Infron · Motif 3 · 46문항 × 3회 · Python TAR. 131,072 output tokens · 기본 1,800초 · 6+1 turns. ACC는 전체 138회, Pass@1은 각 문항의 첫 유효 시도, Pass@3는 3회 내 정답으로 계산합니다.
그림과 같이 , 이고 인 직각삼각형 ABC가 있다. 선분 AB를 2:1로 내분하는 점을 D, 점 A를 중심으로 하고 반지름의 길이가 인 원이 선분 AC와 만나는 점을 E, 직선 AB가 이 원과 만나는 점 중 D가 아닌 점을 F라 하고, 호 EF 위의 점 G를 이 되도록 잡는다. 세 점 C,E,G를 지나는 원 위의 점 H가 를 만족시킬 때, 선분 GH의 길이는?
[원문 그림의 텍스트 전사] 그림의 직선 AB 위 점들은 왼쪽부터 F,A,D,B 순서이고 C는 B 위쪽에 있다. G는 원 A의 위쪽 작은 호 EF 위에 있다. C,E,G,H를 지나는 원에서 H는 E를 포함하지 않는 호 CG 위에 있으며 그림의 왼쪽 위에 있다.
① ② ③ ④ ⑤
최종 답은 선택지 번호 1~5 중 하나를 안에 쓰세요.
Recorded Python call:
import sympy as sp
sqrt = sp.sqrt
# Coordinates
A = sp.Point(3,0)
B = sp.Point(0,0)
C = sp.Point(0,4)
# D divides AB AD:DB = 2:1 => D=(1,0)
D = sp.Point(1,0)
# Circle centered A radius AD=2; E on AC with AE=2
E = sp.Point(sp.Rational(9,5), sp.Rational(8,5))
# F = (5,0)
F = sp.Point(5,0)
# G on circle A, CG=2√6
Gx = (27 + 4*sqrt(15))/10
Gy = (4 + 3*sqrt(15))/10
G = sp.Point(Gx, Gy)
# verify CG
sp.simplify((G.x)**2 + (G.y-4)**2), (2*sqrt(6))**2
Recorded Python call:
import sympy as sp
sqrt=sp.sqrt
Gx=(27+4*sqrt(15))/10
Gy=(4+3*sqrt(15))/10
G=sp.Point(Gx,Gy)
C=sp.Point(0,4)
print(sp.simplify((G.x)**2+(G.y-4)**2))
print(sp.simplify((2*sqrt(6))**2))
print(float(Gx),float(Gy))
Recorded Python call:
import sympy as sp
sqrt=sp.sqrt
C=sp.Point(0,4)
E=sp.Point(sp.Rational(9,5),sp.Rational(8,5))
G=sp.Point((27+4*sqrt(15))/10,(4+3*sqrt(15))/10)
circle=sp.Circle(C,E,G)
O=circle.center
R=circle.radius
print("O=",O)
print("R=",sp.simplify(R))
Recorded Python call:
import sympy as sp
sqrt=sp.sqrt
C=sp.Point(0,4)
E=sp.Point(sp.Rational(9,5),sp.Rational(8,5))
G=sp.Point((27+4*sqrt(15))/10,(4+3*sqrt(15))/10)
O=sp.Point(sp.Rational(9,10)+14*sqrt(15)/25, sp.Rational(14,5)+21*sqrt(15)/50)
R=4*sqrt(15)/5
alpha_cos=sp.Rational(3,5)
alpha_sin=sp.Rational(4,5)
v=sp.Matrix([G.x-C.x, G.y-C.y])
L=sp.sqrt(v.dot(v)) # should 2sqrt6
print("L=",sp.simplify(L))
u=v/L
# rotation matrices
Rplus=sp.Matrix([[alpha_cos,-alpha_sin],[alpha_sin,alpha_cos]])
Rminus=sp.Matrix([[alpha_cos,alpha_sin],[-alpha_sin,alpha_cos]])
for name,Rrot in [("+alpha",Rplus),("-alpha",Rminus)]:
uH=Rrot*u
t=-2*(uH.dot(sp.Matrix([C.x-O.x,C.y-O.y])))
t_s=sp.simplify(t)
H=sp.Point(C.x+t_s*uH[0], C.y+t_s*uH[1])
print(name, "t=",t_s, "H=",sp.simplify(H.x), sp.simplify(H.y))
print("GH=", sp.simplify(G.distance(H)))
# check angle
w=sp.Matrix([H.x-C.x,H.y-C.y])
dot=w.dot(v)
cosang=sp.simplify(dot/(sp.sqrt(w.dot(w))*L))
print("cosang=",cosang)
Recorded Python call:
import sympy as sp
sqrt=sp.sqrt
O=sp.Point(sp.Rational(9,10)+14*sqrt(15)/25, sp.Rational(14,5)+21*sqrt(15)/50)
C=sp.Point(0,4)
E=sp.Point(sp.Rational(9,5),sp.Rational(8,5))
G=sp.Point((27+4*sqrt(15))/10,(4+3*sqrt(15))/10)
H=sp.Point(sp.Rational(27,10)+18*sqrt(15)/25, sp.Rational(3*sqrt(15),10)+sp.Rational(26,5))
import math
def ang(P):
v=(float(P.x-O.x), float(P.y-O.y))
return math.degrees(math.atan2(v[1],v[0]))%360
for name,P in [("C",C),("E",E),("G",G),("H+",H)]:
print(name, ang(P))
Recorded Python call:
import sympy as sp, math
sqrt=sp.sqrt
O=sp.Point(sp.Rational(9,10)+14*sqrt(15)/25, sp.Rational(14,5)+21*sqrt(15)/50)
C=sp.Point(0,4)
E=sp.Point(sp.Rational(9,5),sp.Rational(8,5))
G=sp.Point((27+4*sqrt(15))/10,(4+3*sqrt(15))/10)
H=sp.Point(sp.Rational(27,10)+18*sqrt(15)/25, sp.Rational(3,10)*sqrt(15)+sp.Rational(26,5))
def ang(P):
v=(float(P.x-O.x), float(P.y-O.y))
return math.degrees(math.atan2(v[1],v[0]))%360
for name,P in [("C",C),("E",E),("G",G),("H+",H)]:
print(name, ang(P))
<tool_call>{"name": "python", "arguments": {"code": "import sympy as sp\nsqrt=sp.sqrt\nA=sp.Point(0,0)\nB=sp.Point(3,0)\nC=sp.Point(3,4)\nD=sp.Point(2,0)\nE=sp.Point(sp.Rational(6,5),sp.Rational(8,5))\nGx=(3-4sqrt(15))/10\nGy=(4+3sqrt(15))/10\nG=sp.Point(Gx,Gy)\ncircle=sp.Circle(C,E,G)\nO=circle.center\nR=circle.radius\nprint("O=",O)\nprint("R=",sp.simplify(R))
angle alpha\ncosalpha=sp.Rational(3,5)\nsinalpha=sp.Rational(4,5)\nv=sp.Matrix([G.x-C.x,G.y-C.y])\nL=sp.sqrt(v.dot(v))\nprint("L=",sp.simplify(L))=v/L\nRrot=sp.Matrix([[cosalpha,-sinalpha],[sinalpha,cosalpha]])H=Rrotu\nt=-2(uH.dot(sp.Matrix([C.x-O.x,C.y-O.y])))\nt=sp.simplify(t)\nprint("t=",t)\nH=sp.Point(C.x+tuH[0], C.y+tuH[1])\nprint("H=",sp.simplify(H.x),sp.simplify(H.y))\nprint("GH=",sp.simplify(G.distance(H)))
verify angle\nw=sp.Matrix([H.x-C.x,H.y-C.y])\nprint("cosang=",sp.simplify(w.dot(v)/(sp.sqrt(w.dot(w))*L)))
"}}</tool_call>
Tool-call protocol error
The response ended with malformed or unexecuted tool-call markup.
Comparative Analysis (Pcommon_14) (Average across runs)
Averages use reported measurements only. Missing usage is not treated as zero.