{ "metadata": { "name": "", "signature": "sha256:def85da9352d54bd80cd68c0f8ff7d372c03f6bba940f40da0807f6da9669b0b" }, "nbformat": 3, "nbformat_minor": 0, "worksheets": [ { "cells": [ { "cell_type": "heading", "level": 3, "metadata": {}, "source": [ "Rendering diagrams from a mathematics competition in an IPython notebook" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "We load the following IPython asymptote magics remotely." ] }, { "cell_type": "code", "collapsed": false, "input": [ "# Install the IPython magic extension\n", "%install_ext http://raw.github.com/azjps/ipython-asymptote/master/asymptote.py\n", "# Load the extension\n", "%reload_ext asymptote" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "Installed asymptote.py. To use it, type:\n", " %load_ext asymptote\n" ] } ], "prompt_number": 1 }, { "cell_type": "markdown", "metadata": {}, "source": [ "For ease, we use two asymptote modules written by users of the forum Art of Problem Solving: [`cse5`](http://www.artofproblemsolving.com/Forum/viewtopic.php?t=149650) and [`olympiad`](http://www.artofproblemsolving.com/Forum/viewtopic.php?f=519&t=165767). The first abbreviates some common commands in a more functional style and the second offers a few useful geometry functions. Since they are not underneath version control to my knowledge, I have included them in the github project ipython-asymptote/examples." ] }, { "cell_type": "heading", "level": 3, "metadata": {}, "source": [ "PUMaC Power Round Solution Round 2013" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The following problems are taken from the 2011 Power Round of the [Princeton University Math Competition](https://pumac.princeton.edu/wp-content/uploads/2011/12/azhu_power_revision2.pdf), a week-long open-resources round of a high school mathematics competition." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**General Problem Setting** We are given a scalene triangle $ABC$ in plane, with incircle $\\Gamma$, and we let $D$, $E$, $F$ be the tangency points of $\\Gamma$ with the sides $BC$, $CA$, and $AB$, respectively. Furthermore, let $P$ be an arbitrary point in the interior of the triangle $ABC$ and let $A_{1}B_{1}C_{1}$ be its cevian triangle (that is, $A_{1}$, $B_{1}$, $C_{1}$ are the intersections of the lines $AP$, $BP$, $CP$ with the sides $BC$, $CA$, and $AB$, respectively); from $A_{1}$, $B_{1}$, $C_{1}$ draw the tangents to $\\Gamma$ that are different from the triangle's sides $BC$, $CA$, $AB$, and let their tangency points with the incircle be $X$, $Y$, and $Z$, respectively.\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**Setting up the IPython notebook** For the remainder of this notebook, we'd like to use a common set of settings, such as imports, diagram size, and pre-defined pens. From the above problem description we can also define the points that will be needed for drawing diagrams. We save these to the file `pumac_power_round_2011_config.asy` as follows:" ] }, { "cell_type": "code", "collapsed": false, "input": [ "%%asy --root pumac_power_round_2011_config\n", "import cse5;\n", "import olympiad;\n", "size(100);\n", "\n", "/* Setup for cse5 default pens */\n", "pointpen = black;\n", "pathpen = linewidth(1.5);\n", "dotfactor = 7;\n", "\n", "/* Some other common pens to use */\n", "pen darkgreen = linewidth(1.5),\n", " darkred = linewidth(1) + red,\n", " darkblue = linewidth(1) + rgb(0, 0, 0.7),\n", " faded = linewidth(1) + linetype(\"0 4\"),\n", " dots = linewidth(1)+ dotted;\n", "\n", "/* Define points and paths described in the general problem setting */\n", "pair B = (0,0), C = (10,0), A = (2,7);\n", "path t = A--B--C--cycle, incirc = incircle(A,B,C);\n", "pair I = incenter(A,B,C), D = foot(I,B,C), E = foot(I,C,A), F = foot(I,A,B), P = (4.6,1);\n", "pair A1 = extension(B,C,A,P), B1 = extension(A,C,B,P), C1 = extension(A,B,C,P);\n", "pair X1 = reflect(I,A1)*D, Y1 = reflect(I,B1)*E, Z1 = reflect(I,C1)*F;\n", "pair P1 = extension(A,X1,B,Y1), V = extension(A1,X1,A,C), A2 = extension(A,X1,B,C);\n", "pair A0 = extension(Y1,Z1,E,F), B0 = extension(X1,Z1,D,F), C0 = extension(X1,Y1,D,E);\n", "pair R = (1.5*A + 3.5*C)/5, S = extension(A0,R,A,B), T = extension(C0,R,B,C);\n", "pair P2 = extension(A0,T,B0,R), M = extension(A,B,A0,C), N = extension(A,C0,B,C);\n", "\n", "/* Draw a couple of the common paths */\n", "D(A--B--C--cycle); D(incircle(A,B,C));\n", "\n", "/* Label some of the common points */\n", "D(MP(\"A\", A, NE));\n", "D(MP(\"B\", B));\n", "D(MP(\"C\", C));\n", "D(MP(\"I\", I, W));" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "\n" ] }, { "metadata": {}, "output_type": "pyout", "png": "iVBORw0KGgoAAAANSUhEUgAAAGQAAABVCAYAAAC/xEFcAAAABmJLR0QA/wD/AP+gvaeTAAAACXBI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\nZhOT77etY8svmsAwMvwPOUXzK+D1jpIAAAAASUVORK5CYII=\n", "prompt_number": 2, "text": [ "" ] } ], "prompt_number": 2 }, { "cell_type": "markdown", "metadata": {}, "source": [ "**1)** (a) Extend $A_{1}X$ to intersect $AC$ at $V$. Show that the lines $AA_{1}$, $BV$, and $DE$ are concurrent.\n", "\n", "(b) Let $A_{2}$ be the intersection point of $AX$ with the sideline $BC$. Show that \n", "$$\\frac{A_2B}{A_2C} = \\frac{s-c}{s-b} \\cdot \\frac{A_1B^2}{A_1C^2},$$ where $s = \\frac{a + b + c}{2}$ is the semiperimeter of $\\triangle ABC$. [Hint: Menelaus's Theorem may come in handy.]\n", "\n", "(c) Prove that lines $AX, BY, CZ$ are concurrent." ] }, { "cell_type": "code", "collapsed": false, "input": [ "%%asy pumac_power_round_2011_config.asy\n", "size(500);\n", "\n", "/* Draw lines */\n", "D(A--A1--X1, darkgreen);\n", "D(B--B1--Y1, darkgreen);\n", "D(C--C1--Z1, darkgreen);\n", "D(A--P1--B, darkred);\n", "D(C--Z1, darkred);\n", "D(X1--V);\n", "D(P1--A2, darkred);\n", "\n", "/* Label points */\n", "D(MP(\"P\",P,dir(60)));\n", "D(MP(\"D\",D)); D(MP(\"E\",E,NE)); D(MP(\"F\",F,NW));\n", "D(MP(\"X\",X1,ENE)); D(MP(\"Y\",Y1,1.4*dir(150))); D(MP(\"Z\",Z1,NE)); D(P1);\n", "D(MP(\"A_1\",A1)); D(MP(\"B_1\",B1,NE)); D(MP(\"C_1\",C1,NW));\n", "D(MP(\"V\",V,NE)); D(MP(\"A_2\",A2));" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "\n" ] }, { "metadata": {}, "output_type": "pyout", "png": 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OnHMPM0tDCFHKEAT9c1FcTaynqNKzumhb652Btt7PCWLa1nNyBMzWrdHvsFub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\n/dyZfX/lrS7FsLDz4B7h3DklnDNFRX5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"prompt_number": 3, "text": [ "" ] } ], "prompt_number": 3 }, { "cell_type": "markdown", "metadata": {}, "source": [ "**2)** Show that the lines $B_1C_1$, $EF$, $YZ$ concur at a point, say $A_{0}$. Similarly, the lines $C_1A_1$, $FD$, $ZX$ and $A_1B_1$, $DE$, $XY$ concur at points $B_{0}$ and $C_{0}$, respectively. [Hint: Pascal's theorem might be useful here.]" ] }, { "cell_type": "code", "collapsed": false, "input": [ "%%asy pumac_power_round_2011_config.asy\n", "size(600);\n", "\n", "/* Draw lines */\n", "D(A--A1--X1, darkgreen+faded); D(B--B1--Y1, darkgreen+faded); D(C--C1--Z1, darkgreen+faded);\n", "D(A--P1,darkred+faded); D(P1--B, darkred+faded); D(C--Z1, darkred+faded);\n", "D(Y1--A0--E); D(F--D); D(A1--C1); D(B1--C0--E);\n", "D(D--E--F--cycle,rgb(0,0.7,0)+linewidth(1));\n", "D(A0--B0--C0--cycle, darkblue);\n", "D(A--B0--C, darkblue+linetype(\"3 3\"));\n", "// D(A0--B1,faded); D(C0--X1,faded); D(X1--Z1,faded); D(P1);\n", "\n", "/* Label points */\n", "D(MP(\"D\",D,SSE)); D(MP(\"E\",E,NE)); D(MP(\"F\",F,NW));\n", "D(MP(\"X\",X1,dir(-10))); D(MP(\"Y\",Y1,(0,-2.2))); D(MP(\"Z\",Z1,dir(220))); D(MP(\"P\",P,dir(60)));\n", "D(MP(\"A_1\",A1)); D(MP(\"B_1\",B1,NE)); D(MP(\"C_1\",C1,NW));\n", "D(MP(\"A_0\",A0)); D(MP(\"B_0\",B0,0.8*dir(30))); D(MP(\"C_0\",C0));" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "\n" ] }, { "metadata": {}, "output_type": "pyout", "png": 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NsQvBbqQqvIW2NxXjIGJEJnchBhDWU1sJ/98a4K2aocfSOGY2z5Wc5gpOc5pL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\nMQaNyb4KS2qs4g35+/jlbSSuhGghqmAJUSKhcpXkaS2ZbYK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"prompt_number": 4, "text": [ "" ] } ], "prompt_number": 4 }, { "cell_type": "markdown", "metadata": {}, "source": [ "**3)** Let $R$ a point on $AC$ and let $S$, $T$ be the intersections of $RA_{0}$, $RC_{0}$ with $AB$ and $BC$ respectively. Then, $B_{0}$ lies on $ST$." ] }, { "cell_type": "code", "collapsed": false, "input": [ "%%asy pumac_power_round_2011_config.asy\n", "size(600);\n", "\n", "/* Draw lines */\n", "D(A--A1--X1, darkgreen+faded); D(B--B1--Y1, darkgreen+faded); D(C--C1--Z1, darkgreen+faded);\n", "D(A--P1,darkred+faded); D(P1--B, darkred+faded); D(C--Z1, darkred+faded); D(A0--Y1,faded);\n", "D(A0--E, dots); D(F--D, dots); D(A1--C1, faded); D(B1--C0,faded); D(C0--E, dots);\n", "D(A0--B0--C0, darkblue);\n", "D(A--B0--C, darkblue);\n", "D(A0--R--C0, rgb(0,0.7,0)+linewidth(1));\n", "D(A0--S, rgb(0,0.7,0)+linewidth(1)+linetype(\"3 3\"));\n", "D(C0--T, rgb(0,0.7,0)+linewidth(1)+linetype(\"3 3\"));\n", "D(S+.6*(S-T)--T+.6*(T-S), rgb(0,0.7,0)+linewidth(1)+linetype(\"3 3\"));\n", "\n", "/* Label points */\n", "D(MP(\"D\",D,SSE)); D(MP(\"E\",E,NE)); D(MP(\"F\",F,NW));\n", "D(MP(\"A_0\",A0)); D(MP(\"B_0\",B0,dir(225))); D(MP(\"C_0\",C0));\n", "D(MP(\"R\",R,NE)); D(MP(\"S\",S,1.2*dir(215))); D(MP(\"T\",T,(0,-1.2)));" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "\n" ] }, { "metadata": {}, "output_type": "pyout", "png": 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8w3B596qI7NrC7YjLg0TlsMAi8kcj8Qwmif1UplXZx5gje0andW2c9NveELoOtih6JYDX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\nZTPA6ihYXP3cWGy55UOMAwss6oQ+xTP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"prompt_number": 5, "text": [ "" ] } ], "prompt_number": 5 }, { "cell_type": "markdown", "metadata": {}, "source": [ "**4)** We say that two triangles $\\triangle ABC, \\triangle DEF$ are *perspective* iff $AD, BE, CF$ are concurrent. The point of concurrency is called their *perspector*. Show the following pairs of triangles are perspective:\n", "\n", "(a) $\\triangle A_0B_0C_0$ and $\\triangle DEF$. Their perspector lies on the incircle of $\\triangle ABC$.\n", "\n", "(b) $\\triangle ABC$ and $\\triangle TRS$\n", "\n", "(c) $\\triangle A_0B_0C_0$ and $\\triangle TRS$. The locus of the perspector as $R$ varies along the line $AC$ is a line tangent to the incircle of $\\triangle ABC$." ] }, { "cell_type": "code", "collapsed": false, "input": [ "%%asy pumac_power_round_2011_config.asy\n", "size(600);\n", "\n", "/* Draw lines */\n", "D(A--A1--X1, darkgreen+faded); D(B--B1--Y1, darkgreen+faded); D(C--C1--Z1, darkgreen+faded);\n", "D(A--P1,darkred+faded); D(P1--B, darkred+faded); D(C--Z1, darkred+faded); D(A0--Y1,faded);\n", "D(A0--E, dots); D(F--D, dots); D(A1--C1, faded); D(B1--C0,faded); D(C0--E, dots);\n", "D(A0--B0--C0--cycle, darkblue); D(A--B0--C, darkblue+linetype(\"3 3\"));\n", "D(R--S--T--cycle, rgb(0,0.7,0)+linewidth(1)); D(A0--S, rgb(0,0.7,0)+linewidth(0.7)+linetype(\"3 3\"));\n", "D(C0--T, rgb(0,0.7,0)+linewidth(0.7)+linetype(\"3 3\"));\n", "D(A0--T,linewidth(1)); D(R--P2,linewidth(1)); D(C0--S,linewidth(1));\n", "D(M+1.2*(M-N)--N+(N-M), linewidth(1.5) + linetype(\"2 2\") + rgb(1,0,0), Arrows(6));\n", "// D(A0--B1,faded); D(C0--X1,faded); D(X1--Z1,faded); \n", "\n", "/* Label points */\n", "D(D); D(E); D(F);\n", "D(MP(\"A_0\",A0)); D(MP(\"B_0\",B0,1.5*dir(60))); D(MP(\"C_0\",C0));\n", "D(MP(\"R\",R,NE)); D(MP(\"S\",S,NW)); D(MP(\"T\",T,SE)); D(M); D(N); D(P2);" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "\n" ] }, { "metadata": {}, "output_type": "pyout", "png": 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orHWR7GjacAfA3wPwEQBfBuDtXU+nElI3FFiEhENp9gzaIXgCa2B5TWcFpo5wiEyCOGlX\nMT6nbpf3tKhYfgpsBOuxAH6HHYaE1AcFFiHhUIo9gzqx+55WefYRxAm7CrPTqgdCF4FGs74CwIsB\nfBqAXxaRZ9W6KEI6CgUWIQFQlj2DFrKfwI5cuZGnLseY8SmsSCOBo+N1ngDgjwH8vIh8d81LIqRz\nsMidkADQ2qiRMeZaQdvrwxpSDmB9rWqfadhUeiKDJkazHCLydgDPAPAGY8zX1L0eQroCI1iEhEFh\n9Vdad3Nbfw1iYHQZVFGHBTQzVehjjHkmgJcD+GoR+Y2610NIV6DAIiQMCqm/0k7EE1iz0mtFt+qH\n1EnIodPJMcZ8O4CvAfAFOjz6q6sSqIR0FQosQmqmKHsGHZdyBOBmiWaTwQgskg5jzBsAPAbAZwJ4\nPYAzWjkQUh4UWITUT+70oIgcwJpNXtMC57IIzpS0KpHQa4EYMca8F8s5hgDwwpqWQkjrocAipH5y\npQdVXG3BiquyBVBwIqMqx/ImjdCJEhGh3wHgEMAfAXiZiHx1PasipN2wi5CQGtE6mDMAd2cz/pQD\n2AjYTluL2Ul6RGSQpEZNRH4ewLMAvFzrtAghBcEIFiH1sgVbkJ5KXKkz+21YcXWN4qo6QnV496NU\n68RV5H7Phq3H+hci8mPlrpCQbkGBRUi9pK6/0hOkq6MpvFNw/b4nA5FJEG7uPlUWa4di2xDtAsz6\nOTDG/CNYG4d/op5ZhJACoMAipF5S1V/VKa4AwJjxDB2uw6qTqIjMalMR91ppevDlAJ4hIifsLiQk\nPxRYhNRERnuGPf23cnHlQasGpcx0YYygKvX9VpH1jbA1Wb9MkUVIPiiwCKmPVOlBr6C9TnEFAKz3\nUkJJFyYhibGoMeYHAHwrgCsAflOnAhBCMsAuQkJqQkTOYE1BjxPc11kxXOlCOiwrSbvnQkRE+qG8\nt/p5+1oAfwHgS9hEQUh6GMEipAY0mjBAggiWDoJ2PldBnIBDpWniyo8qhfTe6iSA1wL4MIDbOiWA\nEJICCixC6iGRPYOe2JxDexBRhBC7COsmqct7UYXqVaAi610Afh/AgQp9QkhCKLAIqYeN9Vda/3KA\n8ExEt+pewCaqLtBe5fJelJVCXnIMdr4B4H8A+A8AdjV1SAhJAAUWIfWw1p7Bs2PYL3m2YBZCEntB\nEWqEKqelww0AzwbwFgAjETlihyEhm6HAIqRiEtoznMCmEG9WtKw0BCEa1lFVpCgaGVpUsdOKUXF2\nDcALAHwXbO0gvbII2QAFFiHVszY96KVhdqpZTjqMGXc2grUpQtXkgdDr0IuBHQD/CsC3wIrs27Rx\nIGQ1FFiEVM/K9KAWtW/B1l218mRdFUVFWCKz+xr3nuSov7qApqqPAXwfrNg6hY1kUWQREgMFFiEV\nss6eQU9UewivqP0SIpPg3dzrFEMhDYQuuA7Mpaz3tMtwH7RxICSWO+peACEdI9aeQaMkBwAOkxiP\nBkBr62+ihp9ZhFqTHN7TYIyZi8gObOTq1BizLyJzWBuHvjFmv+41EhIKjGARUi2r6q+cx1BTTlDB\nRGjyEoqVQtGUVYSu0dWbsKJqoKnDq6CNAyEXoMAipFou1V+JyAhWYDWp7qoxEZqogKraSiGpCWnR\nlPlZUlF1Cht1daLrGmjjQMg5nEVISEWoPcOeMeaKd1sfwG0Ax4FaMjSeuucT9kT6bewu1M/uGaxX\n275324nehaOdSKdhBIuQ6ohLD+4CmFNcFUdoZp9tFFfAeYRsBzY1OPBuuwbaOBBCgUVIhVxID/qp\nwdpWlJHQ5hFuGprctZRVVc9XGzLOU4V629wYcwO0cSAdhwKLkApYYc9wAJteCdqSYQX9OkVWaFGq\nJFRZi1Vxam4HwDBq1UAbB9J1KLAIqYYL9gwisgcADU4NzlGhVUOMoEolIEKoBWp5qnAfwF7M+7QP\nK8AORGQ37vGEtBUKLEKq4bz+SqNZjUwNOnRcTisFA0mPCqkplnYj/t9o40A6CbsICakAEVkAuGqM\nmYrIEQBonQqJIWr2Gfp209ATGZRlRFrn89Naq9vQz/mKvx/BCrEmWZIQkgkKLEJKxrdn0ML2I9iT\nUPB1Q1VRt5UCKQaNUA2MMddW/J02DqQzMEVISPn49gx7sONwGi8mRCZb+R5/YYhy418PAsA6vA/1\nQuIStHEgXYICi5Dy2QJwqp1UAzRnHM4mUhW5t3UkTVbqcngvE31PD+HZNsTdhzYOpAtQYBFSIhF7\nhl1YW4a2CIu1zyNGUAUTpWqjL1ZAz2kfQH+TNQNtHEjbocAipFycuegWcN5t1Uqa5E0Vgsgt2rYh\nhOcEXLBt2GjLQBsH0mZY5E5IiWjH4BTANmz06rDmJRVGCB15JFxExM0p3PiZ1zThCexMzsbalxDi\nwwgWIeWyBeBO4NwPiARCQCm13PVYIT0Xj0RRLABQW4drAEYichTo8yEkFRRYhJSE2jPMAPxDtKCw\nPc5NPbSZhE0lb7ow0EjiMRLUYjlUZF2FrVk8ocgiTYcCi5DyGAL4/wD0mxq92jREGfY5NpJARUlr\n8DoKExew08aBtAkKLELKYwvAXbAnmUaQoVB9sOHvJAUttG7YBzBY5YsVB20cSFugwCKkBDx7hich\n4PRgAd5UxwUupxair0GdpE0Vhp5G08/TMVJEsbzH0saBNBp2ERJSAtpy/s8B/GxIXVHs/CNVowL2\nDMCVLNYdKq4OANxss80JaR+MYBFSDk8F8JmoOT1I9/Rm0qZUoYqqU2SIYunjD2GL33d11iEhjYAC\ni5ByeC6A39bOqMqo2uwz7zxC0hlSFbtHoY0DaSIUWIQUjNozfALAD1S0P39octURqmDql/IQ2gk7\nSS1WaGtehzHmGMBcvxtZt0EbB9IoKLAIKZ4vB2DQggLwBFQaoSuLJqZOG7jmY+jIqKzQxoE0CQos\nQorneQB+sawTYJzhZxn7SQjNRkukF1CHYwEcAtjKG3mijQNpChRYhBSIFpV/GoA3FLzNc0KKXBgz\nnhozbkUUCwgv7baIqaELbY1J0XpAN5eziO3RxoEEDQUWIcXyMgCfMMb8RNYNVF2oTkiF5E4T+qht\nww6AA7VGISQYKLAIKZa/CeDX0zwg5AhVEkQmiV26Qyfk195ZN4S8xgQcAxgWae5KGwcSKhRYhBTL\nAMCbNt0p0vnHCBXZSN6B0CHgpQkLtfegjQMJEQosQgpCRF4F4C/j3KabHqXaQGsiWI6Qxuf4hLqu\nlByjhM8MbRxIaFBgEVIAOsz2fwPwYf29S3VUp3UvoGhCfb+MMbMWuLwfw0aaCn8etHEgIUGBRUgx\nvAjAQwD0RWTYsgjVWowZt05gkfJQ8TpDSZFP2jiQUKDAIqQYPqn/vg/25EFIKbShFgslpQl9aONA\n6oYCi5AMxKQ3PhXAR4wxn9Ol6BUAiEyGIpOmp60uEVoNT0vqrxxTVFC7RxsHUicUWIQkIEGR+hPh\njY0J7eRcMn0ArUvDhCaUo3VhDXd5PwUwqEI00saB1AUFFiErSGml8FkAjrz7B3VyLpnWOLk3iTiX\n96ag349Koli6P9o4kMqhwCJEyWql4FIPxpgfKWNdoWPMuEtikhTHKSqMfNLGgVQNBRbpLAVaKTwL\nwB8n2UdbaXMnYd3v4aY0WoNtGyqLYDlo40CqhAKLdIYSzT6fBuCXC9oWCYy6072bhH+DuwpdHVal\nApE2DqQqKLBIa6nC7FNF26cBeHvc3+s+OVdFm+YRkmrQ78YMNTVI0MaBlA0FFmkNMYKqCnHjZqr9\nYgX7IjVRl0VC2uhOA9OFlacJfWjjQMqEAouQfPxt4LyAdiUt8zCKo9VpllBH50RpYLpwBlt0Xhu0\ncSBlQYFFGk3ESqGOk8szAfzapjs15QSdg7Y/v1roQIp5ipoFFkAbB1IOFFikUZRYqJ5lLVsAPgjg\n3rrWEArGjI/rXgNpJFMEEv2kjQMpGgosEjRVFKrnYAjgowBSibwOpAtbS1XvXd79NMXl3X2fQ+nk\no40DKRIKLBIUIUWoErAF4OOw7d6JCUwkFobIpPUno6reu7z7aZjLexBpQgdtHEhRUGCRWgk8QrUS\nFYLuJ2QRWCVbm+9CyCXmCCRN6EMbB5IXCixSOX6UKvAI1Tq2oOaimzoIV9HCGo9GiOMu0ZBUYbCz\nLGnjQPJAgUVKp6lRqg0MAfw2ckSvGiwuVxHsibJIyhbGRdZ5NShVGFwEy0EbB5IVCixSODUZflbN\nFoDfRUdERRKMGXfitSj789ySC5A0BD/HkjYOJAsUWISkRO0ZZgAKO8jygE3KpAEO78GPWqKNA0kL\nBRbJTUciVj5DLK+6C4natOE1E5l0qsi9SSfYwB3eGxOxo40DSQMFVoSeyBt7Iic9kd2eSPBXVXXQ\nMCuFMtiCFViNOcFWRKdejzI+9130SGtaSpQ2DiQpFFiXeRtsuPp7APxsT2TRE5n2RL6/3mXVR0uL\n1DPh2TOcIiDvnkCYiUz4muSg49+tRl3Q0saBbIICK8LCdozswL42D9CbnwLgJQ2oYyiMllgplMEW\ngFPvNSn0hNiktFMMhdalkeLp0jGsCmjjQNZBgRWDiqwbAP5cb/pLAB9Ci40UGaVKjF9/BRQssJos\nZo0Zz7rSSehTRFqvKmEdcC1W8J2Eq6CNA1kFBdYKFsYcA/hiAB8D8AkArwGw1xM5qXVhBcE6qsy4\n+itCABRzMcLvX7OhjQOJgwJrDQv7pXkSgC9eGPONAK4AOHR/b4hLMgBGqIrA2TN4zu2lFrc2seC5\nrnmEInIiIgv9YaomAU06fjUB2jiQKBRYG1gYM1OhhYUxc41suYPTmXYcBl+cySvkQoimB0s9gDZU\nBNcisIwxrnX+bq2LIRsI0OW98d14tHEgPhRYGdGD0xXYL9JJL7AuEl49lQLTg5up5XOnJ7Jp9EJC\nRA40ulXqRVCe7xu/q+e04nWgjQNxUGDlQKNbOwCuaGE8eiKjOkLvrKkql4g9Qx37bgp1CdBL4ldP\nbNuwtiulXgDl+b7V/V1lZ2E50MaBUGAVQCTUPoJNHR6ULbT8K9+GppOaRNSeoTKa9N7W2EW4BeA4\nctsMy2HcjXkNqybgzsLGQxuHbkOBVTALY25iWei4V+S2GaWqlWj9FQkETf9dSg9qquZuAFeM/V5W\nsRZGg8gFaOPQXSiwSmBhzHRhzLWFzcO7tOFe2lA8O/+Covb6qyakCkUmwyrd3EVkD8AJgK1VJ68q\nvzdpLnooxi7R2gsY2jh0EwqsapjDRkDOevaEEAsjVGESY8/gqDQd1iCBXZnAMsbcNMb09Genqv0W\nQWjfb9ZilQttHLoHBVYFuIgWrDv8wB3IGKFqDKvSg0GdIENAa7B44mggrMUqH9o4dAsKrArpAdOF\nMTcWxsx7Iv1FgPYOJJZ16cG6bAlCFjEhr610mpDKDZDOCA3aOHQHCqya0KvFQwC7PZGzHr9kQZLA\nnqEuY81gow3GjA8336u9JIlEhyyQa3J4D/b1KAvaOLQfCqySiVgpXDgpLow5XBhzBfZLRsJknT1D\nsCKHhE3IArlGh/fOlUgUZeMQGRW1EJEzirb6ocAqmCyF6iq0pgDQEznSiNZWWWskqVhnz9C5E0IS\nRCbBj44i4eAuQrtag1qEjYOOiprCjorqwdZ50RKiZiiwclJCofoOrGHiQY+eKSGwyZ6h1tRuoKmm\nENdUOatqsQJ9z2KpKF3Y+fKIvDYO/qgofew2mBmpHQqslJRtpaADpW/Czjl043cGrNGqnjX2DI7a\nr7gDTTXxs4rVF1uBvmexVJgurGsCQDDktHHYArAtIgsA98Metyox1yWrocDaQF1WCiq03EFnAOB2\nT+SkV/LQWnKBTe7tM1To+dQgoiNrCFnHEKxnBJDLxmELwFVND15FwVNESDYosGLwo1QhXG0ujDmF\njWjNYK0deFKvhk3pwTkCElih2APUOI8wSJqUEqyJPgKIBodCWhsHHRU19yLtQ2hEkJ+9eqHAQjMM\nPxfGzBbG7CyM6bmwvY7fCeKk2jYS2DO4kH5Awia8zy1ZXqQ1+WRXssv7AIxgXSKJjYM3KmooIkd6\n8xzATJjtqJ1OCqwYQdXUL3cfdvzOAYVW4ayzZ/AJKooVCuwkvEyDjzNlu7wzgrWCTTYOkVFRN/S2\nY2PMjjEmyfGLlEgnBVZbWNgrnKuwByie0IplU/2VY4rAiroDiajx80iSMgQF1koiHfYlGwAAIABJ\nREFUNg6/KiIHdH9vBnfUvYAqEJG+r+TbpOq1EP6G+90bJr3P2WKX0QPTHpYCYGas2WuULSRrcw5u\n9l4gqcIk4rQzRI9BTaUn0i/yuKLZhD7YRXiOTM7LE5bcAvBqHOA/4SaALwLwUABfUf3qSBpaKbBE\nZOCfZNpwYEvBMayAOOuJ7C9siJksOQJwbIy55vnFXCCBPYPPHIzWxBGU6CTFUMJF2xC2QLvWY3Ss\nqLnMEBc/13NcFobRY8EMF6NzfVyOeE+xqgbtAH8DM3we/ht+F4/BA7y/PLUtor3N9Iwxda8hN/yg\nXUbtHEYLeqFcQETOYNuZV35etHC0r0Wmm7Y3AnBkjLm7wGUWQvRCg5CQkIn08WP4XvwJnoKbWDUm\nJk74RKOjWYTPalGjmLGpNAqrF3wjLJ/vHLYOdCYT2QYwx3VMATwJwFfBPu8bCS8ESQ00VmDx5JEc\nFVtHAG4ubD6/s+hBzB2Q58ZcPoiqCLtpjNno56Tbux92RAVFPgkemWzsLouLslz4njz6D3Ht9x6G\nhXdTlmgO8AN4Cf4KPoIX4t/qLVMz7sb3SMsV3OvsCv0vFKbLRPoAdgHsR18XHauzBeAaRVaYNEZg\nMUqVj55t83VXiddqHOhaG1FRLiL3RyNPWiB+hhSCSQXZTpxY6yoiky1jxjQcVc5Fzb14OJ6MD8Tc\nJUnqqKhoTpSZGddzPBCR2wAOTQcu/DTaPcDy/ZmuO2bIRLYA9M149WujnYV7sMef1r+GTSPYGqyO\n11EVjkauDnWI9BwAeiLDRUeufDTStCsiLmW6h/jOpaT2DD5T2Kv1YAVWDRcotX9fZSJRsRFHETUz\nUeGzFDX34cF4JD56/peluIoTNUUI0mA/gytI2q3bKLx0n/v89WGPKxtFkEattmCjeWuPz8aYfRGZ\nw9o49A1rboMimAgWI1TV0xO5H/ZkcHORIB3WZLRwfQR74OrDnsxuRtPMatY3TXOg0qvIkU607zTn\nxcKTv/M3AADj//gbMXcrQ9QAm0/U800nLFIdIdcvpkUj39HP9WnaMha9KBgCOE6TKtV04wlsA8/G\n2lFSDbUJLAqq+lF35l3YTrrDrhfEe/VUV9PUNNR9ogiyA+reex6OD951J575tvdUXSwcOm079mW1\nbtBmkoEzyGwSKmjc92UO+z2Z5nlfXSG7GWe72NU1HcF+J3fa9BlrKhRY5MIB0g2TXnSsnkhF0psA\nPBDA39pUT6Vh/KXY+Hac4BpeiGfiPd7diqiZSdfWrVDU1M+qRhwe+ywaLZ6ZBlzY6fHBfQ/nsOsu\n5DumF0hbAA7zFvjrReKJ/nqNn7N6qUxg8aDSDLQY/gBWCOyHJrR8YWPG5jQidKZmbOYykV24mgd7\nn2XX4NhM9fct3cZNmcgA34pX4V14LgDgSfhPeDnujez6cgeUzx728bu4Zd5vvrOwJ1sC/B6SUBBb\nonAjtOaQdXYJhe/LNj8M1hWyp96mXf8BaONQO6UKLFopNBOda7gLYLgw5ipwKWLjhMzSDX187tUy\n0L8f+6krFToDqLGnGdur1jXbWCWOLmyzqOesNRT3AvgDfClej5fgU90aEj6+semOshCZDI0Z8+De\nEdKkCl3NUAj1V0nsEgrf5+Tc5Pi0rLpA2jjUT6ECi1fHxSETGbguoxgRcp5WUhEygr3impmxOYwT\nITI5H6FzqNtYJWwubWNxHbOPPhAvftELMX3ds/ABtNCrxrdnALBtjNnX12AEAJuuMOuuwwoRkcme\nMePg0z910eXjZZ0XJGntEgrfvz1+j8y4/I4/2jjUSy6B1bYIlScqVkVoXGHjXEWI+30aSVeNsBQy\nzg7g1NvGAOpv4okjF/VZGSmqq65Gi+H3oMXwsF2HrTox+J2AboSO6yT0aiRm6wpQQ015rKLsE3yX\nvbDadmwsGvW/Oi7bVmCNXUItER0tX5hWeSyXZdnHTdo4VMtKgaVh0114RYhlHZC9iEw0QuPSQk6E\nuJqZOBHiIjSnXlQHuFx340dohrhYVBzd5pBt3Ut6y8/EzbYZlUbtGfTz3/fFkhfWB2IKUptUtFsF\nTBF2k57IYN3xwevWvVK0CC3KLqFoiixkz7R/2jjUwjqBdQQtBMYLHvp/Y/Gl/8S85g3vzVAzEy0o\n3ljLU8YTJcXTEzmBFaf7TY9oicgCEXsGvfI7jl5U6GfYpRmOz1O56pZvjLlS3coJaRZFfk/KsEso\nmjIK2TOtgzYOlbNOYO3hfLTK898BfPh+4EN3AR88A572KcDj/zPwOY+Dtp0bM25EWoQUh1o67MIK\n5p2mmpWqCen3AXh2jPHo3rqIlF5gWOPS65ijpCvzMmE6q1669vrnifSWaZdQNN4cweNQMiGejcOn\nAPgSiqxyWVuDpYp3ZoyZi0xciHMOXD+F/ZBfBXAGPOF/BgbvBp7/GcADJ9CaInhjECjA2osKrX6D\nBdYebDpwx7vNb9HeMsYcrkuRn6exvw9PxdvxVtY6AHrMgDHjLomHTomldcR1FaYx863SLqFoQola\nRXHndAA/DuBJoI1DqWyKYJ1Grw70oOlCssfGjOd6/yGApwP4U/1bH8BvA7d+EUszxSGWwmsKLSgH\nMGetRjvo2c/NNmydVlAHl1XosOabxpjjOBGlEa4Z9GLDu/3yfb9MvhsPxtfgH+N7Qju4Vo3IpA9g\nyIurbrJCYK1MD9Zhl1AGGtWehWD26wv+6PGKNg7lEyuw9Or9CGumnOvB83yuW9xVqpcfd2/qpwP4\nzxfrXCZOePWhNgH671xvowBrGGpWugtg7ny0QsW3Z1gZnZLz4vbDdQd7vR8A3I8X4Fn4e/jb0O7Q\notddFkVHYEQmu8aMOx/N20RXLBv89GDddglF4/n87VdVyB4jmqK/rxRYehttHEpklcByHXlIkicX\nmbjOqukmIeTl0OcArgC4bR9nZvEfgImzTnBfxOgHVyML3UlDNIWeyHBhzFRtHoahucIDF+0ZYv4W\nPVjtGju9fu3J0J1EAOzjFgB3IfLT+CnzGvPeop9DyHTJqiGPSGq7wNKLj78L4DUAXg7gI6jRLqFo\nZCKukavUz/omQaW3DVadT1dskzYOJXFJYOmLPYV6h6QpRBSZOPE0S3pQ1S+eCw0/EMAfwLNO2HQ1\nHRFgPk6MUYDVjDrD34Z9P3ZCElpRe4YN971k3aC3Rw96l9IgXuehG+GT+ABImgHfzyVRu4TPBj7+\nPuAzADzPBB7VToNXyH5YVgd81s9VmsfRxqEcLggsFTu7Xvh2N+7KfhMqerZhI1qpFb0nui6EjwH8\nD2PMO1Ouxa//cmlHl4ac6W1TV0tGykGjWLuwn4sboYisOHuGDfffgkZcvdviriJXmo76nYf+QTnJ\n1WlVUCyQNCSxS6jKXLQqyipkT5um97+rOaOotHEomKjAWnpfWU6dwNKrcpc6vJkkX6v1VUtjxhwi\nRq+InFhyous+Y8zrs25T1+gLMF94uUJ8CrCS0IL407rEloqlvSR+PJGD2HrrhuUsR6y7GtQD9BAr\nnJ3X1VM0BZqNpqMpwjbGLmG67iJFT963sabWsUkUWcie9j1f282c8/Pj2TgAtvi98e9VnZwLLBVQ\nfbN0sh4AuG10tpp2Wrm5UUdpTeK0TmsAK7QKOUlEuk4AG4L+JRc1KOJgpWlP4LL1xAi0oMhFb1lg\neQrbdVjpiTjOniHh4/pQ64Y193EnlEcbs77uSs14h7DmvIkjvk0QXF0ocm/C+5CHyLgZV3qRyi5B\nO9awaPiYLW88WmZH9iI/L2UIcn2/D2CPSbRxyEHPGOObivoRq4Xe59TY+WwLY0zP/c39Py1enda0\nDGESI7oAK4xmZajxiABzwoseYAnpLaM980XFI2Z8e4YMj7Wjm9ZE38Q63f+WMeYbE21z2YU0y5J2\niB64Q4iGdEFgtY2i7RL0hH2GBs3pjCNp1KqMVP+qbZT5HaeNQ34SD3uORLBOXGQrK5qaGwKAMeNS\n20MjnYuOdxtj3lbufi9YUPjeXy4VSQuKCGrxMIQdv1NaVCCJPcOqx3ltz7a4dXW4/nKxe4IDohbO\nOguUPFfKfloz01Vz/pQDU4Sh49klAGqNU6QQ0u/JdtqsRyh4M0iP4wrZy4hglpkGTLkO2jjkII3A\nSl2DlWy78calZRNTQ/BxAL+wzjOk+DVc8gBzRfid9QDzIlrbAA4XJXW0rLNnSLuddUW7emGyv+r7\nsvZAuhRagHYe5llrZL+tTms1mZKjEn66D3BdrSVGKETk9wC8wv8OxJmQhoirkzTj5Xe87shw1fun\njUN2EgussvGMSwHgtGpbhYhdBBDphKnjS9VlDzAdvzMqK22Yxp5hw3YGsEJtlYDa07/nak33hqqX\n0g4eUgdjkwhdqEbtErAsSK9kze7CPJrxaILAkonsAjjFdczLfr3WfY5C+C7SxiEbwQgsnzTGpeWu\nI7ZzcYqlR1fd9S1RDzA/+tUaDzDPR+sQNnWY+3VPa8+wYhvO0G8EYB63rbT1J5sOpps6D8tCn0em\nz3yXzEbrJoldQsXrOdH9V1pfmQfZlefgGXg8cqTnE+0nuRFo7QJL10Ebh5QEKbAcZRfEZ8ETXdFx\nPlHPl9q/FG3yANOI1i60syWPtUMae4YU27Q1GvGFqHsAhmnTkWuvaifndTO1zDxLZ2I42S67zrKN\nJKrXS2mXUCW6tmMAg7qPhVFWRWyrnCMYwjkiLbRxSEfQAsuR17i0bGI6Fy/VNYT4ZYrxAIsO4w5O\ngPXsaz1fGDPTyNY8bUQrqz3Dhm2utG7Qv90Pe0Aq9MCdt/OwsHWsSTEygpWNuNcUOe0SqiRJ9Kqq\nVOGmFHhFjuyJ08khni8ctHFITiMElqNI49KykcuDrgHv6jLkL5BjhQVFMB5gPXsAH8CmDROLizz2\nDGu26RoVBnHb1ZbnQZ6i+g0RLb+G8bjk1EZKp+nJFnC9srqfqii7/qpou4Qq0ejVEYArdaw3lZix\ncwT7JTiyB3+MzwNtHDbTKIHlYw/aGKJA49KyidhFuANm7KDrJnw5V5iwVuoBprYOuwCmC2NubLp/\nVnuGpGiq8FJUoQwvoLjPSJVCKw8henbVTdl2CVVSde1VFrHrfVemZly8QEiXRm/m5582DutprMBy\nqNAaIKA6raSs6FyM7fJp2hcwxoJihhI9wFyqQVOIw1URraLsGdYhIt9pjPnOmNsz1WJlXkfJnYdF\nEnOBEXR3Xl7S2CU07bXQGscDAE8xG6YY+KRJF+b2Z1tOTyj0IqRpx+kioI3DahovsBxVGpeWiSe6\nooOugyuiz0NZHmC95cF9DmAnWgxflD3DOvQ93I7uw4tiFeYjF91+bJF9iQXxSU7+RZiNhiQysnz3\n6rZLqBJNwR8XGb0q8v3XC49UY6nKoOnHcB/aOMTTGoHl0IL4LdgDWGXGpWWyyi7CrJi52IYvbowF\nRZSVFhQ9OS9Y3QJwdXHxtcltz5CEVdYNUpGr9Yr0YU0WD5MRCrYLCT3FGJpdQlVk/XxXcQzzCtn3\ni4pahfa5qxPaOFymdQLLUbdxadms6FyMrdlo40EgjQdYT6Q/BV79d4En3wd8NoAnVxE50ND5O4wx\n74zcXvgVfoK1LLv6Ku48dLV6Zabw677ICNkuoSrSRGjXRaTK6CwsqpA9z+eqjcfhKLRxuEhrBZaP\nGpf2YSNarRJaPjGiC4g50Hfji37RguIr8U9f8dP4wBUA+Aw87i1/gFf+a1RgQSEie1Eh5XVYXa0q\nRbQionVxTFWOq/qNBqlemlALoEeRuxybBE0KZa0v7bZQsV1CE76zrsYQtpEjzg8u0XMoUmCVXchO\nLkMbhyWdEFiOEI1Ly0biB11fEF11X/1Xgb4OJwDwYuDbfgi3fg725BjXAVmYB5hL70atG7QWDGWK\nik3rcoLAG2Zr0+oZhFbaz4yI3IZe4XpXvTer6ppL85mPXLh8BoD/iobYJVSFvka34aXf666b00L2\nkT9HMNN2GLXKBG0cOiawHJ5x6azJBfFZiUlnrC24bcNBwrNneLTrbOrZAwBgfbQiXZvFeYBpV9UM\nWM40k+X4n8JsG/Lg3uMqOg/1ZLztimE1lToMpThWRJ4L4LHwGi5CeI9CRgXzfzHG/LO61wLAzRHM\nVGtYtzBsE123ceikwHJ4dVp9BG5cWiYr7CJWFuU2UXDF2TOoE/wurNjeTzNYWgWYe902eoCJyG5M\nV+EetBA/tNfT1aygoNSK7+auz3vX+/Mp7AG48pNaEruEkCK8Ie37XJSX9DnuiQyiFz4b12jT3lso\neY7g2jU08PhYJl22cei0wPJponFpmazqXESM6ArpBLSKdfYMOuewvyjI2T3eA+wDnw/88GOAl/2/\n8Cwo9Mr/tMqC901ECuJTdR6utIqQya4x4329zxls2sBF81yL903Yq12gPCsLV3fmaqdS2yXU6dlV\n5Xcryfc6GomtVQCqHUnaQnZGrMqnqzYOFFgRVGj1YdOHTAt4eKLrwqDrTemTEARXUnsGdYbfA3AI\nG9Uq0oRQRdetKXDvGHjyg4DfeyIweTHwJf8GeMJ/QQ4PsLIQkT5uwUV751k6sUQmI2PGp5qe3nWR\nRP1MbWP5uXI1aUdFWFkUZZeQ5jPc1BN2lnVrw8K8rlpC4EIN4WnSaGuhvloBHN+aQBdtHCiwVtAW\n49KyWWEXEetI7T2m6hb6LQB7SU/YGtHahX1eVwoWWduwV3F+ZOBCiiWPB1jZeJ2HtnMuYRrGdvJe\ndylZnzmAY9jo1f3GmJ69vyzc/1Ot72J9IbDhs1gFoXp25RUZ+rndRsnzBtd1FRZVyE6qoWs2DhRY\nG/CMS3XQajfrtNIgGwZdx9y/1BSjngj6aUPTPZHhwphpb1mjNi1CbK2wbrgN+xptXKMnwPy5llEP\nsNIsKERkgFuYY0PnYYbOwjMsI1gnxpi7N67jYhq7dLuEIsibYswqjAq2qnBdg9fqSg2mKWQvOqrY\n1ChlCHTJxoECKyFaV+NOKK0zLi0bWTPoesPjch+49cR9M2qVkIae3QZg04a5zQoBbLkaI/29D3vC\nyl17FPUAw+VZkIVYUNh9yQC3zhskcnUeyjI9C8S8DpFoqfsMtc4uoaiTd1kiwItC1FI76BnlrnRk\nDyVKSFbTBRsHCqwMdMW4tExWdC5uLDjOcrUPa89wd94DrtZn7QI4XOTshnHWDZFuNScwSjUg9QTY\nSg+wLPWHavEAxHQe6gXKIE1dWYwor8Uuoe6T9boIbx0NJnpiHBpjrpa5nzge9Vp5wfs+HR+reo4g\nI1bl0HYbBwqsHGir/gg2osWC+Jx4omvtoOvIY9bWt8TZMxSFCi5kjWjp2g79kyXswWaIGusT8niA\naUG8SxGfmvEyfeQK3Vc+boNdAok/0VfcWeguAiqNOnhzBGOjpHWLYJKdNts4UGAVQNeNS8tklV3E\nqkhGjMCaAPiFMr64XsfhHMDOIkN0JeqP5aVfEtVj1UECDzBrQTGRAT6K5+PB+ANcxxz4rO8F/vu3\nGWOOi7BL6ALrhMNqS4xyxIbXar/jp9vLju6ssl8oW1QxalUdbbVxoMAqEBqXVsOKzsVVg64v2DMU\nfdDUAvhdWB+t1AcGZ93gr93zFirFC6pMYj3AnvhbV/Gul34jPrl4AIA/A/ASZLRLqJoaOl4LP6kX\nkUZU4X8bVQ8pt2nnWRZHdtIs2mjjQIFVAnqSGcGeYFinVQExoguwJ/ntdfYMRbfQq8XDNmydVppR\nOtPIOl4E4F9Du7SyrqcuYuwSvhbA43E3/hCvxffj6370L3DfI/8nRJpGNDpWiwVFHYSQ2koi6tTv\nCmWk2mP3F1PI7q8zi8v7xn0yYlU7bbNxoMAqGRqX1oeI/CSARwP4ce/mtW38eVvovYjWLmy67EYS\na4cV1g1uO0F32SS1SxB5zrca8zOvBGBH8fwhruJhOPBraiIeYL7wcgX0pVpQFE0dReh5ibnoeA2A\nL0BFJzwd0wRcb1+HKNlMm2wcKLAqQq/MB7D1KZV2wHSVOHsGSTnoOmabiQRXb+lQfpjkSjtq3eDd\n7g40wVzNZbVLEJnsGTO+KCKXnYenuI55gm2475BGNSa7WHYZ1h4trqmrr7R9qMi/CdsoMvU//4V7\nSy0d2Y/N2Mwqeu0YtQqUNtg4UGBVDI1LqyGpPUOMXQSwoXMxup8kKcaerQVz43diD+gq/hBNCYrI\nLwH4q6hJZBVllxAZ+HxRiExkhD/DGA/EpIh6GxVezvvrOJKC3BgBS3pyb/MJ2uvuWjliqjDPLjfz\n8jqO00aMi5y0QMKi6TYOFFg1ETEuPabQKpY89gyrOheRQHStSjFqRGsX9j3fWWXtELVucNtERZ2F\nZdol+PVVKzvglqN4XIqo4DqbCx5gh57w2kNCD7A2iyqHLJ3aU53Y0tY0qq1HqjmCeejCe9c2mmzj\nQIEVADQuLR4ROYIVJIV8ISOiC8gYyfkpkef8A+C3FkvRNY9egUetG/S2UkRWlXYJTtwkqUVUoWXN\nWNVUspqU0cUIlxVe9/0q8MinAtgP9UKoSOHgtcyv7GLN+l5cMEmdyBAfxXU8GP9n0pmWpJs01caB\nAisgaFxaHFF7hpL2kXrQtc8HRH7kEcDfh6YOewCsIacVPTH1WM5n6ijrQUaWcyKBC3MLqzKqXG02\nGnv/ZV0OYE0myy+y3iBWNPq8i6X3174x43kWt/rQ8E5kh+vsGPIKLGe/gOvFfPbWDoRuQGMB2UwT\nbRwosALEq9Oa07g0PWp7sLfOnqHEffv1So6Vg67V1mEXwKBnBaFrSR8BeJAx5mci2091Jafb8Z3x\na3VHjy10T3ACVKF1bn2SpCB+9RrWF6LnOSF7JqwDqBeeVw7gUpBBCrAqogTRQvaYNTSu65JUR9Ns\nHCiwAkaFljtg07g0ISKyB2veGUQoOUZ0rUzDqc3DNoDDnhXZx57oGhhjZqtOhEntEurEL3TPvA0b\n/bAp9fHmjraivc6y4llQTCPCyzUP1PYdL1NceVGrEYChGWdP26doPqAwaylNsnGgwGoANC5NR5w9\nQ0jEdC6e1z4t7O8nAPBHwA89DPjTuDoYERk+HHj7B2xU5A0A8AjgjvcDP+lOLCF2WIlMdo0ZF1MX\nZ0/Y5zMPl/to9snVu7ByHHuCrL/u+5+lFiuNuEpoShrtEHVCsvBCdka8uksTbBwosBqGb7jIOq3L\nJLVnCI1I917/h4Ev/CrgiU8FfutdwPNga6W+GcBjsYx4fBjAy6EnxujJ5bki97zZmHe636Pu11EB\nVoUgWyewMtf17Mpz8Aw8HADcvLo2dot5Y4hcNNQJLyfIUpuwlp0WlMl5neJxmfVz51Eykf7Cjq0q\nxauLhEXoNg4UWA2FxqXx5LFnCA0Vi28C8CQA+Frg114LfLu5OLdwCBsun2FD4eemk02M4CpckBWS\nIlxv8bANYOo6D73HpI68NI1ol6YKryF0JmT0dZflsPL9PN22a96PbQDz6HtRJxRc7SNkGwcKrIaj\nB1F7UqHQKtyeoW60fuvgC4E7fg248+PAx/4Z8NofB96nd+kD+B0A366/Jy78zFublEWQ9XBrN1rk\nvmJty3b+lOvSlNRy6DotAC5gjxlv3gU+uQ3ccWjMj+3o7f2k0a+1NW+T83FR+1VErXJuI4j6PJKP\nUG0cKLBaAo1LLVXYM9SBiPQfDXzDzDq6z3yjUj24PB3ACwH8FQCvB/CmtK/BKpPUYp7BucAauQsB\nX4BpagcLTfMAgPs96/68zkNrn7EsiG/1STSBuece1PDWXBgjdZ6CvGAVsyzOv/fhxrz09Wv3becI\n9l2qtk6yDIQu+ztAyiNEGwcKrBaidVquTbwzB4c67RmqQKNZcyecevZEOYQdv6Mnw/PCz1cAEFy0\ni8jVUVjEyUZT23Pg+nzTtjalJNOkLOM6D5drEi0eb/eJ1Ou+GiFlYbDIs+8BvuXTASAmBTnE573n\nV/Gqb3oobHq2lIubEIRxCGsgqwnNxoECq8XoyWwImz5sfUF8aPYMZaD1BsfGmLlaOuzqz/HCmBt6\nH1f4eaEmQXIOuo5ZS+L0ytJiwkZDyvg8JhFgvVvnReLTxXVM4wRZG0+iWs93pL9uPPGkEdMykSE+\neNfTcdcHX2vGZh7per5U+9Um2MUYHiHZOFBgdYCuGJeGbs9QFCKy57ts68idkUsb2lQcXLj8GPY1\niRUSyDHoOm57ft0UVkSF0hS6l5WikYkMH/ZBPP0P78J9rgjbCTL3PDbVlIWO55s2wobPQuZ9pChk\nV+HlauPOj0WeP9gsGnEvQrDU9b5RbNVLCDYOFFgdIuKv06o6rabaM2TBGYrGCcneckjv4SuBN34b\n8Cr9006Sg0yMWSmQQHRtHi+z/Hucm3tdeJ2Hs1V1Q27tCSJkwQkyjeruIkGH1ar3MO52b1ZkIU0E\nnsgaRISXO1413paGgqt66rZxoMDqIG00Lm2TPUMStN5sFieaNKK1Cyscrvbsv1tYM7x3w74GAJ4B\nW0APaOeiMebN2dY+GQHXZ6F0b4nIALcwR4LOwzTrrFOQ6Xt2APsdj02TZPYdswavgzIK2VdaPshk\n6EYM+RF5AGhqVJ4F9dVQp40DBVbH8YxLp6HOSEtC2+wZkiAiu+ueb09kuNAT6ytEXvYq4MX32wLQ\njR02m+qrJMeg6ziz0U0nm6oEmGfxAHidh1WsoyhBpuL7AHYweGHdVGU6sheBJ7xmAODS0NERRfWt\nMB0UXMVRl40DBRYBcKEgvpFFqW21Z1iH1jltJxGVPZEjA/wvrwHm3wbcdT+w5RuWuu3lORlLwkHX\nIpPttFEHEbnHeK70VaD1RbYbN2Ywsa6rtshbVGDtizzqfwe+FzY6vb+wjQ8zrxYr9VrPHdKtI/so\nzxzBNfsoXUiITEaPxX0P/h088qGRFOQ2rBANdgi3g55d+ajDxoECi1ygicalbbdnWIceNPpRsRRH\nzwqg3V8C7nw68EUAfhTAS8s80MSIrjnwLx5qzHe9Oud2Kzu5uCHFsBYEa1/P0IBmAAAVO0lEQVTn\nuk56XtTKnTwueX6ljZC5x+PWuTt+o2ug1hGTgnTCaxBqCpIpxvRUbeNAgUVi8YxLAeunFeyVUhfs\nGdbhWzdsuJ/f5Tf8SuDwccDjvht4QVWdl/YA96IfBl7tolG57SLcdtO2y6c9IbnaI9iC+CDERsTb\nKtfImyjP/Zdyz89cxZdBa9KSCLKEa26kENBRRFuwIrYfiYT1Q4uANfV1LpsqbRwosMhGQjcu7Yo9\nwzpUZO6nERk9kdEHgX/3IeCuQ+Dd3wP8nSoOyCKTkd8RJpFB13rzys7FLCeOIiNLSToPvf2WdpLT\nxo5deFGrovaXpZDdF2BFibGyKHI9ngO+WpOcCy/nij9FAClIphgvUoWNAwUWSUyIxqVdsmeIEuM7\n9bXGmB9Iu513ibz0z4GvfwrwMAD7iAi1otHPEdZ9hmLsIuawXZOFfO6KOLlECuKPN9kVFCZ+bNp1\nD/Z1+S5jzE8keEyi53s+R/A2/p35juJq3qLPPUEBf1CCLA/RCJd3HJ3DCq8gLgy7KLjKtnGgwCKp\n8a7Majcu7ZI9QwKvqZXWDQm3P3o68IMvAx7z+8BPPd+Yf5R5sWv3k83NfUXn4jSL6NqUUkxzskkr\ntLKionNP97WvPygwMlea/UJeuiDIVHgN9Nd5pAsSVWcPuiK4yrRxoMAimQnBuLTN9gwZu752ARzm\nOTC+TeStQ+Da7wOf/A7gm/+9MbkK0uPI0kl4eRvShz0huUiXe86XOhfjHpvmNUoRAdrYeehtM1FE\nSy76mh0CeKMx5u3JVp4MXXdhtWV11/8kFWQ9kUctjHlvLYtMiGczcaHLWwWZ+9yXbkGR5yIkdMqy\ncaDAIrnxRmAAwGmVV1ptsmco6oC1yR8rCfsij3o48JofAZ7yDuAdsGnDwtLCZbq5r7CLyDXoOmYf\na98rmYgzLU0sWuJOYLDCagfWnT/3e3BpH8t6sv2yIm95SOsFtimytco7rMkRMs0oIJKCdBcflZVz\nNF1wlWHjQIFFCqVK49Km2zOUdZWfxrohwbb6AHbfCXzzx4EPvRl43Xcb8035t3ux0L1spOBB15Ft\nr7yyT2Px4D3eRaxcx9qrTUbX/LX7sSIQ0TmCWd3qRWSwAOZ5BM+m30OhCWOSonjCawh7IXyqtw/K\nuChuYlF90TYOFFikFKooiG+CPUNdYXUVn4UICMBGtL4QeNOXAPf8IPDBb7ACYKM1xOr1FRfBSlUv\n5ZluosBB15vWh1twtifrZh4O9T7bAN4G4OVFRg0BHQR+C4B9/w4X14GUgid4IREiIb+OXgpyBFvq\n4SJh/jiiQi7WmuDbVaSNAwUWKZUyjUtDtGcI6apNRPaMMYWm4d4qcs+/B77yx4F/+Bjgr/114Cd/\nBXhletuEyVYo3VOOmM5FoGDRJRMZ4IN4Du7Cn0IL4rXIdgvA6JHAT9xnOwNnwPLE6z5XeVNijz+U\n5/7OI/HQogrZQz1JNp1QBJlnQTH3hNc2rPAawEbCcmcqQvwcFWHjQIFFKqFo49JQ7BlCDnvrldhW\nGe3HAPDTIgdbwPYhgB8BfvadwFuS7quIIvcqiIguINK5mFXwyLPlHtyD7//r78EX/fEv4k/wEbwB\nwGE0xbZmXekbICZiPbOuYxbayYzkIwRBphEvICK8sqQgQ7lQzWvjQIFFKke/iEPkMC6ty54hxCut\ndWjtEYpONTl6IoMPAT/0EODLPx14758ADwFwDNvJuPKqL27gcxFk8FtKLZB6y44uR6JB107wwl5o\nDGFfp2PYlJ39PmzoPPS3lcJGwtmqHK4rZC/D/Z6ESdZmgCxoEb67SDl2x3znhwfbGbmp47a29GIe\nGwcKLFIbKrQydbpUZc8QcoQqKUVYNyTlySIveBHwdS8FHns/cD9URETFhytyz1Dzk7lIumRX9dhB\n1wBm0PQflkXr7jW5OPdv2Xk4NeN0KYlVz61o+wXSTcoSZF791ywivJwfWKKu9LKP01ltHCiwSO14\nVzhImjYqyp6hzd4uPkVYNwDJIkIAjhbA1bcBv/oS4C9/E3garPA4/QrgN98EvBG4NQSABa5P21Yk\nrQfjEYB/DODzAPw+rLD6OQBv3STysnQeRvbf16iYnVGZMCq2bntt/E6Q8ihDkPkG19jQpV7GcT2L\njQMFFgkGvZpxnSsrjUvz2DN0QVDFRXF6NjIyMsYcZkibrZwxt2YNI+gJfmHMsb5nI/0ZAA95G/C0\nDwBv/bdlpC+rfF+1TsuJIheFOoZGq5ygiulcXGsXoR5VWwDmaYrSnUAz4/WC2o96tfF7QJpDHkHm\nCa8ZbCRs6t2uNZMXzyVZP+9pbRwosEhwbDIuTWP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