
  $P_3(\cos(\alpha + \beta),  \sin(\alpha + \beta))$ï¼Œ
  $A(1, 0)$ï¼Œåˆ™ \paren
  \begin{choices}
    \item $|\overrightarrow{OP_1}| = |\overrightarrow{OP_2}|$
    \item $|\overrightarrow{AP_1}| = |\overrightarrow{AP_2}|$
    \item $\overrightarrow{OA} \cdot \overrightarrow{OP_3}
      = \overrightarrow{OP_1} \cdot \overrightarrow{OP_2}$
    \item $\overrightarrow{OA} \cdot \overrightarrow{OP_1}
      = \overrightarrow{OP_2} \cdot \overrightarrow{OP_3}$
  \end{choices}
\end{question}

% 11.
\begin{question}
  å·²çŸ¥ç‚¹ $P$ åœ¨åœ† $(x - 5)^2 + (y - 5)^2 = 16$ ä¸Šï¼Œç‚¹ $A(4, 0)$ï¼Œ$B(0, 2)$ï¼Œåˆ™ \paren
  \begin{choices}
    \item ç‚¹ $P$ åˆ°ç›´çº¿ $AB$ çš„è·ç¦»å°äºŽ $10$
    \item ç‚¹ $P$ åˆ°ç›´çº¿ $AB$ çš„è·ç¦»å¤§äºŽ $2$
    \item å½“ $\angle PBA$ æœ€å°æ—¶ï¼Œ$|PB| = 3 \sqrt{2}$
    \item å½“ $\angle PBA$ æœ€å¤§æ—¶ï¼Œ$|PB| = 3 \sqrt{2}$
  \end{choices}
\end{question}

% 12.
\begin{question}
  åœ¨æ­£ä¸‰æ£±æŸ± $ABC$-$A_1 B_1 C_1$ ä¸­ï¼Œ$AB = A A_1 = 1$ï¼Œç‚¹ $P$ æ»¡è¶³
  $\overrightarrow{BP} = \lambda \overrightarrow{BC} + \mu \overrightarrow{BB_1}$ï¼Œ
  å…¶ä¸­ $\lambda \in [0, 1]$ï¼Œ$\mu \in [0, 1]$ï¼Œåˆ™ \paren
  \begin{choices}
    \item å½“ $\lambda = 1$ æ—¶ï¼Œ$\triangle A B_1 P$ çš„å‘¨é•¿ä¸ºå®šå€¼
    \item å½“ $\mu = 1$ æ—¶ï¼Œä¸‰æ£±é”¥ $P$-$A_1 B C$ çš„ä½“ç§¯ä¸ºå®šå€¼
    \item å½“ $\lambda = \frac{1}{2}$ æ—¶ï¼Œæœ‰ä¸”ä»…æœ‰ä¸€ä¸ªç‚¹ $P$ï¼Œä½¿å¾— $A_1 P \perp BP$
    \item å½“ $\mu = \frac{1}{2}$ æ—¶ï¼Œæœ‰ä¸”ä»…æœ‰ä¸€ä¸ªç‚¹ $P$ï¼Œä½¿å¾— $A_1 B \perp \text{å¹³é¢} A B_1 P$
  \end{choices}
\end{question}



\section{å¡«ç©ºé¢˜ï¼šæœ¬é¢˜å…± 4 å°é¢˜ï¼Œæ¯å°é¢˜ 5 åˆ†ï¼Œå…± 20 åˆ†ã€‚}

% 13.
\begin{question}
  å·²çŸ¥å‡½æ•° $f(x) = x^3 (a \cdot 2^x - 2^{-x})$ æ˜¯å¶å‡½æ•°ï¼Œåˆ™ $a = $ \fillin[$1$] ã€‚
\end{question}

\begin{solution}
  ç”± $f(x) = f(-x)$ï¼Œå¾— $a = 1$ã€‚
\end{solution}

% 14.
\begin{question}
  å·²çŸ¥ $O$ ä¸ºåæ&nbsp;‡åŽŸç‚¹ï¼ŒæŠ›ç‰©çº¿ $C \colon y^2 = 2px$ï¼ˆ$p &gt; 0$ï¼‰çš„ç„¦ç‚¹ä¸º $F$ï¼Œ
  $P$ ä¸º $C$ ä¸Šä¸€ç‚¹ï¼Œ$PF$ ä¸Ž $x$ è½´åž‚ç›´ï¼Œ$Q$ ä¸º $x$ è½´ä¸Šä¸€ç‚¹ï¼Œä¸” $PQ \perp OP$ï¼Œ
  è‹¥ $|FQ| = 6$ï¼Œåˆ™ $C$ çš„å‡†çº¿æ–¹ç¨‹ä¸º \fillin[$\dfrac{1}{3}$] ã€‚
\end{question}

% 15.
\begin{question}
  å‡½æ•° $f(x) = |2x - 1| - 2 \ln x$ çš„æœ€å°å€¼ä¸º \fillin[width = 4em][] ã€‚
\end{question}

% 16.
\begin{question}
  æŸæ&nbsp;¡å­¦ç”Ÿåœ¨ç&nbsp;”ç©¶æ°‘é—´å‰ªçº¸è‰ºæœ¯æ—¶ï¼Œå‘çŽ°å‰ªçº¸æ—¶ç»å¸¸ä¼šæ²¿çº¸çš„æŸæ¡å¯¹ç§°è½´æŠŠçº¸å¯¹æŠ˜ã€‚
  è§„æ&nbsp;¼ä¸º \qtyproduct{20 x 12}{dm} çš„é•¿æ–¹å½¢çº¸ï¼Œå¯¹æŠ˜ $1$ æ¬¡å…±å¯ä»¥å¾—åˆ°
  \qtyproduct{10 x 12}{dm}ï¼Œ \qtyproduct{20 x 6}{dm} ä¸¤ç§è§„æ&nbsp;¼çš„å›¾å½¢ï¼Œ
  å®ƒä»¬çš„é¢ç§¯ä¹‹å’Œ $S_1 = \qty{240}{dm^2}$ï¼Œ
  å¯¹æŠ˜ $2$ æ¬¡å…±å¯ä»¥å¾—åˆ° \qtyproduct{5 x 12}{dm}ï¼Œ\qtyproduct{10 x 6}{dm}ï¼Œ
  \qtyproduct{20 x 3}{dm} ä¸‰ç§è§„æ&nbsp;¼çš„å›¾å½¢ï¼Œå®ƒä»¬çš„é¢ç§¯ä¹‹å’Œ $S_2 = \qty{180}{dm^2}$ï¼Œ
  ä»¥æ­¤ç±»æŽ¨ã€‚åˆ™å¯¹æŠ˜ $4$ æ¬¡å…±å¯ä»¥å¾—åˆ°ä¸åŒè§„æ&nbsp;¼å›¾å½¢çš„ç§æ•°ä¸º \fillin ï¼›
  å¦‚æžœå¯¹æŠ˜ $n$ æ¬¡ï¼Œé‚£ä¹ˆ $\sum_{k=1}^n S_k = $ \fillin \unit{dm^2}ã€‚
\end{question}



\section{è§£ç­”é¢˜ï¼šæœ¬é¢˜å…± 6 å°é¢˜ï¼Œå…± 70 åˆ†ã€‚è§£ç­”åº”å†™å‡ºæ–‡å­—è¯´æ˜Žã€è¯æ˜Žè¿‡ç¨‹æˆ–æ¼”ç®—æ­¥éª¤ã€‚}

% 17.
\begin{problem}[points = 10]
  å·²çŸ¥æ•°åˆ— $\{a_n\}$ æ»¡è¶³ $a_1 = 1$ï¼Œ$a_{n+1} =
    \begin{cases}
      a_n + 1, \quad \text{$n$ ä¸ºå¥‡æ•°ï¼Œ} \\
      a_n + 2, \quad \text{$n$ ä¸ºå¶æ•°ã€‚}
    \end{cases}$
  \begin{enumerate}
    \item è®° $b_n = a_{2n}$ï¼Œå†™å‡º $b_1$ï¼Œ$b_2$ï¼Œå¹¶æ±‚æ•°åˆ— $\{b_n\}$ çš„é€šé¡¹å…¬å¼ï¼›
    \item æ±‚ $\{a_n\}$ çš„å‰ $20$ é¡¹å’Œã€‚
  \end{enumerate}
\end{problem}

% 18.
\begin{problem}[points = 12]
  æŸå­¦æ&nbsp;¡ç»„ç»‡â€œä¸€å¸¦ä¸€è·¯â€çŸ¥è¯†ç«žèµ›ï¼Œæœ‰ Aï¼ŒB ä¸¤ç±»é—®é¢˜ã€‚
  æ¯ä½å‚åŠ&nbsp;æ¯”èµ›çš„åŒå­¦å…ˆåœ¨ä¸¤ç±»é—®é¢˜ä¸­é€‰æ‹©ä¸€ç±»å¹¶ä»Žä¸­éšæœºæŠ½å–ä¸€ä¸ªé—®é¢˜å›žç­”ï¼Œ
  è‹¥å›žç­”é”™è¯¯åˆ™è¯¥åŒå­¦æ¯”èµ›ç»“æŸï¼›
  è‹¥å›žç­”æ­£ç¡®åˆ™ä»Žå¦ä¸€ç±»é—®é¢˜ä¸­å†éšæœºæŠ½å–ä¸€ä¸ªé—®é¢˜å›žç­”ï¼Œæ—&nbsp;è®ºå›žç­”æ­£ç¡®ä¸Žå¦ï¼Œè¯¥åŒå­¦æ¯”èµ›ç»“æŸã€‚
  A ç±»é—®é¢˜ä¸­çš„æ¯ä¸ªé—®é¢˜å›žç­”æ­£ç¡®å¾— $20$ åˆ†ï¼Œå¦åˆ™å¾— $0$ åˆ†ï¼›
  B ç±»é—®é¢˜ä¸­çš„æ¯ä¸ªé—®é¢˜å›žç­”æ­£ç¡®å¾— $80$ åˆ†ï¼Œå¦åˆ™å¾— $0$ åˆ†ã€‚

  å·²çŸ¥å°æ˜Žèƒ½æ­£ç¡®å›žç­” A ç±»é—®é¢˜çš„æ¦‚çŽ‡ä¸º $0.8$ï¼Œèƒ½æ­£ç¡®å›žç­” B ç±»é—®é¢˜çš„æ¦‚çŽ‡ä¸º $0.6$ï¼Œ
  ä¸”èƒ½æ­£ç¡®å›žç­”é—®é¢˜çš„æ¦‚çŽ‡ä¸Žå›žç­”æ¬¡åºæ—&nbsp;å…³ã€‚
  \begin{enumerate}
    \item è‹¥å°æ˜Žå…ˆå›žç­” A ç±»é—®é¢˜ï¼Œè®° $X$ ä¸ºå°æ˜Žçš„ç´¯è®¡å¾—åˆ†ï¼Œæ±‚ $X$ çš„åˆ†å¸ƒåˆ—ï¼›
    \item ä¸ºä½¿ç´¯è®¡å¾—åˆ†çš„æœŸæœ›æœ€å¤§ï¼Œå°æ˜Žåº”é€‰æ‹©å…ˆå›žç­”å“ªç±»é—®é¢˜ï¼Ÿå¹¶è¯´æ˜Žç†ç”±ã€‚
  \end{enumerate}
\end{problem}

% 19.
\begin{problem}[points = 12]
  è®° $\triangle ABC$ çš„å†…è§’ $A$ï¼Œ$B$ï¼Œ$C$ çš„å¯¹è¾¹åˆ†åˆ«ä¸º $a$ï¼Œ$b$ï¼Œ$c$ã€‚
  å·²çŸ¥ $b^2 = ac$ï¼Œç‚¹ $D$ åœ¨è¾¹ $AC$ ä¸Šï¼Œ$BD \sin\angle ABC = a \sin C$ã€‚
  \nopagebreak
  \begin{enumerate}
    \item è¯æ˜Žï¼š$BD = b$ï¼›
    \item è‹¥ $AD = 2 DC$ï¼Œæ±‚ $\cos\angle ABC$ã€‚
  \end{enumerate}
\end{problem}

% 20.
\begin{problem}[points = 12]
  \textfigure{
    å¦‚å›¾ï¼Œåœ¨ä¸‰æ£±é”¥ $A$-$BCD$ ä¸­ï¼Œ$\text{å¹³é¢} ABD \perp \text{å¹³é¢} BCD$ï¼Œ
    $AB = AD$ï¼Œ$O$ ä¸º $BD$ çš„ä¸­ç‚¹ã€‚
    \begin{enumerate}
      \item è¯æ˜Žï¼š$OA \perp CD$ï¼›
      \item è‹¥ $\triangle OCD$ æ˜¯å˜é•¿ä¸º $1$ çš„ç­‰è¾¹ä¸‰è§’å½¢ï¼Œç‚¹ $E$ åœ¨æ£± $AD$ ä¸Šï¼Œ
        $DE = 2 EA$ï¼Œä¸”äºŒé¢è§’ $E$-$BC$-$D$ çš„å¤§å°ä¸º \ang{45}ï¼Œ
        æ±‚ä¸‰æ£±é”¥ $A$-$BCD$ çš„ä½“ç§¯ã€‚
    \end{enumerate}
  }{
    \includegraphics[width=3cm]{example-image.png}
  }
\end{problem}

% 21.
\begin{problem}[points = 12]
  åœ¨å¹³é¢ç›´è§’åæ&nbsp;‡ç³» $xOy$ ä¸­ï¼Œå·²çŸ¥ç‚¹ $F_1 (-\sqrt{17}, 0)$ï¼Œ$F_2 (\sqrt{17}, 0)$ï¼Œ
  ç‚¹ $M$ æ»¡è¶³ $|M F_1| - |M F_2| = 2$ã€‚è®° $M$ çš„è½¨è¿¹ä¸º $C$ã€‚
  \begin{enumerate}
    \item æ±‚ $C$ çš„æ–¹ç¨‹ï¼›
    \item è®¾ç‚¹ $T$ åœ¨ç›´çº¿ $x = \frac{1}{2}$ ä¸Šï¼Œè¿‡ $T$ çš„ä¸¤æ¡ç›´çº¿åˆ†åˆ«äº¤ $C$ äºŽ $A$ï¼Œ
      $B$ äº®ç‚¹å’Œ $P$ï¼Œ$Q$ äº®ç‚¹ï¼Œä¸” $|TA| \cdot |TB| = |TP| \cdot |TQ|$ï¼Œ
      æ±‚ç›´çº¿ $AB$ çš„æ–œçŽ‡ä¸Žç›´çº¿ $PQ$ çš„æ–œçŽ‡ä¹‹å’Œã€‚
  \end{enumerate}
\end{problem}

% 22.
\begin{problem}[points = 12]
  å·²çŸ¥å‡½æ•° $f(x) = x (1 - \ln x)$ã€‚
  \begin{enumerate}
    \item è®¨è®º $f(x)$ çš„å•è°ƒæ€§ï¼›
    \item è®¾ $a$ï¼Œ$b$ ä¸ºä¸¤ä¸ªä¸ç›¸ç­‰çš„æ­£æ•°ï¼Œä¸” $b \ln a - a \ln b = a - b$ï¼Œ
      è¯æ˜Žï¼š$2 &lt; \frac{1}{a} + \frac{1}{b} &lt; \eu$ã€‚
  \end{enumerate}
\end{problem}

\begin{solution}
  å‡½æ•°çš„å®šä¹‰åŸŸä¸º $(0, +\infty)$,
  åˆ \[f^{\prime}(x) = 1 - \ln x-1 = -\ln x, \score{2}\]
  å½“ $x \in(0, 1)$ æ—¶, $f^{\prime}(x) &gt; 0$, å½“ $x \in(1, +\infty)$ æ—¶, $f^{\prime}(x) &lt; 0$,
  æ•… $f(x)$ çš„é€’å¢žåŒºé—´ä¸º $(0,1)$, é€’å‡åŒºé—´ä¸º $(1, +\infty)$.
\end{solution}




\chapter{2021 å¹´æ™®é€šé«˜ç­‰å­¦æ&nbsp;¡æ‹›ç”Ÿå…¨å›½ç»Ÿä¸€è€ƒè¯•}

\title{2021 å¹´æ™®é€šé«˜ç­‰å­¦æ&nbsp;¡æ‹›ç”Ÿå…¨å›½ç»Ÿä¸€è€ƒè¯•}

\subject{æ•°å­¦ï¼ˆç†ç§‘ï¼‰}

\secret

\maketitle

æœ¬è¯•å·å…± 4 é¡µï¼Œ22 é¢˜ã€‚å…¨å·æ»¡åˆ† 150 åˆ†ã€‚è€ƒè¯•ç”¨æ—¶ 120 åˆ†é’Ÿã€‚


\begin{notice}
  \item ç­”å·å‰ï¼Œè€ƒç”ŸåŠ¡å¿…å°†è‡ªå·±çš„å§“åã€è€ƒç”Ÿå·ã€è€ƒåœºå·å’Œåº§ä½å·å¡«å†™ç­”é¢˜å¡ä¸Šï¼Œç”¨2Bé“…ç¬”å°†è¯•å·ç±»åž‹ï¼ˆBï¼‰å¡«æ¶‚åœ¨ç­”é¢˜å¡ç›¸åº”ä½ç½®ä¸Šï¼Œå°†æ¡å½¢ç&nbsp;æ¨ªè´´åœ¨ç­”é¢˜å¡å³ä¸Šè§’â€œæ¡å½¢ç&nbsp;ç²˜è´´å¤„â€ã€‚
  \item ä½œç­”é€‰æ‹©é¢˜æ—¶ï¼Œé€‰å‡ºæ¯å°é¢˜ç­”æ¡ˆåŽï¼Œç”¨2Bé“…ç¬”åœ¨ç­”é¢˜å¡ä¸Šå¯¹åº”é¢˜ç›®é€‰é¡¹çš„ç­”æ¡ˆä¿¡æ¯ç‚¹æ¶‚é»‘ï¼›å¦‚éœ€æ”¹åŠ¨ï¼Œç”¨æ©¡çš®æ“¦å¹²å‡€åŽï¼Œå†é€‰æ¶‚å…¶ä»–ç­”æ¡ˆã€‚ç­”æ¡ˆä¸èƒ½ç­”åœ¨è¯•å·ä¸Šã€‚
    å†™åœ¨è¯•å·ã€è‰ç¨¿çº¸å’Œç­”é¢˜å¡ä¸Šçš„éžç­”é¢˜åŒºåŸŸå‡æ—&nbsp;æ•ˆã€‚
  \item éžé€‰æ‹©é¢˜å¿…é¡»ç”¨é»‘è‰²å­—è¿¹çš„é’¢ç¬”æˆ–ç­¾å­—ç¬”ä½œç­”ï¼Œç­”æ¡ˆå¿…é¡»å†™åœ¨ç­”é¢˜å¡å„é¢˜ç›®æŒ‡å®šåŒºåŸŸå†…ç›¸åº”ä½ç½®ä¸Šï¼›å¦‚éœ€æ”¹åŠ¨ï¼Œå…ˆåˆ’æŽ‰åŽŸæ¥çš„ç­”æ¡ˆï¼Œç„¶åŽå†å†™ä¸Šæ–°ç­”æ¡ˆï¼›ä¸å‡†ä½¿ç”¨é“…ç¬”å’Œæ¶‚æ”¹æ¶²ã€‚ä¸æŒ‰ä»¥ä¸Šè¦æ±‚ä½œç­”æ—&nbsp;æ•ˆã€‚
  \item è€ƒç”Ÿå¿…é¡»ä¿æŒç­”é¢˜å¡çš„æ•´æ´ã€‚è€ƒè¯•ç»“æŸåŽï¼Œå°†è¯•å·å’Œç­”é¢˜å¡ä¸€å¹¶äº¤å›žã€‚
\end{notice}



\section{%
  é€‰æ‹©é¢˜ï¼šæœ¬é¢˜å…± 8 å°é¢˜ï¼Œæ¯å°é¢˜ 5 åˆ†ï¼Œå…± 40 åˆ†ã€‚
  åœ¨æ¯å°é¢˜ç»™å‡ºçš„å››ä¸ªé€‰é¡¹ä¸­ï¼Œåªæœ‰ä¸€é¡¹æ˜¯ç¬¦åˆé¢˜ç›®è¦æ±‚çš„ã€‚
}

% 1.
\begin{question}[points = 2]
  è®¾é›†åˆ $A = \{x \mid -2 &lt; x &lt; 4\}$ï¼Œ$B = \{2, 3, 4, 5\}$ï¼Œåˆ™ $A \cap B = $ \paren[]

  \begin{choices}
    \item $\{2\}$
    \item $\{2, 3\}$
    \item $\{3, 4\}$
    \item $\{2, 3, 4\}$
  \end{choices}
\end{question}

% 2.
\begin{question}
  å·²çŸ¥ $z = 2 - \iu$ï¼Œåˆ™ $z (\bar{z} + \iu) = $ \paren
  \begin{choices}
    \item $6 - 2\iu$
    \item $4 - 2\iu$
    \item $6 + 2\iu$
    \item $4 + 2\iu$
  \end{choices}
\end{question}

% 3.
\begin{question}
  å·²çŸ¥åœ†é”¥çš„åº•é¢åŠå¾„ä¸º $\sqrt{2}$ï¼Œå…¶ä¾§é¢å±•å¼€å›¾ä¸ºä¸€ä¸ªåŠåœ†ï¼Œåˆ™è¯¥åœ†é”¥çš„æ¯çº¿é•¿ä¸º \paren
  \begin{choices}
    \item $2$
    \item $2 \sqrt{2}$
    \item $4$
    \item $4 \sqrt{2}$
  \end{choices}
\end{question}

% 4.
\begin{question}
  ä¸‹åˆ—åŒºé—´ä¸­ï¼Œå‡½æ•° $f(x) = 7 \sin \left( x - \frac{\uppi}{6} \right)$ å•è°ƒé€’å¢žçš„åŒºé—´æ˜¯ \paren
  \begin{choices}
    \item $\left( 0               , \frac{\uppi}{2}  \right)$
    \item $\left( \frac{\uppi}{2} , \uppi            \right)$
    \item $\left( \uppi           , \frac{3\uppi}{2} \right)$
    \item $\left( \frac{3\uppi}{2}, 2\uppi           \right)$
  \end{choices}
\end{question}

% 5.
\begin{question}
  å·²çŸ¥ $F_1$ï¼Œ$F_2$ æ˜¯æ¤­åœ† $C \colon \frac{x^2}{9} + \frac{y^2}{4} = 1$ çš„ä¸¤ä¸ªç„¦ç‚¹ï¼Œ
  ç‚¹ $M$ åœ¨ $C$ ä¸Šï¼Œåˆ™ $|M F_1| \cdot |M F_2|$ çš„æœ€å¤§å€¼ä¸º \paren
  \begin{choices}
    \item $13$
    \item $12$
    \item $9$
    \item $6$
  \end{choices}
\end{question}

% 6.
\begin{question}
  è‹¥ $\tan\theta = -2$ï¼Œåˆ™ $\frac{\sin\theta (1 + \sin 2\theta)}{\sin\theta + \cos\theta} = $ \paren
  \begin{choices}
    \item $-\frac{6}{5}$
    \item $-\frac{2}{5}$
    \item $\frac{2}{5}$
    \item $\frac{6}{5}$
  \end{choices}
\end{question}

% 7.
\begin{question}
  è‹¥è¿‡ç‚¹ $(a, b)$ å¯ä»¥ä½œæ›²çº¿ $y = \eu^x$ çš„ä¸¤æ¡åˆ‡çº¿ï¼Œåˆ™ \paren
  \begin{choices}
    \item $\eu^b &lt; a$
    \item $\eu^a &lt; b$
    \item $0 &lt; a &lt; \eu^b$
    \item $0 &lt; b &lt; \eu^a$
  \end{choices}
\end{question}

% 8.
\begin{question}
  æœ‰ $6$ ä¸ªç›¸åŒçš„çƒï¼Œåˆ†åˆ«æ&nbsp;‡æœ‰æ•°å­— $1$ï¼Œ$2$ï¼Œ$3$ï¼Œ$4$ï¼Œ$5$ï¼Œ$6$ï¼Œä»Žä¸­æœ‰æ”¾å›žåœ°éšæœºå–ä¸¤æ¬¡ï¼Œæ¯æ¬¡å– $1$ ä¸ªçƒï¼Œ
  ç”²è¡¨ç¤ºäº‹ä»¶â€œç¬¬ä¸€æ¬¡å–å‡ºçš„çƒçš„æ•°å­—æ˜¯ $1$â€ï¼Œ
  ä¹™è¡¨ç¤ºäº‹ä»¶â€œç¬¬äºŒæ¬¡å–å‡ºçš„çƒçš„æ•°å­—æ˜¯ $2$â€ï¼Œ
  ä¸™è¡¨ç¤ºäº‹ä»¶â€œä¸¤æ¬¡å–å‡ºçš„çƒçš„æ•°å­—ä¹‹å’Œæ˜¯ $8$â€ï¼Œ
  ä¸è¡¨ç¤ºäº‹ä»¶â€œä¸¤æ¬¡å–å‡ºçš„çƒçš„æ•°å­—ä¹‹å’Œæ˜¯ $7$â€ï¼Œåˆ™ \paren
  \begin{choices}
    \item ç”²ä¸Žä¸™ç›¸äº’ç‹¬ç«‹
    \item ç”²ä¸Žä¸ç›¸äº’ç‹¬ç«‹
    \item ä¹™ä¸Žä¸™ç›¸äº’ç‹¬ç«‹
    \item ä¸™ä¸Žä¸ç›¸äº’ç‹¬ç«‹
  \end{choices}
\end{question}



\section{%
  é€‰æ‹©é¢˜ï¼šæœ¬é¢˜å…± 4 å°é¢˜ï¼Œæ¯å°é¢˜ 5 åˆ†ï¼Œå…± 20 åˆ†ã€‚
  åœ¨æ¯å°é¢˜ç»™å‡ºçš„é€‰é¡¹ä¸­ï¼Œæœ‰å¤šé¡¹ç¬¦åˆé¢˜ç›®è¦æ±‚ã€‚
  å…¨éƒ¨é€‰å¯¹çš„å¾— 5 åˆ†ï¼Œéƒ¨åˆ†é€‰å¯¹çš„å¾— 2 åˆ†ï¼Œæœ‰é€‰é”™çš„å¾— 0 åˆ†ã€‚
}

% 9.
\begin{question}
  æœ‰ä¸€ç»„æ&nbsp;·æœ¬æ•°æ® $x_1, x_2, \dots, x_n$ï¼Œç”±è¿™ç»„æ•°æ®å¾—åˆ°æ–°æ&nbsp;·æœ¬æ•°æ® $y_1, y_2, \dots, y_n$ï¼Œ
  å…¶ä¸­ $y_i = x_i + c$ï¼ˆ$i = 1, 2, \dots, n$ï¼‰ï¼Œ$c$ ä¸ºéžé›¶å¸¸æ•°ï¼Œåˆ™ \paren
  \begin{choices}
    \item ä¸¤ç»„æ&nbsp;·æœ¬æ•°æ®çš„æ&nbsp;·æœ¬å¹³å‡æ•°ç›¸åŒ
    \item ä¸¤ç»„æ&nbsp;·æœ¬æ•°æ®çš„æ&nbsp;·æœ¬ä¸­ä½æ•°ç›¸åŒ
    \item ä¸¤ç»„æ&nbsp;·æœ¬æ•°æ®çš„æ&nbsp;·æœ¬æ&nbsp;‡å‡†å·®ç›¸åŒ
    \item ä¸¤ç»„æ&nbsp;·æœ¬æ•°æ®çš„æ&nbsp;·æœ¬æžå·®ç›¸åŒ
  \end{choices}
\end{question}

% 10.
\begin{question}
  å·²çŸ¥ $O$ ä¸ºåæ&nbsp;‡åŽŸç‚¹ï¼Œç‚¹
  $P_1(\cos\alpha,  \sin\alpha)$ï¼Œ
  $P_2(\cos\beta , -\sin\beta)$ï¼Œ
  $P_3(\cos(\alpha + \beta),  \sin(\alpha + \beta))$ï¼Œ
  $A(1, 0)$ï¼Œåˆ™ \paren
  \begin{choices}
    \item $|\overrightarrow{OP_1}| = |\overrightarrow{OP_2}|$
    \item $|\overrightarrow{AP_1}| = |\overrightarrow{AP_2}|$
    \item $\overrightarrow{OA} \cdot \overrightarrow{OP_3}
      = \overrightarrow{OP_1} \cdot \overrightarrow{OP_2}$
    \item $\overrightarrow{OA} \cdot \overrightarrow{OP_1}
      = \overrightarrow{OP_2} \cdot \overrightarrow{OP_3}$
  \end{choices}
\end{question}

% 11.
\begin{question}
  å·²çŸ¥ç‚¹ $P$ åœ¨åœ† $(x - 5)^2 + (y - 5)^2 = 16$ ä¸Šï¼Œç‚¹ $A(4, 0)$ï¼Œ$B(0, 2)$ï¼Œåˆ™ \paren
  \begin{choices}
    \item ç‚¹ $P$ åˆ°ç›´çº¿ $AB$ çš„è·ç¦»å°äºŽ $10$
    \item ç‚¹ $P$ åˆ°ç›´çº¿ $AB$ çš„è·ç¦»å¤§äºŽ $2$
    \item å½“ $\angle PBA$ æœ€å°æ—¶ï¼Œ$|PB| = 3 \sqrt{2}$
    \item å½“ $\angle PBA$ æœ€å¤§æ—¶ï¼Œ$|PB| = 3 \sqrt{2}$
  \end{choices}
\end{question}

% 12.
\begin{question}
  åœ¨æ­£ä¸‰æ£±æŸ± $ABC$-$A_1 B_1 C_1$ ä¸­ï¼Œ$AB = A A_1 = 1$ï¼Œç‚¹ $P$ æ»¡è¶³
  $\overrightarrow{BP} = \lambda \overrightarrow{BC} + \mu \overrightarrow{BB_1}$ï¼Œ
  å…¶ä¸­ $\lambda \in [0, 1]$ï¼Œ$\mu \in [0, 1]$ï¼Œåˆ™ \paren
  \begin{choices}
    \item å½“ $\lambda = 1$ æ—¶ï¼Œ$\triangle A B_1 P$ çš„å‘¨é•¿ä¸ºå®šå€¼
    \item å½“ $\mu = 1$ æ—¶ï¼Œä¸‰æ£±é”¥ $P$-$A_1 B C$ çš„ä½“ç§¯ä¸ºå®šå€¼
    \item å½“ $\lambda = \frac{1}{2}$ æ—¶ï¼Œæœ‰ä¸”ä»…æœ‰ä¸€ä¸ªç‚¹ $P$ï¼Œä½¿å¾— $A_1 P \perp BP$
    \item å½“ $\mu = \frac{1}{2}$ æ—¶ï¼Œæœ‰ä¸”ä»…æœ‰ä¸€ä¸ªç‚¹ $P$ï¼Œä½¿å¾— $A_1 B \perp \text{å¹³é¢} A B_1 P$
  \end{choices}
\end{question}



\section{å¡«ç©ºé¢˜ï¼šæœ¬é¢˜å…± 4 å°é¢˜ï¼Œæ¯å°é¢˜ 5 åˆ†ï¼Œå…± 20 åˆ†ã€‚}

% 13.
\begin{question}
  å·²çŸ¥å‡½æ•° $f(x) = x^3 (a \cdot 2^x - 2^{-x})$ æ˜¯å¶å‡½æ•°ï¼Œåˆ™ $a = $ \fillin[$1$] ã€‚
\end{question}

\begin{solution}
  ç”± $f(x) = f(-x)$ï¼Œå¾— $a = 1$ã€‚
\end{solution}

% 14.
\begin{question}
  å·²çŸ¥ $O$ ä¸ºåæ&nbsp;‡åŽŸç‚¹ï¼ŒæŠ›ç‰©çº¿ $C \colon y^2 = 2px$ï¼ˆ$p &gt; 0$ï¼‰çš„ç„¦ç‚¹ä¸º $F$ï¼Œ
  $P$ ä¸º $C$ ä¸Šä¸€ç‚¹ï¼Œ$PF$ ä¸Ž $x$ è½´åž‚ç›´ï¼Œ$Q$ ä¸º $x$ è½´ä¸Šä¸€ç‚¹ï¼Œä¸” $PQ \perp OP$ï¼Œ
  è‹¥ $|FQ| = 6$ï¼Œåˆ™ $C$ çš„å‡†çº¿æ–¹ç¨‹ä¸º \fillin[$\dfrac{1}{3}$] ã€‚
\end{question}

% 15.
\begin{question}
  å‡½æ•° $f(x) = |2x - 1| - 2 \ln x$ çš„æœ€å°å€¼ä¸º \fillin[width = 4em][] ã€‚
\end{question}

% 16.
\begin{question}
  æŸæ&nbsp;¡å­¦ç”Ÿåœ¨ç&nbsp;”ç©¶æ°‘é—´å‰ªçº¸è‰ºæœ¯æ—¶ï¼Œå‘çŽ°å‰ªçº¸æ—¶ç»å¸¸ä¼šæ²¿çº¸çš„æŸæ¡å¯¹ç§°è½´æŠŠçº¸å¯¹æŠ˜ã€‚
  è§„æ&nbsp;¼ä¸º \qtyproduct{20 x 12}{dm} çš„é•¿æ–¹å½¢çº¸ï¼Œå¯¹æŠ˜ $1$ æ¬¡å…±å¯ä»¥å¾—åˆ°
  \qtyproduct{10 x 12}{dm}ï¼Œ \qtyproduct{20 x 6}{dm} ä¸¤ç§è§„æ&nbsp;¼çš„å›¾å½¢ï¼Œ
  å®ƒä»¬çš„é¢ç§¯ä¹‹å’Œ $S_1 = \qty{240}{dm^2}$ï¼Œ
  å¯¹æŠ˜ $2$ æ¬¡å…±å¯ä»¥å¾—åˆ° \qtyproduct{5 x 12}{dm}ï¼Œ\qtyproduct{10 x 6}{dm}ï¼Œ
  \qtyproduct{20 x 3}{dm} ä¸‰ç§è§„æ&nbsp;¼çš„å›¾å½¢ï¼Œå®ƒä»¬çš„é¢ç§¯ä¹‹å’Œ $S_2 = \qty{180}{dm^2}$ï¼Œ
  ä»¥æ­¤ç±»æŽ¨ã€‚åˆ™å¯¹æŠ˜ $4$ æ¬¡å…±å¯ä»¥å¾—åˆ°ä¸åŒè§„æ&nbsp;¼å›¾å½¢çš„ç§æ•°ä¸º \fillin ï¼›
  å¦‚æžœå¯¹æŠ˜ $n$ æ¬¡ï¼Œé‚£ä¹ˆ $\sum_{k=1}^n S_k = $ \fillin \unit{dm^2}ã€‚
\end{question}



\section{è§£ç­”é¢˜ï¼šæœ¬é¢˜å…± 6 å°é¢˜ï¼Œå…± 70 åˆ†ã€‚è§£ç­”åº”å†™å‡ºæ–‡å­—è¯´æ˜Žã€è¯æ˜Žè¿‡ç¨‹æˆ–æ¼”ç®—æ­¥éª¤ã€‚}

% 17.
\begin{problem}[points = 10]
  å·²çŸ¥æ•°åˆ— $\{a_n\}$ æ»¡è¶³ $a_1 = 1$ï¼Œ$a_{n+1} =
    \begin{cases}
      a_n + 1, \quad \text{$n$ ä¸ºå¥‡æ•°ï¼Œ} \\
      a_n + 2, \quad \text{$n$ ä¸ºå¶æ•°ã€‚}
    \end{cases}$
  \begin{enumerate}
    \item è®° $b_n = a_{2n}$ï¼Œå†™å‡º $b_1$ï¼Œ$b_2$ï¼Œå¹¶æ±‚æ•°åˆ— $\{b_n\}$ çš„é€šé¡¹å…¬å¼ï¼›
    \item æ±‚ $\{a_n\}$ çš„å‰ $20$ é¡¹å’Œã€‚
  \end{enumerate}
\end{problem}

% 18.
\begin{problem}[points = 12]
  æŸå­¦æ&nbsp;¡ç»„ç»‡â€œä¸€å¸¦ä¸€è·¯â€çŸ¥è¯†ç«žèµ›ï¼Œæœ‰ Aï¼ŒB ä¸¤ç±»é—®é¢˜ã€‚
  æ¯ä½å‚åŠ&nbsp;æ¯”èµ›çš„åŒå­¦å…ˆåœ¨ä¸¤ç±»é—®é¢˜ä¸­é€‰æ‹©ä¸€ç±»å¹¶ä»Žä¸­éšæœºæŠ½å–ä¸€ä¸ªé—®é¢˜å›žç­”ï¼Œ
  è‹¥å›žç­”é”™è¯¯åˆ™è¯¥åŒå­¦æ¯”èµ›ç»“æŸï¼›
  è‹¥å›žç­”æ­£ç¡®åˆ™ä»Žå¦ä¸€ç±»é—®é¢˜ä¸­å†éšæœºæŠ½å–ä¸€ä¸ªé—®é¢˜å›žç­”ï¼Œæ—&nbsp;è®ºå›žç­”æ­£ç¡®ä¸Žå¦ï¼Œè¯¥åŒå­¦æ¯”èµ›ç»“æŸã€‚
  A ç±»é—®é¢˜ä¸­çš„æ¯ä¸ªé—®é¢˜å›žç­”æ­£ç¡®å¾— $20$ åˆ†ï¼Œå¦åˆ™å¾— $0$ åˆ†ï¼›
  B ç±»é—®é¢˜ä¸­çš„æ¯ä¸ªé—®é¢˜å›žç­”æ­£ç¡®å¾— $80$ åˆ†ï¼Œå¦åˆ™å¾— $0$ åˆ†ã€‚

  å·²çŸ¥å°æ˜Žèƒ½æ­£ç¡®å›žç­” A ç±»é—®é¢˜çš„æ¦‚çŽ‡ä¸º $0.8$ï¼Œèƒ½æ­£ç¡®å›žç­” B ç±»é—®é¢˜çš„æ¦‚çŽ‡ä¸º $0.6$ï¼Œ
  ä¸”èƒ½æ­£ç¡®å›žç­”é—®é¢˜çš„æ¦‚çŽ‡ä¸Žå›žç­”æ¬¡åºæ—&nbsp;å…³ã€‚
  \begin{enumerate}
    \item è‹¥å°æ˜Žå…ˆå›žç­” A ç±»é—®é¢˜ï¼Œè®° $X$ ä¸ºå°æ˜Žçš„ç´¯è®¡å¾—åˆ†ï¼Œæ±‚ $X$ çš„åˆ†å¸ƒåˆ—ï¼›
    \item ä¸ºä½¿ç´¯è®¡å¾—åˆ†çš„æœŸæœ›æœ€å¤§ï¼Œå°æ˜Žåº”é€‰æ‹©å…ˆå›žç­”å“ªç±»é—®é¢˜ï¼Ÿå¹¶è¯´æ˜Žç†ç”±ã€‚
  \end{enumerate}
\end{problem}

% 19.
\begin{problem}[points = 12]
  è®° $\triangle ABC$ çš„å†…è§’ $A$ï¼Œ$B$ï¼Œ$C$ çš„å¯¹è¾¹åˆ†åˆ«ä¸º $a$ï¼Œ$b$ï¼Œ$c$ã€‚
  å·²çŸ¥ $b^2 = ac$ï¼Œç‚¹ $D$ åœ¨è¾¹ $AC$ ä¸Šï¼Œ$BD \sin\angle ABC = a \sin C$ã€‚
  \nopagebreak
  \begin{enumerate}
    \item è¯æ˜Žï¼š$BD = b$ï¼›
    \item è‹¥ $AD = 2 DC$ï¼Œæ±‚ $\cos\angle ABC$ã€‚
  \end{enumerate}
\end{problem}

% 20.
\begin{problem}[points = 12]
  \textfigure{
    å¦‚å›¾ï¼Œåœ¨ä¸‰æ£±é”¥ $A$-$BCD$ ä¸­ï¼Œ$\text{å¹³é¢} ABD \perp \text{å¹³é¢} BCD$ï¼Œ
    $AB = AD$ï¼Œ$O$ ä¸º $BD$ çš„ä¸­ç‚¹ã€‚
    \begin{enumerate}
      \item è¯æ˜Žï¼š$OA \perp CD$ï¼›
      \item è‹¥ $\triangle OCD$ æ˜¯å˜é•¿ä¸º $1$ çš„ç­‰è¾¹ä¸‰è§’å½¢ï¼Œç‚¹ $E$ åœ¨æ£± $AD$ ä¸Šï¼Œ
        $DE = 2 EA$ï¼Œä¸”äºŒé¢è§’ $E$-$BC$-$D$ çš„å¤§å°ä¸º \ang{45}ï¼Œ
        æ±‚ä¸‰æ£±é”¥ $A$-$BCD$ çš„ä½“ç§¯ã€‚
    \end{enumerate}
  }{
    \includegraphics[width=3cm]{example-image.png}
  }
\end{problem}

% 21.
\begin{problem}[points = 12]
  åœ¨å¹³é¢ç›´è§’åæ&nbsp;‡ç³» $xOy$ ä¸­ï¼Œå·²çŸ¥ç‚¹ $F_1 (-\sqrt{17}, 0)$ï¼Œ$F_2 (\sqrt{17}, 0)$ï¼Œ
  ç‚¹ $M$ æ»¡è¶³ $|M F_1| - |M F_2| = 2$ã€‚è®° $M$ çš„è½¨è¿¹ä¸º $C$ã€‚
  \begin{enumerate}
    \item æ±‚ $C$ çš„æ–¹ç¨‹ï¼›
    \item è®¾ç‚¹ $T$ åœ¨ç›´çº¿ $x = \frac{1}{2}$ ä¸Šï¼Œè¿‡ $T$ çš„ä¸¤æ¡ç›´çº¿åˆ†åˆ«äº¤ $C$ äºŽ $A$ï¼Œ
      $B$ äº®ç‚¹å’Œ $P$ï¼Œ$Q$ äº®ç‚¹ï¼Œä¸” $|TA| \cdot |TB| = |TP| \cdot |TQ|$ï¼Œ
      æ±‚ç›´çº¿ $AB$ çš„æ–œçŽ‡ä¸Žç›´çº¿ $PQ$ çš„æ–œçŽ‡ä¹‹å’Œã€‚
  \end{enumerate}
\end{problem}

% 22.
\begin{problem}[points = 12]
  å·²çŸ¥å‡½æ•° $f(x) = x (1 - \ln x)$ã€‚
  \begin{enumerate}
    \item è®¨è®º $f(x)$ çš„å•è°ƒæ€§ï¼›
    \item è®¾ $a$ï¼Œ$b$ ä¸ºä¸¤ä¸ªä¸ç›¸ç­‰çš„æ­£æ•°ï¼Œä¸” $b \ln a - a \ln b = a - b$ï¼Œ
      è¯æ˜Žï¼š$2 &lt; \frac{1}{a} + \frac{1}{b} &lt; \eu$ã€‚
  \end{enumerate}
\end{problem}

\begin{solution}
  å‡½æ•°çš„å®šä¹‰åŸŸä¸º $(0, +\infty)$,
  åˆ \[f^{\prime}(x) = 1 - \ln x-1 = -\ln x, \score{2}\]
  å½“ $x \in(0, 1)$ æ—¶, $f^{\prime}(x) &gt; 0$, å½“ $x \in(1, +\infty)$ æ—¶, $f^{\prime}(x) &lt; 0$,
  æ•… $f(x)$ çš„é€’å¢žåŒºé—´ä¸º $(0,1)$, é€’å‡åŒºé—´ä¸º $(1, +\infty)$.
\end{solution}


\begin{solution}
  å‡½æ•°çš„å®šä¹‰åŸŸä¸º $(0, +\infty)$,
  åˆ \[f^{\prime}(x) = 1 - \ln x-1 = -\ln x, \score{2}\]
  å½“ $x \in(0, 1)$ æ—¶, $f^{\prime}(x) &gt; 0$, å½“ $x \in(1, +\infty)$ æ—¶, $f^{\prime}(x) &lt; 0$,
  æ•… $f(x)$ çš„é€’å¢žåŒºé—´ä¸º $(0,1)$, é€’å‡åŒºé—´ä¸º $(1, +\infty)$.
\end{solution}


\end{document}
</pre></body></html>