九州大学 文系 2003年度 問2

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解答作成者: 森 宏征

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入試情報

大学名 九州大学
学科・方式 文系
年度 2003年度
問No 問2
学部 文 ・ 教育 ・ 法 ・ 経済
カテゴリ
状態 解答 解説なし ウォッチリスト

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\documentclass[a4paper,12pt,fleqn,dvipdfmx]{jreport} \usepackage{amsmath} \usepackage{amssymb} \usepackage{ascmac} \usepackage{vector3} \setlength{\topmargin}{-25mm} \setlength{\oddsidemargin}{2.5mm} \setlength{\textwidth}{420pt} \setlength{\textheight}{700pt} \usepackage{color} \ExecuteOptions{usename} \usepackage{graphicx} \usepackage{pifont} \usepackage{fancybox} \usepackage{custom_mori} \begin{document} \setlength{\abovedisplayskip}{0.5zw} \setlength{\belowdisplayskip}{0.5zw} \begin{FRAME}  座標平面上で, 不等式 $2\zettaiti{x-4} + \zettaiti{y-5} \leqq 3,\enskip 2\zettaiti{\zettaiti{x} - 4} + \zettaiti{\zettaiti{y} - 5} \leqq 3$ % が表す領域を,それぞれ $A,\ B$ とする. \begin{enumerate} \renewcommand{\labelenumi}{(\arabic{enumi})} \item  領域 $A$ を図示せよ. \item  領域 $B$ を図示せよ. \item  領域 $B$ の点$(x,\ y)$で, $x$ が正の整数であり $y$ が整数であって, $\log_x \zettaiti{y}$ が有理数となる点を, 理由を示してすべて求めよ. \end{enumerate} \end{FRAME} \vskip 2mm \noindent{\color[named]{BurntOrange}\bfseries \Ovalbox{解答}} \begin{enumerate} \renewcommand{\labelenumi}{(\arabic{enumi})} \item  領域 $C$ の定義式を \begin{align*} 2\zettaiti{x} + \zettaiti{y} \leqq 3 \tag*{$\cdott\MARU{1}$} \end{align*} とする. $A$ は $C$ を$x$軸方向に4,$y$軸方向に5,平行移動したものである. \begin{align*} 2\zettaiti{\pm x} + \zettaiti{\pm y} = 2\zettaiti{x} + \zettaiti{y} \quad (複号任意) \end{align*} より $C$ は$x$軸,$y$軸に関して対称な領域である. だから $C$ を $x \geqq 0,\,\,\,y \geqq 0$ に制限した部分 $D$ を$x$軸, $y$軸に関して対称になるように拡張すれば $C$ 全体が得られる.  $x \geqq 0,\,\,\,y \geqq 0$ のとき\MARU{1}は \begin{align*} 2x + y \leqq 3 \end{align*} よって $D$ は下左図, $C$ は下中央図, $A$ は下右図のようになる. \begin{center} \hspace*{-4zw} %WinTpicVersion3.08 \unitlength 0.1in \begin{picture}( 58.5000, 25.8000)( 4.9000,-32.0000) % STR 2 0 3 0 % 3 1390 2003 1390 2020 4 0 % {\small O} \put(13.9000,-20.2000){\makebox(0,0)[rt]{{\small O}}}% % STR 2 0 3 0 % 3 1360 983 1360 1000 4 1600 % $y$ \put(13.6000,-10.0000){\makebox(0,0)[rt]{$y$}}% % STR 2 0 3 0 % 3 2000 2023 2000 2040 4 1600 % $x$ \put(20.0000,-20.4000){\makebox(0,0)[rt]{$x$}}% % VECTOR 2 0 3 0 % 2 1400 2200 1400 1000 % \special{pn 8}% \special{pa 1400 2200}% \special{pa 1400 1000}% \special{fp}% \special{sh 1}% \special{pa 1400 1000}% \special{pa 1380 1068}% \special{pa 1400 1054}% \special{pa 1420 1068}% \special{pa 1400 1000}% \special{fp}% % VECTOR 2 0 3 0 % 2 1200 2000 2000 2000 % \special{pn 8}% \special{pa 1200 2000}% \special{pa 2000 2000}% \special{fp}% \special{sh 1}% \special{pa 2000 2000}% \special{pa 1934 1980}% \special{pa 1948 2000}% \special{pa 1934 2020}% \special{pa 2000 2000}% \special{fp}% % FUNC 2 0 3 0 % 9 1200 1000 2000 2200 1400 2000 1600 2000 1400 1800 1400 1000 1700 2200 0 1 0 0 % -2x+3 \special{pn 8}% \special{pa 1200 1000}% \special{pa 1206 1010}% \special{pa 1210 1020}% \special{pa 1216 1030}% \special{pa 1220 1040}% \special{pa 1226 1050}% \special{pa 1230 1060}% \special{pa 1236 1070}% \special{pa 1240 1080}% \special{pa 1246 1090}% \special{pa 1250 1100}% \special{pa 1256 1110}% \special{pa 1260 1120}% \special{pa 1266 1130}% \special{pa 1270 1140}% \special{pa 1276 1150}% \special{pa 1280 1160}% \special{pa 1286 1170}% \special{pa 1290 1180}% \special{pa 1296 1190}% \special{pa 1300 1200}% \special{pa 1306 1210}% \special{pa 1310 1220}% \special{pa 1316 1230}% \special{pa 1320 1240}% \special{pa 1326 1250}% \special{pa 1330 1260}% \special{pa 1336 1270}% \special{pa 1340 1280}% \special{pa 1346 1290}% \special{pa 1350 1300}% \special{pa 1356 1310}% \special{pa 1360 1320}% \special{pa 1366 1330}% \special{pa 1370 1340}% \special{pa 1376 1350}% \special{pa 1380 1360}% \special{pa 1386 1370}% \special{pa 1390 1380}% \special{pa 1396 1390}% \special{sp -0.045}% \special{pa 1400 1400}% \special{pa 1406 1410}% \special{pa 1410 1420}% \special{pa 1416 1430}% \special{pa 1420 1440}% \special{pa 1426 1450}% \special{pa 1430 1460}% \special{pa 1436 1470}% \special{pa 1440 1480}% \special{pa 1446 1490}% \special{pa 1450 1500}% \special{pa 1456 1510}% \special{pa 1460 1520}% \special{pa 1466 1530}% \special{pa 1470 1540}% \special{pa 1476 1550}% \special{pa 1480 1560}% \special{pa 1486 1570}% \special{pa 1490 1580}% \special{pa 1496 1590}% \special{pa 1500 1600}% \special{pa 1506 1610}% \special{pa 1510 1620}% \special{pa 1516 1630}% \special{pa 1520 1640}% \special{pa 1526 1650}% \special{pa 1530 1660}% \special{pa 1536 1670}% \special{pa 1540 1680}% \special{pa 1546 1690}% \special{pa 1550 1700}% \special{pa 1556 1710}% \special{pa 1560 1720}% \special{pa 1566 1730}% \special{pa 1570 1740}% \special{pa 1576 1750}% \special{pa 1580 1760}% \special{pa 1586 1770}% \special{pa 1590 1780}% \special{pa 1596 1790}% \special{pa 1600 1800}% \special{pa 1606 1810}% \special{pa 1610 1820}% \special{pa 1616 1830}% \special{pa 1620 1840}% \special{pa 1626 1850}% \special{pa 1630 1860}% \special{pa 1636 1870}% \special{pa 1640 1880}% \special{pa 1646 1890}% \special{pa 1650 1900}% \special{pa 1656 1910}% \special{pa 1660 1920}% \special{pa 1666 1930}% \special{pa 1670 1940}% \special{pa 1676 1950}% \special{pa 1680 1960}% \special{pa 1686 1970}% \special{pa 1690 1980}% \special{pa 1696 1990}% \special{pa 1700 2000}% \special{sp}% \special{pa 1700 2000}% \special{pa 1706 2010}% \special{pa 1710 2020}% \special{pa 1716 2030}% \special{pa 1720 2040}% \special{pa 1726 2050}% \special{pa 1730 2060}% \special{pa 1736 2070}% \special{pa 1740 2080}% \special{pa 1746 2090}% \special{pa 1750 2100}% \special{pa 1756 2110}% \special{pa 1760 2120}% \special{pa 1766 2130}% \special{pa 1770 2140}% \special{pa 1776 2150}% \special{pa 1780 2160}% \special{pa 1786 2170}% \special{pa 1790 2180}% \special{pa 1796 2190}% \special{pa 1800 2200}% \special{pa 1800 2200}% \special{sp -0.045}% % LINE 3 0 3 0 % 26 1590 1790 1400 1980 1610 1830 1440 2000 1630 1870 1500 2000 1650 1910 1560 2000 1670 1950 1620 2000 1570 1750 1400 1920 1550 1710 1400 1860 1530 1670 1400 1800 1510 1630 1400 1740 1490 1590 1400 1680 1470 1550 1400 1620 1450 1510 1400 1560 1430 1470 1400 1500 % \special{pn 4}% \special{pa 1590 1790}% \special{pa 1400 1980}% \special{fp}% \special{pa 1610 1830}% \special{pa 1440 2000}% \special{fp}% \special{pa 1630 1870}% \special{pa 1500 2000}% \special{fp}% \special{pa 1650 1910}% \special{pa 1560 2000}% \special{fp}% \special{pa 1670 1950}% \special{pa 1620 2000}% \special{fp}% \special{pa 1570 1750}% \special{pa 1400 1920}% \special{fp}% \special{pa 1550 1710}% \special{pa 1400 1860}% \special{fp}% \special{pa 1530 1670}% \special{pa 1400 1800}% \special{fp}% \special{pa 1510 1630}% \special{pa 1400 1740}% \special{fp}% \special{pa 1490 1590}% \special{pa 1400 1680}% \special{fp}% \special{pa 1470 1550}% \special{pa 1400 1620}% \special{fp}% \special{pa 1450 1510}% \special{pa 1400 1560}% \special{fp}% \special{pa 1430 1470}% \special{pa 1400 1500}% \special{fp}% % STR 2 0 3 0 % 3 3590 2003 3590 2020 4 0 % {\small O} \put(35.9000,-20.2000){\makebox(0,0)[rt]{{\small O}}}% % STR 2 0 3 0 % 3 3560 983 3560 1000 4 2200 % $y$ \put(35.6000,-10.0000){\makebox(0,0)[rt]{$y$}}% % STR 2 0 3 0 % 3 4200 2023 4200 2040 4 2200 % $x$ \put(42.0000,-20.4000){\makebox(0,0)[rt]{$x$}}% % VECTOR 2 0 3 0 % 2 3000 2000 4200 2000 % \special{pn 8}% \special{pa 3000 2000}% \special{pa 4200 2000}% \special{fp}% \special{sh 1}% \special{pa 4200 2000}% \special{pa 4134 1980}% \special{pa 4148 2000}% \special{pa 4134 2020}% \special{pa 4200 2000}% \special{fp}% % VECTOR 2 0 3 0 % 2 3600 2800 3600 1000 % \special{pn 8}% \special{pa 3600 2800}% \special{pa 3600 1000}% \special{fp}% \special{sh 1}% \special{pa 3600 1000}% \special{pa 3580 1068}% \special{pa 3600 1054}% \special{pa 3620 1068}% \special{pa 3600 1000}% \special{fp}% % FUNC 2 0 3 0 % 9 3000 1000 4200 2800 3600 2000 3800 2000 3600 1800 3600 1000 3900 2800 0 1 0 0 % -2x+3 \special{pn 8}% \special{pa 3400 1000}% \special{pa 3400 1000}% \special{pa 3406 1010}% \special{pa 3410 1020}% \special{pa 3416 1030}% \special{pa 3420 1040}% \special{pa 3426 1050}% \special{pa 3430 1060}% \special{pa 3436 1070}% \special{pa 3440 1080}% \special{pa 3446 1090}% \special{pa 3450 1100}% \special{pa 3456 1110}% \special{pa 3460 1120}% \special{pa 3466 1130}% \special{pa 3470 1140}% \special{pa 3476 1150}% \special{pa 3480 1160}% \special{pa 3486 1170}% \special{pa 3490 1180}% \special{pa 3496 1190}% \special{pa 3500 1200}% \special{pa 3506 1210}% \special{pa 3510 1220}% \special{pa 3516 1230}% \special{pa 3520 1240}% \special{pa 3526 1250}% \special{pa 3530 1260}% \special{pa 3536 1270}% \special{pa 3540 1280}% \special{pa 3546 1290}% \special{pa 3550 1300}% \special{pa 3556 1310}% \special{pa 3560 1320}% \special{pa 3566 1330}% \special{pa 3570 1340}% \special{pa 3576 1350}% \special{pa 3580 1360}% \special{pa 3586 1370}% \special{pa 3590 1380}% \special{pa 3596 1390}% \special{sp -0.045}% \special{pa 3600 1400}% \special{pa 3606 1410}% \special{pa 3610 1420}% \special{pa 3616 1430}% \special{pa 3620 1440}% \special{pa 3626 1450}% \special{pa 3630 1460}% \special{pa 3636 1470}% \special{pa 3640 1480}% \special{pa 3646 1490}% \special{pa 3650 1500}% \special{pa 3656 1510}% \special{pa 3660 1520}% \special{pa 3666 1530}% \special{pa 3670 1540}% \special{pa 3676 1550}% \special{pa 3680 1560}% \special{pa 3686 1570}% \special{pa 3690 1580}% \special{pa 3696 1590}% \special{pa 3700 1600}% \special{pa 3706 1610}% \special{pa 3710 1620}% \special{pa 3716 1630}% \special{pa 3720 1640}% \special{pa 3726 1650}% \special{pa 3730 1660}% \special{pa 3736 1670}% \special{pa 3740 1680}% \special{pa 3746 1690}% \special{pa 3750 1700}% \special{pa 3756 1710}% \special{pa 3760 1720}% \special{pa 3766 1730}% \special{pa 3770 1740}% \special{pa 3776 1750}% \special{pa 3780 1760}% \special{pa 3786 1770}% \special{pa 3790 1780}% \special{pa 3796 1790}% \special{pa 3800 1800}% \special{pa 3806 1810}% \special{pa 3810 1820}% \special{pa 3816 1830}% \special{pa 3820 1840}% \special{pa 3826 1850}% \special{pa 3830 1860}% \special{pa 3836 1870}% \special{pa 3840 1880}% \special{pa 3846 1890}% \special{pa 3850 1900}% \special{pa 3856 1910}% \special{pa 3860 1920}% \special{pa 3866 1930}% \special{pa 3870 1940}% \special{pa 3876 1950}% \special{pa 3880 1960}% \special{pa 3886 1970}% \special{pa 3890 1980}% \special{pa 3896 1990}% \special{pa 3900 2000}% \special{sp}% \special{pa 3900 2000}% \special{pa 3906 2010}% \special{pa 3910 2020}% \special{pa 3916 2030}% \special{pa 3920 2040}% \special{pa 3926 2050}% \special{pa 3930 2060}% \special{pa 3936 2070}% \special{pa 3940 2080}% \special{pa 3946 2090}% \special{pa 3950 2100}% \special{pa 3956 2110}% \special{pa 3960 2120}% \special{pa 3966 2130}% \special{pa 3970 2140}% \special{pa 3976 2150}% \special{pa 3980 2160}% \special{pa 3986 2170}% \special{pa 3990 2180}% \special{pa 3996 2190}% \special{pa 4000 2200}% \special{pa 4006 2210}% \special{pa 4010 2220}% \special{pa 4016 2230}% \special{pa 4020 2240}% \special{pa 4026 2250}% \special{pa 4030 2260}% \special{pa 4036 2270}% \special{pa 4040 2280}% \special{pa 4046 2290}% \special{pa 4050 2300}% \special{pa 4056 2310}% \special{pa 4060 2320}% \special{pa 4066 2330}% \special{pa 4070 2340}% \special{pa 4076 2350}% \special{pa 4080 2360}% \special{pa 4086 2370}% \special{pa 4090 2380}% \special{pa 4096 2390}% \special{pa 4100 2400}% \special{pa 4106 2410}% \special{pa 4110 2420}% \special{pa 4116 2430}% \special{pa 4120 2440}% \special{pa 4126 2450}% \special{pa 4130 2460}% \special{pa 4136 2470}% \special{pa 4140 2480}% \special{pa 4146 2490}% \special{pa 4150 2500}% \special{pa 4156 2510}% \special{pa 4160 2520}% \special{pa 4166 2530}% \special{pa 4170 2540}% \special{pa 4176 2550}% \special{pa 4180 2560}% \special{pa 4186 2570}% \special{pa 4190 2580}% \special{pa 4196 2590}% \special{pa 4200 2600}% \special{sp -0.045}% % FUNC 2 0 3 0 % 10 3000 1000 4200 2800 3600 2000 3800 2000 3600 1800 3300 1000 3600 2800 0 1 0 0 0 0 % 2x+3 \special{pn 8}% \special{pa 3000 2600}% \special{pa 3006 2590}% \special{pa 3010 2580}% \special{pa 3016 2570}% \special{pa 3020 2560}% \special{pa 3026 2550}% \special{pa 3030 2540}% \special{pa 3036 2530}% \special{pa 3040 2520}% \special{pa 3046 2510}% \special{pa 3050 2500}% \special{pa 3056 2490}% \special{pa 3060 2480}% \special{pa 3066 2470}% \special{pa 3070 2460}% \special{pa 3076 2450}% \special{pa 3080 2440}% \special{pa 3086 2430}% \special{pa 3090 2420}% \special{pa 3096 2410}% \special{pa 3100 2400}% \special{pa 3106 2390}% \special{pa 3110 2380}% \special{pa 3116 2370}% \special{pa 3120 2360}% \special{pa 3126 2350}% \special{pa 3130 2340}% \special{pa 3136 2330}% \special{pa 3140 2320}% \special{pa 3146 2310}% \special{pa 3150 2300}% \special{pa 3156 2290}% \special{pa 3160 2280}% \special{pa 3166 2270}% \special{pa 3170 2260}% \special{pa 3176 2250}% \special{pa 3180 2240}% \special{pa 3186 2230}% \special{pa 3190 2220}% \special{pa 3196 2210}% \special{pa 3200 2200}% \special{pa 3206 2190}% \special{pa 3210 2180}% \special{pa 3216 2170}% \special{pa 3220 2160}% \special{pa 3226 2150}% \special{pa 3230 2140}% \special{pa 3236 2130}% \special{pa 3240 2120}% \special{pa 3246 2110}% \special{pa 3250 2100}% \special{pa 3256 2090}% \special{pa 3260 2080}% \special{pa 3266 2070}% \special{pa 3270 2060}% \special{pa 3276 2050}% \special{pa 3280 2040}% \special{pa 3286 2030}% \special{pa 3290 2020}% \special{pa 3296 2010}% \special{sp -0.045}% \special{pa 3300 2000}% \special{pa 3306 1990}% \special{pa 3310 1980}% \special{pa 3316 1970}% \special{pa 3320 1960}% \special{pa 3326 1950}% \special{pa 3330 1940}% \special{pa 3336 1930}% \special{pa 3340 1920}% \special{pa 3346 1910}% \special{pa 3350 1900}% \special{pa 3356 1890}% \special{pa 3360 1880}% \special{pa 3366 1870}% \special{pa 3370 1860}% \special{pa 3376 1850}% \special{pa 3380 1840}% \special{pa 3386 1830}% \special{pa 3390 1820}% \special{pa 3396 1810}% \special{pa 3400 1800}% \special{pa 3406 1790}% \special{pa 3410 1780}% \special{pa 3416 1770}% \special{pa 3420 1760}% \special{pa 3426 1750}% \special{pa 3430 1740}% \special{pa 3436 1730}% \special{pa 3440 1720}% \special{pa 3446 1710}% \special{pa 3450 1700}% \special{pa 3456 1690}% \special{pa 3460 1680}% \special{pa 3466 1670}% \special{pa 3470 1660}% \special{pa 3476 1650}% \special{pa 3480 1640}% \special{pa 3486 1630}% \special{pa 3490 1620}% \special{pa 3496 1610}% \special{pa 3500 1600}% \special{pa 3506 1590}% \special{pa 3510 1580}% \special{pa 3516 1570}% \special{pa 3520 1560}% \special{pa 3526 1550}% \special{pa 3530 1540}% \special{pa 3536 1530}% \special{pa 3540 1520}% \special{pa 3546 1510}% \special{pa 3550 1500}% \special{pa 3556 1490}% \special{pa 3560 1480}% \special{pa 3566 1470}% \special{pa 3570 1460}% \special{pa 3576 1450}% \special{pa 3580 1440}% \special{pa 3586 1430}% \special{pa 3590 1420}% \special{pa 3596 1410}% \special{pa 3600 1400}% \special{sp}% \special{pa 3600 1400}% \special{pa 3606 1390}% \special{pa 3610 1380}% \special{pa 3616 1370}% \special{pa 3620 1360}% \special{pa 3626 1350}% \special{pa 3630 1340}% \special{pa 3636 1330}% \special{pa 3640 1320}% \special{pa 3646 1310}% \special{pa 3650 1300}% \special{pa 3656 1290}% \special{pa 3660 1280}% \special{pa 3666 1270}% \special{pa 3670 1260}% \special{pa 3676 1250}% \special{pa 3680 1240}% \special{pa 3686 1230}% \special{pa 3690 1220}% \special{pa 3696 1210}% \special{pa 3700 1200}% \special{pa 3706 1190}% \special{pa 3710 1180}% \special{pa 3716 1170}% \special{pa 3720 1160}% \special{pa 3726 1150}% \special{pa 3730 1140}% \special{pa 3736 1130}% \special{pa 3740 1120}% \special{pa 3746 1110}% \special{pa 3750 1100}% \special{pa 3756 1090}% \special{pa 3760 1080}% \special{pa 3766 1070}% \special{pa 3770 1060}% \special{pa 3776 1050}% \special{pa 3780 1040}% \special{pa 3786 1030}% \special{pa 3790 1020}% \special{pa 3796 1010}% \special{pa 3800 1000}% \special{pa 3800 1000}% \special{sp -0.045}% % FUNC 2 0 3 0 % 10 3000 1000 4200 2800 3600 2000 3800 2000 3600 1800 3300 1000 3600 2800 0 1 0 0 0 0 % -2x-3 \special{pn 8}% \special{pa 3000 1400}% \special{pa 3006 1410}% \special{pa 3010 1420}% \special{pa 3016 1430}% \special{pa 3020 1440}% \special{pa 3026 1450}% \special{pa 3030 1460}% \special{pa 3036 1470}% \special{pa 3040 1480}% \special{pa 3046 1490}% \special{pa 3050 1500}% \special{pa 3056 1510}% \special{pa 3060 1520}% \special{pa 3066 1530}% \special{pa 3070 1540}% \special{pa 3076 1550}% \special{pa 3080 1560}% \special{pa 3086 1570}% \special{pa 3090 1580}% \special{pa 3096 1590}% \special{pa 3100 1600}% \special{pa 3106 1610}% \special{pa 3110 1620}% \special{pa 3116 1630}% \special{pa 3120 1640}% \special{pa 3126 1650}% \special{pa 3130 1660}% \special{pa 3136 1670}% \special{pa 3140 1680}% \special{pa 3146 1690}% \special{pa 3150 1700}% \special{pa 3156 1710}% \special{pa 3160 1720}% \special{pa 3166 1730}% \special{pa 3170 1740}% \special{pa 3176 1750}% \special{pa 3180 1760}% \special{pa 3186 1770}% \special{pa 3190 1780}% \special{pa 3196 1790}% \special{pa 3200 1800}% \special{pa 3206 1810}% \special{pa 3210 1820}% \special{pa 3216 1830}% \special{pa 3220 1840}% \special{pa 3226 1850}% \special{pa 3230 1860}% \special{pa 3236 1870}% \special{pa 3240 1880}% \special{pa 3246 1890}% \special{pa 3250 1900}% \special{pa 3256 1910}% \special{pa 3260 1920}% \special{pa 3266 1930}% \special{pa 3270 1940}% \special{pa 3276 1950}% \special{pa 3280 1960}% \special{pa 3286 1970}% \special{pa 3290 1980}% \special{pa 3296 1990}% \special{sp -0.045}% \special{pa 3300 2000}% \special{pa 3306 2010}% \special{pa 3310 2020}% \special{pa 3316 2030}% \special{pa 3320 2040}% \special{pa 3326 2050}% \special{pa 3330 2060}% \special{pa 3336 2070}% \special{pa 3340 2080}% \special{pa 3346 2090}% \special{pa 3350 2100}% \special{pa 3356 2110}% \special{pa 3360 2120}% \special{pa 3366 2130}% \special{pa 3370 2140}% \special{pa 3376 2150}% \special{pa 3380 2160}% \special{pa 3386 2170}% \special{pa 3390 2180}% \special{pa 3396 2190}% \special{pa 3400 2200}% \special{pa 3406 2210}% \special{pa 3410 2220}% \special{pa 3416 2230}% \special{pa 3420 2240}% \special{pa 3426 2250}% \special{pa 3430 2260}% \special{pa 3436 2270}% \special{pa 3440 2280}% \special{pa 3446 2290}% \special{pa 3450 2300}% \special{pa 3456 2310}% \special{pa 3460 2320}% \special{pa 3466 2330}% \special{pa 3470 2340}% \special{pa 3476 2350}% \special{pa 3480 2360}% \special{pa 3486 2370}% \special{pa 3490 2380}% \special{pa 3496 2390}% \special{pa 3500 2400}% \special{pa 3506 2410}% \special{pa 3510 2420}% \special{pa 3516 2430}% \special{pa 3520 2440}% \special{pa 3526 2450}% \special{pa 3530 2460}% \special{pa 3536 2470}% \special{pa 3540 2480}% \special{pa 3546 2490}% \special{pa 3550 2500}% \special{pa 3556 2510}% \special{pa 3560 2520}% \special{pa 3566 2530}% \special{pa 3570 2540}% \special{pa 3576 2550}% \special{pa 3580 2560}% \special{pa 3586 2570}% \special{pa 3590 2580}% \special{pa 3596 2590}% \special{pa 3600 2600}% \special{sp}% \special{pa 3600 2600}% \special{pa 3606 2610}% \special{pa 3610 2620}% \special{pa 3616 2630}% \special{pa 3620 2640}% \special{pa 3626 2650}% \special{pa 3630 2660}% \special{pa 3636 2670}% \special{pa 3640 2680}% \special{pa 3646 2690}% \special{pa 3650 2700}% \special{pa 3656 2710}% \special{pa 3660 2720}% \special{pa 3666 2730}% \special{pa 3670 2740}% \special{pa 3676 2750}% \special{pa 3680 2760}% \special{pa 3686 2770}% \special{pa 3690 2780}% \special{pa 3696 2790}% \special{pa 3700 2800}% \special{pa 3700 2800}% \special{sp -0.045}% % FUNC 2 0 3 0 % 10 3000 1000 4200 2800 3600 2000 3800 2000 3600 1800 3600 1000 3900 2800 0 1 0 0 0 0 % 2x-3 \special{pn 8}% \special{pa 3500 2800}% \special{pa 3500 2800}% \special{pa 3506 2790}% \special{pa 3510 2780}% \special{pa 3516 2770}% \special{pa 3520 2760}% \special{pa 3526 2750}% \special{pa 3530 2740}% \special{pa 3536 2730}% \special{pa 3540 2720}% \special{pa 3546 2710}% \special{pa 3550 2700}% \special{pa 3556 2690}% \special{pa 3560 2680}% \special{pa 3566 2670}% \special{pa 3570 2660}% \special{pa 3576 2650}% \special{pa 3580 2640}% \special{pa 3586 2630}% \special{pa 3590 2620}% \special{pa 3596 2610}% \special{sp -0.045}% \special{pa 3600 2600}% \special{pa 3606 2590}% \special{pa 3610 2580}% \special{pa 3616 2570}% \special{pa 3620 2560}% \special{pa 3626 2550}% \special{pa 3630 2540}% \special{pa 3636 2530}% \special{pa 3640 2520}% \special{pa 3646 2510}% \special{pa 3650 2500}% \special{pa 3656 2490}% \special{pa 3660 2480}% \special{pa 3666 2470}% \special{pa 3670 2460}% \special{pa 3676 2450}% \special{pa 3680 2440}% \special{pa 3686 2430}% \special{pa 3690 2420}% \special{pa 3696 2410}% \special{pa 3700 2400}% \special{pa 3706 2390}% \special{pa 3710 2380}% \special{pa 3716 2370}% \special{pa 3720 2360}% \special{pa 3726 2350}% \special{pa 3730 2340}% \special{pa 3736 2330}% \special{pa 3740 2320}% \special{pa 3746 2310}% \special{pa 3750 2300}% \special{pa 3756 2290}% \special{pa 3760 2280}% \special{pa 3766 2270}% \special{pa 3770 2260}% \special{pa 3776 2250}% \special{pa 3780 2240}% \special{pa 3786 2230}% \special{pa 3790 2220}% \special{pa 3796 2210}% \special{pa 3800 2200}% \special{pa 3806 2190}% \special{pa 3810 2180}% \special{pa 3816 2170}% \special{pa 3820 2160}% \special{pa 3826 2150}% \special{pa 3830 2140}% \special{pa 3836 2130}% \special{pa 3840 2120}% \special{pa 3846 2110}% \special{pa 3850 2100}% \special{pa 3856 2090}% \special{pa 3860 2080}% \special{pa 3866 2070}% \special{pa 3870 2060}% \special{pa 3876 2050}% \special{pa 3880 2040}% \special{pa 3886 2030}% \special{pa 3890 2020}% \special{pa 3896 2010}% \special{pa 3900 2000}% \special{sp}% \special{pa 3900 2000}% \special{pa 3906 1990}% \special{pa 3910 1980}% \special{pa 3916 1970}% \special{pa 3920 1960}% \special{pa 3926 1950}% \special{pa 3930 1940}% \special{pa 3936 1930}% \special{pa 3940 1920}% \special{pa 3946 1910}% \special{pa 3950 1900}% \special{pa 3956 1890}% \special{pa 3960 1880}% \special{pa 3966 1870}% \special{pa 3970 1860}% \special{pa 3976 1850}% \special{pa 3980 1840}% \special{pa 3986 1830}% \special{pa 3990 1820}% \special{pa 3996 1810}% \special{pa 4000 1800}% \special{pa 4006 1790}% \special{pa 4010 1780}% \special{pa 4016 1770}% \special{pa 4020 1760}% \special{pa 4026 1750}% \special{pa 4030 1740}% \special{pa 4036 1730}% \special{pa 4040 1720}% \special{pa 4046 1710}% \special{pa 4050 1700}% \special{pa 4056 1690}% \special{pa 4060 1680}% \special{pa 4066 1670}% \special{pa 4070 1660}% \special{pa 4076 1650}% \special{pa 4080 1640}% \special{pa 4086 1630}% \special{pa 4090 1620}% \special{pa 4096 1610}% \special{pa 4100 1600}% \special{pa 4106 1590}% \special{pa 4110 1580}% \special{pa 4116 1570}% \special{pa 4120 1560}% \special{pa 4126 1550}% \special{pa 4130 1540}% \special{pa 4136 1530}% \special{pa 4140 1520}% \special{pa 4146 1510}% \special{pa 4150 1500}% \special{pa 4156 1490}% \special{pa 4160 1480}% \special{pa 4166 1470}% \special{pa 4170 1460}% \special{pa 4176 1450}% \special{pa 4180 1440}% \special{pa 4186 1430}% \special{pa 4190 1420}% \special{pa 4196 1410}% \special{pa 4200 1400}% \special{sp -0.045}% % LINE 3 0 3 0 % 38 3770 2250 3540 2480 3710 2370 3560 2520 3650 2490 3580 2560 3830 2130 3520 2440 3890 2010 3500 2400 3880 1960 3480 2360 3860 1920 3460 2320 3840 1880 3440 2280 3820 1840 3420 2240 3800 1800 3400 2200 3780 1760 3380 2160 3760 1720 3360 2120 3740 1680 3340 2080 3720 1640 3320 2040 3700 1600 3300 2000 3680 1560 3360 1880 3660 1520 3420 1760 3640 1480 3480 1640 3620 1440 3540 1520 % \special{pn 4}% \special{pa 3770 2250}% \special{pa 3540 2480}% \special{fp}% \special{pa 3710 2370}% \special{pa 3560 2520}% \special{fp}% \special{pa 3650 2490}% \special{pa 3580 2560}% \special{fp}% \special{pa 3830 2130}% \special{pa 3520 2440}% \special{fp}% \special{pa 3890 2010}% \special{pa 3500 2400}% \special{fp}% \special{pa 3880 1960}% \special{pa 3480 2360}% \special{fp}% \special{pa 3860 1920}% \special{pa 3460 2320}% \special{fp}% \special{pa 3840 1880}% \special{pa 3440 2280}% \special{fp}% \special{pa 3820 1840}% \special{pa 3420 2240}% \special{fp}% \special{pa 3800 1800}% \special{pa 3400 2200}% \special{fp}% \special{pa 3780 1760}% \special{pa 3380 2160}% \special{fp}% \special{pa 3760 1720}% \special{pa 3360 2120}% \special{fp}% \special{pa 3740 1680}% \special{pa 3340 2080}% \special{fp}% \special{pa 3720 1640}% \special{pa 3320 2040}% \special{fp}% \special{pa 3700 1600}% \special{pa 3300 2000}% \special{fp}% \special{pa 3680 1560}% \special{pa 3360 1880}% \special{fp}% \special{pa 3660 1520}% \special{pa 3420 1760}% \special{fp}% \special{pa 3640 1480}% \special{pa 3480 1640}% \special{fp}% \special{pa 3620 1440}% \special{pa 3540 1520}% \special{fp}% % STR 2 0 3 0 % 3 1420 1290 1420 1390 2 0 % {\small 3} \put(14.2000,-13.9000){\makebox(0,0)[lb]{{\small 3}}}% % STR 2 0 3 0 % 3 1700 1860 1700 1960 2 0 % $\frac{3}{2}$ \put(17.0000,-19.6000){\makebox(0,0)[lb]{$\frac{3}{2}$}}% % STR 2 0 3 0 % 3 3660 1340 3660 1440 2 0 % {\small 3} \put(36.6000,-14.4000){\makebox(0,0)[lb]{{\small 3}}}% % STR 2 0 3 0 % 3 3640 2560 3640 2660 2 0 % {\small $-3$} \put(36.4000,-26.6000){\makebox(0,0)[lb]{{\small $-3$}}}% % CIRCLE 3 0 3 0 % 4 4140 2180 4130 2480 4140 1570 3560 1750 % \special{pn 4}% \special{ar 4140 2180 300 300 3.7795537 4.7123890}% % STR 2 0 3 0 % 3 4160 1820 4160 1920 2 0 % $\frac{3}{2}$ \put(41.6000,-19.2000){\makebox(0,0)[lb]{$\frac{3}{2}$}}% % CIRCLE 3 0 3 0 % 4 3070 2140 3120 2400 3640 1790 3070 1620 % \special{pn 4}% \special{ar 3070 2140 266 266 4.7123890 5.7325098}% % STR 2 0 3 0 % 3 2820 1820 2820 1920 2 0 % $-\frac{3}{2}$ \put(28.2000,-19.2000){\makebox(0,0)[lb]{$-\frac{3}{2}$}}% % STR 2 0 3 0 % 3 1000 690 1000 790 2 0 % {\footnotesize$D$} \put(10.0000,-7.9000){\makebox(0,0)[lb]{{\footnotesize$D$}}}% % STR 2 0 3 0 % 3 3000 690 3000 790 2 0 % {\footnotesize$C$} \put(30.0000,-7.9000){\makebox(0,0)[lb]{{\footnotesize$C$}}}% % POLYGON 2 0 3 0 % 5 5800 1400 5500 2000 5800 2600 6100 2000 5800 1400 % \special{pn 8}% \special{pa 5800 1400}% \special{pa 5500 2000}% \special{pa 5800 2600}% \special{pa 6100 2000}% \special{pa 5800 1400}% \special{fp}% % LINE 3 0 3 0 % 60 6000 1800 5600 2200 6010 1820 5610 2220 6020 1840 5620 2240 6030 1860 5630 2260 6040 1880 5640 2280 6050 1900 5650 2300 6060 1920 5660 2320 6070 1940 5670 2340 6080 1960 5680 2360 6090 1980 5690 2380 6095 2005 5700 2400 6070 2060 5710 2420 6040 2120 5720 2440 6010 2180 5730 2460 5980 2240 5740 2480 5950 2300 5750 2500 5920 2360 5760 2520 5890 2420 5770 2540 5860 2480 5780 2560 5830 2540 5790 2580 5990 1780 5590 2180 5980 1760 5580 2160 5970 1740 5570 2140 5960 1720 5560 2120 5950 1700 5550 2100 5940 1680 5540 2080 5930 1660 5530 2060 5920 1640 5520 2040 5910 1620 5510 2020 5900 1600 5505 1995 % \special{pn 4}% \special{pa 6000 1800}% \special{pa 5600 2200}% \special{fp}% \special{pa 6010 1820}% \special{pa 5610 2220}% \special{fp}% \special{pa 6020 1840}% \special{pa 5620 2240}% \special{fp}% \special{pa 6030 1860}% \special{pa 5630 2260}% \special{fp}% \special{pa 6040 1880}% \special{pa 5640 2280}% \special{fp}% \special{pa 6050 1900}% \special{pa 5650 2300}% \special{fp}% \special{pa 6060 1920}% \special{pa 5660 2320}% \special{fp}% \special{pa 6070 1940}% \special{pa 5670 2340}% \special{fp}% \special{pa 6080 1960}% \special{pa 5680 2360}% \special{fp}% \special{pa 6090 1980}% \special{pa 5690 2380}% \special{fp}% \special{pa 6096 2006}% \special{pa 5700 2400}% \special{fp}% \special{pa 6070 2060}% \special{pa 5710 2420}% \special{fp}% \special{pa 6040 2120}% \special{pa 5720 2440}% \special{fp}% \special{pa 6010 2180}% \special{pa 5730 2460}% \special{fp}% \special{pa 5980 2240}% \special{pa 5740 2480}% \special{fp}% \special{pa 5950 2300}% \special{pa 5750 2500}% \special{fp}% \special{pa 5920 2360}% \special{pa 5760 2520}% \special{fp}% \special{pa 5890 2420}% \special{pa 5770 2540}% \special{fp}% \special{pa 5860 2480}% \special{pa 5780 2560}% \special{fp}% \special{pa 5830 2540}% \special{pa 5790 2580}% \special{fp}% \special{pa 5990 1780}% \special{pa 5590 2180}% \special{fp}% \special{pa 5980 1760}% \special{pa 5580 2160}% \special{fp}% \special{pa 5970 1740}% \special{pa 5570 2140}% \special{fp}% \special{pa 5960 1720}% \special{pa 5560 2120}% \special{fp}% \special{pa 5950 1700}% \special{pa 5550 2100}% \special{fp}% \special{pa 5940 1680}% \special{pa 5540 2080}% \special{fp}% \special{pa 5930 1660}% \special{pa 5530 2060}% \special{fp}% \special{pa 5920 1640}% \special{pa 5520 2040}% \special{fp}% \special{pa 5910 1620}% \special{pa 5510 2020}% \special{fp}% \special{pa 5900 1600}% \special{pa 5506 1996}% \special{fp}% % LINE 3 0 3 1 % 18 5890 1580 5535 1935 5880 1560 5565 1875 5870 1540 5595 1815 5860 1520 5625 1755 5850 1500 5655 1695 5840 1480 5685 1635 5830 1460 5715 1575 5820 1440 5745 1515 5810 1420 5775 1455 % \special{pn 4}% \special{pa 5890 1580}% \special{pa 5536 1936}% \special{fp}% \special{pa 5880 1560}% \special{pa 5566 1876}% \special{fp}% \special{pa 5870 1540}% \special{pa 5596 1816}% \special{fp}% \special{pa 5860 1520}% \special{pa 5626 1756}% \special{fp}% \special{pa 5850 1500}% \special{pa 5656 1696}% \special{fp}% \special{pa 5840 1480}% \special{pa 5686 1636}% \special{fp}% \special{pa 5830 1460}% \special{pa 5716 1576}% \special{fp}% \special{pa 5820 1440}% \special{pa 5746 1516}% \special{fp}% \special{pa 5810 1420}% \special{pa 5776 1456}% \special{fp}% % VECTOR 2 0 3 0 % 2 4800 3000 6310 3000 % \special{pn 8}% \special{pa 4800 3000}% \special{pa 6310 3000}% \special{fp}% \special{sh 1}% \special{pa 6310 3000}% \special{pa 6244 2980}% \special{pa 6258 3000}% \special{pa 6244 3020}% \special{pa 6310 3000}% \special{fp}% % VECTOR 2 0 3 0 % 2 5000 3200 5000 1200 % \special{pn 8}% \special{pa 5000 3200}% \special{pa 5000 1200}% \special{fp}% \special{sh 1}% \special{pa 5000 1200}% \special{pa 4980 1268}% \special{pa 5000 1254}% \special{pa 5020 1268}% \special{pa 5000 1200}% \special{fp}% % POLYLINE 2 2 3 0 % 4 5000 2000 5800 2000 5800 3000 5800 3000 % \special{pn 8}% \special{pa 5000 2000}% \special{pa 5800 2000}% \special{pa 5800 3000}% \special{pa 5800 3000}% \special{dt 0.045}% % DOT 0 2 3 0 % 1 5800 2000 % \special{pn 20}% \special{sh 1}% \special{ar 5800 2000 10 10 0 6.28318530717959E+0000}% % LINE 2 2 3 0 % 2 5000 2600 5800 2600 % \special{pn 8}% \special{pa 5000 2600}% \special{pa 5800 2600}% \special{dt 0.045}% % LINE 2 2 3 0 % 2 5000 1400 5800 1400 % \special{pn 8}% \special{pa 5000 1400}% \special{pa 5800 1400}% \special{dt 0.045}% % LINE 2 2 3 0 % 2 5800 1400 5800 2000 % \special{pn 8}% \special{pa 5800 1400}% \special{pa 5800 2000}% \special{dt 0.045}% % STR 2 0 3 0 % 3 5040 1200 5040 1300 2 0 % $y$ \put(50.4000,-13.0000){\makebox(0,0)[lb]{$y$}}% % STR 2 0 3 0 % 3 6340 2930 6340 3030 2 0 % $x$ \put(63.4000,-30.3000){\makebox(0,0)[lb]{$x$}}% % STR 2 0 3 0 % 3 5750 3070 5750 3170 2 0 % 4 \put(57.5000,-31.7000){\makebox(0,0)[lb]{4}}% % STR 2 0 3 0 % 3 4890 1960 4890 2060 2 0 % 5 \put(48.9000,-20.6000){\makebox(0,0)[lb]{5}}% % STR 2 0 3 0 % 3 4900 1350 4900 1450 2 0 % 8 \put(49.0000,-14.5000){\makebox(0,0)[lb]{8}}% % STR 2 0 3 0 % 3 4890 2550 4890 2650 2 0 % 2 \put(48.9000,-26.5000){\makebox(0,0)[lb]{2}}% % LINE 2 2 3 0 % 2 5500 3000 5500 2000 % \special{pn 8}% \special{pa 5500 3000}% \special{pa 5500 2000}% \special{dt 0.045}% % LINE 2 2 3 0 % 2 6100 3000 6100 2000 % \special{pn 8}% \special{pa 6100 3000}% \special{pa 6100 2000}% \special{dt 0.045}% % STR 2 0 3 0 % 3 5450 3130 5450 3230 2 0 % $\frac{5}{2}$ \put(54.5000,-32.3000){\makebox(0,0)[lb]{$\frac{5}{2}$}}% % STR 2 0 3 0 % 3 6020 3130 6020 3230 2 0 % $\frac{11}{2}$ \put(60.2000,-32.3000){\makebox(0,0)[lb]{$\frac{11}{2}$}}% % LINE 2 2 3 0 % 2 5800 2000 6100 2000 % \special{pn 8}% \special{pa 5800 2000}% \special{pa 6100 2000}% \special{dt 0.045}% % STR 2 0 3 0 % 3 4870 3030 4870 3130 2 0 % {\small O} \put(48.7000,-31.3000){\makebox(0,0)[lb]{{\small O}}}% % STR 2 0 3 0 % 3 5000 690 5000 790 2 0 % {\footnotesize$A$} \put(50.0000,-7.9000){\makebox(0,0)[lb]{{\footnotesize$A$}}}% \end{picture}% %\input{q2003s2f_zu_6} \end{center} \item  $B$ の定義式を, \begin{align*} 2\zettaiti{\zettaiti{x} - 4} + \zettaiti{\zettaiti{y} - 5} \leqq 3 \tag*{$\cdott\MARU{2}$} \end{align*} とする. \begin{align*} 2\zettaiti{\zettaiti{\pm x} - 4} + \zettaiti{\zettaiti{\pm y} - 5} = 2\zettaiti{\zettaiti{x} - 4} + \zettaiti{\zettaiti{y} - 5} \quad (複号同順) \end{align*} より $B$ は$x$軸,$y$軸に関して対称な領域である. だから $B$ を $x \geqq 0,\,\,\,y \geqq 0$ に制限した部分 $E$ を$x$軸, $y$軸に関して対称になるように拡張すれば $B$ 全体が得られる.  $E$ の定義式は, \begin{align*} 2\zettaiti{x - 4} + \zettaiti{y - 5} \leqq 3 \end{align*} これは $A$ の定義式に一致する. すなわち $E = A$ である. ゆえに $B$ は下図の斜線部分のようになる. \begin{center} %WinTpicVersion3.08 \unitlength 0.1in \begin{picture}( 24.4000, 31.4000)( 10.4000,-34.8000) % POLYGON 2 0 3 0 % 5 1680 760 1440 1240 1680 1720 1920 1240 1680 760 % \special{pn 8}% \special{pa 1680 760}% \special{pa 1440 1240}% \special{pa 1680 1720}% \special{pa 1920 1240}% \special{pa 1680 760}% \special{fp}% % LINE 3 0 3 0 % 54 1443 1246 1683 766 1453 1264 1693 784 1461 1282 1701 802 1470 1301 1710 821 1478 1318 1718 838 1488 1336 1728 856 1498 1354 1738 874 1506 1371 1746 891 1515 1390 1755 910 1523 1408 1763 928 1533 1426 1773 946 1542 1445 1782 965 1552 1462 1792 982 1560 1480 1800 1000 1568 1498 1808 1018 1578 1515 1818 1035 1587 1534 1827 1054 1597 1552 1837 1072 1605 1570 1845 1090 1614 1589 1854 1109 1622 1606 1862 1126 1632 1624 1872 1144 1642 1642 1882 1162 1650 1659 1890 1179 1659 1678 1899 1198 1667 1696 1907 1216 1677 1714 1917 1234 % \special{pn 4}% \special{pa 1444 1246}% \special{pa 1684 766}% \special{fp}% \special{pa 1454 1264}% \special{pa 1694 784}% \special{fp}% \special{pa 1462 1282}% \special{pa 1702 802}% \special{fp}% \special{pa 1470 1302}% \special{pa 1710 822}% \special{fp}% \special{pa 1478 1318}% \special{pa 1718 838}% \special{fp}% \special{pa 1488 1336}% \special{pa 1728 856}% \special{fp}% \special{pa 1498 1354}% \special{pa 1738 874}% \special{fp}% \special{pa 1506 1372}% \special{pa 1746 892}% \special{fp}% \special{pa 1516 1390}% \special{pa 1756 910}% \special{fp}% \special{pa 1524 1408}% \special{pa 1764 928}% \special{fp}% \special{pa 1534 1426}% \special{pa 1774 946}% \special{fp}% \special{pa 1542 1446}% \special{pa 1782 966}% \special{fp}% \special{pa 1552 1462}% \special{pa 1792 982}% \special{fp}% \special{pa 1560 1480}% \special{pa 1800 1000}% \special{fp}% \special{pa 1568 1498}% \special{pa 1808 1018}% \special{fp}% \special{pa 1578 1516}% \special{pa 1818 1036}% \special{fp}% \special{pa 1588 1534}% \special{pa 1828 1054}% \special{fp}% \special{pa 1598 1552}% \special{pa 1838 1072}% \special{fp}% \special{pa 1606 1570}% \special{pa 1846 1090}% \special{fp}% \special{pa 1614 1590}% \special{pa 1854 1110}% \special{fp}% \special{pa 1622 1606}% \special{pa 1862 1126}% \special{fp}% \special{pa 1632 1624}% \special{pa 1872 1144}% \special{fp}% \special{pa 1642 1642}% \special{pa 1882 1162}% \special{fp}% \special{pa 1650 1660}% \special{pa 1890 1180}% \special{fp}% \special{pa 1660 1678}% \special{pa 1900 1198}% \special{fp}% \special{pa 1668 1696}% \special{pa 1908 1216}% \special{fp}% \special{pa 1678 1714}% \special{pa 1918 1234}% \special{fp}% % POLYGON 2 0 3 0 % 5 2960 760 2720 1240 2960 1720 3200 1240 2960 760 % \special{pn 8}% \special{pa 2960 760}% \special{pa 2720 1240}% \special{pa 2960 1720}% \special{pa 3200 1240}% \special{pa 2960 760}% \special{fp}% % LINE 3 0 3 0 % 54 2723 1246 2963 766 2733 1264 2973 784 2741 1282 2981 802 2750 1301 2990 821 2758 1318 2998 838 2768 1336 3008 856 2778 1354 3018 874 2786 1371 3026 891 2795 1390 3035 910 2803 1408 3043 928 2813 1426 3053 946 2822 1445 3062 965 2832 1462 3072 982 2840 1480 3080 1000 2848 1498 3088 1018 2858 1515 3098 1035 2867 1534 3107 1054 2877 1552 3117 1072 2885 1570 3125 1090 2894 1589 3134 1109 2902 1606 3142 1126 2912 1624 3152 1144 2922 1642 3162 1162 2930 1659 3170 1179 2939 1678 3179 1198 2947 1696 3187 1216 2957 1714 3197 1234 % \special{pn 4}% \special{pa 2724 1246}% \special{pa 2964 766}% \special{fp}% \special{pa 2734 1264}% \special{pa 2974 784}% \special{fp}% \special{pa 2742 1282}% \special{pa 2982 802}% \special{fp}% \special{pa 2750 1302}% \special{pa 2990 822}% \special{fp}% \special{pa 2758 1318}% \special{pa 2998 838}% \special{fp}% \special{pa 2768 1336}% \special{pa 3008 856}% \special{fp}% \special{pa 2778 1354}% \special{pa 3018 874}% \special{fp}% \special{pa 2786 1372}% \special{pa 3026 892}% \special{fp}% \special{pa 2796 1390}% \special{pa 3036 910}% \special{fp}% \special{pa 2804 1408}% \special{pa 3044 928}% \special{fp}% \special{pa 2814 1426}% \special{pa 3054 946}% \special{fp}% \special{pa 2822 1446}% \special{pa 3062 966}% \special{fp}% \special{pa 2832 1462}% \special{pa 3072 982}% \special{fp}% \special{pa 2840 1480}% \special{pa 3080 1000}% \special{fp}% \special{pa 2848 1498}% \special{pa 3088 1018}% \special{fp}% \special{pa 2858 1516}% \special{pa 3098 1036}% \special{fp}% \special{pa 2868 1534}% \special{pa 3108 1054}% \special{fp}% \special{pa 2878 1552}% \special{pa 3118 1072}% \special{fp}% \special{pa 2886 1570}% \special{pa 3126 1090}% \special{fp}% \special{pa 2894 1590}% \special{pa 3134 1110}% \special{fp}% \special{pa 2902 1606}% \special{pa 3142 1126}% \special{fp}% \special{pa 2912 1624}% \special{pa 3152 1144}% \special{fp}% \special{pa 2922 1642}% \special{pa 3162 1162}% \special{fp}% \special{pa 2930 1660}% \special{pa 3170 1180}% \special{fp}% \special{pa 2940 1678}% \special{pa 3180 1198}% \special{fp}% \special{pa 2948 1696}% \special{pa 3188 1216}% \special{fp}% \special{pa 2958 1714}% \special{pa 3198 1234}% \special{fp}% % POLYGON 2 0 3 0 % 5 1680 2360 1440 2840 1680 3320 1920 2840 1680 2360 % \special{pn 8}% \special{pa 1680 2360}% \special{pa 1440 2840}% \special{pa 1680 3320}% \special{pa 1920 2840}% \special{pa 1680 2360}% \special{fp}% % LINE 3 0 3 0 % 54 1443 2846 1683 2366 1453 2864 1693 2384 1461 2882 1701 2402 1470 2901 1710 2421 1478 2918 1718 2438 1488 2936 1728 2456 1498 2954 1738 2474 1506 2971 1746 2491 1515 2990 1755 2510 1523 3008 1763 2528 1533 3026 1773 2546 1542 3045 1782 2565 1552 3062 1792 2582 1560 3080 1800 2600 1568 3098 1808 2618 1578 3115 1818 2635 1587 3134 1827 2654 1597 3152 1837 2672 1605 3170 1845 2690 1614 3189 1854 2709 1622 3206 1862 2726 1632 3224 1872 2744 1642 3242 1882 2762 1650 3259 1890 2779 1659 3278 1899 2798 1667 3296 1907 2816 1677 3314 1917 2834 % \special{pn 4}% \special{pa 1444 2846}% \special{pa 1684 2366}% \special{fp}% \special{pa 1454 2864}% \special{pa 1694 2384}% \special{fp}% \special{pa 1462 2882}% \special{pa 1702 2402}% \special{fp}% \special{pa 1470 2902}% \special{pa 1710 2422}% \special{fp}% \special{pa 1478 2918}% \special{pa 1718 2438}% \special{fp}% \special{pa 1488 2936}% \special{pa 1728 2456}% \special{fp}% \special{pa 1498 2954}% \special{pa 1738 2474}% \special{fp}% \special{pa 1506 2972}% \special{pa 1746 2492}% \special{fp}% \special{pa 1516 2990}% \special{pa 1756 2510}% \special{fp}% \special{pa 1524 3008}% \special{pa 1764 2528}% \special{fp}% \special{pa 1534 3026}% \special{pa 1774 2546}% \special{fp}% \special{pa 1542 3046}% \special{pa 1782 2566}% \special{fp}% \special{pa 1552 3062}% \special{pa 1792 2582}% \special{fp}% \special{pa 1560 3080}% \special{pa 1800 2600}% \special{fp}% \special{pa 1568 3098}% \special{pa 1808 2618}% \special{fp}% \special{pa 1578 3116}% \special{pa 1818 2636}% \special{fp}% \special{pa 1588 3134}% \special{pa 1828 2654}% \special{fp}% \special{pa 1598 3152}% \special{pa 1838 2672}% \special{fp}% \special{pa 1606 3170}% \special{pa 1846 2690}% \special{fp}% \special{pa 1614 3190}% \special{pa 1854 2710}% \special{fp}% \special{pa 1622 3206}% \special{pa 1862 2726}% \special{fp}% \special{pa 1632 3224}% \special{pa 1872 2744}% \special{fp}% \special{pa 1642 3242}% \special{pa 1882 2762}% \special{fp}% \special{pa 1650 3260}% \special{pa 1890 2780}% \special{fp}% \special{pa 1660 3278}% \special{pa 1900 2798}% \special{fp}% \special{pa 1668 3296}% \special{pa 1908 2816}% \special{fp}% \special{pa 1678 3314}% \special{pa 1918 2834}% \special{fp}% % POLYGON 2 0 3 0 % 5 2960 2360 2720 2840 2960 3320 3200 2840 2960 2360 % \special{pn 8}% \special{pa 2960 2360}% \special{pa 2720 2840}% \special{pa 2960 3320}% \special{pa 3200 2840}% \special{pa 2960 2360}% \special{fp}% % LINE 3 0 3 0 % 54 2723 2846 2963 2366 2733 2864 2973 2384 2741 2882 2981 2402 2750 2901 2990 2421 2758 2918 2998 2438 2768 2936 3008 2456 2778 2954 3018 2474 2786 2971 3026 2491 2795 2990 3035 2510 2803 3008 3043 2528 2813 3026 3053 2546 2822 3045 3062 2565 2832 3062 3072 2582 2840 3080 3080 2600 2848 3098 3088 2618 2858 3115 3098 2635 2867 3134 3107 2654 2877 3152 3117 2672 2885 3170 3125 2690 2894 3189 3134 2709 2902 3206 3142 2726 2912 3224 3152 2744 2922 3242 3162 2762 2930 3259 3170 2779 2939 3278 3179 2798 2947 3296 3187 2816 2957 3314 3197 2834 % \special{pn 4}% \special{pa 2724 2846}% \special{pa 2964 2366}% \special{fp}% \special{pa 2734 2864}% \special{pa 2974 2384}% \special{fp}% \special{pa 2742 2882}% \special{pa 2982 2402}% \special{fp}% \special{pa 2750 2902}% \special{pa 2990 2422}% \special{fp}% \special{pa 2758 2918}% \special{pa 2998 2438}% \special{fp}% \special{pa 2768 2936}% \special{pa 3008 2456}% \special{fp}% \special{pa 2778 2954}% \special{pa 3018 2474}% \special{fp}% \special{pa 2786 2972}% \special{pa 3026 2492}% \special{fp}% \special{pa 2796 2990}% \special{pa 3036 2510}% \special{fp}% \special{pa 2804 3008}% \special{pa 3044 2528}% \special{fp}% \special{pa 2814 3026}% \special{pa 3054 2546}% \special{fp}% \special{pa 2822 3046}% \special{pa 3062 2566}% \special{fp}% \special{pa 2832 3062}% \special{pa 3072 2582}% \special{fp}% \special{pa 2840 3080}% \special{pa 3080 2600}% \special{fp}% \special{pa 2848 3098}% \special{pa 3088 2618}% \special{fp}% \special{pa 2858 3116}% \special{pa 3098 2636}% \special{fp}% \special{pa 2868 3134}% \special{pa 3108 2654}% \special{fp}% \special{pa 2878 3152}% \special{pa 3118 2672}% \special{fp}% \special{pa 2886 3170}% \special{pa 3126 2690}% \special{fp}% \special{pa 2894 3190}% \special{pa 3134 2710}% \special{fp}% \special{pa 2902 3206}% \special{pa 3142 2726}% \special{fp}% \special{pa 2912 3224}% \special{pa 3152 2744}% \special{fp}% \special{pa 2922 3242}% \special{pa 3162 2762}% \special{fp}% \special{pa 2930 3260}% \special{pa 3170 2780}% \special{fp}% \special{pa 2940 3278}% \special{pa 3180 2798}% \special{fp}% \special{pa 2948 3296}% \special{pa 3188 2816}% \special{fp}% \special{pa 2958 3314}% \special{pa 3198 2834}% \special{fp}% % VECTOR 2 0 3 0 % 2 1200 2040 3440 2040 % \special{pn 8}% \special{pa 1200 2040}% \special{pa 3440 2040}% \special{fp}% \special{sh 1}% \special{pa 3440 2040}% \special{pa 3374 2020}% \special{pa 3388 2040}% \special{pa 3374 2060}% \special{pa 3440 2040}% \special{fp}% % VECTOR 2 0 3 0 % 2 2320 3480 2320 400 % \special{pn 8}% \special{pa 2320 3480}% \special{pa 2320 400}% \special{fp}% \special{sh 1}% \special{pa 2320 400}% \special{pa 2300 468}% \special{pa 2320 454}% \special{pa 2340 468}% \special{pa 2320 400}% \special{fp}% % LINE 2 2 3 0 % 2 1680 760 2960 760 % \special{pn 8}% \special{pa 1680 760}% \special{pa 2960 760}% \special{dt 0.045}% % LINE 2 2 3 0 % 4 2960 760 2960 3320 1680 3320 1680 760 % \special{pn 8}% \special{pa 2960 760}% \special{pa 2960 3320}% \special{dt 0.045}% \special{pa 1680 3320}% \special{pa 1680 760}% \special{dt 0.045}% % LINE 2 2 3 0 % 2 1680 3320 2960 3320 % \special{pn 8}% \special{pa 1680 3320}% \special{pa 2960 3320}% \special{dt 0.045}% % LINE 2 2 3 0 % 2 1680 1720 2960 1720 % \special{pn 8}% \special{pa 1680 1720}% \special{pa 2960 1720}% \special{dt 0.045}% % LINE 2 2 3 0 % 2 2960 2360 1680 2360 % \special{pn 8}% \special{pa 2960 2360}% \special{pa 1680 2360}% \special{dt 0.045}% % LINE 2 2 3 0 % 2 1440 1240 3200 1240 % \special{pn 8}% \special{pa 1440 1240}% \special{pa 3200 1240}% \special{dt 0.045}% % LINE 2 2 3 0 % 6 3200 1240 3200 2840 3200 2840 1440 2840 1440 2840 1440 1240 % \special{pn 8}% \special{pa 3200 1240}% \special{pa 3200 2840}% \special{dt 0.045}% \special{pa 3200 2840}% \special{pa 1440 2840}% \special{dt 0.045}% \special{pa 1440 2840}% \special{pa 1440 1240}% \special{dt 0.045}% % LINE 2 2 3 0 % 2 1920 1240 1920 2840 % \special{pn 8}% \special{pa 1920 1240}% \special{pa 1920 2840}% \special{dt 0.045}% % LINE 2 2 3 0 % 2 2720 2840 2720 1240 % \special{pn 8}% \special{pa 2720 2840}% \special{pa 2720 1240}% \special{dt 0.045}% % STR 2 0 3 0 % 3 2360 410 2360 510 2 0 % $y$ \put(23.6000,-5.1000){\makebox(0,0)[lb]{$y$}}% % STR 2 0 3 0 % 3 3480 1970 3480 2070 2 0 % $x$ \put(34.8000,-20.7000){\makebox(0,0)[lb]{$x$}}% % STR 2 0 3 0 % 3 2990 1910 2990 2010 2 0 % {\small 4} \put(29.9000,-20.1000){\makebox(0,0)[lb]{{\small 4}}}% % STR 2 0 3 0 % 3 1140 1830 1140 1930 2 0 % {\small $-4$} \put(11.4000,-19.3000){\makebox(0,0)[lb]{{\small $-4$}}}% % STR 2 0 3 0 % 3 3230 2170 3230 2270 2 0 % $\frac{11}{2}$ \put(32.3000,-22.7000){\makebox(0,0)[lb]{$\frac{11}{2}$}}% % STR 2 0 3 0 % 3 1130 2180 1130 2280 2 0 % $-\frac{11}{2}$ \put(11.3000,-22.8000){\makebox(0,0)[lb]{$-\frac{11}{2}$}}% % STR 2 0 3 0 % 3 2180 2070 2180 2170 2 0 % {\small O} \put(21.8000,-21.7000){\makebox(0,0)[lb]{{\small O}}}% % STR 2 0 3 0 % 3 2200 620 2200 720 2 0 % {\small 8} \put(22.0000,-7.2000){\makebox(0,0)[lb]{{\small 8}}}% % STR 2 0 3 0 % 3 2200 1100 2200 1200 2 0 % {\small 5} \put(22.0000,-12.0000){\makebox(0,0)[lb]{{\small 5}}}% % STR 2 0 3 0 % 3 2200 1570 2200 1670 2 0 % {\small 2} \put(22.0000,-16.7000){\makebox(0,0)[lb]{{\small 2}}}% % STR 2 0 3 0 % 3 2100 2400 2100 2500 2 0 % {\small $-2$} \put(21.0000,-25.0000){\makebox(0,0)[lb]{{\small $-2$}}}% % STR 2 0 3 0 % 3 2100 2700 2100 2800 2 0 % {\small $-5$} \put(21.0000,-28.0000){\makebox(0,0)[lb]{{\small $-5$}}}% % STR 2 0 3 0 % 3 2100 3180 2100 3280 2 0 % {\small $-8$} \put(21.0000,-32.8000){\makebox(0,0)[lb]{{\small $-8$}}}% % STR 2 0 3 0 % 3 2610 1910 2610 2010 2 0 % $\frac{5}{2}$ \put(26.1000,-20.1000){\makebox(0,0)[lb]{$\frac{5}{2}$}}% % STR 2 0 3 0 % 3 1040 1660 1040 1760 2 0 % $-\frac{5}{2}$ \put(10.4000,-17.6000){\makebox(0,0)[lb]{$-\frac{5}{2}$}}% % CIRCLE 3 0 3 0 % 4 1360 2260 1350 2650 1990 1820 1360 1210 % \special{pn 4}% \special{ar 1360 2260 390 390 4.7123890 5.6735254}% % CIRCLE 3 0 3 0 % 4 1300 2430 1440 3150 2530 1660 1300 1060 % \special{pn 4}% \special{ar 1300 2430 734 734 4.7123890 5.7238555}% \end{picture}% %\input{q2003s2f_zu_3} \end{center} \item  (2)の結果より $B$ の点で $x$ が正の整数となるのは \begin{align*} x = 3,\ 4,\ 5 \tag*{$\cdott\MARU{3}$} \end{align*} のみである. $y$ が正の整数の場合に限定する. $m,\ n$ を整数とするとき, \begin{align*} \log_x y = \frac{n}{m} \enskip \Longleftrightarrow \enskip y = x^{\frac{n}{m}} \enskip \Longleftrightarrow \enskip y^m = (x^\frac{n}{m})^m = x^n \end{align*} \parbox{280pt}{ これより, %$x$ の素因数分解を考えれば, \begin{align*} & x\>が素因数\>p\>をもつ \\ \Longrightarrow \,\,\,{} & y^m\>は素因数\>p\>をもつ \\ \Longrightarrow \,\,\,{} & y\>は素因数\>p\>をもつ \tag*{$\cdott{\color[named]{MidnightBlue}(*)}$} \end{align*} \MARU{3},\enskip{\color[named]{MidnightBlue}$(*)$} より 題意をみたす可能性のある点は \begin{align*} (x,\ y) = (4,\ 2),\ (4,\ 4),\ (4,\ 8),\ (5,\ 5) \end{align*} の4点のみ.このとき \begin{gather*} (x,\ y) = (4,\ 2) \enskip \Longrightarrow \,\,\, \log_x y = \log_4 4^{\frac{1}{2}} = \frac{1}{2} \end{gather*} } \parbox{160pt}{ \hspace*{1zw} %WinTpicVersion3.08 \unitlength 0.1in \begin{picture}( 15.4000, 20.7000)( 8.0000,-24.0000) % POLYGON 2 2 3 0 % 5 1800 600 1500 1200 1800 1800 2100 1200 1800 600 % \special{pn 8}% \special{pa 1800 600}% \special{pa 1500 1200}% \special{pa 1800 1800}% \special{pa 2100 1200}% \special{pa 1800 600}% \special{dt 0.045}% % VECTOR 2 0 3 0 % 2 800 2200 2310 2200 % \special{pn 8}% \special{pa 800 2200}% \special{pa 2310 2200}% \special{fp}% \special{sh 1}% \special{pa 2310 2200}% \special{pa 2244 2180}% \special{pa 2258 2200}% \special{pa 2244 2220}% \special{pa 2310 2200}% \special{fp}% % VECTOR 2 0 3 0 % 2 1000 2400 1000 400 % \special{pn 8}% \special{pa 1000 2400}% \special{pa 1000 400}% \special{fp}% \special{sh 1}% \special{pa 1000 400}% \special{pa 980 468}% \special{pa 1000 454}% \special{pa 1020 468}% \special{pa 1000 400}% \special{fp}% % POLYLINE 2 2 3 0 % 4 1000 1200 1800 1200 1800 2200 1800 2200 % \special{pn 8}% \special{pa 1000 1200}% \special{pa 1800 1200}% \special{pa 1800 2200}% \special{pa 1800 2200}% \special{dt 0.045}% % DOT 0 2 3 0 % 1 1800 1200 % \special{pn 20}% \special{sh 1}% \special{ar 1800 1200 10 10 0 6.28318530717959E+0000}% % LINE 2 2 3 0 % 2 1000 1800 1800 1800 % \special{pn 8}% \special{pa 1000 1800}% \special{pa 1800 1800}% \special{dt 0.045}% % LINE 2 2 3 0 % 2 1000 600 1800 600 % \special{pn 8}% \special{pa 1000 600}% \special{pa 1800 600}% \special{dt 0.045}% % LINE 2 2 3 0 % 2 1800 600 1800 1200 % \special{pn 8}% \special{pa 1800 600}% \special{pa 1800 1200}% \special{dt 0.045}% % STR 2 0 3 0 % 3 1040 400 1040 500 2 0 % $y$ \put(10.4000,-5.0000){\makebox(0,0)[lb]{$y$}}% % STR 2 0 3 0 % 3 2340 2130 2340 2230 2 0 % $x$ \put(23.4000,-22.3000){\makebox(0,0)[lb]{$x$}}% % STR 2 0 3 0 % 3 1750 2270 1750 2370 2 0 % 4 \put(17.5000,-23.7000){\makebox(0,0)[lb]{4}}% % STR 2 0 3 0 % 3 890 1160 890 1260 2 0 % {\small 5} \put(8.9000,-12.6000){\makebox(0,0)[lb]{{\small 5}}}% % STR 2 0 3 0 % 3 900 550 900 650 2 0 % {\small 8} \put(9.0000,-6.5000){\makebox(0,0)[lb]{{\small 8}}}% % STR 2 0 3 0 % 3 890 1750 890 1850 2 0 % {\small 2} \put(8.9000,-18.5000){\makebox(0,0)[lb]{{\small 2}}}% % LINE 2 2 3 0 % 2 1800 1200 2100 1200 % \special{pn 8}% \special{pa 1800 1200}% \special{pa 2100 1200}% \special{dt 0.045}% % DOT 0 0 3 0 % 1 1800 1400 % \special{pn 20}% \special{sh 1}% \special{ar 1800 1400 10 10 0 6.28318530717959E+0000}% % DOT 0 0 3 0 % 1 1800 1600 % \special{pn 20}% \special{sh 1}% \special{ar 1800 1600 10 10 0 6.28318530717959E+0000}% % DOT 0 0 3 0 % 1 1800 1800 % \special{pn 20}% \special{sh 1}% \special{ar 1800 1800 10 10 0 6.28318530717959E+0000}% % DOT 0 0 3 0 % 1 1800 1000 % \special{pn 20}% \special{sh 1}% \special{ar 1800 1000 10 10 0 6.28318530717959E+0000}% % DOT 0 0 3 0 % 1 1800 800 % \special{pn 20}% \special{sh 1}% \special{ar 1800 800 10 10 0 6.28318530717959E+0000}% % DOT 0 0 3 0 % 1 1800 600 % \special{pn 20}% \special{sh 1}% \special{ar 1800 600 10 10 0 6.28318530717959E+0000}% % DOT 0 0 3 0 % 1 1600 1200 % \special{pn 20}% \special{sh 1}% \special{ar 1600 1200 10 10 0 6.28318530717959E+0000}% % DOT 0 0 3 0 % 1 2000 1200 % \special{pn 20}% \special{sh 1}% \special{ar 2000 1200 10 10 0 6.28318530717959E+0000}% % LINE 2 2 3 0 % 2 2000 1200 2000 2200 % \special{pn 8}% \special{pa 2000 1200}% \special{pa 2000 2200}% \special{dt 0.045}% % LINE 2 2 3 0 % 2 1600 1200 1600 2200 % \special{pn 8}% \special{pa 1600 1200}% \special{pa 1600 2200}% \special{dt 0.045}% % LINE 2 2 3 0 % 2 1000 1600 1800 1600 % \special{pn 8}% \special{pa 1000 1600}% \special{pa 1800 1600}% \special{dt 0.045}% % LINE 2 2 3 0 % 2 1000 1400 1800 1400 % \special{pn 8}% \special{pa 1000 1400}% \special{pa 1800 1400}% \special{dt 0.045}% % LINE 2 2 3 0 % 2 1000 1000 1800 1000 % \special{pn 8}% \special{pa 1000 1000}% \special{pa 1800 1000}% \special{dt 0.045}% % LINE 2 2 3 0 % 2 1000 800 1800 800 % \special{pn 8}% \special{pa 1000 800}% \special{pa 1800 800}% \special{dt 0.045}% % STR 2 0 3 0 % 3 890 1550 890 1650 2 0 % {\small 3} \put(8.9000,-16.5000){\makebox(0,0)[lb]{{\small 3}}}% % STR 2 0 3 0 % 3 890 1350 890 1450 2 0 % {\small 4} \put(8.9000,-14.5000){\makebox(0,0)[lb]{{\small 4}}}% % STR 2 0 3 0 % 3 890 950 890 1050 2 0 % {\small 6} \put(8.9000,-10.5000){\makebox(0,0)[lb]{{\small 6}}}% % STR 2 0 3 0 % 3 890 750 890 850 2 0 % {\small 7} \put(8.9000,-8.5000){\makebox(0,0)[lb]{{\small 7}}}% % STR 2 0 3 0 % 3 1560 2270 1560 2370 2 0 % 3 \put(15.6000,-23.7000){\makebox(0,0)[lb]{3}}% % STR 2 0 3 0 % 3 1960 2270 1960 2370 2 0 % 5 \put(19.6000,-23.7000){\makebox(0,0)[lb]{5}}% % STR 2 0 3 0 % 3 870 2230 870 2330 2 0 % O \put(8.7000,-23.3000){\makebox(0,0)[lb]{O}}% \end{picture}% %\input{q2003s2f_zu_4} } \vspace*{-3mm} \begin{gather*} (x,\ y) = (4,\ 4) \,\,\, \Longrightarrow \,\,\, \log_x y = \log_4 4 = 1 \\ (x,\ y) = (4,\ 8) \,\,\, \Longrightarrow \,\,\, \log_x y = \log_4 8 = \frac{\log_2 2^3}{\log_2 2^2} = \frac{3}{2} \\ (x,\ y) = (5,\ 5) \,\,\, \Longrightarrow \,\,\, \log_x y = \log_5 5 = 1 \end{gather*} より $\log_x y$ はすべて有理数になる. $B$ は$x$軸に関して対称だから, 求めるべき点は \begin{align*} (x,\ y) = \textcolor{red}{\boldsymbol{ (4,\ \pm 2),\ (4,\ \pm 4),\ (4,\ \pm 8),\ (5,\ \pm 5) }} \tag*{$\Ans$} \end{align*} \end{enumerate} \end{document}