added 2024 math

data/2024/math.json

@@ -0,0 +1,324 @@
+{
+  "1": {
+    "id": 1,
+    "name": "1",
+    "problem": "\\[\n1. \\left( \\frac{4}{2 ^{\\sqrt{2}}} \\right)^{2 + \\sqrt{2}} \\text{\uc758 \uac12\uc740? [2\uc810]}\n\\]\n\\begin{itemize}\n    \\item[1] $\\frac{1}{4}$\n    \\item[2] $\\frac{1}{2}$\n    \\item[3] $1$\n    \\item[4] $2$\n    \\item[5] $4$\n\\end{itemize}\n",
+    "answer": "1",
+    "score": 2
+  },
+  "2": {
+    "id": 2,
+    "name": "2",
+    "problem": "\\[\n2. \\lim_{x \\to \\infty} \\frac{\\sqrt{x^2 - 2} + 3x}{x+5} \\text{\uc758 \uac12\uc740? [2\uc810]}\n\\]\n\\begin{itemize}\n  \\item[1] 1\n  \\item[2] 2\n  \\item[3] 3\n  \\item[4] 4\n  \\item[5] 5\n\\end{itemize}\n",
+    "answer": "4",
+    "score": 2
+  },
+  "3": {
+    "id": 3,
+    "name": "3",
+    "problem": "3. \uacf5\ube44\uac00 \uc591\uc218\uc778 \ub4f1\ube44\uc218\uc5f4 $\\{a_n\\}$\uc774\n\\[\na_2 + a_4 = 30, \\quad a_4 + a_6 = \\frac{15}{2}\n\\]\n\ub97c \ub9cc\uc871\uc2dc\ud0ac \ub54c, $a_1$\uc758 \uac12\uc740? [3\uc810]\n\\begin{itemize}\n    \\item[1] 48\n    \\item[2] 56\n    \\item[3] 64\n    \\item[4] 72\n    \\item[5] 80\n\\end{itemize}\n",
+    "answer": "2",
+    "score": 3
+  },
+  "4": {
+    "id": 4,
+    "name": "4",
+    "problem": "4. \ub2e4\ud56d\ud568\uc218 $f(x)$\uc5d0 \ub300\ud558\uc5ec \ud568\uc218 $g(x)$\ub97c\n\\[\ng(x) = x^2 f(x)\n\\]\n\ub77c \ud558\uc790. $f(2) = 1$, $f'(2) = 3$\uc77c \ub54c, $g'(2)$\uc758 \uac12\uc740? [3\uc810]\n\\begin{itemize}\n    \\item[1] 12\n    \\item[2] 14\n    \\item[3] 16\n    \\item[4] 18\n    \\item[5] 20\n\\end{itemize}\n",
+    "answer": "1",
+    "score": 3
+  },
+  "5": {
+    "id": 5,
+    "name": "5",
+    "problem": "5. \\(\\tan \\theta < 0\\)\uc774\uace0 \\(\\cos \\left(\\frac{\\pi}{2} + \\theta \\right) = \\frac{\\sqrt{5}}{5}\\)\uc77c \ub54c, \\(\\cos \\theta\\)\uc758 \uac12\uc740? [3\uc810]\n\\begin{itemize}\n    \\item[1] \\(- \\frac{2\\sqrt{5}}{5}\\)\n    \\item[2] \\(- \\frac{\\sqrt{5}}{5}\\)\n    \\item[3] 0\n    \\item[4] \\(\\frac{\\sqrt{5}}{5}\\)\n    \\item[5] \\(\\frac{2\\sqrt{5}}{5}\\)\n\\end{itemize}\n",
+    "answer": "4",
+    "score": 3
+  },
+  "6": {
+    "id": 6,
+    "name": "6",
+    "problem": "6. \ud568\uc218 \\( f(x) = 2x^3 - 9x^2 + ax + 5 \\)\ub294 \\( x = 1 \\)\uc5d0\uc11c \uadf9\ub300\uc774\uace0, \\( x = b \\)\uc5d0\uc11c \uadf9\uc18c\uc774\ub2e4. \\( a + b \\)\uc758 \uac12\uc740? (\ub2e8, \\( a, b \\)\ub294 \uc0c1\uc218\uc774\ub2e4.) [3\uc810]\n\\begin{itemize}\n    \\item[1] 12\n    \\item[2] 14\n    \\item[3] 16\n    \\item[4] 18\n    \\item[5] 20\n\\end{itemize}\n",
+    "answer": "4",
+    "score": 3
+  },
+  "7": {
+    "id": 7,
+    "name": "7",
+    "problem": "7. \ubaa8\ub4e0 \ud56d\uc774 \uc591\uc218\uc774\uace0 \uccab\uc9f8\ud56d\uacfc \uacf5\ucc28\uac00 \uac19\uc740 \ub4f1\ucc28\uc218\uc5f4 $\\{a_n\\}$\uc774 \n\\[\n\\sum_{k=1}^{15} \\frac{1}{\\sqrt{a_k} + \\sqrt{a_{k+1}}} = 2\n\\]\n\ub97c \ub9cc\uc871\uc2dc\ud0ac \ub54c, $a_4$\uc758 \uac12\uc740? \\textbf{[3\uc810]}\n\\begin{itemize}\n    \\item[1] 6\n    \\item[2] 7\n    \\item[3] 8\n    \\item[4] 9\n    \\item[5] 10\n\\end{itemize}\n",
+    "answer": "5",
+    "score": 3
+  },
+  "8": {
+    "id": 8,
+    "name": "8",
+    "problem": "8. \uc810 $(0, 4)$\uc5d0\uc11c \uace1\uc120 $y = x^3 - x + 2$\uc5d0 \uadf8\uc740 \uc811\uc120\uc758 $x$\uc808\ud3b8\uc740? [3\uc810]\n\\begin{itemize}\n    \\item[1] $-\\frac{1}{2}$\n    \\item[2] $-1$\n    \\item[3] $-\\frac{3}{2}$\n    \\item[4] $-2$\n    \\item[5] $-\\frac{5}{2}$\n\\end{itemize}\n",
+    "answer": "2",
+    "score": 3
+  },
+  "9": {
+    "id": 9,
+    "name": "9",
+    "problem": "9. \ud568\uc218\n\\[\nf(x) = a - \\sqrt{3} \\tan 2x\n\\]\n\uac00 \ub2eb\ud78c\uad6c\uac04 \\(\\left[ -\\frac{\\pi}{6}, b \\right]\\) \uc5d0\uc11c \ucd5c\ub313\uac12 7, \ucd5c\uc19f\uac12 3\uc744 \uac00\uc9c8 \ub54c, \\(a \\times b\\)\uc758 \uac12\uc740? (\ub2e8, \\(a, b\\)\ub294 \uc0c1\uc218\uc774\ub2e4.) [4\uc810]\n\\begin{itemize}\n    \\item[1] \\(\\frac{\\pi}{2}\\)\n    \\item[2] \\(\\frac{5\\pi}{12}\\)\n    \\item[3] \\(\\frac{\\pi}{3}\\)\n    \\item[4] \\(\\frac{\\pi}{4}\\)\n    \\item[5] \\(\\frac{\\pi}{6}\\)\n\\end{itemize}\n",
+    "answer": "4",
+    "score": 4
+  },
+  "10": {
+    "id": 10,
+    "name": "10",
+    "problem": "10. \ub450 \uace1\uc120 \\(y = x^3 + x^2\\), \\(y = -x^2 + k\\)\uc640 \\(y\\) \ucd95\uc73c\ub85c \ub458\ub7ec\uc2f8\uc778 \ubd80\ubd84\uc758 \ub113\uc774\ub97c \\(A\\), \ub450 \uace1\uc120 \\(y = x^3 + x^2\\), \\(y = -x^2 + k\\)\uc640 \uc9c1\uc120 \\(x = 2\\)\ub85c \ub458\ub7ec\uc2f8\uc778 \ubd80\ubd84\uc758 \ub113\uc774\ub97c \\(B\\)\ub77c \ud558\uc790. \\(A = B\\)\uc77c \ub54c, \uc0c1\uc218 \\(k\\)\uc758 \uac12\uc740? (\ub2e8, \\(4 < k < 5\\)) [4\uc810]\n\\begin{itemize}\n    \\item[1] \\(\\frac{25}{6}\\)\n    \\item[2] \\(\\frac{13}{3}\\)\n    \\item[3] \\(\\frac{9}{2}\\)\n    \\item[4] \\(\\frac{14}{3}\\)\n    \\item[5] \\(\\frac{29}{6}\\)\n\\end{itemize}\n",
+    "answer": "2",
+    "score": 4
+  },
+  "11": {
+    "id": 11,
+    "name": "11",
+    "problem": "11. \uadf8\ub9bc\uacfc \uac19\uc774 \uc0ac\uac01\ud615 ABCD\uac00 \ud55c \uc6d0\uc5d0 \ub0b4\uc811\ud558\uace0 \\\\\n\\[\n\\overline{AB} = 5, \\quad \\overline{AC} = 3\\sqrt{5}, \\quad \\overline{AD} = 7, \\quad \\angle BAC = \\angle CAD\n\\]\n\uc77c \ub54c, \uc774 \uc6d0\uc758 \ubc18\uc9c0\ub984\uc758 \uae38\uc774\ub294? \\textbf{[4\uc810]}\\\\\n\\begin{itemize}\n    \\item[1] \\(\\frac{5\\sqrt{2}}{2}\\)\n    \\item[2] \\(\\frac{8\\sqrt{5}}{5}\\)\n    \\item[3] \\(\\frac{5\\sqrt{5}}{3}\\)\n    \\item[4] \\(\\frac{8\\sqrt{2}}{3}\\)\n    \\item[5] \\(\\frac{9\\sqrt{3}}{4}\\)\n\\end{itemize}\n",
+    "answer": "1",
+    "score": 4
+  },
+  "12": {
+    "id": 12,
+    "name": "12",
+    "problem": "12. \uc2e4\uc218 \uc804\uccb4\uc758 \uc9d1\ud569\uc5d0\uc11c \uc5f0\uc18d\uc778 \ud568\uc218 \\( f(x) \\)\uac00 \ub2e4\uc74c \uc870\uac74\uc744 \ub9cc\uc871\uc2dc\ud0a8\ub2e4.\n\\[\n\\boxed{\nn-1 \\leq x < n \\text{\uc77c \ub54c}, \\, |f(x)| = |6(x-n+1)(x-n)| \\, \\text{\uc774\ub2e4}. \\, (\\text{\ub2e8}, n \\, \\text{\uc740 \uc790\uc5f0\uc218\uc774\ub2e4.})\n}\n\\]\n\uc5f4\ub9b0\uad6c\uac04 \\( (0, 4) \\)\uc5d0\uc11c \uc815\uc758\ub41c \ud568\uc218\n\\[\ng(x) = \\int_0^x f(t)dt - \\int_x^4 f(t)dt\n\\]\n\uac00 \\( x = 2 \\)\uc5d0\uc11c \ucd5c\uc19f\uac12 0\uc744 \uac00\uc9c8 \ub54c, \\( \\int_\\frac{1}{2}^4 f(x)dx \\)\uc758 \uac12\uc740? [4\uc810]\n\\begin{itemize}\n    \\item[1] \\( -\\frac{3}{2} \\)\n    \\item[2] \\( -\\frac{1}{2} \\)\n    \\item[3] \\( \\frac{1}{2} \\)\n    \\item[4] \\( \\frac{3}{2} \\)\n    \\item[5] \\( \\frac{5}{2} \\)\n\\end{itemize}\n",
+    "answer": "3",
+    "score": 4
+  },
+  "13": {
+    "id": 13,
+    "name": "13",
+    "problem": "13. \uc790\uc5f0\uc218 $m(m \\geq 2)$\uc5d0 \ub300\ud558\uc5ec $m^{12}$\uc758 $n$\uc81c\uacf1\uadfc \uc911\uc5d0\uc11c \uc815\uc218\uac00 \uc874\uc7ac\ud558\ub3c4\ub85d \ud558\ub294 2 \uc774\uc0c1\uc758 \uc790\uc5f0\uc218 $n$\uc758 \uac1c\uc218\ub97c $f(m)$\uc774\ub77c \ud560 \ub54c,\n\\[\n\\sum_{m=2}^{9} f(m) \\text{\uc758 \uac12\uc740? [4\uc810]} \n\\]\n\\begin{itemize}\n    \\item[1] 37\n    \\item[2] 42\n    \\item[3] 47\n    \\item[4] 52\n    \\item[5] 57\n\\end{itemize}\n",
+    "answer": "1",
+    "score": 4
+  },
+  "14": {
+    "id": 14,
+    "name": "14",
+    "problem": "14. \ub2e4\ud56d\ud568\uc218 $f(x)$\uc5d0 \ub300\ud558\uc5ec \ud568\uc218 $g(x)$\ub97c \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.\n\\[\ng(x) = \\begin{cases} \nx & (x < -1 \\text{ \ub610\ub294 } x > 1) \\\\\nf(x) & (-1 \\leq x \\leq 1)\n\\end{cases}\n\\]\n\ud568\uc218 $h(x) = \\lim_{t \\to 0^+} g(x + t) \\times \\lim_{t \\to 2^+} g(x + t)$\uc5d0 \ub300\ud558\uc5ec \\\\\n\\textless \ubcf4\uae30\\textgreater \uc5d0\uc11c \uc633\uc740 \uac83\ub9cc\uc744 \uc788\ub294 \ub300\ub85c \uace0\ub978 \uac83\uc740? [4\uc810]\n\\textless \ubcf4\uae30\\textgreater \\\\\n\\fbox{\n  \\parbox{\\textwidth}{\n    \u3131. $h(1) = 3$ \\\\\n    \u3134. \ud568\uc218 $h(x)$\ub294 \uc2e4\uc218 \uc804\uccb4\uc758 \uc9d1\ud569\uc5d0\uc11c \uc5f0\uc18d\uc774\ub2e4. \\\\\n    \u3137. \ud568\uc218 $g(x)$\uac00 \ub2eb\ud78c\uad6c\uac04 $[-1, 1]$\uc5d0\uc11c \uac10\uc18c\ud558\uace0 $g(-1) = -2$\uc774\uba74 \ud568\uc218 $h(x)$\ub294 \uc2e4\uc218 \uc804\uccb4\uc758 \uc9d1\ud569\uc5d0\uc11c \ucd5c\uc19f\uac12\uc744 \uac16\ub294\ub2e4.\n  }\n}\n\\begin{itemize}\n\\item[1] \u3131\n\\item[2] \u3134\n\\item[3] \u3131, \u3134\n\\item[4] \u3131, \u3137\n\\item[5] \u3134, \u3137\n\\end{itemize}\n",
+    "answer": "1",
+    "score": 4
+  },
+  "15": {
+    "id": 15,
+    "name": "15",
+    "problem": "\\textbf{15.} \ubaa8\ub4e0 \ud56d\uc774 \uc790\uc5f0\uc218\uc774\uace0 \ub2e4\uc74c \uc870\uac74\uc744 \ub9cc\uc871\uc2dc\ud0a4\ub294 \ubaa8\ub4e0 \uc218\uc5f4 $\\{a_n\\}$\uc5d0 \ub300\ud558\uc5ec $a_9$\uc758 \ucd5c\ub313\uac12\uacfc \ucd5c\uc19f\uac12\uc744 \uac01\uac01 $M, m$\uc774\ub77c \ud560 \ub54c, $M+m$\uc758 \uac12\uc740? \\textbf{[4\uc810]}\n\\[\n\\text{(\uac00)} \\quad a_7 = 40\n\\]\n\\text{(\ub098)} \\quad \ubaa8\ub4e0 \uc790\uc5f0\uc218 $n$\uc5d0 \ub300\ud558\uc5ec \n\\[\na_{n+2} = \n\\begin{cases} \na_{n+1} + a_n & \\text{(}a_{n+1}\\text{\uc774 3\uc758 \ubc30\uc218\uac00 \uc544\ub2cc \uacbd\uc6b0)}\\\\\n\\frac{1}{3} a_{n+1} & \\text{(}a_{n+1}\\text{\uc774 3\uc758 \ubc30\uc218\uc778 \uacbd\uc6b0)}\n\\end{cases}\n\\]\n\\begin{itemize}\n  \\item[1] 216\n  \\item[2] 218\n  \\item[3] 220\n  \\item[4] 222\n  \\item[5] 224\n\\end{itemize}\n",
+    "answer": "3",
+    "score": 4
+  },
+  "16": {
+    "id": 16,
+    "name": "16",
+    "problem": "\\text{16. \ubc29\uc815\uc2dd}\n\\[\n\\log_2{(3x+2)} = 2 + \\log_2{(x-2)}\n\\]\n\\text{\ub97c \ub9cc\uc871\uc2dc\ud0a4\ub294 \uc2e4\uc218 } \\( x \\) \\text{\uc758 \uac12\uc744 \uad6c\ud558\uc2dc\uc624. [3\uc810]}\n",
+    "answer": "2",
+    "score": 3
+  },
+  "17": {
+    "id": 17,
+    "name": "17",
+    "problem": "17. \ud568\uc218 $f(x)$\uc5d0 \ub300\ud558\uc5ec $f'(x) = 4x^3 - 2x$\uc774\uace0 $f(0) = 3$\uc77c \ub54c, $f(2)$\uc758 \uac12\uc744 \uad6c\ud558\uc2dc\uc624. [3\uc810]\n",
+    "answer": "8",
+    "score": 3
+  },
+  "18": {
+    "id": 18,
+    "name": "18",
+    "problem": "18. \ub450 \uc218\uc5f4 $\\{a_n\\}$, $\\{b_n\\}$\uc5d0 \ub300\ud558\uc5ec\n\\[\n\\sum_{k=1}^{5} (3a_k + 5) = 55, \\quad \\sum_{k=1}^{5} (a_k + b_k) = 32\n\\]\n\uc77c \ub54c, $\\sum_{k=1}^{5} b_k$\uc758 \uac12\uc744 \uad6c\ud558\uc2dc\uc624. [3\uc810]\n",
+    "answer": "9",
+    "score": 3
+  },
+  "19": {
+    "id": 19,
+    "name": "19",
+    "problem": "19. \ubc29\uc815\uc2dd $2x^3 - 6x^2 + k = 0$\uc758 \uc11c\ub85c \ub2e4\ub978 \uc591\uc758 \uc2e4\uadfc\uc758 \uac1c\uc218\uac00 2\uac00 \ub418\ub3c4\ub85d \ud558\ub294 \uc815\uc218 $k$\uc758 \uac1c\uc218\ub97c \uad6c\ud558\uc2dc\uc624. [3\uc810]\n",
+    "answer": "32",
+    "score": 3
+  },
+  "20": {
+    "id": 20,
+    "name": "20",
+    "problem": "\\textbf{20.} \uc218\uc9c1\uc120 \uc704\ub97c \uc6c0\uc9c1\uc774\ub294 \uc810 P\uc758 \uc2dc\uac01 \\(t(t\\geq0)\\)\uc5d0\uc11c\uc758 \uc18d\ub3c4 \\(v(t)\\)\uc640 \uac00\uc18d\ub3c4 \\(a(t)\\)\uac00 \ub2e4\uc74c \uc870\uac74\uc744 \ub9cc\uc871\uc2dc\ud0a8\ub2e4.\n\\[\n\\text{(\uac00)} \\quad 0 \\leq t \\leq 2 \\text{\uc77c \ub54c}, \\quad v(t) = 2t^3 - 8t \\text{\uc774\ub2e4.}\n\\]\n\\[\n\\text{(\ub098)} \\quad t \\geq 2 \\text{\uc77c \ub54c}, \\quad a(t) = 6t + 4\\text{\uc774\ub2e4.}\n\\]\n\uc2dc\uac01 \\( t=0 \\)\uc5d0\uc11c \\( t=3 \\)\uae4c\uc9c0 \uc810 P\uac00 \uc6c0\uc9c1\uc778 \uac70\ub9ac\ub97c \uad6c\ud558\uc2dc\uc624. \\hfill [4\uc810]\n",
+    "answer": "25",
+    "score": 4
+  },
+  "21": {
+    "id": 21,
+    "name": "21",
+    "problem": "21. \uc790\uc5f0\uc218 \\(n\\)\uc5d0 \ub300\ud558\uc5ec \ud568\uc218 \\(f(x)\\)\ub97c\n\\[\nf(x) =\n\\begin{cases} \n    |3^{x+2}-n| & (x<0) \\\\ \n    | \\log_2 (x+4) -n| & (x \\geq 0)\n\\end{cases}\n\\]\n\uc774\ub77c \ud558\uc790. \uc2e4\uc218 \\(t\\)\uc5d0 \ub300\ud558\uc5ec \\(x\\)\uc5d0 \ub300\ud55c \ubc29\uc815\uc2dd \\(f(x) = t\\)\uc758 \uc11c\ub85c \ub2e4\ub978 \uc2e4\uadfc\uc758 \uac1c\uc218\ub97c \\(g(t)\\)\ub77c \ud560 \ub54c, \ud568\uc218 \\(g(t)\\)\uc758 \ucd5c\ub313\uac12\uc774 4\uac00 \ub418\ub3c4\ub85d \ud558\ub294 \ubaa8\ub4e0 \uc790\uc5f0\uc218 \\(n\\)\uc758 \uac12\uc758 \ud569\uc744 \uad6c\ud558\uc2dc\uc624. [4\uc810]\n",
+    "answer": "10",
+    "score": 4
+  },
+  "22": {
+    "id": 22,
+    "name": "22",
+    "problem": "22. \ucd5c\uace0\ucc28\ud56d\uc758 \uacc4\uc218\uac00 1\uc778 \uc0bc\ucc28\ud568\uc218 \\( f(x) \\)\uc640 \uc2e4\uc218 \uc804\uccb4\uc758 \uc9d1\ud569\uc5d0\uc11c \uc5f0\uc18d\uc778 \ud568\uc218 \\( g(x) \\)\uac00 \ub2e4\uc74c \uc870\uac74\uc744 \ub9cc\uc871\uc2dc\ud0ac \ub54c, \\( f(4) \\)\uc758 \uac12\uc744 \uad6c\ud558\uc2dc\uc624. [4\uc810]\n\\[\n\\begin{aligned}\n\\text{(\uac00)} & \\quad \\text{\ubaa8\ub4e0 \uc2e4\uc218 } x \\text{\uc5d0 \ub300\ud558\uc5ec} \\\\\n& \\quad f(x) = f(1) + (x - 1)f'(g(x)) \\text{\uc774\ub2e4.} \\\\\n\\text{(\ub098)} & \\quad \\text{\ud568\uc218 } g(x) \\text{\uc758 \ucd5c\uc19f\uac12\uc740 } \\frac{5}{2} \\text{\uc774\ub2e4.} \\\\\n\\text{(\ub2e4)} & \\quad f(0) = -3, \\, f(g(1)) = 6\n\\end{aligned}\n\\]\n",
+    "answer": "483",
+    "score": 4
+  },
+  "23": {
+    "id": 23,
+    "name": "23_prob",
+    "problem": "23. \ub2e4\ud56d\uc2dd $(x^3 + 3)^5$ \uc758 \uc804\uac1c\uc2dd\uc5d0\uc11c $x^9$\uc758 \uacc4\uc218\ub294? [2\uc810]\n\\begin{itemize}\n    \\item[1] 30\n    \\item[2] 60\n    \\item[3] 90\n    \\item[4] 120\n    \\item[5] 150\n\\end{itemize}\n",
+    "answer": "3",
+    "score": 2
+  },
+  "24": {
+    "id": 24,
+    "name": "23_calc",
+    "problem": "\\[\n23. \\lim_{x \\to 0} \\frac{\\ln(x+1)}{\\sqrt{x+4} - 2} \\text{\uc758 \uac12\uc740? [2\uc810]}\n\\]\n\\begin{itemize}\n    \\item[1] 1\n    \\item[2] 2\n    \\item[3] 3\n    \\item[4] 4\n    \\item[5] 5\n\\end{itemize}\n",
+    "answer": "3",
+    "score": 2
+  },
+  "25": {
+    "id": 25,
+    "name": "23_geom",
+    "problem": "23. \uc88c\ud45c\uacf5\uac04\uc758 \uc810 A(2, 2, -1)\uc744 \\(x\\)\ucd95\uc5d0 \ub300\ud558\uc5ec \ub300\uce6d\uc774\ub3d9\ud55c \uc810\uc744 B\ub77c \ud558\uc790. \uc810 C(-2, 1, 1)\uc5d0 \ub300\ud558\uc5ec \uc120\ubd84 BC\uc758 \uae38\uc774\ub294? \\hfill [2\uc810]\n\\begin{itemize}\n    \\item[1] 1\n    \\item[2] 2\n    \\item[3] 3\n    \\item[4] 4\n    \\item[5] 5\n\\end{itemize}\n",
+    "answer": "4",
+    "score": 2
+  },
+  "26": {
+    "id": 26,
+    "name": "24_prob",
+    "problem": "24. \uc22b\uc790 1, 2, 3, 4, 5 \uc911\uc5d0\uc11c \uc911\ubcf5\uc744 \ud5c8\ub77d\ud558\uc5ec 4\uac1c\ub97c \ud0dd\ud574 \uc77c\ub82c\ub85c \ub098\uc5f4\ud558\uc5ec \ub9cc\ub4e4 \uc218 \uc788\ub294 \ub124 \uc790\ub9ac\uc758 \uc790\uc5f0\uc218 \uc911 4000 \uc774\uc0c1\uc778 \ud640\uc218\uc758 \uac1c\uc218\ub294? [3\uc810]\n\\begin{itemize}\n  \\item[1] 125\n  \\item[2] 150\n  \\item[3] 175\n  \\item[4] 200\n  \\item[5] 225\n\\end{itemize}\n",
+    "answer": "4",
+    "score": 3
+  },
+  "27": {
+    "id": 27,
+    "name": "24_calc",
+    "problem": "\\[\n24. \\lim_{n \\to \\infty} \\frac{1}{n} \\sum_{k=1}^{n} \\sqrt{1 + \\frac{3k}{n}} \\text{\uc758 \uac12\uc740? [3\uc810]}\n\\]\n\\begin{itemize}\n    \\item[1] $\\frac{4}{3}$\n    \\item[2] $\\frac{13}{9}$\n    \\item[3] $\\frac{14}{9}$\n    \\item[4] $\\frac{5}{3}$\n    \\item[5] $\\frac{16}{9}$\n\\end{itemize}\n",
+    "answer": "2",
+    "score": 3
+  },
+  "28": {
+    "id": 28,
+    "name": "24_geom",
+    "problem": "24. \ucd08\uc810\uc774 $F\\left(\\frac{1}{3}, 0\\right)$\uc774\uace0 \uc900\uc120\uc774 $x = -\\frac{1}{3}$\uc778 \ud3ec\ubb3c\uc120\uc774 \uc810 $(a, 2)$\ub97c \uc9c0\ub0a0 \ub54c, $a$\uc758 \uac12\uc740? [3\uc810]\n\\begin{itemize}\n    \\item[1] 1\n    \\item[2] 2\n    \\item[3] 3\n    \\item[4] 4\n    \\item[5] 5\n\\end{itemize}\n",
+    "answer": "3",
+    "score": 3
+  },
+  "29": {
+    "id": 29,
+    "name": "25_prob",
+    "problem": "\\textbf{25.} \ud770\uc0c9 \ub9c8\uc2a4\ud06c 5\uac1c, \uac80\uc740\uc0c9 \ub9c8\uc2a4\ud06c 9\uac1c\uac00 \ub4e4\uc5b4 \uc788\ub294 \uc0c1\uc790\uac00 \uc788\ub2e4. \uc774 \uc0c1\uc790\uc5d0\uc11c \uc784\uc758\ub85c 3\uac1c\uc758 \ub9c8\uc2a4\ud06c\ub97c \ub3d9\uc2dc\uc5d0 \uaebc\ub0bc \ub54c, \uaebc\ub0b8 3\uac1c\uc758 \ub9c8\uc2a4\ud06c \uc911\uc5d0\uc11c \uc801\uc5b4\ub3c4 \ud55c \uac1c\uac00 \ud770\uc0c9 \ub9c8\uc2a4\ud06c\uc77c \ud655\ub960\uc740? [3\uc810]\n\\begin{itemize}\n    \\item[1] $\\frac{8}{13}$\n    \\item[2] $\\frac{17}{26}$\n    \\item[3] $\\frac{9}{13}$\n    \\item[4] $\\frac{19}{26}$\n    \\item[5] $\\frac{10}{13}$\n\\end{itemize}\n",
+    "answer": "5",
+    "score": 3
+  },
+  "30": {
+    "id": 30,
+    "name": "25_calc",
+    "problem": "25. \ub4f1\ube44\uc218\uc5f4 $\\{a_n\\}$\uc5d0 \ub300\ud558\uc5ec $\\lim_{n \\to \\infty} \\frac{a_{n+1}}{3^n + 2^{2n-1}} = 3$\uc77c \ub54c, $a_2$\uc758 \uac12\uc740? \\hspace{3mm}[3\uc810]\n\\begin{itemize}\n    \\item[1] 16\n    \\item[2] 18\n    \\item[3] 20\n    \\item[4] 22\n    \\item[5] 24\n\\end{itemize}\n",
+    "answer": "4",
+    "score": 3
+  },
+  "31": {
+    "id": 31,
+    "name": "25_geom",
+    "problem": " 25. \ud0c0\uc6d0 $\\dfrac{x^2}{a^2} + \\dfrac{y^2}{b^2} = 1$ \uc704\uc758 \uc810 $(2, 1)$\uc5d0\uc11c\uc758 \uc811\uc120\uc758 \uae30\uc6b8\uae30\uac00 $-\\dfrac{1}{2}$\uc77c \ub54c, \uc774 \ud0c0\uc6d0\uc758 \ub450 \ucd08\uc810 \uc0ac\uc774\uc758 \uac70\ub9ac\ub294?\\\\\n(\ub2e8, $a$, $b$\ub294 \uc591\uc218\uc774\ub2e4.) [3\uc810]\n\\begin{itemize}\n    \\item[1] $2 \\sqrt{3}$\n    \\item[2] $4$\n    \\item[3] $2 \\sqrt{5}$\n    \\item[4] $2 \\sqrt{6}$\n    \\item[5] $2 \\sqrt{7}$\n\\end{itemize}\n",
+    "answer": "2",
+    "score": 3
+  },
+  "32": {
+    "id": 32,
+    "name": "26_prob",
+    "problem": "26. \uc8fc\uba38\ub2c8\uc5d0 1\uc774 \uc801\ud78c \ud770 \uacf5 1\uac1c, 2\uac00 \uc801\ud78c \ud770 \uacf5 1\uac1c, 1\uc774 \uc801\ud78c \uac80\uc740 \uacf5 1\uac1c, 2\uac00 \uc801\ud78c \uac80\uc740 \uacf5 3\uac1c\uac00 \ub4e4\uc5b4 \uc788\ub2e4.  \n\uc774 \uc8fc\uba38\ub2c8\uc5d0\uc11c \uc784\uc758\ub85c 3\uac1c\uc758 \uacf5\uc744 \ub3d9\uc2dc\uc5d0 \uaebc\ub0b4\ub294 \uc2dc\ud589\uc744 \ud55c\ub2e4.  \n\uc774 \uc2dc\ud589\uc5d0\uc11c \uaebc\ub0b8 3\uac1c\uc758 \uacf5 \uc911\uc5d0\uc11c \ud770 \uacf5\uc774 1\uac1c\uc774\uace0 \uac80\uc740 \uacf5\uc774 2\uac1c\uc778 \uc0ac\uac74\uc744 A, \uaebc\ub0b8 3\uac1c\uc758 \uacf5\uc5d0 \uc801\ud600 \uc788\ub294 \uc218\ub97c \ubaa8\ub450 \uacf1\ud55c \uac12\uc774 8\uc778 \uc0ac\uac74\uc744 B\ub77c \ud560 \ub54c, $P(A \\cup B)$\uc758 \uac12\uc740? [3\uc810]\n\\begin{itemize}\n  \\item[1] $\\frac{11}{20}$\n  \\item[2] $\\frac{3}{5}$\n  \\item[3] $\\frac{13}{20}$\n  \\item[4] $\\frac{7}{10}$\n  \\item[5] $\\frac{3}{4}$\n\\end{itemize}\n",
+    "answer": "2",
+    "score": 3
+  },
+  "33": {
+    "id": 33,
+    "name": "26_calc",
+    "problem": "26. \uadf8\ub9bc\uacfc \uac19\uc774 \uace1\uc120 $y=\\sqrt{\\sec^2x + \\tan x} \\ \\left( 0 \\leq x \\leq \\frac{\\pi}{3} \\right)$ \uc640 $x$\ucd95, $y$\ucd95 \ubc0f \uc9c1\uc120 $x=\\frac{\\pi}{3}$\ub85c \ub458\ub7ec\uc2f8\uc778 \ubd80\ubd84\uc744 \ubc11\uba74\uc73c\ub85c \ud558\ub294 \uc785\uccb4\ub3c4\ud615\uc774 \uc788\ub2e4. \uc774 \uc785\uccb4\ub3c4\ud615\uc744 $x$\ucd95\uc5d0 \uc218\uc9c1\uc778 \ud3c9\uba74\uc73c\ub85c \uc790\ub978 \ub2e8\uba74\uc774 \ubaa8\ub450 \uc815\uc0ac\uac01\ud615\uc77c \ub54c, \uc774 \uc785\uccb4\ub3c4\ud615\uc758 \ubd80\ud53c\ub294? [3\uc810]\n\\begin{itemize}\n    \\item[1] $\\frac{\\sqrt{3}}{2} + \\frac{\\ln 2}{2}$\n    \\item[2] $\\frac{\\sqrt{3}}{2} + \\ln 2$\n    \\item[3] $\\sqrt{3} + \\frac{\\ln 2}{2}$\n    \\item[4] $\\sqrt{3} + \\ln 2$\n    \\item[5] $\\frac{\\sqrt{3}}{2} + 2 \\ln 2$\n\\end{itemize}\n",
+    "answer": "3",
+    "score": 3
+  },
+  "34": {
+    "id": 34,
+    "name": "26_geom",
+    "problem": "26. \uc88c\ud45c\ud3c9\uba74\uc5d0\uc11c \uc138 \ubca1\ud130\n\\[\n\\vec{a} = (2, 4), \\quad \\vec{b} = (2, 8), \\quad \\vec{c} = (1, 0)\n\\]\n\uc5d0 \ub300\ud558\uc5ec \ub450 \ubca1\ud130 \\(\\vec{p}, \\vec{q}\\)\uac00\n\\[\n(\\vec{p} - \\vec{a}) \\cdot (\\vec{p} - \\vec{b}) = 0, \\quad \\vec{q} = \\frac{1}{2} \\vec{a} + t \\vec{c} \\quad (t\ub294 \\, \uc2e4\uc218)\n\\]\n\ub97c \ub9cc\uc871\uc2dc\ud0ac \ub54c, \\(\\left| \\vec{p} - \\vec{q} \\right|\\)\uc758 \ucd5c\uc18c\uac12\uc740? [3\uc810]\n\\begin{itemize}\n    \\item[1] \\(\\frac{3}{2}\\)\n    \\item[2] 2\n    \\item[3] \\(\\frac{5}{2}\\)\n    \\item[4] 3\n    \\item[5] \\(\\frac{7}{2}\\)\n\\end{itemize}\n",
+    "answer": "5",
+    "score": 3
+  },
+  "35": {
+    "id": 35,
+    "name": "27_prob",
+    "problem": "27. \uc5b4\ub290 \ud68c\uc0ac\uc5d0\uc11c \uc0dd\uc0b0\ud558\ub294 \uc0d8\ud50c 1\uac1c\uc758 \uc6a9\ub7c9\uc740 \uc815\uaddc\ubd84\ud3ec \\( N(\\mu, \\sigma^2) \\) \ub97c \ub530\ub978\ub2e4\uace0 \ud55c\ub2e4. \uc774 \ud68c\uc0ac\uc5d0\uc11c \uc0dd\uc0b0\ud558\ub294 \uc0d8\ud50c \uc911\uc5d0\uc11c 16\uac1c\ub97c \uc784\uc758\ucd94\ucd9c\ud558\uc5ec \uc5bb\uc740 \ud45c\ubcf8\ud3c9\uade0\uc744 \uc774\uc6a9\ud558\uc5ec \uad6c\ud55c \\( m \\) \uc5d0 \ub300\ud55c \uc2e0\ub8b0\ub3c4 95\\%\uc758 \uc2e0\ub8b0\uad6c\uac04\uc774 \\( 746.1 \\leq m \\leq 755.9 \\)\uc774\ub2e4. \uc774 \ud68c\uc0ac\uc5d0\uc11c \uc0dd\uc0b0\ud558\ub294 \uc0d8\ud50c \uc911\uc5d0\uc11c \\( n \\) \uac1c\ub97c \uc784\uc758\ucd94\ucd9c\ud558\uc5ec \uc5bb\uc740 \ud45c\ubcf8\ud3c9\uade0\uc744 \uc774\uc6a9\ud558\uc5ec \uad6c\ud558\ub294 \\( m \\) \uc5d0 \ub300\ud55c \uc2e0\ub8b0\ub3c4 99\\%\uc758 \uc2e0\ub8b0\uad6c\uac04\uc774 \\( a \\leq m \\leq b \\)\uc77c \ub54c, \\( b-a \\)\uc758 \uac12\uc774 6 \uc774\ud558\uac00 \ub418\uae30 \uc704\ud55c \uc790\uc5f0\uc218 \\( n \\)\uc758 \ucd5c\uc18c\uac12\uc740? (\ub2e8, \uc6a9\ub7c9\uc758 \ub2e8\uc704\ub294 mL\uc774\uace0, \\( Z \\)\uac00 \ud45c\uc900\uc815\uaddc\ubd84\ud3ec\ub97c \ub530\ub974\ub294 \ud655\ub960\ubcc0\uc218\uc77c \ub54c, \\( P(|Z| \\leq 1.96) = 0.95, P(|Z| \\leq 2.58) = 0.99 \\) \ub85c \uacc4\uc0b0\ud55c\ub2e4.) [3\uc810]\n\\begin{itemize}\n    \\item[1] 70\n    \\item[2] 74\n    \\item[3] 78\n    \\item[4] 82\n    \\item[5] 86\n\\end{itemize}\n",
+    "answer": "2",
+    "score": 3
+  },
+  "36": {
+    "id": 36,
+    "name": "27_calc",
+    "problem": "27. \uadf8\ub9bc\uacfc \uac19\uc774 \uc911\uc2ec\uc774 $O$, \ubc18\uc9c0\ub984\uc758 \uae38\uc774\uac00 $1$\uc774\uace0 \uc911\uc2ec\uac01\uc758 \ud06c\uae30\uac00 $\\frac{\\pi}{2}$\uc778 \ubd80\ucc44\uaf34 $OA_1B_1$\uc774 \uc788\ub2e4. \ud638 $A_1B_1$ \uc704\uc5d0 \uc810 $P_1$, \uc120\ubd84 $OA_1$ \uc704\uc5d0 \uc810 $C_1$, \uc120\ubd84 $OB_1$ \uc704\uc5d0 \uc810 $D_1$\uc744 \uc0ac\uac01\ud615 $OC_1P_1D_1$\uc774 $OC_1 : OD_1 = 3:4$\uc778 \uc9c1\uc0ac\uac01\ud615\uc774 \ub418\ub3c4\ub85d \uc7a1\ub294\ub2e4.\n\ubd80\ucc44\uaf34 $OA_1B_1$\uc758 \ub0b4\ubd80\uc5d0 \uc810 $Q_1$\uc744 $P_1Q_1 = A_1Q_1$, $\\angle P_1Q_1A_1 = \\frac{\\pi}{2}$\uac00 \ub418\ub3c4\ub85d \uc7a1\uace0, \uc774\ub4f1\ubcc0\uc0bc\uac01\ud615 $P_1Q_1A_1$\uc5d0 \uc0c9\uce60\ud558\uc5ec \uc5bb\uc740 \uadf8\ub9bc\uc744 $R_1$\uc774\ub77c \ud558\uc790.\n\uadf8\ub9bc $R_1$\uc5d0\uc11c \uc120\ubd84 $OA_1$ \uc704\uc758 \uc810 $A_2$\uc640 \uc120\ubd84 $OB_1$ \uc704\uc758 \uc810 $B_2$\ub97c $OQ_1 = OA_2 = OB_2$\uac00 \ub418\ub3c4\ub85d \uc7a1\uace0, \uc911\uc2ec\uc774 $O$, \ubc18\uc9c0\ub984\uc758 \uae38\uc774\uac00 $OQ_1$, \uc911\uc2ec\uac01\uc758 \ud06c\uae30\uac00 $\\frac{\\pi}{2}$\uc778 \ubd80\ucc44\uaf34 $OA_2B_2$\ub97c \uadf8\ub9b0\ub2e4. \uadf8\ub9bc $R_1$\uc744 \uc5bb\uc740 \uac83\uacfc \uac19\uc740 \ubc29\ubc95\uc73c\ub85c \ub124 \uc810 $P_2, C_2, D_2, Q_2$\ub97c \uc7a1\uace0, \uc774\ub4f1\ubcc0\uc0bc\uac01\ud615 $P_2Q_2A_2$\uc5d0 \uc0c9\uce60\ud558\uc5ec \uc5bb\uc740 \uadf8\ub9bc\uc744 $R_2$\ub77c \ud558\uc790.\n\uc774\uc640 \uac19\uc740 \uacfc\uc815\uc744 \uacc4\uc18d\ud558\uc5ec $n$\ubc88\uc9f8 \uc5bb\uc740 \uadf8\ub9bc $R_n$\uc5d0 \uc0c9\uce60\ub418\uc5b4 \uc788\ub294 \ubd80\ubd84\uc758 \ub113\uc774\ub97c $S_n$\uc774\ub77c \ud560 \ub54c, $\\lim_{n \\to \\infty} S_n$\uc758 \uac12\uc740? [3\uc810]\n\\begin{itemize}\n    \\item[1] $\\frac{9}{40}$\n    \\item[2] $\\frac{1}{4}$\n    \\item[3] $\\frac{11}{40}$\n    \\item[4] $\\frac{3}{10}$\n    \\item[5] $\\frac{13}{40}$\n\\end{itemize}\n",
+    "answer": "1",
+    "score": 3
+  },
+  "37": {
+    "id": 37,
+    "name": "27_geom",
+    "problem": "27. \uc88c\ud45c\uacf5\uac04\uc5d0 \uc9c1\uc120 AB\ub97c \ud3ec\ud568\ud558\ub294 \ud3c9\uba74 $\\alpha$\uac00 \uc788\ub2e4. \ud3c9\uba74 $\\alpha$ \uc704\uc5d0 \uc788\uc9c0 \uc54a\uc740 \uc810 C\uc5d0 \ub300\ud558\uc5ec \uc9c1\uc120 AB\uc640 \uc9c1\uc120 AC\uac00 \uc774\ub8e8\ub294 \uc608\uac01\uc758 \ud06c\uae30\ub97c $\\theta_1$\uc774\ub77c \ud560 \ub54c $\\sin \\theta_1 = \\frac{4}{5}$\uc774\uace0, \uc9c1\uc120 AC\uc640 \ud3c9\uba74 $\\alpha$\uac00 \uc774\ub8e8\ub294 \uc608\uac01\uc758 \ud06c\uae30\ub294 $\\frac{\\pi}{2} - \\theta_1$\uc774\ub2e4. \ud3c9\uba74 ABC\uc640 \ud3c9\uba74 $\\alpha$\uac00 \uc774\ub8e8\ub294 \uc608\uac01\uc758 \ud06c\uae30\ub97c $\\theta_2$\ub77c \ud560 \ub54c, $\\cos \\theta_2$\uc758 \uac12\uc740? [3\uc810]\n\\begin{itemize}\n    \\item[1] $\\frac{\\sqrt{7}}{4}$\n    \\item[2] $\\frac{\\sqrt{7}}{5}$\n    \\item[3] $\\frac{\\sqrt{7}}{6}$\n    \\item[4] $\\frac{\\sqrt{7}}{7}$\n    \\item[5] $\\frac{\\sqrt{7}}{8}$\n\\end{itemize}\n",
+    "answer": "3",
+    "score": 3
+  },
+  "38": {
+    "id": 38,
+    "name": "28_prob",
+    "problem": "28. \uc5f0\uc18d\ud655\ub960\ubcc0\uc218 \\( X \\) \uac00 \uac16\ub294 \uac12\uc758 \ubc94\uc704\ub294 \\( 0 \\leq X \\leq a \\) \uc774\uace0, \\( X \\)\uc758 \ud655\ub960\ubc00\ub3c4\ud568\uc218\uc758 \uadf8\ub798\ud504\uac00 \uadf8\ub9bc\uacfc \uac19\ub2e4.\\\\\n\\begin{center}\n\\begin{tikzpicture}\n    % Draw axes\n    \\draw[->] (0,0) -- (5,0) node[right] {$x$};\n    \\draw[->] (0,0) -- (0,4) node[above] {$y$};\n    % Label points\n    \\node at (1,-0.3) {$O$};\n    \\node at (3,-0.3) {$b$};\n    \\node at (5,-0.3) {$a$};\n    \\node at (-0.3,3) {$c$};\n    % Draw the function\n    \\draw[thick] (0,0) -- (3,3) -- (5,0);\n    % Dotted lines for the heights\n    \\draw[dashed] (3,0) -- (3,3);\n    \\draw[dashed] (5,0) -- (5,0);\n\\end{tikzpicture}\n\\end{center}\n\\( P(X \\leq b) - P(X \\geq b) = \\frac{1}{4}, \\quad P(X \\leq \\sqrt{5}) = \\frac{1}{2} \\)\uc77c \ub54c,\\\\\n\\( a + b + c \\)\uc758 \uac12\uc740? (\ub2e8, \\(a, b, c\\)\ub294 \uc0c1\uc218\uc774\ub2e4.) [4\uc810] \n\\begin{itemize}\n  \\item[1] \\(\\frac{11}{2}\\)\n  \\item[2] 6\n  \\item[3] \\(\\frac{13}{2}\\)\n  \\item[4] 7\n  \\item[5] \\(\\frac{15}{2}\\)\n\\end{itemize}\n",
+    "answer": "4",
+    "score": 4
+  },
+  "39": {
+    "id": 39,
+    "name": "28_calc",
+    "problem": "28. \uadf8\ub9bc\uacfc \uac19\uc774 \uc911\uc2ec\uc774 $O$\uc774\uace0 \uae38\uc774\uac00 2\uc778 \uc120\ubd84 $AB$\ub97c \uc9c0\ub984\uc73c\ub85c \ud558\ub294 \ubc18\uc6d0 \uc704\uc5d0 $\\angle AOC = \\frac{\\pi}{2}$\uc778 \uc810 $C$\uac00 \uc788\ub2e4. \ud638 $BC$ \uc704\uc5d0 \uc810 $P$\uc640 \ud638 $CA$ \uc704\uc5d0 \uc810 $Q$\ub97c $PB = QC$\uac00 \ub418\ub3c4\ub85d \uc7a1\uace0, \uc120\ubd84 $AP$ \uc704\uc5d0 \uc810 $R$\uc744 $\\angle CQR = \\frac{\\pi}{2}$\uac00 \ub418\ub3c4\ub85d \uc7a1\ub294\ub2e4.\\\\\n\uc120\ubd84 $AP$\uc640 \uc120\ubd84 $CO$\uc758 \uad50\uc810\uc744 $S$\ub77c \ud558\uc790. $\\angle PAB = \\theta$\uc77c \ub54c, \uc0bc\uac01\ud615 $POB$\uc758 \ub113\uc774\ub97c $f(\\theta)$, \uc0ac\uac01\ud615 $CQRS$\uc758 \ub113\uc774\ub97c $g(\\theta)$\ub77c \ud558\uc790. \\\\\n\\[\n\\lim_{\\theta \\to 0^{+}} \\frac{3f(\\theta) - 2g(\\theta)}{\\theta^2}\n\\]\n\uc758 \uac12\uc740? (\ub2e8, $0 < \\theta < \\frac{\\pi}{4}$) [4\uc810] \n\\begin{itemize}\n    \\item[1] 1\n    \\item[2] 2\n    \\item[3] 3\n    \\item[4] 4\n    \\item[5] 5\n\\end{itemize}\n",
+    "answer": "2",
+    "score": 4
+  },
+  "40": {
+    "id": 40,
+    "name": "28_geom",
+    "problem": "28. \ub450 \ucd08\uc810\uc774 $F(c,0), F'(-c,0)(c>0)$\uc778 \uc30d\uace1\uc120 $C$\uc640 y\ucd95 \uc704\uc758 \uc810 $A$\uac00 \uc788\ub2e4. \uc30d\uace1\uc120 $C$\uac00 \uc120\ubd84 $AF$\uc640 \ub9cc\ub098\ub294 \uc810\uc744 $P$, \uc120\ubd84 $AF'$\uacfc \ub9cc\ub098\ub294 \uc810\uc744 $P'$\uc774\ub77c \ud558\uc790. \\\\\n\uc9c1\uc120 $AF$\ub294 \uc30d\uace1\uc120 $C$\uc758 \ud55c \uc810\uadfc\uc120\uacfc \ud3c9\ud589\ud558\uace0 \\\\\n\\[\n\\frac{AP}{PP'} = \\frac{5}{6}, \\quad PF = 1\n\\]\n\uc77c \ub54c, \uc30d\uace1\uc120 $C$\uc758 \uc8fc\ucd95\uc758 \uae38\uc774\ub294? \\textbf{[4\uc810]} \\\\\n\\begin{itemize}\n    \\item[1] $\\frac{13}{6}$\n    \\item[2] $9/4$\n    \\item[3] $7/3$\n    \\item[4] $\\frac{29}{12}$\n    \\item[5] $\\frac{5}{2}$\n\\end{itemize}\n",
+    "answer": "5",
+    "score": 4
+  },
+  "41": {
+    "id": 41,
+    "name": "29_prob",
+    "problem": "\\textbf{29.} \uc55e\uba74\uc5d0\ub294 1\ubd80\ud130 6\uae4c\uc9c0\uc758 \uc790\uc5f0\uc218\uac00 \ud558\ub098\uc529 \uc801\ud600 \uc788\uace0 \ub4b7\uba74\uc5d0\ub294 \ubaa8\ub450 0\uc774 \ud558\ub098\uc529 \uc801\ud600 \uc788\ub294 6\uc7a5\uc758 \uce74\ub4dc\uac00 \uc788\ub2e4. \uc774 6\uc7a5\uc758 \uce74\ub4dc\ub97c \uadf8\ub9bc\uacfc \uac19\uc774 6 \uc774\ud558\uc758 \uc790\uc5f0\uc218 $k$\uc5d0 \ub300\ud558\uc5ec $k$\ubc88\uc9f8 \uc790\ub9ac\uc5d0 \uc790\uc5f0\uc218 $k$\uac00 \ubcf4\uc774\ub3c4\ub85d \ub193\uc5ec \uc788\ub2e4. \\\\\n\\[\n\\begin{array}{|c|c|c|c|c|c|}\n\\hline\n\\text{1\ubc88\uc9f8 \uc790\ub9ac} & \\text{2\ubc88\uc9f8 \uc790\ub9ac} & \\text{3\ubc88\uc9f8 \uc790\ub9ac} & \\text{4\ubc88\uc9f8 \uc790\ub9ac} & \\text{5\ubc88\uc9f8 \uc790\ub9ac} & \\text{6\ubc88\uc9f8 \uc790\ub9ac} \\\\\n\\hline\n1 & 2 & 3 & 4 & 5 & 6 \\\\\n\\hline\n\\end{array}\n\\]\n\uc774 6\uc7a5\uc758 \uce74\ub4dc\uc640 \ud55c \uac1c\uc758 \uc8fc\uc0ac\uc704\ub97c \uc0ac\uc6a9\ud558\uc5ec \ub2e4\uc74c \uc2dc\ud589\uc744 \ud55c\ub2e4. \\\\\n\\framebox{\n\\parbox{\\textwidth}{\n\uc8fc\uc0ac\uc704\ub97c \ud55c \ubc88 \ub358\uc838 \ub098\uc628 \ub208\uc758 \uc218\uac00 $k$\uc774\uba74 $k$\ubc88\uc9f8 \uc790\ub9ac\uc5d0 \ub193\uc5ec \uc788\ub294 \uce74\ub4dc\ub97c \ud55c \ubc88 \ub4a4\uc9d1\uc5b4 \uc81c\uc790\ub9ac\uc5d0 \ub193\ub294\ub2e4.\n}\n} \\\\\n\uc704\uc758 \uc2dc\ud589\uc744 3\ubc88 \ubc18\ubcf5\ud55c \ud6c4 6\uc7a5\uc758 \uce74\ub4dc\uc5d0 \ubcf4\uc774\ub294 \ubaa8\ub4e0 \uc218\uc758 \ud569\uc774 \uc9dd\uc218\uc77c \ub54c, \uc8fc\uc0ac\uc704\uc758 1\uc758 \ub208\uc774 \ud55c \ubc88\ub9cc \ub098\uc654\uc744 \ud655\ub960\uc744 $\\frac{p}{q}$\uc774\ub2e4. $p+q$\uc758 \uac12\uc744 \uad6c\ud558\uc2dc\uc624. (\ub2e8, $p$\uc640 $q$\ub294 \uc11c\ub85c\uc18c\uc778 \uc790\uc5f0\uc218\uc774\ub2e4.) [4\uc810]\n",
+    "answer": "196",
+    "score": 4
+  },
+  "42": {
+    "id": 42,
+    "name": "29_calc",
+    "problem": "\\textbf{29.} \uc138 \uc0c1\uc218 \\(a, b, c\\)\uc5d0 \ub300\ud558\uc5ec \ud568\uc218 \\(f(x) = ae^{2x} + be^x + c\\)\uac00 \ub2e4\uc74c \uc870\uac74\uc744 \ub9cc\uc871\uc2dc\ud0a8\ub2e4.\n\\[\n(\uac00)\\ \\lim_{x \\to -\\infty} \\frac{f(x) + 6}{e^x} = 1\n\\]\n\\[\n(\ub098)\\ f(\\ln 2) = 0\n\\]\n\ud568\uc218 \\(f(x)\\)\uc758 \uc5ed\ud568\uc218\ub97c \\(g(x)\\)\ub77c \ud560 \ub54c,\n\\[\n\\int_0^{14} g(x) dx = p + q \\ln 2 \uc774\ub2e4. \\ p+q\uc758 \uac12\uc744 \uad6c\ud558\uc2dc\uc624.\n\\]\n(\ub2e8, \\(p, q\\)\ub294 \uc720\ub9ac\uc218\uc774\uace0, \\(\\ln 2\\)\ub294 \ubb34\ub9ac\uc218\uc774\ub2e4.) [4\uc810]\n",
+    "answer": "162",
+    "score": 4
+  },
+  "43": {
+    "id": 43,
+    "name": "29_geom",
+    "problem": "29.\\ \ud3c9\uba74\\ \\(\\alpha\\) \uc704\uc5d0\\ \\(\\overline{AB} = \\overline{CD} = \\overline{AD} = 2\\),\\ \\(\\angle ABC = \\angle BCD = \\frac{\\pi}{3}\\)\\ \uc778\\ \uc0ac\ub2e4\ub9ac\uaf34\\ \\(ABCD\\)\\ \uac00\\ \uc788\ub2e4.\\ \ub2e4\uc74c\\ \uc870\uac74\uc744\\ \ub9cc\uc871\uc2dc\ud0a4\ub294\\ \ud3c9\uba74\\ \\(\\alpha\\) \uc704\uc758\\ \ub450\\ \uc810\\ \\(P, Q\\)\uc5d0\\ \ub300\ud558\uc5ec\\ \\(CP \\cdot DQ\\)\uc758\\ \uac12\uc744\\ \uad6c\ud558\uc2dc\uc624.\\ [4\uc810]\n\\begin{itemize}\n    \\item[(\uac00)] \\(\\overrightarrow{AC} = 2(\\overrightarrow{AD} + \\overrightarrow{BP})\\)\n    \\item[(\ub098)] \\(\\overrightarrow{AC} \\cdot \\overrightarrow{PQ} = 6\\)\n    \\item[(\ub2e4)] \\(2 \\times \\angle BQA = \\angle PBQ < \\frac{\\pi}{2}\\)\n\\end{itemize}\n\\begin{center}\n\\begin{tikzpicture}\n    \\draw (0,0) -- (2,0) -- (2.5,1.5) -- (-0.5,1.5) -- cycle;\n    \\node[below] at (0,0) {B};\n    \\node[below] at (2,0) {C};\n    \\node[above] at (2.5,1.5) {D};\n    \\node[above] at (-0.5,1.5) {A};\n\\end{tikzpicture}\n\\end{center}\n",
+    "answer": "11",
+    "score": 4
+  },
+  "44": {
+    "id": 44,
+    "name": "30_prob",
+    "problem": "30. \uc9d1\ud569 $X=\\{x \\mid x \\text{\ub294 10 \uc774\ud558\uc758 \uc790\uc5f0\uc218}\\}$\uc5d0 \ub300\ud558\uc5ec \ub2e4\uc74c \uc870\uac74\uc744 \ub9cc\uc871\uc2dc\ud0a4\ub294 \ud568\uc218 $f: X \\rightarrow X$\uc758 \uac1c\uc218\ub97c \uad6c\ud558\uc2dc\uc624. [4\uc810]\n\\begin{quote}\n\\textbf{(\uac00)} 9 \uc774\ud558\uc758 \ubaa8\ub4e0 \uc790\uc5f0\uc218 $x$\uc5d0 \ub300\ud558\uc5ec $f(x) \\leq f(x+1)$ \uc774\ub2e4.\n\\textbf{(\ub098)} $1 \\leq x \\leq 5$\uc77c \ub54c $f(x) \\leq x$\uc774\uace0, \\\\\n$6 \\leq x \\leq 10$\uc77c \ub54c $f(x) \\geq x$\uc774\ub2e4.\n\\textbf{(\ub2e4)} $f(6) = f(5) + 6$\n\\end{quote}\n",
+    "answer": "673",
+    "score": 4
+  },
+  "45": {
+    "id": 45,
+    "name": "30_calc",
+    "problem": "30. \ucd5c\uace0\ucc28\ud56d\uc758 \uacc4\uc218\uac00 \uc591\uc218\uc778 \uc0bc\ucc28\ud568\uc218 $f(x)$\uc640\\\\\n\ud568\uc218 $g(x) = e^{\\sin \\pi x} - 1$\uc5d0 \ub300\ud558\uc5ec \uc2e4\uc218 \uc804\uccb4\uc758 \uc9d1\ud569\uc5d0\uc11c \uc815\uc758\ub41c \ud569\uc131\ud568\uc218 $h(x) = g(f(x))$\uac00 \ub2e4\uc74c \uc870\uac74\uc744 \ub9cc\uc871\uc2dc\ud0a8\ub2e4.\n\\begin{itemize}\n    \\item[(\uac00)] \ud568\uc218 $h(x)$\ub294 $x = 0$\uc5d0\uc11c \uadf9\ub313\uac12 $0$\uc744 \uac16\ub294\ub2e4.\n    \\item[(\ub098)] \uc5f4\ub9b0\uad6c\uac04 $(0, 3)$\uc5d0\uc11c \ubc29\uc815\uc2dd $h(x) = 1$\uc758 \uc11c\ub85c \ub2e4\ub978 \uc2e4\uadfc\uc758 \uac1c\uc218\ub294 7\uc774\ub2e4.\n\\end{itemize}\n$f(3) = \\frac{1}{2}, f'(3) = 0$\uc77c \ub54c, $f(2) = \\frac{q}{p}$\uc774\ub2e4. $p + q$\uc758 \uac12\uc744 \uad6c\ud558\uc2dc\uc624. (\ub2e8, $p$\uc640 $q$\ub294 \uc11c\ub85c\uc18c\uc778 \uc790\uc5f0\uc218\uc774\ub2e4.) [4\uc810]\n",
+    "answer": "125",
+    "score": 4
+  },
+  "46": {
+    "id": 46,
+    "name": "30_geom",
+    "problem": "\\textbf{30.} \uc88c\ud45c\uacf5\uac04\uc5d0 \uc815\uc0ac\uba74\uccb4 $ABCD$ \uac00 \uc788\ub2e4. \uc815\uc0bc\uac01\ud615 $BCD$ \uc758 \uc678\uc2ec\uc744 \uc911\uc2ec\uc73c\ub85c \ud558\uace0 \uc810 $B$\ub97c \uc9c0\ub098\ub294 \uad6c\ub97c $S$\ub77c \ud558\uc790. \\\\\n\uad6c $S$\uc640 \uc120\ubd84 $AB$\uac00 \ub9cc\ub098\ub294 \uc810 \uc911 $B$\uac00 \uc544\ub2cc \uc810\uc744 $P$, \\\\\n\uad6c $S$\uc640 \uc120\ubd84 $AC$\uac00 \ub9cc\ub098\ub294 \uc810 \uc911 $C$\uac00 \uc544\ub2cc \uc810\uc744 $Q$, \\\\\n\uad6c $S$\uc640 \uc120\ubd84 $AD$\uac00 \ub9cc\ub098\ub294 \uc810 \uc911 $D$\uac00 \uc544\ub2cc \uc810\uc744 $R$ \ud558\uace0, \\\\\n\uc810 $P$\uc5d0\uc11c \uad6c $S$\uc5d0 \uc811\ud558\ub294 \ud3c9\uba74\uc744 $\\alpha$\ub77c \ud558\uc790. \\\\\n\uad6c $S$\uc758 \ubc18\uc9c0\ub984\uc758 \uae38\uc774\uac00 6\uc77c \ub54c, \uc0bc\uac01\ud615 $PQR$\uc758 \ud3c9\uba74 $\\alpha$ \uc704\ub85c\uc758 \uc815\uc0ac\uc601\uc758 \ub113\uc774\ub294 $k$\uc774\ub2e4. $k^2$\uc758 \uac12\uc744 \uad6c\ud558\uc2dc\uc624. \\hfill [4\uc810]\n",
+    "answer": "147",
+    "score": 4
+  }
+}