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  1. data/.gitignore +1 -0
  2. data/json/2022.zip +3 -0
  3. data/json/2022/{math/answer_score_comment.json β†’ answer_score_comment.json} +13 -13
  4. data/json/2022/math.json +46 -46
  5. data/json/2022/math/math_29_prob.txt +0 -12
  6. data/json/2022/{math/math_1.txt β†’ math_1.txt} +0 -0
  7. data/json/2022/{math/math_10.txt β†’ math_10.txt} +0 -0
  8. data/json/2022/{math/math_11.txt β†’ math_11.txt} +0 -0
  9. data/json/2022/{math/math_12.txt β†’ math_12.txt} +0 -0
  10. data/json/2022/{math/math_13.txt β†’ math_13.txt} +0 -0
  11. data/json/2022/{math/math_14.txt β†’ math_14.txt} +0 -0
  12. data/json/2022/{math/math_15.txt β†’ math_15.txt} +0 -0
  13. data/json/2022/{math/math_16.txt β†’ math_16.txt} +0 -0
  14. data/json/2022/{math/math_17.txt β†’ math_17.txt} +0 -0
  15. data/json/2022/{math/math_18.txt β†’ math_18.txt} +0 -0
  16. data/json/2022/{math/math_19.txt β†’ math_19.txt} +0 -0
  17. data/json/2022/{math/math_2.txt β†’ math_2.txt} +0 -0
  18. data/json/2022/{math/math_20.txt β†’ math_20.txt} +0 -0
  19. data/json/2022/{math/math_21.txt β†’ math_21.txt} +0 -0
  20. data/json/2022/{math/math_22.txt β†’ math_22.txt} +0 -0
  21. data/json/2022/{math/math_23_calc.txt β†’ math_23_calc.txt} +0 -0
  22. data/json/2022/{math/math_23_geom.txt β†’ math_23_geom.txt} +0 -0
  23. data/json/2022/{math/math_23_prob.txt β†’ math_23_prob.txt} +0 -0
  24. data/json/2022/{math/math_24_calc.txt β†’ math_24_calc.txt} +0 -0
  25. data/json/2022/{math/math_24_geom.txt β†’ math_24_geom.txt} +0 -0
  26. data/json/2022/{math/math_24_prob.txt β†’ math_24_prob.txt} +0 -0
  27. data/json/2022/{math/math_25_calc.txt β†’ math_25_calc.txt} +0 -0
  28. data/json/2022/{math/math_25_geom.txt β†’ math_25_geom.txt} +0 -0
  29. data/json/2022/{math/math_25_prob.txt β†’ math_25_prob.txt} +0 -0
  30. data/json/2022/{math/math_26_calc.txt β†’ math_26_calc.txt} +0 -0
  31. data/json/2022/{math/math_26_geom.txt β†’ math_26_geom.txt} +0 -0
  32. data/json/2022/{math/math_26_prob.txt β†’ math_26_prob.txt} +0 -0
  33. data/json/2022/{math/math_27_calc.txt β†’ math_27_calc.txt} +0 -0
  34. data/json/2022/{math/math_27_geom.txt β†’ math_27_geom.txt} +0 -0
  35. data/json/2022/{math/math_27_prob.txt β†’ math_27_prob.txt} +0 -0
  36. data/json/2022/{math/math_28_calc.txt β†’ math_28_calc.txt} +0 -0
  37. data/json/2022/{math/math_28_geom.txt β†’ math_28_geom.txt} +0 -0
  38. data/json/2022/{math/math_28_prob.txt β†’ math_28_prob.txt} +0 -0
  39. data/json/2022/{math/math_29_calc.txt β†’ math_29_calc.txt} +0 -0
  40. data/json/2022/{math/math_29_geom.txt β†’ math_29_geom.txt} +0 -0
  41. data/json/2022/math_29_prob.txt +17 -0
  42. data/json/2022/{math/math_3.txt β†’ math_3.txt} +0 -0
  43. data/json/2022/{math/math_30_calc.txt β†’ math_30_calc.txt} +0 -0
  44. data/json/2022/{math/math_30_geom.txt β†’ math_30_geom.txt} +0 -0
  45. data/json/2022/{math/math_30_prob.txt β†’ math_30_prob.txt} +0 -0
  46. data/json/2022/math_4.txt +17 -0
  47. data/json/2022/{math/math_5.txt β†’ math_5.txt} +0 -0
  48. data/json/2022/{math/math_6.txt β†’ math_6.txt} +0 -0
  49. data/json/2022/{math/math_7.txt β†’ math_7.txt} +0 -0
  50. data/json/2022/{math/math_8.txt β†’ math_8.txt} +0 -0
data/.gitignore ADDED
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data/json/2022.zip ADDED
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data/json/2022/{math/answer_score_comment.json β†’ answer_score_comment.json} RENAMED
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  "name":"4",
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  "answer":"4",
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  "score":"3",
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- "comment":"Removed figure and the statement referring to the figure. Need the figure to solve the problem"
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  },
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  "5":{
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  "name":"9",
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  "name":"11",
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  "name":"14",
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  "answer":"3",
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  "score":"4",
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- "comment":"<보기> changed to 'μ•„λž˜ γ„±,γ„΄,γ„·, 쀑'"
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  },
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  "15":{
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  "name":"15",
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  "name":"26_prob",
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  "27":{
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  "name":"29_prob",
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  "answer":"31",
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  "score":"4",
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- "comment":"Removed figure and the statement referring to the figure. Need the figure to solve the problem"
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  "30":{
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  "name":"30_prob",
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  "name":"29_calc",
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  "answer":"11",
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  "38":{
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  "name":"30_calc",
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  "name":"26_geom",
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  "answer":"2",
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- "comment":"Removed figure"
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  },
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  "43":{
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  "name":"27_geom",
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  "answer":"4",
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  "score":"3",
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- "comment":"Removed figure and the statement referring to the figure"
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  "44":{
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  "name":"28_geom",
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  "45":{
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  "name":"29_geom",
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  "answer":"100",
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  "46":{
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  "name":"30_geom",
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  "answer":"23",
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  }
 
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  "name":"4",
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+ "comment":"Removed figure and the statement referring to the figure. The figure is needed to solve the problem, so we paraphrased the figure into text."
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  },
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  "5":{
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  "name":"5",
 
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  "name":"9",
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  "answer":"4",
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  "10":{
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  "name":"10",
 
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  "12":{
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  "name":"12",
 
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  "name":"14",
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  "answer":"3",
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  "score":"4",
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+ "comment":"<보기> changed to 'μ•„λž˜ γ„±,γ„΄,γ„·, 쀑'."
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  },
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  "15":{
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  "name":"15",
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  "27":{
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+ "comment":"Removed figure and the statement referring to the figure. The figure is needed to solve the problem, so we paraphrased the figure into text."
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  "30":{
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  "name":"30_prob",
 
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  "name":"29_calc",
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  "answer":"11",
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  "38":{
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  "name":"30_calc",
 
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  "43":{
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  "name":"27_geom",
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  "answer":"4",
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  "score":"3",
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+ "comment":"Removed figure and the statement referring to the figure."
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  },
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  "44":{
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  "name":"28_geom",
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  "answer":"5",
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  "45":{
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  "name":"29_geom",
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  "answer":"100",
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  "46":{
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  "name":"30_geom",
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data/json/2022/math.json CHANGED
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1
- {"id": 1, "name": "1", "problem": "1. $\\left(2^{\\sqrt{3}} \\times 4\\right)^{\\sqrt{3} - 2}$ 의 값은? [2점] \\begin{itemize} \\item[1] \\frac{1}{4} \\item[2] \\frac{1}{2} \\item[3] 1 \\item[4] 2 \\item[5] 4 \\end{itemize}", "answer": 2, "score": 2, "review": null, "incomplete": false}
2
- {"id": 2, "name": "2", "problem": "2. ν•¨μˆ˜ $f(x) = x^3 + 3x^2 + x - 1$ 에 λŒ€ν•˜μ—¬ $f'(1)$의 값은? [2점] \\begin{itemize} \\item[1] 6 \\item[2] 7 \\item[3] 8 \\item[4] 9 \\item[5] 10 \\end{itemize}", "answer": 5, "score": 2, "review": null, "incomplete": false}
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- {"id": 3, "name": "3", "problem": "3. λ“±μ°¨μˆ˜μ—΄ $\\{a_n\\}$에 λŒ€ν•˜μ—¬ \\[ a_2 = 6, \\quad a_4 + a_6 = 36 \\] 일 λ•Œ, $a_{10}$의 값은? [3점] \\begin{itemize} \\item[1] 30 \\item[2] 32 \\item[3] 34 \\item[4] 36 \\item[5] 38 \\end{itemize}", "answer": 5, "score": 3, "review": null, "incomplete": false}
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- {"id": 4, "name": "4", "problem": "4. ν•¨μˆ˜ $( y = f(x) )$의 κ·Έλž˜ν”„κ°€ κ·Έλ¦Όκ³Ό κ°™λ‹€.\n\n\\[ \\lim_{x \\to -1-} f(x) + \\lim_{x \\to 2} f(x) \\text{의 값은? [3점]} \\]\n\n\\begin{itemize} \\item[1] 1 \\item[2] 2 \\item[3] 3 \\item[4] 4 \\item[5] 5 \\end{itemize}", "answer": 4, "score": 3, "review": "Removed figure and the statement referring to the figure. The figure is needed to solve the problem.", "incomplete": true}
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- {"id": 5, "name": "5", "problem": "5. 첫째항이 1인 μˆ˜μ—΄ $\\{a_n\\}$이 λͺ¨λ“  μžμ—°μˆ˜ $n$에 λŒ€ν•˜μ—¬ \\[ a_{n+1} = \\begin{cases} 2a_n & (a_n < 7) \\\\ a_n - 7 & (a_n \\geq 7) \\end{cases} \\] 일 λ•Œ, $\\sum_{k=1}^{8} a_k$의 값은? [3점] \\begin{itemize} \\item[1] 30 \\item[2] 32 \\item[3] 34 \\item[4] 36 \\item[5] 38 \\end{itemize}", "answer": 1, "score": 3, "review": null, "incomplete": false}
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- {"id": 6, "name": "6", "problem": "6. 방정식 $( 2x^3 - 3x^2 - 12x + k = 0 )$이 μ„œλ‘œ λ‹€λ₯Έ μ„Έ 싀근을 갖도둝 ν•˜λŠ” μ •μˆ˜ $k$의 κ°œμˆ˜λŠ”? [3점] \\begin{itemize} \\item[1] 20 \\item[2] 23 \\item[3] 26 \\item[4] 29 \\item[5] 32 \\end{itemize}", "answer": 3, "score": 3, "review": null, "incomplete": false}
7
- {"id": 7, "name": "7", "problem": "7. $( \\pi < \\theta < \\frac{3}{2}\\pi )$인 $\\theta$에 λŒ€ν•˜μ—¬ $\\tan \\theta - \\frac{6}{\\tan \\theta} = 1$일 λ•Œ, $ \\sin \\theta + \\cos \\theta $의 값은? [3점] \\begin{itemize} \\item[1] -\\frac{2\\sqrt{10}}{5} \\item[2] -\\frac{\\sqrt{10}}{5} \\item[3] 0 \\item[4] \\frac{\\sqrt{10}}{5} \\item[5] \\frac{2\\sqrt{10}}{5} \\end{itemize}", "answer": 1, "score": 3, "review": null, "incomplete": false}
8
- {"id": 8, "name": "8", "problem": "8. 곑선 $( y = x^2 - 5x )$와 직선 $( y = x )$둜 λ‘˜λŸ¬μ‹ΈμΈ λΆ€λΆ„μ˜ 넓이λ₯Ό 직선 $( x = k )$κ°€ 이등뢄할 λ•Œ, μƒμˆ˜ $k$의 값은? [3점] \\begin{itemize} \\item[1] 3 \\item[2] \\frac{13}{4} \\item[3] \\frac{7}{2} \\item[4] \\frac{15}{4} \\item[5] 4 \\end{itemize}", "answer": 1, "score": 3, "review": null, "incomplete": false}
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- {"id": 9, "name": "9", "problem": "9. 직선 $( y = 2x + k )$ κ°€ 두 ν•¨μˆ˜ \\[ y = \\left( \\frac{2}{3} \\right)^{x+3} + 1, \\quad y = \\left( \\frac{2}{3} \\right)^{x+1} + \\frac{8}{3} \\] 의 κ·Έλž˜ν”„μ™€ λ§Œλ‚˜λŠ” 점을 각각 $( \\mathrm{P} )$, $( \\mathrm{Q} )$라 ν•˜μž. $\\overline{\\mathrm{PQ}} = \\sqrt{5}$일 λ•Œ, μƒμˆ˜ $k$의 값은? [4점] \\begin{itemize} \\item[1] \\frac{31}{6} \\item[2] \\frac{16}{3} \\item[3] \\frac{11}{2} \\item[4] \\frac{17}{3} \\item[5] \\frac{35}{6} \\end{itemize}", "answer": 4, "score": 4, "review": "Removed figure.", "incomplete": false}
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- {"id": 10, "name": "10", "problem": "10. μ‚Όμ°¨ν•¨μˆ˜ $( f(x) )$에 λŒ€ν•˜μ—¬ 곑선 $( y = f(x) )$ μœ„μ˜ 점 $( 0, 0 )$μ—μ„œμ˜ μ ‘μ„ κ³Ό 곑선 $( y = x f(x) )$ μœ„μ˜ 점 $( 1, 2 )$μ—μ„œμ˜ 접선이 μΌμΉ˜ν•  λ•Œ, $f'(2)$의 값은? [4점] \\begin{itemize} \\item[1] -18 \\item[2] -17 \\item[3] -16 \\item[4] -15 \\item[5] -14 \\end{itemize}", "answer": 5, "score": 4, "review": null, "incomplete": false}
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- {"id": 11, "name": "11", "problem": "11. μ–‘μˆ˜ $a$에 λŒ€ν•˜μ—¬ 집합 $\\left\\{ x \\ \\middle| \\ -\\frac{a}{2} < x \\leq a, \\ x \\neq \\frac{a}{2} \\right\\}$ μ—μ„œ μ •μ˜λœ ν•¨μˆ˜ \\[ f(x) = \\tan \\frac{\\pi x}{a} \\] κ°€ μžˆλ‹€. κ·Έλ¦Όκ³Ό 같이 ν•¨μˆ˜ $y = f(x)$의 κ·Έλž˜ν”„ μœ„μ˜ μ„Έ 점 $( \\mathrm{O, A, B} )$λ₯Ό μ§€λ‚˜λŠ” 직선이 μžˆλ‹€. 점 $( \\mathrm{A} )$λ₯Ό μ§€λ‚˜κ³  $x$좕에 ν‰ν–‰ν•œ 직선이 ν•¨μˆ˜ $y = f(x)$의 κ·Έλž˜ν”„μ™€ λ§Œλ‚˜λŠ” 점 쀑 $( \\mathrm{A} )$κ°€ μ•„λ‹Œ 점을 $( \\mathrm{C} )$라 ν•˜μž. μ‚Όκ°ν˜• $( \\mathrm{ABC} )$κ°€ μ •μ‚Όκ°ν˜•μΌ λ•Œ, μ‚Όκ°ν˜• $( \\mathrm{ABC} )$의 λ„“μ΄λŠ”? (단, $( \\mathrm{O} )$λŠ” 원점이닀.) [4점] \\begin{itemize} \\item[1] \\frac{3\\sqrt{3}}{2} \\item[2] \\frac{17\\sqrt{3}}{12} \\item[3] \\frac{4\\sqrt{3}}{3} \\item[4] \\frac{5\\sqrt{3}}{4} \\item[5] \\frac{7\\sqrt{3}}{6} \\end{itemize}", "answer": 3, "score": 4, "review": "Removed figure and the statement referring to the figure.", "incomplete": false}
12
- {"id": 12, "name": "12", "problem": "12. μ‹€μˆ˜ μ „μ²΄μ˜ μ§‘ν•©μ—μ„œ 연속인 ν•¨μˆ˜ $f(x)$κ°€ λͺ¨λ“  μ‹€μˆ˜ $x$에 λŒ€ν•˜μ—¬ \\[ \\{f(x)\\}^3 - \\{f(x)\\}^2 - x^2 f(x) + x^2 = 0 \\] 을 λ§Œμ‘±μ‹œν‚¨λ‹€. ν•¨μˆ˜ $f(x)$의 μ΅œλŒ“κ°’μ΄ 1이고 μ΅œμ†Ÿκ°’μ΄ 0일 λ•Œ, \\[ f\\left( -\\frac{4}{3} \\right) + f(0) + f\\left( \\frac{1}{2} \\right) \\] 의 값은? [4점] \\begin{itemize} \\item[1] \\frac{1}{2} \\item[2] 1 \\item[3] \\frac{3}{2} \\item[4] 2 \\item[5] \\frac{5}{2} \\end{itemize}", "answer": 3, "score": 4, "review": null, "incomplete": false}
13
- {"id": 13, "name": "13", "problem": "13. 두 μƒμˆ˜ $( a, b \\ (1 < a < b) )$에 λŒ€ν•˜μ—¬ μ’Œν‘œν‰λ©΄ μœ„μ˜ 두 점 $(a, \\log_2 a), \\ (b, \\log_2 b)$λ₯Ό μ§€λ‚˜λŠ” μ§μ„ μ˜ $y$절편과 두 점 $(a, \\log_4 a), \\ (b, \\log_4 b)$λ₯Ό μ§€λ‚˜λŠ” μ§μ„ μ˜ $y$절편이 κ°™λ‹€. ν•¨μˆ˜ $f(x) = a^{bx} + b^{ax}$에 λŒ€ν•˜μ—¬ $f(1) = 40$일 λ•Œ, $f(2)$의 값은? [4점] \\begin{itemize} \\item[1] 760 \\item[2] 800 \\item[3] 840 \\item[4] 880 \\item[5] 920 \\end{itemize}", "answer": 2, "score": 4, "review": null, "incomplete": false}
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- {"id": 14, "name": "14", "problem": "14. μˆ˜μ§μ„  μœ„λ₯Ό μ›€μ§μ΄λŠ” 점 $\\mathrm{P}$의 μ‹œκ° $t$μ—μ„œμ˜ μœ„μΉ˜ $x(t)$κ°€ 두 μƒμˆ˜ $a$, $b$에 λŒ€ν•˜μ—¬ \\[ x(t) = t(t - 1)(at + b) \\quad (a \\neq 0) \\] 이닀. 점 $\\mathrm{P}$의 μ‹œκ° $t$μ—μ„œμ˜ 속도 $v(t)$κ°€ $\\int_0^1 |v(t)| \\, dt = 2$λ₯Ό λ§Œμ‘±μ‹œν‚¬ λ•Œ, μ•„λž˜ γ„±, γ„΄, γ„· μ€‘μ—μ„œ μ˜³μ€ κ²ƒλ§Œμ„ μžˆλŠ” λŒ€λ‘œ κ³ λ₯Έ 것은? [4점]\n\n\\begin{itemize} \\item[γ„±.] $\\int_0^1 v(t) \\, dt = 0$ \\item[γ„΄.] $|x(t_1)| > 1$인 $t_1$이 열린ꡬ간 $(0, 1)$에 μ‘΄μž¬ν•œλ‹€. \\item[γ„·.] $0 \\leq t \\leq 1$인 λͺ¨λ“  $t$에 λŒ€ν•˜μ—¬ $|x(t)| < 1$이면 $x(t_2) = 0$인 $t_2$κ°€ 열린ꡬ간 $(0, 1)$에 μ‘΄μž¬ν•œλ‹€. \\end{itemize}\n\n\\begin{itemize} \\item[1] γ„± \\item[2] γ„±, γ„΄ \\item[3] γ„±, γ„· \\item[4] γ„΄, γ„· \\item[5] γ„±, γ„΄, γ„· \\end{itemize}", "answer": 3, "score": 4, "review": "<보기> changed to 'μ•„λž˜ γ„±,γ„΄,γ„·, 쀑'", "incomplete": false}
15
- {"id": 15, "name": "15", "problem": "15. 두 점 $( \\mathrm{O}_1, \\mathrm{O}_2 )$λ₯Ό 각각 μ€‘μ‹¬μœΌλ‘œ ν•˜κ³  λ°˜μ§€λ¦„μ˜ 길이가 $(\\overline{\\mathrm{O}_1\\mathrm{O}_2} )$인 두 원 $( C_1, C_2 )$κ°€ μžˆλ‹€. κ·Έλ¦Όκ³Ό 같이 원 $( C_1 )$ μœ„μ˜ μ„œλ‘œ λ‹€λ₯Έ μ„Έ 점 $( \\mathrm{A}, \\mathrm{B}, \\mathrm{C} )$와 원 $( C_2 )$ μœ„μ˜ 점 $( \\mathrm{D} )$κ°€ μ£Όμ–΄μ Έ 있고, μ„Έ 점 $( \\mathrm{A}, \\mathrm{O}_1, \\mathrm{O}_2 )$와 μ„Έ 점 $( \\mathrm{C}, \\mathrm{O}_2, \\mathrm{D} )$κ°€ 각각 ν•œ 직선 μœ„μ— μžˆλ‹€.\n\nμ΄λ•Œ $(\\angle \\mathrm{B}\\mathrm{O}_1\\mathrm{A} = \\theta_1)$, $(\\angle \\mathrm{O}_2\\mathrm{O}_1\\mathrm{C} = \\theta_2)$, $(\\angle \\mathrm{O}_1\\mathrm{O}_2\\mathrm{D} = \\theta_3)$이라 ν•˜μž.\n\nλ‹€μŒμ€ $( \\overline{\\mathrm{A}\\mathrm{B}} : \\overline{\\mathrm{O}_1\\mathrm{D}} = 1 : 2\\sqrt{2} )$이고 $( \\theta_3 = \\theta_1 + \\theta_2 )$일 λ•Œ, μ„ λΆ„ $( \\mathrm{A}\\mathrm{B} )$와 μ„ λΆ„ $( \\mathrm{C}\\mathrm{D} )$의 길이의 λΉ„λ₯Ό κ΅¬ν•˜λŠ” 과정이닀.\n\n\\[ \\begin{aligned} &\\angle \\mathrm{C}\\mathrm{O}_2\\mathrm{O}_1 + \\angle \\mathrm{O}_1\\mathrm{O}_2\\mathrm{D} = \\pi \\text{μ΄λ―€λ‘œ } \\theta_3 = \\frac{\\pi}{2} + \\frac{\\theta_2}{2} \\text{이고} \\\\ &\\theta_3 = \\theta_1 + \\theta_2 \\text{μ—μ„œ } 2\\theta_1 + \\theta_2 = \\pi \\text{μ΄λ―€λ‘œ } \\angle \\mathrm{C}\\mathrm{O}_1\\mathrm{B} = \\theta_1 \\text{이닀.} \\\\ &\\text{μ΄λ•Œ } \\angle \\mathrm{O}_2\\mathrm{O}_1\\mathrm{B} = \\theta_1 + \\theta_2 = \\theta_3 \\text{μ΄λ―€λ‘œ μ‚Όκ°ν˜• } \\mathrm{O}_1\\mathrm{O}_2\\mathrm{B} \\text{와 μ‚Όκ°ν˜• } \\mathrm{O}_2\\mathrm{O}_1\\mathrm{D} \\text{λŠ” 합동이닀.} \\\\ &\\overline{\\mathrm{A}\\mathrm{B}} = k \\text{라 ν•  λ•Œ} \\\\ &\\overline{\\mathrm{B}\\mathrm{O}_2} = \\overline{\\mathrm{O}_1\\mathrm{D}}= 2\\sqrt{2}k \\text{μ΄λ―€λ‘œ } \\overline{\\mathrm{A}\\mathrm{O}_2} = \\text{(κ°€)이고,} \\\\ &\\angle \\mathrm{B}\\mathrm{O}_2\\mathrm{A} = \\frac{\\theta_1}{2} \\text{μ΄λ―€λ‘œ } \\cos \\frac{\\theta_1}{2} = \\text{(λ‚˜) 이닀.} \\\\ &\\text{μ‚Όκ°ν˜• } \\mathrm{O}_2\\mathrm{B}\\mathrm{C} \\text{μ—μ„œ} \\\\ &\\overline{\\mathrm{B}\\mathrm{C}} = k, \\overline{\\mathrm{B}\\mathrm{O}_2} = 2\\sqrt{2}k, \\angle \\mathrm{C}\\mathrm{O}_2\\mathrm{B} = \\frac{\\theta_1}{2} \\text{μ΄λ―€λ‘œ} \\\\ &\\text{코사인법칙에 μ˜ν•˜μ—¬ } \\overline{\\mathrm{O}_2\\mathrm{C}} = \\text{(λ‹€) 이닀.} \\\\ &\\overline{\\mathrm{C}\\mathrm{D}} = \\overline{\\mathrm{O}_2\\mathrm{D}} + \\overline{\\mathrm{O}_2\\mathrm{C}} = \\overline{\\mathrm{O}_1\\mathrm{O}_2} + \\overline{\\mathrm{O}_2\\mathrm{C}} \\text{μ΄λ―€λ‘œ} \\\\ &\\overline{\\mathrm{A}\\mathrm{B}} : \\overline{\\mathrm{C}\\mathrm{D}} = k : \\left(\\frac{\\text{(κ°€)}}{2} + \\text{(λ‹€)}\\right) \\text{이닀.} \\end{aligned} \\]\n\nμœ„μ˜ (κ°€), (λ‹€)에 μ•Œλ§žμ€ 식을 각각 $( f(k), g(k) )$라 ν•˜κ³ , (λ‚˜)에 μ•Œλ§žμ€ 수λ₯Ό $( p )$라 ν•  λ•Œ, $( f(p) \\times g(p) )$의 값은? [4점]\n\n\\begin{itemize} \\item[1] \\frac{169}{27} \\item[2] \\frac{56}{9} \\item[3] \\frac{167}{27} \\item[4] \\frac{166}{27} \\item[5] \\frac{55}{9} \\end{itemize}", "answer": 2, "score": 4, "review": "Removed figure and the statement referring to the figure.", "incomplete": false}
16
- {"id": 16, "name": "16", "problem": "16. $\\log_2 120 - \\frac{1}{\\log_{15} 2}$ 의 값을 κ΅¬ν•˜μ‹œμ˜€. [3점]", "answer": 3, "score": 3, "review": null, "incomplete": false}
17
- {"id": 17, "name": "17", "problem": "17. ν•¨μˆ˜ $f(x)$에 λŒ€ν•˜μ—¬ $f'(x) = 3x^2 + 2x$이고 $f(0) = 2$일 λ•Œ, $f(1)$의 값을 κ΅¬ν•˜μ‹œμ˜€. [3점]", "answer": 4, "score": 3, "review": null, "incomplete": false}
18
- {"id": 18, "name": "18", "problem": "18. μˆ˜μ—΄ $\\{a_n\\}$에 λŒ€ν•˜μ—¬ \\[ \\sum_{k=1}^{10} a_k - \\sum_{k=1}^{7} \\frac{a_k}{2} = 56, \\quad \\sum_{k=1}^{10} 2a_k - \\sum_{k=1}^{8} a_k = 100 \\] 일 λ•Œ, $a_8$의 값을 κ΅¬ν•˜μ‹œμ˜€. [3점]", "answer": 12, "score": 3, "review": null, "incomplete": false}
19
- {"id": 19, "name": "19", "problem": "19. ν•¨μˆ˜ $f(x) = x^3 + ax^2 - (a^2 - 8a)x + 3$이 μ‹€μˆ˜ μ „μ²΄μ˜ μ§‘ν•©μ—μ„œ μ¦κ°€ν•˜λ„λ‘ ν•˜λŠ” μ‹€μˆ˜ $a$의 μ΅œλŒ“κ°’μ„ κ΅¬ν•˜μ‹œμ˜€. [3점]", "answer": 6, "score": 3, "review": null, "incomplete": false}
20
- {"id": 20, "name": "20", "problem": "20. μ‹€μˆ˜ μ „μ²΄μ˜ μ§‘ν•©μ—μ„œ λ―ΈλΆ„κ°€λŠ₯ν•œ ν•¨μˆ˜ $( f(x) )$κ°€ λ‹€μŒ 쑰건을 λ§Œμ‘±μ‹œν‚¨λ‹€.\n\n\\begin{itemize} \\item[(κ°€)] λ‹«νžŒκ΅¬κ°„ $[0, 1]$μ—μ„œ $f(x) = x$이닀. \\item[(λ‚˜)] μ–΄λ–€ μƒμˆ˜ $a, b$에 λŒ€ν•˜μ—¬ ꡬ간 $[0, \\infty)$μ—μ„œ $f(x+1) - x f(x) = ax + b$이닀. \\end{itemize}\n\n\\[ 60 \\times \\int_1^2 f(x) \\, dx \\] 의 값을 κ΅¬ν•˜μ‹œμ˜€. [4점]", "answer": 110, "score": 4, "review": null, "incomplete": false}
21
- {"id": 21, "name": "21", "problem": "21. μˆ˜μ—΄ $\\{a_n\\}$이 λ‹€μŒ 쑰건을 λ§Œμ‘±μ‹œν‚¨λ‹€.\n\n\\begin{itemize} \\item[(κ°€)] $( |a_1| = 2 )$ \\item[(λ‚˜)] λͺ¨λ“  μžμ—°μˆ˜ $( n )$에 λŒ€ν•˜μ—¬ $( |a_{n+1}| = 2|a_n| )$이닀. \\item[(λ‹€)] $\\sum_{n=1}^{10} a_n = -14$ \\end{itemize}\n\n$a_1 + a_3 + a_5 + a_7 + a_9$의 값을 κ΅¬ν•˜μ‹œμ˜€. [4점]", "answer": 678, "score": 4, "review": null, "incomplete": false}
22
- {"id": 22, "name": "22", "problem": "22. μ΅œκ³ μ°¨ν•­μ˜ κ³„μˆ˜κ°€ $\\frac{1}{2}$ 인 μ‚Όμ°¨ν•¨μˆ˜ $f(x)$와 μ‹€μˆ˜ $t$에 λŒ€ν•˜μ—¬ 방정식 $f'(x) = 0$이 λ‹«νžŒκ΅¬κ°„ $[t, t+2]$μ—μ„œ κ°–λŠ” μ‹€κ·Όμ˜ 개수λ₯Ό $g(t)$라 ν•  λ•Œ, ν•¨μˆ˜ $g(t)$λŠ” λ‹€μŒ 쑰건을 λ§Œμ‘±μ‹œν‚¨λ‹€.\n\n\\begin{itemize} \\item[(κ°€)] λͺ¨λ“  μ‹€μˆ˜ $( a )$에 λŒ€ν•˜μ—¬ $( \\lim_{t \\to a+} g(t) + \\lim_{t \\to a-} g(t) \\leq 2 )$이닀. \\item[(λ‚˜)] $( g(f(1)) = g(f(4)) = 2, \\ g(f(0)) = 1 )$ \\end{itemize}\n\n$f(5)$의 값을 κ΅¬ν•˜μ‹œμ˜€. [4점]", "answer": 9, "score": 4, "review": null, "incomplete": false}
23
- {"id": 23, "name": "23_prob", "problem": "23. 닀항식 $(x+2)^7$의 μ „κ°œμ‹μ—μ„œ $x^5$의 κ³„μˆ˜λŠ”? [2점] \\begin{itemize} \\item[1] 42 \\item[2] 56 \\item[3] 70 \\item[4] 84 \\item[5] 98 \\end{itemize}", "answer": 4, "score": 2, "review": null, "incomplete": false}
24
- {"id": 24, "name": "24_prob", "problem": "24. ν™•λ₯ λ³€μˆ˜ $X$κ°€ 이항뢄포 $\\mathrm{B}\\left(n, \\frac{1}{3}\\right)$을 λ”°λ₯΄κ³  $\\mathrm{V}(2X) = 40$일 λ•Œ, $n$의 값은? [3점] \\begin{itemize} \\item[1] 30 \\item[2] 35 \\item[3] 40 \\item[4] 45 \\item[5] 50 \\end{itemize}", "answer": 4, "score": 3, "review": null, "incomplete": false}
25
- {"id": 25, "name": "25_prob", "problem": "25. λ‹€μŒ 쑰건을 λ§Œμ‘±μ‹œν‚€λŠ” μžμ—°μˆ˜ $a, \\ b, \\ c, \\ d, \\ e$의 λͺ¨λ“  μˆœμ„œμŒ $(a, b, c, d, e)$의 κ°œμˆ˜λŠ”? [3점]\n\n\\begin{itemize} \\item[(κ°€)] $a + b + c + d + e = 12$ \\item[(λ‚˜)] $\\left| a^2 - b^2 \\right| = 5$ \\end{itemize}\n\n\\begin{itemize} \\item[1] 30 \\item[2] 32 \\item[3] 34 \\item[4] 36 \\item[5] 38 \\end{itemize}", "answer": 1, "score": 3, "review": null, "incomplete": false}
26
- {"id": 26, "name": "26_prob", "problem": "26. $( 1 )$λΆ€ν„° $( 10 )$κΉŒμ§€ μžμ—°μˆ˜κ°€ ν•˜λ‚˜μ”© μ ν˜€ μžˆλŠ” $( 10 )$μž₯의 μΉ΄λ“œκ°€ λ“€μ–΄ μžˆλŠ” μ£Όλ¨Έλ‹ˆκ°€ μžˆλ‹€. 이 μ£Όλ¨Έλ‹ˆμ—μ„œ μž„μ˜λ‘œ μΉ΄λ“œ $( 3 )$μž₯을 λ™μ‹œμ— κΊΌλ‚Ό λ•Œ, κΊΌλ‚Έ μΉ΄λ“œμ— μ ν˜€ μžˆλŠ” μ„Έ μžμ—°μˆ˜ μ€‘μ—μ„œ κ°€μž₯ μž‘μ€ μˆ˜κ°€ $( 4 )$ μ΄ν•˜μ΄κ±°λ‚˜ $( 7 )$ 이상일 ν™•λ₯ μ€? [3점]\n\n\\begin{itemize} \\item[1] \\frac{4}{5} \\item[2] \\frac{5}{6} \\item[3] \\frac{13}{15} \\item[4] \\frac{9}{10} \\item[5] \\frac{14}{15} \\end{itemize}", "answer": 3, "score": 3, "review": "Removed figure.", "incomplete": false}
27
- {"id": 27, "name": "27_prob", "problem": "27. μ–΄λŠ μžλ™μ°¨ νšŒμ‚¬μ—μ„œ μƒμ‚°ν•˜λŠ” μ „κΈ° μžλ™μ°¨μ˜ 1회 μΆ©μ „ μ£Όν–‰ κ±°λ¦¬λŠ” 평균이 $m$이고 ν‘œμ€€νŽΈμ°¨κ°€ $\\sigma$인 μ •κ·œλΆ„ν¬λ₯Ό λ”°λ₯Έλ‹€κ³  ν•œλ‹€.\n\n이 μžλ™μ°¨ νšŒμ‚¬μ—μ„œ μƒμ‚°ν•œ μ „κΈ° μžλ™μ°¨ 100λŒ€λ₯Ό μž„μ˜μΆ”μΆœν•˜μ—¬ 얻은 1회 μΆ©μ „ μ£Όν–‰ 거리의 ν‘œλ³Έν‰κ· μ΄ $\\overline{x_1}$일 λ•Œ, λͺ¨ν‰κ·  $m$에 λŒ€ν•œ 신뒰도 95\\%의 신뒰ꡬ간이 $a \\le m \\le b$이닀.\n\n이 μžλ™μ°¨ νšŒμ‚¬μ—μ„œ μƒμ‚°ν•œ μ „κΈ° μžλ™μ°¨ 400λŒ€λ₯Ό μž„μ˜μΆ”μΆœν•˜μ—¬ 얻은 1회 μΆ©μ „ μ£Όν–‰ 거리의 ν‘œλ³Έν‰κ· μ΄ $\\overline{x_2}$일 λ•Œ, λͺ¨ν‰κ·  $m$에 λŒ€ν•œ 신뒰도 99\\%의 신뒰ꡬ간이 $c \\le m \\le d$이닀.\n\n$\\overline{x_1} - \\overline{x_2} = 1.34$이고 $a = c$일 λ•Œ, $b - a$의 값은? (단, μ£Όν–‰ 거리의 λ‹¨μœ„λŠ” km이고, $Z$κ°€ ν‘œμ€€μ •κ·œλΆ„ν¬λ₯Ό λ”°λ₯΄λŠ” ν™•λ₯ λ³€μˆ˜μΌ λ•Œ $\\mathrm{P}(|Z| \\le 1.96) = 0.95$, $\\mathrm{P}(|Z| \\le 2.58) = 0.99$둜 κ³„μ‚°ν•œλ‹€.) [3점]\n\n\\begin{itemize} \\item[1] 5.88 \\item[2] 7.84 \\item[3] 9.80 \\item[4] 11.76 \\item[5] 13.72 \\end{itemize}", "answer": 2, "score": 3, "review": null, "incomplete": false}
28
- {"id": 28, "name": "28_prob", "problem": "28. 두 집합 $X = \\{1, 2, 3, 4, 5\\}$, $Y = \\{1, 2, 3, 4\\}$에 λŒ€ν•˜μ—¬ λ‹€μŒ 쑰건을 λ§Œμ‘±μ‹œν‚€λŠ” $X$μ—μ„œ $Y$둜의 ν•¨μˆ˜ $f$의 κ°œμˆ˜λŠ”? [4점]\n\n\\begin{itemize} \\item[(κ°€)] 집합 $X$의 λͺ¨λ“  μ›μ†Œ $x$에 λŒ€ν•˜μ—¬ $f(x) \\geq \\sqrt{x}$이닀. \\item[(λ‚˜)] ν•¨μˆ˜ $f$의 μΉ˜μ—­μ˜ μ›μ†Œμ˜ κ°œμˆ˜λŠ” 3이닀. \\end{itemize}\n\n\\begin{itemize} \\item[1] 128 \\item[2] 138 \\item[3] 148 \\item[4] 158 \\item[5] 168 \\end{itemize}", "answer": 1, "score": 4, "review": null, "incomplete": false}
29
- {"id": 29, "name": "29_prob", "problem": "29. 두 연속확λ₯ λ³€μˆ˜ $( X )$와 $( Y )$κ°€ κ°–λŠ” κ°’μ˜ λ²”μœ„λŠ” $( 0 \\leq X \\leq 6 )$, $( 0 \\leq Y \\leq 6 )$이고, $( X )$와 $( Y )$의 ν™•λ₯ λ°€λ„ν•¨μˆ˜λŠ” 각각 $( f(x), g(x) )$이닀. ν™•λ₯ λ³€μˆ˜ $( X )$의 ν™•λ₯ λ°€λ„ν•¨μˆ˜ $( f(x) )$의 κ·Έλž˜ν”„λŠ” κ·Έλ¦Όκ³Ό κ°™λ‹€.\n\n\\[ 0 \\leq x \\leq 6\\ \\text{인 λͺ¨λ“  } x \\text{에 λŒ€ν•˜μ—¬} \\]\n\\[ f(x) + g(x) = k \\quad (k \\text{λŠ” μƒμˆ˜}) \\]\nλ₯Ό λ§Œμ‘±μ‹œν‚¬ λ•Œ, $( \\mathrm{P}(6k \\leq Y \\leq 15k) = \\frac{q}{p} )$이닀. $( p + q )$의 값을 κ΅¬ν•˜μ‹œμ˜€. (단, $( p )$와 $( q )$λŠ” μ„œλ‘œμ†ŒμΈ μžμ—°μˆ˜μ΄λ‹€.) [4점]", "answer": 31, "score": 4, "review": "Removed figure and the statement referring to the figure. The figure is needed to solve the problem.", "incomplete": true}
30
- {"id": 30, "name": "30_prob", "problem": "30. 흰 곡과 검은 곡이 각각 10개 이상 λ“€μ–΄ μžˆλŠ” λ°”κ΅¬λ‹ˆμ™€ λΉ„μ–΄ μžˆλŠ” μ£Όλ¨Έλ‹ˆκ°€ μžˆλ‹€. ν•œ 개의 μ£Όμ‚¬μœ„λ₯Ό μ‚¬μš©ν•˜μ—¬ λ‹€μŒ μ‹œν–‰μ„ ν•œλ‹€.\n\n\\[ \\begin{array}{|c|} \\hline \\text{μ£Όμ‚¬μœ„λ₯Ό ν•œ 번 던져} \\\\ \\text{λ‚˜μ˜¨ 눈의 μˆ˜κ°€ 5 이상이면} \\\\ \\text{λ°”κ΅¬λ‹ˆμ— μžˆλŠ” 흰 곡 2개λ₯Ό μ£Όλ¨Έλ‹ˆμ— λ„£κ³ ,} \\\\ \\text{λ‚˜μ˜¨ 눈의 μˆ˜κ°€ 4 μ΄ν•˜μ΄λ©΄} \\\\ \\text{λ°”κ΅¬λ‹ˆμ— μžˆλŠ” 검은 곡 1개λ₯Ό μ£Όλ¨Έλ‹ˆμ— λ„£λŠ”λ‹€.} \\\\ \\hline \\end{array} \\]\n\nμœ„μ˜ μ‹œν–‰μ„ 5번 λ°˜λ³΅ν•  λ•Œ, $( n(1 \\leq n \\leq 5) )$번째 μ‹œν–‰ ν›„ μ£Όλ¨Έλ‹ˆμ— λ“€μ–΄ μžˆλŠ” 흰 곡과 검은 곡의 개수λ₯Ό 각각 $( a_n )$, $( b_n )$이라 ν•˜μž. $( a_5 + b_5 \\geq 7 )$일 λ•Œ, $( a_k = b_k )$인 μžμ—°μˆ˜ $( k(1 \\leq k \\leq 5) )$κ°€ μ‘΄μž¬ν•  ν™•λ₯ μ„ $( \\frac{q}{p} )$이닀. $( p + q )$의 값을 κ΅¬ν•˜μ‹œμ˜€. (단, $(p)$와 $(q)$λŠ” μ„œλ‘œμ†ŒμΈ μžμ—°μˆ˜μ΄λ‹€.) [4점]", "answer": 191, "score": 4, "review": null, "incomplete": false}
31
- {"id": 31, "name": "23_calc", "problem": "23. \\[ \\lim_{n \\to \\infty} \\frac{\\frac{5}{n} + \\frac{3}{n^2}}{\\frac{1}{n} - \\frac{2}{n^3}} \\text{의 값은? [2점]} \\] \\begin{itemize} \\item[1] 1 \\item[2] 2 \\item[3] 3 \\item[4] 4 \\item[5] 5 \\end{itemize}", "answer": 5, "score": 2, "review": null, "incomplete": false}
32
- {"id": 32, "name": "24_calc", "problem": "24. μ‹€μˆ˜ μ „μ²΄μ˜ μ§‘ν•©μ—μ„œ λ―ΈλΆ„κ°€λŠ₯ν•œ ν•¨μˆ˜ $f(x)$κ°€ λͺ¨λ“  μ‹€μˆ˜ $x$에 λŒ€ν•˜μ—¬ \\[ f(x^3 + x) = e^x \\] 을 λ§Œμ‘±μ‹œν‚¬ λ•Œ, $f'(2)$의 값은? [3점] \\begin{itemize} \\item[1] e \\item[2] \\frac{e}{2} \\item[3] \\frac{e}{3} \\item[4] \\frac{e}{4} \\item[5] \\frac{e}{5} \\end{itemize}", "answer": 4, "score": 3, "review": null, "incomplete": false}
33
- {"id": 33, "name": "25_calc", "problem": "25. λ“±λΉ„μˆ˜μ—΄ $\\{a_n\\}$에 λŒ€ν•˜μ—¬ \\[ \\sum_{n=1}^{\\infty} (a_{2n-1} - a_{2n}) = 3, \\quad \\sum_{n=1}^{\\infty} a_n^2 = 6 \\] 일 λ•Œ, $\\sum_{n=1}^{\\infty} a_n$ 의 값은? [3점] \\begin{itemize} \\item[1] 1 \\item[2] 2 \\item[3] 3 \\item[4] 4 \\item[5] 5 \\end{itemize}", "answer": 2, "score": 3, "review": null, "incomplete": false}
34
- {"id": 34, "name": "26_calc", "problem": "26. \\[ \\lim_{n \\to \\infty} \\sum_{k=1}^{n} \\frac{k^2 + 2kn}{k^3 + 3k^2 n + n^3} \\text{의 값은?} \\quad [3 \\text{점}] \\] \\begin{itemize} \\item[1] \\ln 5 \\item[2] \\frac{\\ln 5}{2} \\item[3] \\frac{\\ln 5}{3} \\item[4] \\frac{\\ln 5}{4} \\item[5] \\frac{\\ln 5}{5} \\end{itemize}", "answer": 3, "score": 3, "review": null, "incomplete": false}
35
- {"id": 35, "name": "27_calc", "problem": "27. μ’Œν‘œν‰λ©΄ μœ„λ₯Ό μ›€μ§μ΄λŠ” 점 $\\mathrm{P}$의 μ‹œκ° $t \\ (t>0)$μ—μ„œμ˜ μœ„μΉ˜κ°€ 곑선 $y = x^2$κ³Ό 직선 $y = t^2 x - \\frac{\\ln t}{8}$κ°€ λ§Œλ‚˜λŠ” μ„œλ‘œ λ‹€λ₯Έ 두 점의 쀑점일 λ•Œ, μ‹œκ° $t=1$μ—μ„œ $t=e$κΉŒμ§€ 점 $\\mathrm{P}$κ°€ 움직인 κ±°λ¦¬λŠ”? [3점] \\begin{itemize} \\item[1] \\frac{e^4}{2} - \\frac{3}{8} \\item[2] \\frac{e^4}{2} - \\frac{5}{16} \\item[3] \\frac{e^4}{2} - \\frac{1}{4} \\item[4] \\frac{e^4}{2} - \\frac{3}{16} \\item[5] \\frac{e^4}{2} - \\frac{1}{8} \\end{itemize}", "answer": 1, "score": 3, "review": null, "incomplete": false}
36
- {"id": 36, "name": "28_calc", "problem": "28. ν•¨μˆ˜ $( f(x) = 6\\pi (x - 1)^2 )$에 λŒ€ν•˜μ—¬ ν•¨μˆ˜ $( g(x) )$λ₯Ό \\[ g(x) = 3f(x) + 4\\cos f(x) \\] 라 ν•˜μž. $( 0 < x < 2 )$μ—μ„œ ν•¨μˆ˜ $( g(x) )$κ°€ κ·Ήμ†Œκ°€ λ˜λŠ” $( x )$의 κ°œμˆ˜λŠ”? [4점] \\begin{itemize} \\item[1] 6 \\item[2] 7 \\item[3] 8 \\item[4] 9 \\item[5] 10 \\end{itemize}", "answer": 2, "score": 4, "review": null, "incomplete": false}
37
- {"id": 37, "name": "29_calc", "problem": "29. κ·Έλ¦Όκ³Ό 같이 길이가 2인 μ„ λΆ„ $(\\mathrm{AB})$λ₯Ό μ§€λ¦„μœΌλ‘œ ν•˜λŠ” λ°˜μ›μ΄ μžˆλ‹€. 호 $(\\mathrm{AB})$ μœ„μ— 두 점 $(\\mathrm{P})$, $(\\mathrm{Q})$λ₯Ό $(\\angle \\mathrm{PAB} = \\theta)$, $(\\angle \\mathrm{QBA} = 2\\theta)$κ°€ λ˜λ„λ‘ 작고, 두 μ„ λΆ„ $(\\mathrm{AP})$, $(\\mathrm{BQ})$의 ꡐ점을 $(\\mathrm{R})$라 ν•˜μž. μ„ λΆ„ $(\\mathrm{AB})$ μœ„μ˜ 점 $(\\mathrm{S})$, μ„ λΆ„ $(\\mathrm{BR})$ μœ„μ˜ 점 $(\\mathrm{T})$, μ„ λΆ„ $(\\mathrm{AR})$ μœ„μ˜ 점 $(\\mathrm{U})$λ₯Ό μ„ λΆ„ $(\\mathrm{UT})$κ°€ μ„ λΆ„ $(\\mathrm{AB})$에 ν‰ν–‰ν•˜κ³  μ‚Όκ°ν˜• $(\\mathrm{STU})$κ°€ μ •μ‚Όκ°ν˜•μ΄ λ˜λ„λ‘ μž‘λŠ”λ‹€. 두 μ„ λΆ„ $(\\mathrm{AR})$, $(\\mathrm{QR})$와 호 $(\\mathrm{AQ})$둜 λ‘˜λŸ¬μ‹ΈμΈ λΆ€λΆ„μ˜ 넓이λ₯Ό $(f(\\theta))$, μ‚Όκ°ν˜• $(\\mathrm{STU})$의 넓이λ₯Ό $(g(\\theta))$라 ν•  λ•Œ,\n\\[ \\lim_{\\theta \\to 0+} \\frac{g(\\theta)}{\\theta \\times f(\\theta)} = \\frac{q}{p} \\sqrt{3} \\]\n이닀. $(p + q)$의 값을 κ΅¬ν•˜μ‹œμ˜€.\n\n(단, $(0 < \\theta < \\frac{\\pi}{6})$이고, $(p)$와 $(q)$λŠ” μ„œλ‘œμ†ŒμΈ μžμ—°μˆ˜μ΄λ‹€.) [4점]", "answer": 11, "score": 4, "review": "Removed figure and the statement referring to the figure.", "incomplete": false}
38
- {"id": 38, "name": "30_calc", "problem": "30. μ‹€μˆ˜ μ „μ²΄μ˜ μ§‘ν•©μ—μ„œ μ¦κ°€ν•˜κ³  λ―ΈλΆ„κ°€λŠ₯ν•œ ν•¨μˆ˜ $f(x)$κ°€ λ‹€μŒ 쑰건을 λ§Œμ‘±μ‹œν‚¨λ‹€.\n\n\\begin{itemize} \\item[(κ°€)] $f(1) = 1$, \\quad $\\int_{1}^{2} f(x) \\, dx = \\frac{5}{4}$ \\item[(λ‚˜)] ν•¨μˆ˜ $f(x)$의 μ—­ν•¨μˆ˜λ₯Ό $g(x)$라 ν•  λ•Œ, $x \\geq 1$인 λͺ¨λ“  μ‹€μˆ˜ $x$에 λŒ€ν•˜μ—¬ $g(2x) = 2f(x)$이닀. \\end{itemize}\n\n\\[ \\int_{1}^{8} x f'(x) \\, dx = \\frac{q}{p} \\text{일 λ•Œ, } p+q \\text{의 값을 κ΅¬ν•˜μ‹œμ˜€.} \\]\n(단, $p$와 $q$λŠ” μ„œλ‘œμ†ŒμΈ μžμ—°μˆ˜μ΄λ‹€.) [4점]", "answer": 143, "score": 4, "review": null, "incomplete": false}
39
- {"id": 39, "name": "23_geom", "problem": "23. μ’Œν‘œκ³΅κ°„μ˜ 점 $\\mathrm{A}(2, 1, 3)$을 $xy$ 평면에 λŒ€ν•˜μ—¬ λŒ€μΉ­μ΄λ™ν•œ 점을 $\\mathrm{P}$라 ν•˜κ³ , 점 $\\mathrm{A}$λ₯Ό $yz$ 평면에 λŒ€ν•˜μ—¬ λŒ€μΉ­μ΄λ™ν•œ 점을 $\\mathrm{Q}$라 ν•  λ•Œ, μ„ λΆ„ $\\mathrm{PQ}$의 κΈΈμ΄λŠ”? [2점]\n\n\\begin{itemize} \\item[1] 5 \\sqrt{2} \\item[2] 2 \\sqrt{13} \\item[3] 3 \\sqrt{6} \\item[4] 2 \\sqrt{14} \\item[5] 2 \\sqrt{15} \\end{itemize}", "answer": 2, "score": 2, "review": null, "incomplete": false}
40
- {"id": 40, "name": "24_geom", "problem": "24. ν•œ 초점의 μ’Œν‘œκ°€ $\\left( 3\\sqrt{2}, 0 \\right)$ 인 μŒκ³‘μ„  $\\frac{x^2}{a^2} - \\frac{y^2}{6} = 1$ 의 μ£ΌμΆ•μ˜ κΈΈμ΄λŠ”? (단, $a$ λŠ” μ–‘μˆ˜μ΄λ‹€.) [3점]\n\n\\begin{itemize} \\item[1] 3\\sqrt{3} \\item[2] \\frac{7\\sqrt{3}}{2} \\item[3] 4\\sqrt{3} \\item[4] \\frac{9\\sqrt{3}}{2} \\item[5] 5\\sqrt{3} \\end{itemize}", "answer": 3, "score": 3, "review": null, "incomplete": false}
41
- {"id": 41, "name": "25_geom", "problem": "25. μ’Œν‘œν‰λ©΄μ—μ„œ 두 직선 \\[ \\frac{x+1}{2} = y - 3, \\quad x - 2 = \\frac{y - 5}{3} \\] κ°€ μ΄λ£¨λŠ” 예각의 크기λ₯Ό $\\theta$라 ν•  λ•Œ, $\\cos \\theta$의 값은? [3점]\n\n\\begin{itemize} \\item[1] \\frac{1}{2} \\item[2] \\frac{\\sqrt{5}}{4} \\item[3] \\frac{\\sqrt{6}}{4} \\item[4] \\frac{\\sqrt{7}}{4} \\item[5] \\frac{\\sqrt{2}}{2} \\end{itemize}", "answer": 5, "score": 3, "review": null, "incomplete": false}
42
- {"id": 42, "name": "26_geom", "problem": "26. 두 초점이 $( \\mathrm{F}, \\mathrm{F'} )$인 타원 $\\frac{x^2}{64} + \\frac{y^2}{16} = 1$ μœ„μ˜ 점 쀑 제1사뢄면에 μžˆλŠ” 점 $( \\mathrm{A} )$κ°€ μžˆλ‹€. 두 직선 $( \\mathrm{AF}, \\mathrm{AF'} )$에 λ™μ‹œμ— μ ‘ν•˜κ³  쀑심이 $y$μΆ• μœ„μ— μžˆλŠ” 원 쀑 μ€‘μ‹¬μ˜ $y$μ’Œν‘œκ°€ 음수인 것을 $( C )$라 ν•˜μž. 원 $( C )$의 쀑심을 $( \\mathrm{B} )$라 ν•  λ•Œ μ‚¬κ°ν˜• $( \\mathrm{AFBF'} )$의 넓이가 72이닀. 원 $( C )$의 λ°˜μ§€λ¦„μ˜ κΈΈμ΄λŠ”? [3점]\n\n\\begin{itemize} \\item[1] \\frac{17}{2} \\item[2] 9 \\item[3] \\frac{19}{2} \\item[4] 10 \\item[5] \\frac{21}{2} \\end{itemize}", "answer": 2, "score": 3, "review": "Removed figure.", "incomplete": false}
43
- {"id": 43, "name": "27_geom", "problem": "27. κ·Έλ¦Όκ³Ό 같이 ν•œ λͺ¨μ„œλ¦¬μ˜ 길이가 4인 μ •μœ‘λ©΄μ²΄ $\\mathrm{ABCD - EFGH}$ κ°€ μžˆλ‹€. μ„ λΆ„ $\\mathrm{AD}$ 의 쀑점을 $\\mathrm{M}$이라 ν•  λ•Œ, μ‚Όκ°ν˜• $\\mathrm{MEG}$ 의 λ„“μ΄λŠ”? [3점]\n\n\\begin{itemize} \\item[1] \\frac{21}{2} \\item[2] 11 \\item[3] \\frac{23}{2} \\item[4] 12 \\item[5] \\frac{25}{2} \\end{itemize}", "answer": 4, "score": 3, "review": "Removed figure and the statement referring to the figure.", "incomplete": false}
44
- {"id": 44, "name": "28_geom", "problem": "28. 두 μ–‘μˆ˜ $( a )$, $( p )$에 λŒ€ν•˜μ—¬ 포물선 $( (y - a)^2 = 4px )$의 μ΄ˆμ μ„ $( \\mathrm{F}_1 )$이라 ν•˜κ³ , 포물선 $( y^2 = -4x )$의 μ΄ˆμ μ„ $( \\mathrm{F}_2 )$라 ν•˜μž. μ„ λΆ„ $( \\mathrm{F}_1 \\mathrm{F}_2 )$κ°€ 두 포물선과 λ§Œλ‚˜λŠ” 점을 각각 $( \\mathrm{P} )$, $( \\mathrm{Q} )$라 ν•  λ•Œ, $( \\overline{\\mathrm{F}_1 \\mathrm{F}_2} = 3 )$, $( \\overline{\\mathrm{P}\\mathrm{Q}} = 1 )$이닀. $( a^2 + p^2 )$의 값은? [4점]\n\n\\begin{itemize} \\item[1] 6 \\item[2] \\frac{25}{4} \\item[3] \\frac{13}{2} \\item[4] \\frac{27}{4} \\item[5] 7 \\end{itemize}", "answer": 5, "score": 4, "review": "Removed figure.", "incomplete": false}
45
- {"id": 45, "name": "29_geom", "problem": "29. μ’Œν‘œν‰λ©΄μ—μ„œ $\\overline{\\mathrm{OA}} = \\sqrt{2}$, $\\overline{\\mathrm{OB}} = 2\\sqrt{2}$이고\n\\[ \\cos(\\angle \\mathrm{AOB}) = \\frac{1}{4} \\]\n인 ν‰ν–‰μ‚¬λ³€ν˜• $\\mathrm{OACB}$에 λŒ€ν•˜μ—¬ 점 $\\mathrm{P}$κ°€ λ‹€μŒ 쑰건을 λ§Œμ‘±μ‹œν‚¨λ‹€.\n\n\\begin{itemize} \\item[(κ°€)] $\\overrightarrow{\\mathrm{OP}} = s \\overrightarrow{\\mathrm{OA}} + t \\overrightarrow{\\mathrm{OB}} \\quad (0 \\leq s \\leq 1, \\ 0 \\leq t \\leq 1)$ \\item[(λ‚˜)] $\\overrightarrow{\\mathrm{OP}} \\cdot \\overrightarrow{\\mathrm{OB}} + \\overrightarrow{\\mathrm{BP}} \\cdot \\overrightarrow{\\mathrm{BC}} = 2$ \\end{itemize}\n\n점 $\\mathrm{O}$λ₯Ό μ€‘μ‹¬μœΌλ‘œ ν•˜κ³  점 $\\mathrm{A}$λ₯Ό μ§€λ‚˜λŠ” 원 μœ„λ₯Ό μ›€μ§μ΄λŠ” 점 $\\mathrm{X}$에 λŒ€ν•˜μ—¬ $|3\\overrightarrow{\\mathrm{OP}} - \\overrightarrow{\\mathrm{OX}}|$의 μ΅œλŒ“κ°’κ³Ό μ΅œμ†Ÿκ°’μ„ 각각 $M$, $m$이라 ν•˜μž. $M \\times m = a\\sqrt{6} + b$일 λ•Œ, $a^2 + b^2$의 값을 κ΅¬ν•˜μ‹œμ˜€. (단, $a$와 $b$λŠ” μœ λ¦¬μˆ˜μ΄λ‹€.) [4점]", "answer": 100, "score": 4, "review": "Removed figure.", "incomplete": false}
46
- {"id": 46, "name": "30_geom", "problem": "30. μ’Œν‘œκ³΅κ°„μ— 쀑심이 $\\mathrm{C}(2, \\sqrt{5}, 5)$이고 점 $\\mathrm{P}(0, 0, 1)$을 μ§€λ‚˜λŠ” ꡬ \\[ S: (x - 2)^2 + (y - \\sqrt{5})^2 + (z - 5)^2 = 25 \\] κ°€ μžˆλ‹€. ꡬ $S$κ°€ 평면 $\\mathrm{OPC}$와 λ§Œλ‚˜μ„œ μƒκΈ°λŠ” 원 μœ„λ₯Ό μ›€μ§μ΄λŠ” 점 $\\mathrm{Q}$, ꡬ $S$ μœ„λ₯Ό μ›€μ§μ΄λŠ” 점 $\\mathrm{R}$에 λŒ€ν•˜μ—¬ 두 점 $\\mathrm{Q}, \\mathrm{R}$의 $xy$평면 μœ„λ‘œμ˜ μ •μ‚¬μ˜μ„ 각각 $\\mathrm{Q}_1, \\mathrm{R}_1$이라 ν•˜μž.\n\nμ‚Όκ°ν˜• $\\mathrm{O}\\mathrm{Q}_1\\mathrm{R}_1$의 넓이가 μ΅œλŒ€κ°€ λ˜λ„λ‘ ν•˜λŠ” 두 점 $\\mathrm{Q}, \\mathrm{R}$에 λŒ€ν•˜μ—¬ μ‚Όκ°ν˜• $\\mathrm{O}\\mathrm{Q}_1\\mathrm{R}_1$의 평면 $\\mathrm{PQR}$ μœ„λ‘œμ˜ μ •μ‚¬μ˜μ˜ λ„“μ΄λŠ” $\\frac{q}{p} \\sqrt{6}$이닀. $p+q$의 값을 κ΅¬ν•˜μ‹œμ˜€.\n\n(단, $\\mathrm{O}$λŠ” 원점이고 μ„Έ 점 $\\mathrm{O}, \\mathrm{Q}_1, \\mathrm{R}_1$은 ν•œ 직선 μœ„μ— μžˆμ§€ μ•ŠμœΌλ©°, $p$와 $q$λŠ” μ„œλ‘œμ†ŒμΈ μžμ—°μˆ˜μ΄λ‹€.) [4점]", "answer": 23, "score": 4, "review": "Removed figure.", "incomplete": false}
 
1
+ {"id":1,"name":"1","problem":"1. $\\left(2^{\\sqrt{3}} \\times 4\\right)^{\\sqrt{3} - 2}$ 의 값은? [2점] \\begin{itemize} \\item[1] \\frac{1}{4} \\item[2] \\frac{1}{2} \\item[3] 1 \\item[4] 2 \\item[5] 4 \\end{itemize}","answer":2,"score":2,"review":null}
2
+ {"id":2,"name":"2","problem":"2. ν•¨μˆ˜ $f(x) = x^3 + 3x^2 + x - 1$ 에 λŒ€ν•˜μ—¬ $f'(1)$의 값은? [2점] \\begin{itemize} \\item[1] 6 \\item[2] 7 \\item[3] 8 \\item[4] 9 \\item[5] 10 \\end{itemize}","answer":5,"score":2,"review":null}
3
+ {"id":3,"name":"3","problem":"3. λ“±μ°¨μˆ˜μ—΄ $\\{a_n\\}$에 λŒ€ν•˜μ—¬ \\[ a_2 = 6, \\quad a_4 + a_6 = 36 \\] 일 λ•Œ, $a_{10}$의 값은? [3점] \\begin{itemize} \\item[1] 30 \\item[2] 32 \\item[3] 34 \\item[4] 36 \\item[5] 38 \\end{itemize}","answer":5,"score":3,"review":null}
4
+ {"id":4,"name":"4","problem":"4. ν•¨μˆ˜ $( y = f(x) )$κ°€ λ‹€μŒκ³Ό 같이 μ •μ˜λ˜μ–΄ μžˆλ‹€.\n\n\\[\nf(x) =\n\\begin{cases}\n-x+2, & x < -1, \\\\\n2, & x = -1, \\\\\n(3*x+3)/2, & -1 < x < 1, \\\\\n1, & 1 \\leq x < 2, \\\\\n3, & x = 2, \\\\\n1, & x \\geq 2.\n\\end{cases}\n\\]\n\n\\[ \\lim_{x \\to -1-} f(x) + \\lim_{x \\to 2} f(x) \\text{의 값은? [3점]} \\]\n\n\\begin{itemize} \\item[1] 1 \\item[2] 2 \\item[3] 3 \\item[4] 4 \\item[5] 5 \\end{itemize}","answer":4,"score":3,"review":"Removed figure and the statement referring to the figure. The figure is needed to solve the problem, so we paraphrased the figure into text."}
5
+ {"id":5,"name":"5","problem":"5. 첫째항이 1인 μˆ˜μ—΄ $\\{a_n\\}$이 λͺ¨λ“  μžμ—°μˆ˜ $n$에 λŒ€ν•˜μ—¬ \\[ a_{n+1} = \\begin{cases} 2a_n & (a_n < 7) \\\\ a_n - 7 & (a_n \\geq 7) \\end{cases} \\] 일 λ•Œ, $\\sum_{k=1}^{8} a_k$의 값은? [3점] \\begin{itemize} \\item[1] 30 \\item[2] 32 \\item[3] 34 \\item[4] 36 \\item[5] 38 \\end{itemize}","answer":1,"score":3,"review":null}
6
+ {"id":6,"name":"6","problem":"6. 방정식 $( 2x^3 - 3x^2 - 12x + k = 0 )$이 μ„œλ‘œ λ‹€λ₯Έ μ„Έ 싀근을 갖도둝 ν•˜λŠ” μ •μˆ˜ $k$의 κ°œμˆ˜λŠ”? [3점] \\begin{itemize} \\item[1] 20 \\item[2] 23 \\item[3] 26 \\item[4] 29 \\item[5] 32 \\end{itemize}","answer":3,"score":3,"review":null}
7
+ {"id":7,"name":"7","problem":"7. $( \\pi < \\theta < \\frac{3}{2}\\pi )$인 $\\theta$에 λŒ€ν•˜μ—¬ $\\tan \\theta - \\frac{6}{\\tan \\theta} = 1$일 λ•Œ, $ \\sin \\theta + \\cos \\theta $의 값은? [3점] \\begin{itemize} \\item[1] -\\frac{2\\sqrt{10}}{5} \\item[2] -\\frac{\\sqrt{10}}{5} \\item[3] 0 \\item[4] \\frac{\\sqrt{10}}{5} \\item[5] \\frac{2\\sqrt{10}}{5} \\end{itemize}","answer":1,"score":3,"review":null}
8
+ {"id":8,"name":"8","problem":"8. 곑선 $( y = x^2 - 5x )$와 직선 $( y = x )$둜 λ‘˜λŸ¬μ‹ΈμΈ λΆ€λΆ„μ˜ 넓이λ₯Ό 직선 $( x = k )$κ°€ 이등뢄할 λ•Œ, μƒμˆ˜ $k$의 값은? [3점] \\begin{itemize} \\item[1] 3 \\item[2] \\frac{13}{4} \\item[3] \\frac{7}{2} \\item[4] \\frac{15}{4} \\item[5] 4 \\end{itemize}","answer":1,"score":3,"review":null}
9
+ {"id":9,"name":"9","problem":"9. 직선 $( y = 2x + k )$ κ°€ 두 ν•¨μˆ˜ \\[ y = \\left( \\frac{2}{3} \\right)^{x+3} + 1, \\quad y = \\left( \\frac{2}{3} \\right)^{x+1} + \\frac{8}{3} \\] 의 κ·Έλž˜ν”„μ™€ λ§Œλ‚˜λŠ” 점을 각각 $( \\mathrm{P} )$, $( \\mathrm{Q} )$라 ν•˜μž. $\\overline{\\mathrm{PQ}} = \\sqrt{5}$일 λ•Œ, μƒμˆ˜ $k$의 값은? [4점] \\begin{itemize} \\item[1] \\frac{31}{6} \\item[2] \\frac{16}{3} \\item[3] \\frac{11}{2} \\item[4] \\frac{17}{3} \\item[5] \\frac{35}{6} \\end{itemize}","answer":4,"score":4,"review":"Removed figure."}
10
+ {"id":10,"name":"10","problem":"10. μ‚Όμ°¨ν•¨μˆ˜ $( f(x) )$에 λŒ€ν•˜μ—¬ 곑선 $( y = f(x) )$ μœ„μ˜ 점 $( 0, 0 )$μ—μ„œμ˜ μ ‘μ„ κ³Ό 곑선 $( y = x f(x) )$ μœ„μ˜ 점 $( 1, 2 )$μ—μ„œμ˜ 접선이 μΌμΉ˜ν•  λ•Œ, $f'(2)$의 값은? [4점] \\begin{itemize} \\item[1] -18 \\item[2] -17 \\item[3] -16 \\item[4] -15 \\item[5] -14 \\end{itemize}","answer":5,"score":4,"review":null}
11
+ {"id":11,"name":"11","problem":"11. μ–‘μˆ˜ $a$에 λŒ€ν•˜μ—¬ 집합 $\\left\\{ x \\ \\middle| \\ -\\frac{a}{2} < x \\leq a, \\ x \\neq \\frac{a}{2} \\right\\}$ μ—μ„œ μ •μ˜λœ ν•¨μˆ˜ \\[ f(x) = \\tan \\frac{\\pi x}{a} \\] κ°€ μžˆλ‹€. κ·Έλ¦Όκ³Ό 같이 ν•¨μˆ˜ $y = f(x)$의 κ·Έλž˜ν”„ μœ„μ˜ μ„Έ 점 $( \\mathrm{O, A, B} )$λ₯Ό μ§€λ‚˜λŠ” 직선이 μžˆλ‹€. 점 $( \\mathrm{A} )$λ₯Ό μ§€λ‚˜κ³  $x$좕에 ν‰ν–‰ν•œ 직선이 ν•¨μˆ˜ $y = f(x)$의 κ·Έλž˜ν”„μ™€ λ§Œλ‚˜λŠ” 점 쀑 $( \\mathrm{A} )$κ°€ μ•„λ‹Œ 점을 $( \\mathrm{C} )$라 ν•˜μž. μ‚Όκ°ν˜• $( \\mathrm{ABC} )$κ°€ μ •μ‚Όκ°ν˜•μΌ λ•Œ, μ‚Όκ°ν˜• $( \\mathrm{ABC} )$의 λ„“μ΄λŠ”? (단, $( \\mathrm{O} )$λŠ” 원점이닀.) [4점] \\begin{itemize} \\item[1] \\frac{3\\sqrt{3}}{2} \\item[2] \\frac{17\\sqrt{3}}{12} \\item[3] \\frac{4\\sqrt{3}}{3} \\item[4] \\frac{5\\sqrt{3}}{4} \\item[5] \\frac{7\\sqrt{3}}{6} \\end{itemize}","answer":3,"score":4,"review":"Removed figure and the statement referring to the figure."}
12
+ {"id":12,"name":"12","problem":"12. μ‹€μˆ˜ μ „μ²΄μ˜ μ§‘ν•©μ—μ„œ 연속인 ν•¨μˆ˜ $f(x)$κ°€ λͺ¨λ“  μ‹€μˆ˜ $x$에 λŒ€ν•˜μ—¬ \\[ \\{f(x)\\}^3 - \\{f(x)\\}^2 - x^2 f(x) + x^2 = 0 \\] 을 λ§Œμ‘±μ‹œν‚¨λ‹€. ν•¨μˆ˜ $f(x)$의 μ΅œλŒ“κ°’μ΄ 1이고 μ΅œμ†Ÿκ°’μ΄ 0일 λ•Œ, \\[ f\\left( -\\frac{4}{3} \\right) + f(0) + f\\left( \\frac{1}{2} \\right) \\] 의 값은? [4점] \\begin{itemize} \\item[1] \\frac{1}{2} \\item[2] 1 \\item[3] \\frac{3}{2} \\item[4] 2 \\item[5] \\frac{5}{2} \\end{itemize}","answer":3,"score":4,"review":null}
13
+ {"id":13,"name":"13","problem":"13. 두 μƒμˆ˜ $( a, b \\ (1 < a < b) )$에 λŒ€ν•˜μ—¬ μ’Œν‘œν‰λ©΄ μœ„μ˜ 두 점 $(a, \\log_2 a), \\ (b, \\log_2 b)$λ₯Ό μ§€λ‚˜λŠ” μ§μ„ μ˜ $y$절편과 두 점 $(a, \\log_4 a), \\ (b, \\log_4 b)$λ₯Ό μ§€λ‚˜λŠ” μ§μ„ μ˜ $y$절편이 κ°™λ‹€. ν•¨μˆ˜ $f(x) = a^{bx} + b^{ax}$에 λŒ€ν•˜μ—¬ $f(1) = 40$일 λ•Œ, $f(2)$의 값은? [4점] \\begin{itemize} \\item[1] 760 \\item[2] 800 \\item[3] 840 \\item[4] 880 \\item[5] 920 \\end{itemize}","answer":2,"score":4,"review":null}
14
+ {"id":14,"name":"14","problem":"14. μˆ˜μ§μ„  μœ„λ₯Ό μ›€μ§μ΄λŠ” 점 $\\mathrm{P}$의 μ‹œκ° $t$μ—μ„œμ˜ μœ„μΉ˜ $x(t)$κ°€ 두 μƒμˆ˜ $a$, $b$에 λŒ€ν•˜μ—¬ \\[ x(t) = t(t - 1)(at + b) \\quad (a \\neq 0) \\] 이닀. 점 $\\mathrm{P}$의 μ‹œκ° $t$μ—μ„œμ˜ 속도 $v(t)$κ°€ $\\int_0^1 |v(t)| \\, dt = 2$λ₯Ό λ§Œμ‘±μ‹œν‚¬ λ•Œ, μ•„λž˜ γ„±, γ„΄, γ„· μ€‘μ—μ„œ μ˜³μ€ κ²ƒλ§Œμ„ μžˆλŠ” λŒ€λ‘œ κ³ λ₯Έ 것은? [4점]\n\n\\begin{itemize} \\item[γ„±.] $\\int_0^1 v(t) \\, dt = 0$ \\item[γ„΄.] $|x(t_1)| > 1$인 $t_1$이 열린ꡬ간 $(0, 1)$에 μ‘΄μž¬ν•œλ‹€. \\item[γ„·.] $0 \\leq t \\leq 1$인 λͺ¨λ“  $t$에 λŒ€ν•˜μ—¬ $|x(t)| < 1$이면 $x(t_2) = 0$인 $t_2$κ°€ 열린ꡬ간 $(0, 1)$에 μ‘΄μž¬ν•œλ‹€. \\end{itemize}\n\n\\begin{itemize} \\item[1] γ„± \\item[2] γ„±, γ„΄ \\item[3] γ„±, γ„· \\item[4] γ„΄, γ„· \\item[5] γ„±, γ„΄, γ„· \\end{itemize}","answer":3,"score":4,"review":"<보기> changed to 'μ•„λž˜ γ„±,γ„΄,γ„·, 쀑'."}
15
+ {"id":15,"name":"15","problem":"15. 두 점 $( \\mathrm{O}_1, \\mathrm{O}_2 )$λ₯Ό 각각 μ€‘μ‹¬μœΌλ‘œ ν•˜κ³  λ°˜μ§€λ¦„μ˜ 길이가 $(\\overline{\\mathrm{O}_1\\mathrm{O}_2} )$인 두 원 $( C_1, C_2 )$κ°€ μžˆλ‹€. κ·Έλ¦Όκ³Ό 같이 원 $( C_1 )$ μœ„μ˜ μ„œλ‘œ λ‹€λ₯Έ μ„Έ 점 $( \\mathrm{A}, \\mathrm{B}, \\mathrm{C} )$와 원 $( C_2 )$ μœ„μ˜ 점 $( \\mathrm{D} )$κ°€ μ£Όμ–΄μ Έ 있고, μ„Έ 점 $( \\mathrm{A}, \\mathrm{O}_1, \\mathrm{O}_2 )$와 μ„Έ 점 $( \\mathrm{C}, \\mathrm{O}_2, \\mathrm{D} )$κ°€ 각각 ν•œ 직선 μœ„μ— μžˆλ‹€.\n\nμ΄λ•Œ $(\\angle \\mathrm{B}\\mathrm{O}_1\\mathrm{A} = \\theta_1)$, $(\\angle \\mathrm{O}_2\\mathrm{O}_1\\mathrm{C} = \\theta_2)$, $(\\angle \\mathrm{O}_1\\mathrm{O}_2\\mathrm{D} = \\theta_3)$이라 ν•˜μž.\n\nλ‹€μŒμ€ $( \\overline{\\mathrm{A}\\mathrm{B}} : \\overline{\\mathrm{O}_1\\mathrm{D}} = 1 : 2\\sqrt{2} )$이고 $( \\theta_3 = \\theta_1 + \\theta_2 )$일 λ•Œ, μ„ λΆ„ $( \\mathrm{A}\\mathrm{B} )$와 μ„ λΆ„ $( \\mathrm{C}\\mathrm{D} )$의 길이의 λΉ„λ₯Ό κ΅¬ν•˜λŠ” 과정이닀.\n\n\\[ \\begin{aligned} &\\angle \\mathrm{C}\\mathrm{O}_2\\mathrm{O}_1 + \\angle \\mathrm{O}_1\\mathrm{O}_2\\mathrm{D} = \\pi \\text{μ΄λ―€λ‘œ } \\theta_3 = \\frac{\\pi}{2} + \\frac{\\theta_2}{2} \\text{이고} \\\\ &\\theta_3 = \\theta_1 + \\theta_2 \\text{μ—μ„œ } 2\\theta_1 + \\theta_2 = \\pi \\text{μ΄λ―€λ‘œ } \\angle \\mathrm{C}\\mathrm{O}_1\\mathrm{B} = \\theta_1 \\text{이닀.} \\\\ &\\text{μ΄λ•Œ } \\angle \\mathrm{O}_2\\mathrm{O}_1\\mathrm{B} = \\theta_1 + \\theta_2 = \\theta_3 \\text{μ΄λ―€λ‘œ μ‚Όκ°ν˜• } \\mathrm{O}_1\\mathrm{O}_2\\mathrm{B} \\text{와 μ‚Όκ°ν˜• } \\mathrm{O}_2\\mathrm{O}_1\\mathrm{D} \\text{λŠ” 합동이닀.} \\\\ &\\overline{\\mathrm{A}\\mathrm{B}} = k \\text{라 ν•  λ•Œ} \\\\ &\\overline{\\mathrm{B}\\mathrm{O}_2} = \\overline{\\mathrm{O}_1\\mathrm{D}}= 2\\sqrt{2}k \\text{μ΄λ―€λ‘œ } \\overline{\\mathrm{A}\\mathrm{O}_2} = \\text{(κ°€)이고,} \\\\ &\\angle \\mathrm{B}\\mathrm{O}_2\\mathrm{A} = \\frac{\\theta_1}{2} \\text{μ΄λ―€λ‘œ } \\cos \\frac{\\theta_1}{2} = \\text{(λ‚˜) 이닀.} \\\\ &\\text{μ‚Όκ°ν˜• } \\mathrm{O}_2\\mathrm{B}\\mathrm{C} \\text{μ—μ„œ} \\\\ &\\overline{\\mathrm{B}\\mathrm{C}} = k, \\overline{\\mathrm{B}\\mathrm{O}_2} = 2\\sqrt{2}k, \\angle \\mathrm{C}\\mathrm{O}_2\\mathrm{B} = \\frac{\\theta_1}{2} \\text{μ΄λ―€λ‘œ} \\\\ &\\text{코사인법칙에 μ˜ν•˜μ—¬ } \\overline{\\mathrm{O}_2\\mathrm{C}} = \\text{(λ‹€) 이닀.} \\\\ &\\overline{\\mathrm{C}\\mathrm{D}} = \\overline{\\mathrm{O}_2\\mathrm{D}} + \\overline{\\mathrm{O}_2\\mathrm{C}} = \\overline{\\mathrm{O}_1\\mathrm{O}_2} + \\overline{\\mathrm{O}_2\\mathrm{C}} \\text{μ΄λ―€λ‘œ} \\\\ &\\overline{\\mathrm{A}\\mathrm{B}} : \\overline{\\mathrm{C}\\mathrm{D}} = k : \\left(\\frac{\\text{(κ°€)}}{2} + \\text{(λ‹€)}\\right) \\text{이닀.} \\end{aligned} \\]\n\nμœ„μ˜ (κ°€), (λ‹€)에 μ•Œλ§žμ€ 식을 각각 $( f(k), g(k) )$라 ν•˜κ³ , (λ‚˜)에 μ•Œλ§žμ€ 수λ₯Ό $( p )$라 ν•  λ•Œ, $( f(p) \\times g(p) )$의 값은? [4점]\n\n\\begin{itemize} \\item[1] \\frac{169}{27} \\item[2] \\frac{56}{9} \\item[3] \\frac{167}{27} \\item[4] \\frac{166}{27} \\item[5] \\frac{55}{9} \\end{itemize}","answer":2,"score":4,"review":"Removed figure and the statement referring to the figure."}
16
+ {"id":16,"name":"16","problem":"16. $\\log_2 120 - \\frac{1}{\\log_{15} 2}$ 의 값을 κ΅¬ν•˜μ‹œμ˜€. [3점]","answer":3,"score":3,"review":null}
17
+ {"id":17,"name":"17","problem":"17. ν•¨μˆ˜ $f(x)$에 λŒ€ν•˜μ—¬ $f'(x) = 3x^2 + 2x$이고 $f(0) = 2$일 λ•Œ, $f(1)$의 값을 κ΅¬ν•˜μ‹œμ˜€. [3점]","answer":4,"score":3,"review":null}
18
+ {"id":18,"name":"18","problem":"18. μˆ˜μ—΄ $\\{a_n\\}$에 λŒ€ν•˜μ—¬ \\[ \\sum_{k=1}^{10} a_k - \\sum_{k=1}^{7} \\frac{a_k}{2} = 56, \\quad \\sum_{k=1}^{10} 2a_k - \\sum_{k=1}^{8} a_k = 100 \\] 일 λ•Œ, $a_8$의 값을 κ΅¬ν•˜μ‹œμ˜€. [3점]","answer":12,"score":3,"review":null}
19
+ {"id":19,"name":"19","problem":"19. ν•¨μˆ˜ $f(x) = x^3 + ax^2 - (a^2 - 8a)x + 3$이 μ‹€μˆ˜ μ „μ²΄μ˜ μ§‘ν•©μ—μ„œ μ¦κ°€ν•˜λ„λ‘ ν•˜λŠ” μ‹€μˆ˜ $a$의 μ΅œλŒ“κ°’μ„ κ΅¬ν•˜μ‹œμ˜€. [3점]","answer":6,"score":3,"review":null}
20
+ {"id":20,"name":"20","problem":"20. μ‹€μˆ˜ μ „μ²΄μ˜ μ§‘ν•©μ—μ„œ λ―ΈλΆ„κ°€λŠ₯ν•œ ν•¨μˆ˜ $( f(x) )$κ°€ λ‹€μŒ 쑰건을 λ§Œμ‘±μ‹œν‚¨λ‹€.\n\n\\begin{itemize} \\item[(κ°€)] λ‹«νžŒκ΅¬κ°„ $[0, 1]$μ—μ„œ $f(x) = x$이닀. \\item[(λ‚˜)] μ–΄λ–€ μƒμˆ˜ $a, b$에 λŒ€ν•˜μ—¬ ꡬ간 $[0, \\infty)$μ—μ„œ $f(x+1) - x f(x) = ax + b$이닀. \\end{itemize}\n\n\\[ 60 \\times \\int_1^2 f(x) \\, dx \\] 의 값을 κ΅¬ν•˜μ‹œμ˜€. [4점]","answer":110,"score":4,"review":null}
21
+ {"id":21,"name":"21","problem":"21. μˆ˜μ—΄ $\\{a_n\\}$이 λ‹€μŒ 쑰건을 λ§Œμ‘±μ‹œν‚¨λ‹€.\n\n\\begin{itemize} \\item[(κ°€)] $( |a_1| = 2 )$ \\item[(λ‚˜)] λͺ¨λ“  μžμ—°μˆ˜ $( n )$에 λŒ€ν•˜μ—¬ $( |a_{n+1}| = 2|a_n| )$이닀. \\item[(λ‹€)] $\\sum_{n=1}^{10} a_n = -14$ \\end{itemize}\n\n$a_1 + a_3 + a_5 + a_7 + a_9$의 값을 κ΅¬ν•˜μ‹œμ˜€. [4점]","answer":678,"score":4,"review":null}
22
+ {"id":22,"name":"22","problem":"22. μ΅œκ³ μ°¨ν•­μ˜ κ³„μˆ˜κ°€ $\\frac{1}{2}$ 인 μ‚Όμ°¨ν•¨μˆ˜ $f(x)$와 μ‹€μˆ˜ $t$에 λŒ€ν•˜μ—¬ 방정식 $f'(x) = 0$이 λ‹«νžŒκ΅¬κ°„ $[t, t+2]$μ—μ„œ κ°–λŠ” μ‹€κ·Όμ˜ 개수λ₯Ό $g(t)$라 ν•  λ•Œ, ν•¨μˆ˜ $g(t)$λŠ” λ‹€μŒ 쑰건을 λ§Œμ‘±μ‹œν‚¨λ‹€.\n\n\\begin{itemize} \\item[(κ°€)] λͺ¨λ“  μ‹€μˆ˜ $( a )$에 λŒ€ν•˜μ—¬ $( \\lim_{t \\to a+} g(t) + \\lim_{t \\to a-} g(t) \\leq 2 )$이닀. \\item[(λ‚˜)] $( g(f(1)) = g(f(4)) = 2, \\ g(f(0)) = 1 )$ \\end{itemize}\n\n$f(5)$의 값을 κ΅¬ν•˜μ‹œμ˜€. [4점]","answer":9,"score":4,"review":null}
23
+ {"id":23,"name":"23_prob","problem":"23. 닀항식 $(x+2)^7$의 μ „κ°œμ‹μ—μ„œ $x^5$의 κ³„μˆ˜λŠ”? [2점] \\begin{itemize} \\item[1] 42 \\item[2] 56 \\item[3] 70 \\item[4] 84 \\item[5] 98 \\end{itemize}","answer":4,"score":2,"review":null}
24
+ {"id":24,"name":"24_prob","problem":"24. ν™•λ₯ λ³€μˆ˜ $X$κ°€ 이항뢄포 $\\mathrm{B}\\left(n, \\frac{1}{3}\\right)$을 λ”°λ₯΄κ³  $\\mathrm{V}(2X) = 40$일 λ•Œ, $n$의 값은? [3점] \\begin{itemize} \\item[1] 30 \\item[2] 35 \\item[3] 40 \\item[4] 45 \\item[5] 50 \\end{itemize}","answer":4,"score":3,"review":null}
25
+ {"id":25,"name":"25_prob","problem":"25. λ‹€μŒ 쑰건을 λ§Œμ‘±μ‹œν‚€λŠ” μžμ—°μˆ˜ $a, \\ b, \\ c, \\ d, \\ e$의 λͺ¨λ“  μˆœμ„œμŒ $(a, b, c, d, e)$의 κ°œμˆ˜λŠ”? [3점]\n\n\\begin{itemize} \\item[(κ°€)] $a + b + c + d + e = 12$ \\item[(λ‚˜)] $\\left| a^2 - b^2 \\right| = 5$ \\end{itemize}\n\n\\begin{itemize} \\item[1] 30 \\item[2] 32 \\item[3] 34 \\item[4] 36 \\item[5] 38 \\end{itemize}","answer":1,"score":3,"review":null}
26
+ {"id":26,"name":"26_prob","problem":"26. $( 1 )$λΆ€ν„° $( 10 )$κΉŒμ§€ μžμ—°μˆ˜κ°€ ν•˜λ‚˜μ”© μ ν˜€ μžˆλŠ” $( 10 )$μž₯의 μΉ΄λ“œκ°€ λ“€μ–΄ μžˆλŠ” μ£Όλ¨Έλ‹ˆκ°€ μžˆλ‹€. 이 μ£Όλ¨Έλ‹ˆμ—μ„œ μž„μ˜λ‘œ μΉ΄λ“œ $( 3 )$μž₯을 λ™μ‹œμ— κΊΌλ‚Ό λ•Œ, κΊΌλ‚Έ μΉ΄λ“œμ— μ ν˜€ μžˆλŠ” μ„Έ μžμ—°μˆ˜ μ€‘μ—μ„œ κ°€μž₯ μž‘μ€ μˆ˜κ°€ $( 4 )$ μ΄ν•˜μ΄κ±°λ‚˜ $( 7 )$ 이상일 ν™•λ₯ μ€? [3점]\n\n\\begin{itemize} \\item[1] \\frac{4}{5} \\item[2] \\frac{5}{6} \\item[3] \\frac{13}{15} \\item[4] \\frac{9}{10} \\item[5] \\frac{14}{15} \\end{itemize}","answer":3,"score":3,"review":"Removed figure."}
27
+ {"id":27,"name":"27_prob","problem":"27. μ–΄λŠ μžλ™μ°¨ νšŒμ‚¬μ—μ„œ μƒμ‚°ν•˜λŠ” μ „κΈ° μžλ™μ°¨μ˜ 1회 μΆ©μ „ μ£Όν–‰ κ±°λ¦¬λŠ” 평균이 $m$이고 ν‘œμ€€νŽΈμ°¨κ°€ $\\sigma$인 μ •κ·œλΆ„ν¬λ₯Ό λ”°λ₯Έλ‹€κ³  ν•œλ‹€.\n\n이 μžλ™μ°¨ νšŒμ‚¬μ—μ„œ μƒμ‚°ν•œ μ „κΈ° μžλ™μ°¨ 100λŒ€λ₯Ό μž„μ˜μΆ”μΆœν•˜μ—¬ 얻은 1회 μΆ©μ „ μ£Όν–‰ 거리의 ν‘œλ³Έν‰κ· μ΄ $\\overline{x_1}$일 λ•Œ, λͺ¨ν‰κ·  $m$에 λŒ€ν•œ 신뒰도 95\\%의 신뒰ꡬ간이 $a \\le m \\le b$이닀.\n\n이 μžλ™μ°¨ νšŒμ‚¬μ—μ„œ μƒμ‚°ν•œ μ „κΈ° μžλ™μ°¨ 400λŒ€λ₯Ό μž„μ˜μΆ”μΆœν•˜μ—¬ 얻은 1회 μΆ©μ „ μ£Όν–‰ 거리의 ν‘œλ³Έν‰κ· μ΄ $\\overline{x_2}$일 λ•Œ, λͺ¨ν‰κ·  $m$에 λŒ€ν•œ 신뒰도 99\\%의 신뒰ꡬ간이 $c \\le m \\le d$이닀.\n\n$\\overline{x_1} - \\overline{x_2} = 1.34$이고 $a = c$일 λ•Œ, $b - a$의 값은? (단, μ£Όν–‰ 거리의 λ‹¨μœ„λŠ” km이고, $Z$κ°€ ν‘œμ€€μ •κ·œλΆ„ν¬λ₯Ό λ”°λ₯΄λŠ” ν™•λ₯ λ³€μˆ˜μΌ λ•Œ $\\mathrm{P}(|Z| \\le 1.96) = 0.95$, $\\mathrm{P}(|Z| \\le 2.58) = 0.99$둜 κ³„μ‚°ν•œλ‹€.) [3점]\n\n\\begin{itemize} \\item[1] 5.88 \\item[2] 7.84 \\item[3] 9.80 \\item[4] 11.76 \\item[5] 13.72 \\end{itemize}","answer":2,"score":3,"review":null}
28
+ {"id":28,"name":"28_prob","problem":"28. 두 집합 $X = \\{1, 2, 3, 4, 5\\}$, $Y = \\{1, 2, 3, 4\\}$에 λŒ€ν•˜μ—¬ λ‹€μŒ 쑰건을 λ§Œμ‘±μ‹œν‚€λŠ” $X$μ—μ„œ $Y$둜의 ν•¨μˆ˜ $f$의 κ°œμˆ˜λŠ”? [4점]\n\n\\begin{itemize} \\item[(κ°€)] 집합 $X$의 λͺ¨λ“  μ›μ†Œ $x$에 λŒ€ν•˜μ—¬ $f(x) \\geq \\sqrt{x}$이닀. \\item[(λ‚˜)] ν•¨μˆ˜ $f$의 μΉ˜μ—­μ˜ μ›μ†Œμ˜ κ°œμˆ˜λŠ” 3이닀. \\end{itemize}\n\n\\begin{itemize} \\item[1] 128 \\item[2] 138 \\item[3] 148 \\item[4] 158 \\item[5] 168 \\end{itemize}","answer":1,"score":4,"review":null}
29
+ {"id":29,"name":"29_prob","problem":"29. 두 연속확λ₯ λ³€μˆ˜ $( X )$와 $( Y )$κ°€ κ°–λŠ” κ°’μ˜ λ²”μœ„λŠ” $( 0 \\leq X \\leq 6 )$, $( 0 \\leq Y \\leq 6 )$이고, $( X )$와 $( Y )$의 ν™•λ₯ λ°€λ„ν•¨μˆ˜λŠ” 각각 $( f(x), g(x) )$이닀. ν™•λ₯ λ³€μˆ˜ $( X )$의 ν™•λ₯ λ°€λ„ν•¨μˆ˜ $( f(x) )$κ°€ λ‹€μŒκ³Ό 같이 μ •μ˜λ˜μ–΄ μžˆλ‹€.\n\n\\[\nf(x) =\n\\begin{cases}\n0, & x < 0, \\\\\n\\frac{1}{12}x, & 0 \\leq x < 3, \\\\\n\\frac{1}{4}, & 3 \\leq x \\leq 5, \\\\\n\\frac{1}{4}(6-x), & 5 < x \\leq 6, \\\\\n0, & x > 6.\n\\end{cases}\n\\]\n\n\n\\[ 0 \\leq x \\leq 6\\ \\text{인 λͺ¨λ“  } x \\text{에 λŒ€ν•˜μ—¬} \\]\n\\[ f(x) + g(x) = k \\quad (k \\text{λŠ” μƒμˆ˜}) \\]\nλ₯Ό λ§Œμ‘±μ‹œν‚¬ λ•Œ, $( \\mathrm{P}(6k \\leq Y \\leq 15k) = \\frac{q}{p} )$이닀. $( p + q )$의 값을 κ΅¬ν•˜μ‹œμ˜€. (단, $( p )$와 $( q )$λŠ” μ„œλ‘œμ†ŒμΈ μžμ—°μˆ˜μ΄λ‹€.) [4점]","answer":31,"score":4,"review":"Removed figure and the statement referring to the figure. The figure is needed to solve the problem, so we paraphrased the figure into text."}
30
+ {"id":30,"name":"30_prob","problem":"30. 흰 곡과 검은 곡이 각각 10개 이상 λ“€μ–΄ μžˆλŠ” λ°”κ΅¬λ‹ˆμ™€ λΉ„μ–΄ μžˆλŠ” μ£Όλ¨Έλ‹ˆκ°€ μžˆλ‹€. ν•œ 개의 μ£Όμ‚¬μœ„λ₯Ό μ‚¬μš©ν•˜μ—¬ λ‹€μŒ μ‹œν–‰μ„ ν•œλ‹€.\n\n\\[ \\begin{array}{|c|} \\hline \\text{μ£Όμ‚¬μœ„λ₯Ό ν•œ 번 던져} \\\\ \\text{λ‚˜μ˜¨ 눈의 μˆ˜κ°€ 5 이상이면} \\\\ \\text{λ°”κ΅¬λ‹ˆμ— μžˆλŠ” 흰 곡 2개λ₯Ό μ£Όλ¨Έλ‹ˆμ— λ„£κ³ ,} \\\\ \\text{λ‚˜μ˜¨ 눈의 μˆ˜κ°€ 4 μ΄ν•˜μ΄λ©΄} \\\\ \\text{λ°”κ΅¬λ‹ˆμ— μžˆλŠ” 검은 곡 1개λ₯Ό μ£Όλ¨Έλ‹ˆμ— λ„£λŠ”λ‹€.} \\\\ \\hline \\end{array} \\]\n\nμœ„μ˜ μ‹œν–‰μ„ 5번 λ°˜λ³΅ν•  λ•Œ, $( n(1 \\leq n \\leq 5) )$번째 μ‹œν–‰ ν›„ μ£Όλ¨Έλ‹ˆμ— λ“€μ–΄ μžˆλŠ” 흰 곡과 검은 곡의 개수λ₯Ό 각각 $( a_n )$, $( b_n )$이라 ν•˜μž. $( a_5 + b_5 \\geq 7 )$일 λ•Œ, $( a_k = b_k )$인 μžμ—°μˆ˜ $( k(1 \\leq k \\leq 5) )$κ°€ μ‘΄μž¬ν•  ν™•λ₯ μ„ $( \\frac{q}{p} )$이닀. $( p + q )$의 값을 κ΅¬ν•˜μ‹œμ˜€. (단, $(p)$와 $(q)$λŠ” μ„œλ‘œμ†ŒμΈ μžμ—°μˆ˜μ΄λ‹€.) [4점]","answer":191,"score":4,"review":null}
31
+ {"id":31,"name":"23_calc","problem":"23. \\[ \\lim_{n \\to \\infty} \\frac{\\frac{5}{n} + \\frac{3}{n^2}}{\\frac{1}{n} - \\frac{2}{n^3}} \\text{의 값은? [2점]} \\] \\begin{itemize} \\item[1] 1 \\item[2] 2 \\item[3] 3 \\item[4] 4 \\item[5] 5 \\end{itemize}","answer":5,"score":2,"review":null}
32
+ {"id":32,"name":"24_calc","problem":"24. μ‹€μˆ˜ μ „μ²΄μ˜ μ§‘ν•©μ—μ„œ λ―ΈλΆ„κ°€λŠ₯ν•œ ν•¨μˆ˜ $f(x)$κ°€ λͺ¨λ“  μ‹€μˆ˜ $x$에 λŒ€ν•˜μ—¬ \\[ f(x^3 + x) = e^x \\] 을 λ§Œμ‘±μ‹œν‚¬ λ•Œ, $f'(2)$의 값은? [3점] \\begin{itemize} \\item[1] e \\item[2] \\frac{e}{2} \\item[3] \\frac{e}{3} \\item[4] \\frac{e}{4} \\item[5] \\frac{e}{5} \\end{itemize}","answer":4,"score":3,"review":null}
33
+ {"id":33,"name":"25_calc","problem":"25. λ“±λΉ„μˆ˜μ—΄ $\\{a_n\\}$에 λŒ€ν•˜μ—¬ \\[ \\sum_{n=1}^{\\infty} (a_{2n-1} - a_{2n}) = 3, \\quad \\sum_{n=1}^{\\infty} a_n^2 = 6 \\] 일 λ•Œ, $\\sum_{n=1}^{\\infty} a_n$ 의 값은? [3점] \\begin{itemize} \\item[1] 1 \\item[2] 2 \\item[3] 3 \\item[4] 4 \\item[5] 5 \\end{itemize}","answer":2,"score":3,"review":null}
34
+ {"id":34,"name":"26_calc","problem":"26. \\[ \\lim_{n \\to \\infty} \\sum_{k=1}^{n} \\frac{k^2 + 2kn}{k^3 + 3k^2 n + n^3} \\text{의 값은?} \\quad [3 \\text{점}] \\] \\begin{itemize} \\item[1] \\ln 5 \\item[2] \\frac{\\ln 5}{2} \\item[3] \\frac{\\ln 5}{3} \\item[4] \\frac{\\ln 5}{4} \\item[5] \\frac{\\ln 5}{5} \\end{itemize}","answer":3,"score":3,"review":null}
35
+ {"id":35,"name":"27_calc","problem":"27. μ’Œν‘œν‰λ©΄ μœ„λ₯Ό μ›€μ§μ΄λŠ” 점 $\\mathrm{P}$의 μ‹œκ° $t \\ (t>0)$μ—μ„œμ˜ μœ„μΉ˜κ°€ 곑선 $y = x^2$κ³Ό 직선 $y = t^2 x - \\frac{\\ln t}{8}$κ°€ λ§Œλ‚˜λŠ” μ„œλ‘œ λ‹€λ₯Έ 두 점의 쀑점일 λ•Œ, μ‹œκ° $t=1$μ—μ„œ $t=e$κΉŒμ§€ 점 $\\mathrm{P}$κ°€ 움직인 κ±°λ¦¬λŠ”? [3점] \\begin{itemize} \\item[1] \\frac{e^4}{2} - \\frac{3}{8} \\item[2] \\frac{e^4}{2} - \\frac{5}{16} \\item[3] \\frac{e^4}{2} - \\frac{1}{4} \\item[4] \\frac{e^4}{2} - \\frac{3}{16} \\item[5] \\frac{e^4}{2} - \\frac{1}{8} \\end{itemize}","answer":1,"score":3,"review":null}
36
+ {"id":36,"name":"28_calc","problem":"28. ν•¨μˆ˜ $( f(x) = 6\\pi (x - 1)^2 )$에 λŒ€ν•˜μ—¬ ν•¨μˆ˜ $( g(x) )$λ₯Ό \\[ g(x) = 3f(x) + 4\\cos f(x) \\] 라 ν•˜μž. $( 0 < x < 2 )$μ—μ„œ ν•¨μˆ˜ $( g(x) )$κ°€ κ·Ήμ†Œκ°€ λ˜λŠ” $( x )$의 κ°œμˆ˜λŠ”? [4점] \\begin{itemize} \\item[1] 6 \\item[2] 7 \\item[3] 8 \\item[4] 9 \\item[5] 10 \\end{itemize}","answer":2,"score":4,"review":null}
37
+ {"id":37,"name":"29_calc","problem":"29. κ·Έλ¦Όκ³Ό 같이 길이가 2인 μ„ λΆ„ $(\\mathrm{AB})$λ₯Ό μ§€λ¦„μœΌλ‘œ ν•˜λŠ” λ°˜μ›μ΄ μžˆλ‹€. 호 $(\\mathrm{AB})$ μœ„μ— 두 점 $(\\mathrm{P})$, $(\\mathrm{Q})$λ₯Ό $(\\angle \\mathrm{PAB} = \\theta)$, $(\\angle \\mathrm{QBA} = 2\\theta)$κ°€ λ˜λ„λ‘ 작고, 두 μ„ λΆ„ $(\\mathrm{AP})$, $(\\mathrm{BQ})$의 ꡐ점을 $(\\mathrm{R})$라 ν•˜μž. μ„ λΆ„ $(\\mathrm{AB})$ μœ„μ˜ 점 $(\\mathrm{S})$, μ„ λΆ„ $(\\mathrm{BR})$ μœ„μ˜ 점 $(\\mathrm{T})$, μ„ λΆ„ $(\\mathrm{AR})$ μœ„μ˜ 점 $(\\mathrm{U})$λ₯Ό μ„ λΆ„ $(\\mathrm{UT})$κ°€ μ„ λΆ„ $(\\mathrm{AB})$에 ν‰ν–‰ν•˜κ³  μ‚Όκ°ν˜• $(\\mathrm{STU})$κ°€ μ •μ‚Όκ°ν˜•μ΄ λ˜λ„λ‘ μž‘λŠ”λ‹€. 두 μ„ λΆ„ $(\\mathrm{AR})$, $(\\mathrm{QR})$와 호 $(\\mathrm{AQ})$둜 λ‘˜λŸ¬μ‹ΈμΈ λΆ€λΆ„μ˜ 넓이λ₯Ό $(f(\\theta))$, μ‚Όκ°ν˜• $(\\mathrm{STU})$의 넓이λ₯Ό $(g(\\theta))$라 ν•  λ•Œ,\n\\[ \\lim_{\\theta \\to 0+} \\frac{g(\\theta)}{\\theta \\times f(\\theta)} = \\frac{q}{p} \\sqrt{3} \\]\n이닀. $(p + q)$의 값을 κ΅¬ν•˜μ‹œμ˜€.\n\n(단, $(0 < \\theta < \\frac{\\pi}{6})$이고, $(p)$와 $(q)$λŠ” μ„œλ‘œμ†ŒμΈ μžμ—°μˆ˜μ΄λ‹€.) [4점]","answer":11,"score":4,"review":"Removed figure and the statement referring to the figure."}
38
+ {"id":38,"name":"30_calc","problem":"30. μ‹€μˆ˜ μ „μ²΄μ˜ μ§‘ν•©μ—μ„œ μ¦κ°€ν•˜κ³  λ―ΈλΆ„κ°€λŠ₯ν•œ ν•¨μˆ˜ $f(x)$κ°€ λ‹€μŒ 쑰건을 λ§Œμ‘±μ‹œν‚¨λ‹€.\n\n\\begin{itemize} \\item[(κ°€)] $f(1) = 1$, \\quad $\\int_{1}^{2} f(x) \\, dx = \\frac{5}{4}$ \\item[(λ‚˜)] ν•¨μˆ˜ $f(x)$의 μ—­ν•¨μˆ˜λ₯Ό $g(x)$라 ν•  λ•Œ, $x \\geq 1$인 λͺ¨λ“  μ‹€μˆ˜ $x$에 λŒ€ν•˜μ—¬ $g(2x) = 2f(x)$이닀. \\end{itemize}\n\n\\[ \\int_{1}^{8} x f'(x) \\, dx = \\frac{q}{p} \\text{일 λ•Œ, } p+q \\text{의 값을 κ΅¬ν•˜μ‹œμ˜€.} \\]\n(단, $p$와 $q$λŠ” μ„œλ‘œμ†ŒμΈ μžμ—°μˆ˜μ΄λ‹€.) [4점]","answer":143,"score":4,"review":null}
39
+ {"id":39,"name":"23_geom","problem":"23. μ’Œν‘œκ³΅κ°„μ˜ 점 $\\mathrm{A}(2, 1, 3)$을 $xy$ 평면에 λŒ€ν•˜μ—¬ λŒ€μΉ­μ΄λ™ν•œ 점을 $\\mathrm{P}$라 ν•˜κ³ , 점 $\\mathrm{A}$λ₯Ό $yz$ 평면에 λŒ€ν•˜μ—¬ λŒ€μΉ­μ΄λ™ν•œ 점을 $\\mathrm{Q}$라 ν•  λ•Œ, μ„ λΆ„ $\\mathrm{PQ}$의 κΈΈμ΄λŠ”? [2점]\n\n\\begin{itemize} \\item[1] 5 \\sqrt{2} \\item[2] 2 \\sqrt{13} \\item[3] 3 \\sqrt{6} \\item[4] 2 \\sqrt{14} \\item[5] 2 \\sqrt{15} \\end{itemize}","answer":2,"score":2,"review":null}
40
+ {"id":40,"name":"24_geom","problem":"24. ν•œ 초점의 μ’Œν‘œκ°€ $\\left( 3\\sqrt{2}, 0 \\right)$ 인 μŒκ³‘μ„  $\\frac{x^2}{a^2} - \\frac{y^2}{6} = 1$ 의 μ£ΌμΆ•μ˜ κΈΈμ΄λŠ”? (단, $a$ λŠ” μ–‘μˆ˜μ΄λ‹€.) [3점]\n\n\\begin{itemize} \\item[1] 3\\sqrt{3} \\item[2] \\frac{7\\sqrt{3}}{2} \\item[3] 4\\sqrt{3} \\item[4] \\frac{9\\sqrt{3}}{2} \\item[5] 5\\sqrt{3} \\end{itemize}","answer":3,"score":3,"review":null}
41
+ {"id":41,"name":"25_geom","problem":"25. μ’Œν‘œν‰λ©΄μ—μ„œ 두 직선 \\[ \\frac{x+1}{2} = y - 3, \\quad x - 2 = \\frac{y - 5}{3} \\] κ°€ μ΄λ£¨λŠ” 예각의 크기λ₯Ό $\\theta$라 ν•  λ•Œ, $\\cos \\theta$의 값은? [3점]\n\n\\begin{itemize} \\item[1] \\frac{1}{2} \\item[2] \\frac{\\sqrt{5}}{4} \\item[3] \\frac{\\sqrt{6}}{4} \\item[4] \\frac{\\sqrt{7}}{4} \\item[5] \\frac{\\sqrt{2}}{2} \\end{itemize}","answer":5,"score":3,"review":null}
42
+ {"id":42,"name":"26_geom","problem":"26. 두 초점이 $( \\mathrm{F}, \\mathrm{F'} )$인 타원 $\\frac{x^2}{64} + \\frac{y^2}{16} = 1$ μœ„μ˜ 점 쀑 제1사뢄면에 μžˆλŠ” 점 $( \\mathrm{A} )$κ°€ μžˆλ‹€. 두 직선 $( \\mathrm{AF}, \\mathrm{AF'} )$에 λ™μ‹œμ— μ ‘ν•˜κ³  쀑심이 $y$μΆ• μœ„μ— μžˆλŠ” 원 쀑 μ€‘μ‹¬μ˜ $y$μ’Œν‘œκ°€ 음수인 것을 $( C )$라 ν•˜μž. 원 $( C )$의 쀑심을 $( \\mathrm{B} )$라 ν•  λ•Œ μ‚¬κ°ν˜• $( \\mathrm{AFBF'} )$의 넓이가 72이닀. 원 $( C )$의 λ°˜μ§€λ¦„μ˜ κΈΈμ΄λŠ”? [3점]\n\n\\begin{itemize} \\item[1] \\frac{17}{2} \\item[2] 9 \\item[3] \\frac{19}{2} \\item[4] 10 \\item[5] \\frac{21}{2} \\end{itemize}","answer":2,"score":3,"review":"Removed figure."}
43
+ {"id":43,"name":"27_geom","problem":"27. κ·Έλ¦Όκ³Ό 같이 ν•œ λͺ¨μ„œλ¦¬μ˜ 길이가 4인 μ •μœ‘λ©΄μ²΄ $\\mathrm{ABCD - EFGH}$ κ°€ μžˆλ‹€. μ„ λΆ„ $\\mathrm{AD}$ 의 쀑점을 $\\mathrm{M}$이라 ν•  λ•Œ, μ‚Όκ°ν˜• $\\mathrm{MEG}$ 의 λ„“μ΄λŠ”? [3점]\n\n\\begin{itemize} \\item[1] \\frac{21}{2} \\item[2] 11 \\item[3] \\frac{23}{2} \\item[4] 12 \\item[5] \\frac{25}{2} \\end{itemize}","answer":4,"score":3,"review":"Removed figure and the statement referring to the figure."}
44
+ {"id":44,"name":"28_geom","problem":"28. 두 μ–‘μˆ˜ $( a )$, $( p )$에 λŒ€ν•˜μ—¬ 포물선 $( (y - a)^2 = 4px )$의 μ΄ˆμ μ„ $( \\mathrm{F}_1 )$이라 ν•˜κ³ , 포물선 $( y^2 = -4x )$의 μ΄ˆμ μ„ $( \\mathrm{F}_2 )$라 ν•˜μž. μ„ λΆ„ $( \\mathrm{F}_1 \\mathrm{F}_2 )$κ°€ 두 포물선과 λ§Œλ‚˜λŠ” 점을 각각 $( \\mathrm{P} )$, $( \\mathrm{Q} )$라 ν•  λ•Œ, $( \\overline{\\mathrm{F}_1 \\mathrm{F}_2} = 3 )$, $( \\overline{\\mathrm{P}\\mathrm{Q}} = 1 )$이닀. $( a^2 + p^2 )$의 값은? [4점]\n\n\\begin{itemize} \\item[1] 6 \\item[2] \\frac{25}{4} \\item[3] \\frac{13}{2} \\item[4] \\frac{27}{4} \\item[5] 7 \\end{itemize}","answer":5,"score":4,"review":"Removed figure."}
45
+ {"id":45,"name":"29_geom","problem":"29. μ’Œν‘œν‰λ©΄μ—μ„œ $\\overline{\\mathrm{OA}} = \\sqrt{2}$, $\\overline{\\mathrm{OB}} = 2\\sqrt{2}$이고\n\\[ \\cos(\\angle \\mathrm{AOB}) = \\frac{1}{4} \\]\n인 ν‰ν–‰μ‚¬λ³€ν˜• $\\mathrm{OACB}$에 λŒ€ν•˜μ—¬ 점 $\\mathrm{P}$κ°€ λ‹€μŒ 쑰건을 λ§Œμ‘±μ‹œν‚¨λ‹€.\n\n\\begin{itemize} \\item[(κ°€)] $\\overrightarrow{\\mathrm{OP}} = s \\overrightarrow{\\mathrm{OA}} + t \\overrightarrow{\\mathrm{OB}} \\quad (0 \\leq s \\leq 1, \\ 0 \\leq t \\leq 1)$ \\item[(λ‚˜)] $\\overrightarrow{\\mathrm{OP}} \\cdot \\overrightarrow{\\mathrm{OB}} + \\overrightarrow{\\mathrm{BP}} \\cdot \\overrightarrow{\\mathrm{BC}} = 2$ \\end{itemize}\n\n점 $\\mathrm{O}$λ₯Ό μ€‘μ‹¬μœΌλ‘œ ν•˜κ³  점 $\\mathrm{A}$λ₯Ό μ§€λ‚˜λŠ” 원 μœ„λ₯Ό μ›€μ§μ΄λŠ” 점 $\\mathrm{X}$에 λŒ€ν•˜μ—¬ $|3\\overrightarrow{\\mathrm{OP}} - \\overrightarrow{\\mathrm{OX}}|$의 μ΅œλŒ“κ°’κ³Ό μ΅œμ†Ÿκ°’μ„ 각각 $M$, $m$이라 ν•˜μž. $M \\times m = a\\sqrt{6} + b$일 λ•Œ, $a^2 + b^2$의 값을 κ΅¬ν•˜μ‹œμ˜€. (단, $a$와 $b$λŠ” μœ λ¦¬μˆ˜μ΄λ‹€.) [4점]","answer":100,"score":4,"review":"Removed figure."}
46
+ {"id":46,"name":"30_geom","problem":"30. μ’Œν‘œκ³΅κ°„μ— 쀑심이 $\\mathrm{C}(2, \\sqrt{5}, 5)$이고 점 $\\mathrm{P}(0, 0, 1)$을 μ§€λ‚˜λŠ” ꡬ \\[ S: (x - 2)^2 + (y - \\sqrt{5})^2 + (z - 5)^2 = 25 \\] κ°€ μžˆλ‹€. ꡬ $S$κ°€ 평면 $\\mathrm{OPC}$와 λ§Œλ‚˜μ„œ μƒκΈ°λŠ” 원 μœ„λ₯Ό μ›€μ§μ΄λŠ” 점 $\\mathrm{Q}$, ꡬ $S$ μœ„λ₯Ό μ›€μ§μ΄λŠ” 점 $\\mathrm{R}$에 λŒ€ν•˜μ—¬ 두 점 $\\mathrm{Q}, \\mathrm{R}$의 $xy$평면 μœ„λ‘œμ˜ μ •μ‚¬μ˜μ„ 각각 $\\mathrm{Q}_1, \\mathrm{R}_1$이라 ν•˜μž.\n\nμ‚Όκ°ν˜• $\\mathrm{O}\\mathrm{Q}_1\\mathrm{R}_1$의 넓이가 μ΅œλŒ€κ°€ λ˜λ„λ‘ ν•˜λŠ” 두 점 $\\mathrm{Q}, \\mathrm{R}$에 λŒ€ν•˜μ—¬ μ‚Όκ°ν˜• $\\mathrm{O}\\mathrm{Q}_1\\mathrm{R}_1$의 평면 $\\mathrm{PQR}$ μœ„λ‘œμ˜ μ •μ‚¬μ˜μ˜ λ„“μ΄λŠ” $\\frac{q}{p} \\sqrt{6}$이닀. $p+q$의 값을 κ΅¬ν•˜μ‹œμ˜€.\n\n(단, $\\mathrm{O}$λŠ” 원점이고 μ„Έ 점 $\\mathrm{O}, \\mathrm{Q}_1, \\mathrm{R}_1$은 ν•œ 직선 μœ„μ— μžˆμ§€ μ•ŠμœΌλ©°, $p$와 $q$λŠ” μ„œλ‘œμ†ŒμΈ μžμ—°μˆ˜μ΄λ‹€.) [4점]","answer":23,"score":4,"review":"Removed figure."}
data/json/2022/math/math_29_prob.txt DELETED
@@ -1,12 +0,0 @@
1
- 29. 두 연속확λ₯ λ³€μˆ˜ \( X \)와 \( Y \)κ°€ κ°–λŠ” κ°’μ˜ λ²”μœ„λŠ” \( 0 \leq X \leq 6 \),
2
- \( 0 \leq Y \leq 6 \)이고, \( X \)와 \( Y \)의 ν™•λ₯ λ°€λ„ν•¨μˆ˜λŠ” 각각 \( f(x), g(x) \)이닀.
3
- ν™•λ₯ λ³€μˆ˜ \( X \)의 ν™•λ₯ λ°€λ„ν•¨μˆ˜ \( f(x) \)의 κ·Έλž˜ν”„λŠ” κ·Έλ¦Όκ³Ό κ°™λ‹€.
4
-
5
- \[
6
- 0 \leq x \leq 6\ \text{인 λͺ¨λ“  } x \text{에 λŒ€ν•˜μ—¬}
7
- \]
8
- \[
9
- f(x) + g(x) = k \quad (k \text{λŠ” μƒμˆ˜})
10
- \]
11
- λ₯Ό λ§Œμ‘±μ‹œν‚¬ λ•Œ, \( \mathrm{P}(6k \leq Y \leq 15k) = \frac{q}{p} \) 이닀. \( p + q \)의 값을 κ΅¬ν•˜μ‹œμ˜€.
12
- (단, \( p \)와 \( q \)λŠ” μ„œλ‘œμ†ŒμΈ μžμ—°μˆ˜μ΄λ‹€.) [4점]
 
 
 
 
 
 
 
 
 
 
 
 
 
data/json/2022/{math/math_1.txt β†’ math_1.txt} RENAMED
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data/json/2022/{math/math_10.txt β†’ math_10.txt} RENAMED
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data/json/2022/{math/math_27_prob.txt β†’ math_27_prob.txt} RENAMED
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data/json/2022/{math/math_28_calc.txt β†’ math_28_calc.txt} RENAMED
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data/json/2022/{math/math_29_calc.txt β†’ math_29_calc.txt} RENAMED
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data/json/2022/math_29_prob.txt ADDED
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1
+ 29. 두 연속확λ₯ λ³€μˆ˜ $( X )$와 $( Y )$κ°€ κ°–λŠ” κ°’μ˜ λ²”μœ„λŠ” $( 0 \leq X \leq 6 )$, $( 0 \leq Y \leq 6 )$이고, $( X )$와 $( Y )$의 ν™•λ₯ λ°€λ„ν•¨μˆ˜λŠ” 각각 $( f(x), g(x) )$이닀. ν™•λ₯ λ³€μˆ˜ $( X )$의 ν™•λ₯ λ°€λ„ν•¨μˆ˜ $( f(x) )$κ°€ λ‹€μŒκ³Ό 같이 μ •μ˜λ˜μ–΄ μžˆλ‹€.
2
+
3
+ \[
4
+ f(x) =
5
+ \begin{cases}
6
+ 0, & x < 0, \\
7
+ \frac{1}{12}x, & 0 \leq x < 3, \\
8
+ \frac{1}{4}, & 3 \leq x \leq 5, \\
9
+ \frac{1}{4}(6-x), & 5 < x \leq 6, \\
10
+ 0, & x > 6.
11
+ \end{cases}
12
+ \]
13
+
14
+
15
+ \[ 0 \leq x \leq 6\ \text{인 λͺ¨λ“  } x \text{에 λŒ€ν•˜μ—¬} \]
16
+ \[ f(x) + g(x) = k \quad (k \text{λŠ” μƒμˆ˜}) \]
17
+ λ₯Ό λ§Œμ‘±μ‹œν‚¬ λ•Œ, $( \mathrm{P}(6k \leq Y \leq 15k) = \frac{q}{p} )$이닀. $( p + q )$의 값을 κ΅¬ν•˜μ‹œμ˜€. (단, $( p )$와 $( q )$λŠ” μ„œλ‘œμ†ŒμΈ μžμ—°μˆ˜μ΄λ‹€.) [4점]
data/json/2022/{math/math_3.txt β†’ math_3.txt} RENAMED
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data/json/2022/{math/math_30_calc.txt β†’ math_30_calc.txt} RENAMED
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data/json/2022/{math/math_30_geom.txt β†’ math_30_geom.txt} RENAMED
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data/json/2022/math_4.txt ADDED
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1
+ 4. ν•¨μˆ˜ $( y = f(x) )$κ°€ λ‹€μŒκ³Ό 같이 μ •μ˜λ˜μ–΄ μžˆλ‹€.
2
+
3
+ \[
4
+ f(x) =
5
+ \begin{cases}
6
+ -x+2, & x < -1, \\
7
+ 2, & x = -1, \\
8
+ (3*x+3)/2, & -1 < x < 1, \\
9
+ 1, & 1 \leq x < 2, \\
10
+ 3, & x = 2, \\
11
+ 1, & x \geq 2.
12
+ \end{cases}
13
+ \]
14
+
15
+ \[ \lim_{x \to -1-} f(x) + \lim_{x \to 2} f(x) \text{의 값은? [3점]} \]
16
+
17
+ \begin{itemize} \item[1] 1 \item[2] 2 \item[3] 3 \item[4] 4 \item[5] 5 \end{itemize}
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