train/precalculus/1029.json

{
    "problem": "Regular decagon $P_1 P_2 \\dotsb P_{10}$ is drawn in the coordinate plane with $P_1$ at $(1,0)$ and $P_6$ at $(3,0).$  If $P_n$ is the point $(x_n,y_n),$ compute the numerical value of the product\n\\[(x_1 + y_1 i)(x_2 + y_2 i)(x_3 + y_3 i) \\dotsm (x_{10} + y_{10} i).\\]",
    "level": "Level 5",
    "type": "Precalculus",
    "solution": "Let $p_k$ denote the complex number corresponding to the point $P_k,$ for $1 \\le k \\le 10.$  Since the $P_k$ form a regular decagon centered at 2, the $p_k$ are the roots of\n\\[(z - 2)^{10} = 1.\\]Hence,\n\\[(z - p_1)(z - p_2)(z - p_3) \\dotsm (z - p_{10}) = (z - 2)^{10} - 1.\\]By Vieta's formulas, $p_1 p_2 p_3 \\dotsm p_{10} = 2^{10} - 1 = \\boxed{1023}.$\n\n[asy]\nunitsize(1.5 cm);\n\nint i;\npair[] P;\n\nfor (i = 1; i <= 10; ++i) {\n  P[i] = (2,0) + dir(180 - 36*(i - 1));\n  draw(((2,0) + dir(180 - 36*(i - 1)))--((2,0) + dir(180 - 36*i)));\n}\n\ndraw((-1,0)--(4,0));\ndraw((0,-1.5)--(0,1.5));\n\nlabel(\"$P_1$\", P[1], NW);\nlabel(\"$P_2$\", P[2], dir(180 - 36));\nlabel(\"$P_3$\", P[3], dir(180 - 2*36));\nlabel(\"$P_4$\", P[4], dir(180 - 3*36));\nlabel(\"$P_5$\", P[5], dir(180 - 4*36));\nlabel(\"$P_6$\", P[6], NE);\nlabel(\"$P_7$\", P[7], dir(180 - 6*36));\nlabel(\"$P_8$\", P[8], dir(180 - 7*36));\nlabel(\"$P_9$\", P[9], dir(180 - 8*36));\nlabel(\"$P_{10}$\", P[10], dir(180 - 9*36));\n\ndot(\"$2$\", (2,0), S);\n[/asy]"
}