{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# ============================================================\n",
    "#         Четвърта национална студентска Олимпиада \n",
    "#     по Компютърна Математика \"акад. Стефан Додунеков\"\n",
    "#              Бургаски Свободен Университет\n",
    "#                 13-15 ноемвери 2015 г.\n",
    "# ============================================================"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# ------------------------------------------------------------\n",
    "#   Работа с програмни модули в Python\n",
    "# ------------------------------------------------------------\n",
    "# import math           # math.sin(math.pi/5)\n",
    "# import math as m      # m.sin(m.pi/5)\n",
    "# from math import sin, cos, exp, log, pi, e # sin(pi/5)\n",
    "# from math import *    # използваме ДИРЕКТНО имената \n",
    "                        # на ВСИЧКИ обекти от модула math.\n",
    "# ------------------------------------------------------------\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Populating the interactive namespace from numpy and matplotlib\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "WARNING: pylab import has clobbered these variables: ['ldexp', 'isclose', 'isnan', 'degrees', 'pi', 'tanh', 'modf', 'nan', 'log10', 'floor', 'log', 'trunc', 'copysign', 'sqrt', 'cosh', 'fmod', 'tan', 'isfinite', 'ceil', 'cos', 'gamma', 'exp', 'log2', 'inf', 'fabs', 'hypot', 'isinf', 'expm1', 'log1p', 'radians', 'sinh', 'sin', 'e', 'frexp']\n",
      "`%matplotlib` prevents importing * from pylab and numpy\n"
     ]
    }
   ],
   "source": [
    "#   Задачи за група Б\n",
    "# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n",
    "from math  import * \n",
    "import pylab as pl\n",
    "import scipy as sp\n",
    "import numpy as np\n",
    "import sympy as sy\n",
    "%pylab inline   \n",
    "#-------------------------------------------------------------"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4.4263636316651169"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# ------------------------------------------------------------\n",
    "# Задача 1\n",
    "sqrt(5 + (3)**(1/3)) + (5 + sqrt(3))**(1/3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.5322601508095928"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# ------------------------------------------------------------\n",
    "# Задача 2\n",
    "# Вариант 1 - директно като формула (хамалско решение)\n",
    "1 + (1/(sqrt(2) + 1/(sqrt(3) + 1/(sqrt(4) + \n",
    "                  1/(sqrt(5) + 1/sqrt(6))))))\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.5322601508095928"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Вариант 2 - с for-цикъл\n",
    "z = sqrt(5) + 1/sqrt(6)\n",
    "for n in range(4,0,-1):\n",
    "    z = sqrt(n) + 1/z\n",
    "z"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.5321825076095439"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Обобщение за произволна стойност на n\n",
    "n = 1001\n",
    "z = sqrt(n+1) + 1/sqrt(n+2)\n",
    "for i in range(n,0,-1):\n",
    "    z = sqrt(i) + 1/z\n",
    "z"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "8.0646600508771371e-11"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# ------------------------------------------------------------\n",
    "# Задача 3\n",
    "# Вариант 1 - директно заместване\n",
    "n = 2015; s = sqrt\n",
    "s(n + s(n + s(n + s(n + s(n + s(n + s(n))))))) - \\\n",
    "      s(n + s(n + s(n + s(n + s(n + s(n))))))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "8.0646600508771371e-11"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Вариант 2 - с for-цикъл\n",
    "sq = sqrt(2015)\n",
    "for i in range(5):\n",
    "    sq = sqrt(2015 + sq)\n",
    "sqrt(2015 + sq) - sq"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.2912859970626636"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# ------------------------------------------------------------\n",
    "# Задача 4\n",
    "s = 0\n",
    "for i in range(1,51):\n",
    "    s = s + 1/i**i\n",
    "s"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "2.792117542621514 2.718281828459045\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "0.07383571416246903"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# ------------------------------------------------------------\n",
    "# Задача 5\n",
    "s = 0\n",
    "for i in range(1,2015):\n",
    "    n1 = i\n",
    "    n2 = i + 1\n",
    "    s = s + (n1/n2)**n1/n2\n",
    "print(s, e)\n",
    "s - e\n",
    "# Забележка: Тъй като при директно прилагане на формулата\n",
    "# за i-тия член n1**n1/n2**n2 се получава препълване, тя бе\n",
    "# преобразувана в еквивалентния вид ((n1/n2)**n1)/n2.\n",
    "# Така се избягна препълването!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# ------------------------------------------------------------\n",
    "# Задача 6\n",
    "n = sy.Symbol('n')\n",
    "sy.Sum(1/n**2, (n, 1, sy.oo)).evalf() < sqrt(e)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(1.64493406684823, 1.6487212707001282)"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sy.Sum(1/n**2, (n, 1, sy.oo)).evalf(), sqrt(e)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.00378720385190179"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sqrt(e) - sy.Sum(1/n**2, (n, 1, sy.oo)).evalf() "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "x"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# ------------------------------------------------------------\n",
    "# Задача 7\n",
    "x = sy.Symbol('x')\n",
    "expr = (x**4 + 7*x**3 + 21*x**2 + 63*x + 108)/ \\\n",
    "       (x**5 + 4*x**4 - 81*x - 324) *          \\\n",
    "       (3*x**5 + 10*x**4 - 81*x**2 - 270*x)/   \\\n",
    "       (3*x**3 + 19*x**2 + 57*x + 90)\n",
    "sy.simplify(expr)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(n**2 + 3*n + 1)**2"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# ------------------------------------------------------------\n",
    "# Задача 8\n",
    "n = sy.Symbol('n', integer = True)\n",
    "sy.factor(n*(n+1)*(n+2)*(n+3) + 1) # Факторизация на изразза"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[2, -1 + sqrt(2), -sqrt(2) - 1]"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# ------------------------------------------------------------\n",
    "# Задача 9\n",
    "x = sy.Symbol('x')\n",
    "sy.solve(x**3 - 5*x + 2, x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[0, 5]"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# ------------------------------------------------------------\n",
    "# Задача 10\n",
    "x = sy.Symbol('x')\n",
    "eq = sy.sqrt(1 + x*sy.sqrt(x**2 + 24)) - x - 1\n",
    "sol = sy.solve(eq, x); sol"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(0, 0)"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Проверка\n",
    "eq.subs(x,sol[0]), eq.subs(x,sol[1])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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9hM2bzVZG5UIMggbikkMA3DcIYfZllOEincfcZNiokM41a+yFjJo3B1q14ion\nU9eQbg8hm84OHXhUcV2dvt+xkT8AxCA0OMaM4cqSdetsK9FP3PIHaWzmEWwaBMD83Mom50JI07Qp\nGzad/YzEQ4gxcckhAFxeeM45wOuv29WTizD7MkqDoPOYmzQIhXTaNgjpPILJHILOkFEunbrnVhYP\nQYgM18NGQREPwR87d3LFmc1R3aZHK0fhIQD6J8oRDyHGxCmHALCHMHMmV+W4RtB9uX07sGsXz1Mc\nBbpzCKb6GeXTmS45jXJGrvqkDYKJa6imho2eztLNXDp1J5bFQxAio0MHYPBg4L33bCvRR7rltc2b\nW1B69OCbSdQG2na4CDDrIWzaxOd6FKOwdZeebt4sBiG2xCmHkGbSJDfzCEH3ZdThIp3HvKSESz9X\nr9a2ycPk0+mSQTBxDZkoOc2l04SHICEjITImTnTTIAQlrvmDNDbyCC4ZBBOYngchE/EQhMPELYcA\nAKNG8VNIZaUVOTkJui/nzYvWIOg+5qYMQj6dLhkEE9eQicZ2UeYQxEMQIqNRI+5tVAxeAhGwYEH8\nPQSTE+VkwwWD0KULh3Z0DupKs2EDj3OIAvEQhMPEMYcAuBk2CrIvv/gCaN2a+95Hhe5jbqp9hes5\nhKZNeUKjk05KaN+2CYMQRQ6hpgbYvdtOObAYhAbMhAlAMgns329bSTjinj8Aos8h1NZyqKZ79+h+\nMxem8ghx9RC2buWHm0YW7s5iEDQQxxwCwPXZJ53kVvlpkH0Zdf4A0H/MS0u5ykh36CSXzg0bOCTh\nQiPAbt2AN95Iat/uhg3R5RDat9fXz8jWoDRADEKDx9XyUz9UVAAjRthWEY6WLTl0YnqO4TQuhIvS\ndO3KN1PdROkhNGkCnHACN+oLi61BaYAYBC3ENYcAcB7BpTYWQfbl3LlAWZl+LfkwccxNhI1y6XTJ\nIHTrBhx/fELrNom4yiiqHAKgL48gHoJgjREjuO1D1BUuuti5ky/8gQNtKwlPlJVGrhkE3Z7Rzp38\n1N6ypd7t5kNXHkE8hJgT1xwCwIkrl6qN/O7LdP4g6kniTRxzE5VGuXS6ZhDmzUtq3aapcFG+4y4e\nglAUuGQQ/GIjXGSKKCuNXDMIunMIUeYP0oiHIACIdw4B4PLTd991o/zU776cO9dOQllyCPro1g3Y\nvTuhdZumDEKhHIIOgyAegmCVtm2BYcOAd96xrcQ/xVBhlKahegidO/NT8aFD+rZpouS0ELomyREP\nIebEOYfnE57aAAAYcUlEQVSQxpWwkZ99efAgsHQpj6WIGhPHvGNH/j/pnIoxm87du9kb1DlPQBhK\nSoBWrZJaewHZyiGIhyAUBXEcj7BwISdiW7SwrUQPSvH/x3Sl0dq13G7bpbkjOnTQW2lkI4egK6ks\nHkLMiXsOAeDE7K5d9iZ7T+NnX9rKHwDmjrnusFE2natXs+FxiUGDEloNgokxCED+464jqUxkr7Ed\nIAZBSKFU/LqfVlQUT4VRmijyCKtWAX37mv0Nv3Trxk/1uoirh7B7Nzf8s+X1ikHQQDHkEAAOG9ke\ntexnX9r0EEwdc90GIZvOVavc8xAOHEjGImSU77i3b8/5n9ra4Nu3mT8AxCAIGXz968CsWcC+fbaV\nFKauDvj8c/EQguBiyKh9e6CqSs+29u3jV9Tto0tKuJ9RmDEVNsNFgBgELRRDDgHg5mplZXbLT73u\nyxUruEVwlHMgZGLqmOtOKmfT6WLI6OyzE9oMQrrk1ETSvNBxD5tHsDVTWhpjBkEpdY9Sap1S6rPU\n69yMz+5WSi1XSi1WSk0wpUHwj2vN7nLx8cfAmDG2VeinVy++oR08aGb7RG6GjHr0ANat07MtG/mD\nNGHzCMXuITxARCNTrzcAQCk1GMDlAAYDmAjgYaVcKoDzT7HkEAD74xG86rRtEEwd8yZN+Oaoa67r\n+jq3bOE5EE44Qc/2dVFZmYyFQSh03MVDyE+2G/1FAJ4jokNEVAlgOYAifNaLJ8OHA3v3AsuX21aS\nnzlzgFNOsa3CDCbzCC6GiwDg+OOBAweAPXvCb8tUyakXxEPIz81KqQql1F+VUulnku4A1masU5Va\nFluKJYcAcNzVppfgRefBg9zldNQo83pyYfKY6zQI9XW6GC4CgLPOSqBHDz2JZZMeguQQ8qCUmqGU\nmpfxmp/69wIADwMoJaIyABsB/FaHYME8ro9anj+fb2qtWtlWYgaTHsLq1W56CACHytauLbxeIeKc\nQ6iuZqNii5IwXyair3tc9f8BeCX1dxWAnhmf9Ugty8qUKVPQp08fAECbNm1QVlZ22Eqn43m236eX\nuaIn2/v6WvOtP358AlOm8Dy3zZtHq7eiogK333573vWXLEnglFPisz/9vi8tTeC998zsz/ffBy68\nUP/+CPs+mUyipAR46y0+/8Jsb8OGBLp2tXN+btkCbNwYfPsrVgCdO/v/fjKZxBNPPAEAh++XgSAi\nIy8AXTL+vgPAM6m/hwCYC6ApgL4AVgBQObZBcaC8vNy2hIL41ZhIEE2dakZLPrzovOYaokceMa8l\nHyaPeUUF0dCherZVX+dZZxFNn65n2zopLy+nH/2I6Oc/D7+t4cOJPvkk/HayUei4z5pFNHZs8O33\n6EFUWRn8+2lS907f923F39WPUuopAGUA6gBUAriOiKpTn90N4DsAagDcRkTTc2yDTOkT8vO733Hz\nuL/+1baSYznpJOCpp4CRI20rMcPu3dwSeu9e/bX0ffsCM2YA/fvr3a4OHnmEBxs++mi47XTuzKPY\nu3XTo8sPq1YBZ5/NoTm/EAHNm/P0n82bh9OhlAIR+T57jBkEHYhBsMeqVcBXvsIdKBs5NHxx924e\ndLRjB5doFismbmo1NZx3SffLcY1XXgH+/Gdg2rTg2zh4kP+PX37JI4ejZu9erhL68kv/xnzbNjbY\nO3eG1xHUIDh0qceXzHiyq/jVWFrKCbKPPjKjJxeFdH72GU/mY9sYmD7mAwfyXA9hydS5Zg0nW100\nBslkUsvgtPXr2ZiaMgaFjnvLlnxuBrmpV1ezdpuIQRByctFFwMsv21ZxNLYHpEXFkCHAokV6t+ly\nhRGgp8po3Tqe68EmXbrwWAi/iEEoEtJZf5cJovHCC6M3CIV0fvyxGwPSTB/zIUOAxYvDbydTp6tj\nEADW2aEDN6Xbuzf4dtatY8NiCi/HPYxBiHraz/qIQRByMno0u77LltlWwhBxN9bTT7etxDyDB+v3\nEFw2CADH3Hv2BL74Ivg21q41axC80LWreAgNmmLMIQCcTI7aS8inc+VKoHFjIEyZtS5MH3NdIaNM\nnS6HjNI6e/fmXEdQTIeMvBz3Ll2CTfYjBkFwHpfyCO+9B4wb59ZcwKbo3p0rVbZt07fNlSvdNQhp\neveOv4cgOYQGTrHmEADga18DFizQM3m4F/LpTBsEFzB9zJXSk0dI6yTihoUDB4bXZoK0zrAGwbSH\nYDKHsHGjGATBcZo145nUwtSG6+Ldd4Ezz7StIjp05hGqq7kcsn17PdszRa9e4Q2CbQ8hTA5BkspF\nQLHmENJceCEwdao+LfnIpXPDBmD7dn5qdoEojrmOPEJa59KlwKBB4TWZQkcO4eBB7hZqsrGd5BCE\nBs955wEzZ3JM2xbvvcfVRS6NmjaNrtJTgCvFXA0XZRImZLRhA99QGzfWq8kvQQwCkRsGQVpXCJ4Y\nPx644Qbg0kvt/P4tt3A44Qc/sPP7Nli1CkgkwlXdpPn+97mlwl13hd+WSWpqeLTv3r3+R6PPmsXn\nx+zZZrR5pa6OexHt2uW9J9HmzezB6SoikNYVglGuuAJ44QV7v+9SQjkqevcGtm7l3kNhcT1klKZJ\nE35KDjJRzhdf8D6zTaNG3INq/Xrv36mq4soy24hB0ECx5xAA4OKLgTfeCDeK1AvZdO7YwSWTLnU3\njeKYN27MN/ElS4JvI245BCB42CiKcRZej3v37v6M2vr1YhCEGNGhAzB2LPDqq9H/9vvvc7sKF5uy\nmUZHpVFNDYed+vXTo8k0paUcLvNLZaUbgxYB+G7UJx5CEVHM4xAyueIK4Pnnw2vJRzadb73FPeZd\nIqpjHrbSKJFIYOVKvkE1a6ZPl24y92fQKUSj8BC8HncxCELRM3ky35x1xLT9MGMGj4VoiOgoPV2w\ngCcVigsuGwSv9OjhL2RUVWVnQp/6iEHQQEPIIQBAu3bAGWfwRCamqK+zqorjq6NGmfvNIER1zE86\nCZg/P/j3k8lkLAxC5v4MYhAOHeJzpVcvvbrq4yeH4MdDkByCEEsuv9x82CiTt97i9hm2a8tt0a8f\nVxqFKUeMg0HIJIhBqKriCZ1cCYvFNWQk4xAEX+zezb1ili/nC9A03/oWt6u47jrzv+UqZ5wB/Pzn\nwFlnBfv+iScC//hHfIwCEdC6NTeqa9PG23eSSeC//5vLk11gzRqegtarUejYkQ23roFpMg5BiITj\nj+dWFk8/bf63Dh3iUtfzzjP/Wy4zYgTPrxyE/fu5hDMOo5TTKOXfS3ApfwBw+4xNm/gcLsSBAzzv\nSBQPWIUQg6CBhpJDSDNlCvDEE9o2dxSZOmfP5piw7WZl2YjymIcxCH//exL9+7tfslt/fwYxCFGU\nnHo97ulGgtXVhdfdsIHbXbjQlsUBCULcSCR4sFhFhdnfmTYNuOACs78RB8rKghuE1avjEyrKxK9B\nWLnSvdngvM4R7Ur+ABCDoIWGMg4hTaNGwNVXA48/rm2Th8nU+corwPnn6/8NHUR5zIcO5Rvevn3+\nv7t/fwLDhunXpJv6+3PAAH9Tty5dyrkS0/g57n378mC5QlRWutFyAxCDIATk6quBZ5/llsMmWLyY\n46qjR5vZfpxo1ozHIwTxEj75hEd5xw0/nV6J3GzN4XXEtUv5DzEIGmhoOQSAT/YhQ/TPk5DW+eKL\nwGWXuRFXzUbUx3zsWODDD/19p64OmDMn6VQPqFzU35/plh1eigzXrweOOw5o29aMtkz8HPe+fcUg\nCA2IG24AHnrIzLZffJHHPAjMaaf5NwjLlnHZZrt2ZjSZpF07boPtZbSvi94BwA9Nq1cXXk8MQpHR\n0HIIaS69lE/mzz7Tt81EIoHFi3l2tNNO07dd3UR9zIN4CJ98AowblzCiRzfZ9ufgwd7CRkuWRJM/\nAPwddwkZCQ2KkhLg5puB3/9e73afegq48kp3w0U26NePZ6zz0x/nk0/inYPx2sfJVQ+hVy8OZ9XU\n5F7n0CFex3TLDa/IJaeBhphDSPPd73J5aJA5ZLPx9ttJPPUUcM01erZniqiPuVL+vYSPPwZKSpLG\nNOkk2/70ahCi9BD8HPcmTXiAWr7S07VreXSyK+NExCAIoWjbFvjmN4GHH9azvTlzuH576FA92ysm\nxo3jFg1e2LsXmDePwy5xxY9BcNFDAAqHjVwKFwHSy0jQwLJl3G9n9WpOBIZh8mRg4sSG3bsoF599\nxr2dvMTVZ8wA/ud/3OntE4QtW4D+/bmxX67w4datfEPdscPNEOO11wJjxgDf+172zx97jI+R7pH/\n0stIsMbAgdx47Y9/DLedFSt4drRvf1uPrmKjrIwnY/fSMC2Z5BHlcaZDB/ZAV6zIvc5nn3FrDxeN\nAcAeQj79rnkIju7GeNGQcwhpfvpT4Le/BXbtCr6NBx8EJkxIhvYyosDGMW/UiFuBv/124XXTBiEO\n5yaQW+eYMZwLyUXUiXO/+3PIEGDhwtyfuxbuEoMgaGHwYGDSJODee4N9v7qaO6hOnqxXV7Exfjww\nfXr+dXbvBj7/3O2yXa8UMgiffure5EmZnHxy/gmO5s3jdVxBcgiCNjZs4JN79mzuReOHW2/lJ2Dd\nJazFRlUV7+MNG3JPBvP88xyTfv31SKUZ4b33gB/8IHd1Vd++wJtvutveu66O53ZYt+7YuR2+/JI7\nou7axRVJOpEcgmCdrl2Bu+7iEcx1dd6/t3Ilewc/+pE5bcVC9+7AsGE8T0QuXnwR+MY3otNkkpEj\n+Qk7W8+s9Exy/ftHr8srjRpxxdyCBcd+tmgRGzLdxiAMoQyCUuoypdQCpVStUmpkvc/uVkotV0ot\nVkpNyFg+Uik1Tym1TClVFM+DcYjTRqXx9tuBPXuAP/3J2/pEXIHxwx8CnTrFY18CdnVeeSXw3HPZ\nP9u7lyuMLrqI38d9f7Zsyd7mp58e+9n773PjvigTykH257Bh2cNG8+e7FS4CwnsI8wFcDOCdzIVK\nqcEALgcwGMBEAA8rpdLuyyMAvkNEAwEMVEqdE1KD4BAlJcDf/sYlj7NnF17/4Yc55v2f/2leW7Fw\n6aUcDtqx49jPpk7lAWzt20evyxQTJ/Lgx/q8+irnrVwnVx7BRYMAIgr9AlAOYGTG+7sA/DDj/esA\nTgXQBcCijOVXAngkz3ZJiCevvkrUpQvRokW513nzTaLOnYmWLo1OV7EwZQrRPfccvay2lmjYMKKp\nU61IMsbs2URDhx69rK6OqFu3eJw7M2cSnX76scvHjyd67TUzv5m6d/q+l5tytroDyBywXZVa1h1A\nZhX1utQyociYNAm47z4ufcwW7372WR5k9eKL7iYEXeYnP+FxH9u3H1n28svsobk6qVBQxozhQWqZ\nM6hVVHA4KQ7nzsiRXPWVOcFRbS2PoRg+3J6ubJQUWkEpNQNA58xFAAjAj4noFVPC0kyZMgV9UpOl\ntmnTBmVlZYc7Dqbjebbfp5e5oifb+/pao/j9nj2TuOsu4KabEujXD+jXjz9ftCiB9euBe+9NorYW\nAI58v6KiArfffnvk+8fvexv7M/N9v37AuHFJnHsu8O67CaxbB1x3XRJ33gkodWT9YtmfF1wA/PrX\nSVx1Fb9/8UVg+PAkeLxFdHqD7s8RI4CHHkpizBh+P2cO0Lp1EsuWAd266dl/T6SGO/cJM7l0ELei\n/guFQ0Zv4EjIaHHG8qIIGZWXl9uWUBCbGvfvJ3rmGaKbbiK64w6i558nOngw+7px2JdEbuisqSG6\n9FKiXr2I2rUj+vOfj13HBZ1eKKRz/nyijh2JqquJ1qwhat+eaNWqaLRlEnR//vKXRLfdduT9T35C\n9MMf6tGUDQQMGWkZh6CUKgfwfSL6NPV+CICnU0agO4AZAAYQESmlPgRwK4A5AF4F8CARZS2ik3EI\ngpCf2loe7dq8ObfILma+/32uNqqpAc4+G/jZz2wr8s7cudwEcskSfj9yJI+5OfNMM78XdBxCKIOg\nlJoM4CEAHQDsAFBBRBNTn90N4DsAagDcRkTTU8tHAXgCQHMArxHRbXm2LwZBEAQAXM786KOcJ7nu\nOqBFC9uKvFNXx2NIpk3j2eBGjQI2beL/iwmCGgQtISNTL0jISBtx0EgkOnUjOvUSRuezzxKVlhIN\nGUL0m9/o05QNBAwZGbJPgiAIQiZXXsmty4mAO++0rSY70stIEAShyJBeRoIgCEIoxCBoILOG2lXi\noBEQnboRnXqJi86giEEQBEEQAEgOQRAEoeiQHIIgCIIQCjEIGohDXDEOGgHRqRvRqZe46AyKGARB\nEAQBgOQQBEEQig7JIQiCIAihEIOggTjEFeOgERCduhGdeomLzqCIQRAEQRAASA5BEASh6JAcgiAI\nghAKMQgaiENcMQ4aAdGpG9Gpl7joDIoYBEEQBAGA5BAEQRCKDskhCIIgCKEQg6CBOMQV46AREJ26\nEZ16iYvOoIhBEARBEABIDkEQBKHokByCIAiCEAoxCBqIQ1wxDhoB0akb0amXuOgMihgEQRAEAYDk\nEARBEIoOySEIgiAIoRCDoIE4xBXjoBEQnboRnXqJi86giEEQBEEQAEgOQRAEoeiQHIIgCIIQCjEI\nGohDXDEOGgHRqRvRqZe46AyKGARBEAQBgOQQBEEQig7JIQiCIAihCGUQlFKXKaUWKKVqlVIjM5b3\nVkp9qZT6LPV6OOOzkUqpeUqpZUqp34f5fVeIQ1wxDhoB0akb0amXuOgMSlgPYT6AiwG8k+WzFUQ0\nMvW6MWP5IwC+Q0QDAQxUSp0TUoN1KioqbEsoSBw0AqJTN6JTL3HRGZRQBoGIlhLRcgDZYlXHLFNK\ndQFwPBHNSS16CsDkMBpcYMeOHbYlFCQOGgHRqRvRqZe46AyKyRxCn1S4qFwpdUZqWXcA6zLWWZda\nJgiCIFimpNAKSqkZADpnLgJAAH5MRK/k+Np6AL2IaHsqt/B/SqkhodU6SmVlpW0JBYmDRkB06kZ0\n6iUuOoOipexUKVUO4E4i+izf52BDUU5Eg1PLrwTwVSK6Icf3pOZUEAQhAEHKTgt6CD44/ONKqQ4A\nthFRnVKqFEB/AKuIaIdSaqdSagyAOQD+HcCDuTYY5D8kCIIgBCNs2elkpdRaAGMBTFNKvZ766EwA\n85RSnwF4AcB1RJTOxtwE4DEAywAsJ6I3wmgQBEEQ9OD0SGVBEAQhOqyPVFZKnauUWpIaqPbDHOs8\nqJRarpSqUEqVRa0xpSGvTqXUVUqpz1OvWUqpk13UmbHeKUqpGqXUJVHqy/h9L8c9oZSamxr8WB61\nxpSGQse9tVJqaurcnK+UmmJB42NKqWql1Lw867hwDeXV6dA1VHB/ptazfQ15Oe7+riEisvYCG6QV\nAHoDaAKgAsCJ9daZCODV1N+nAvjQUZ1jAZyQ+vtcV3VmrPc2gGkALnFRJ4ATACwE0D31voOjOu8G\ncG9aI4CtAEoi1nkGgDIA83J8bv0a8qjT+jXkRWfGuWHtGvK4P31fQ7Y9hDHgPMIXRFQD4DkAF9Vb\n5yLwADYQ0UcATlBKdUa0FNRJRB8S0c7U2w9hZ3yFl/0JALcA+AeATVGKy8CLzqsAvEREVQBARFsi\n1gh400kAjk/9fTyArUR0KEKNIKJZALbnWcWFa6igTkeuIS/7E7B/DXnR6fsasm0QugNYm/E+20C1\n+utUZVnHNF50ZnItgNfzfG6KgjqVUt0ATCaiR5B9hHkUeNmfAwG0Sw1snKOU+rfI1B3Bi84/Ahii\nlFoP4HMAt0WkzQ8uXEN+sXUNFcSRa8gLvq8hnWWnAgCl1FkArgG7cy7yewCZsXBXT+gSACMBfA1A\nSwCzlVKziWiFXVnHcA6AuUT0NaVUPwAzlFLDiGiPbWFxRa4hbfi+hmwbhCoAvTLe90gtq79OzwLr\nmMaLTiilhgH4C4BziaiQy2kCLzpHA3hOKaXAMe+JSqkaIpoakUbAm851ALYQ0X4A+5VS7wIYDo7p\nR4UXndcAuBcAiGilUmo1gBMBfBKJQm+4cA15woFryAsuXENe8H8N2UiGZCQ9GuNI0q4pOGk3uN46\nk3AkITYWdpK1XnT2ArAcwFiX92e99R+HnaSyl/15IoAZqXWPA3fWHeKgzj8BuCf1d2dwaKadhX3a\nB8D8HJ9Zv4Y86rR+DXnRWW89K9eQx/3p+xqy6iEQUa1S6mYA08H5jMeIaLFS6jr+mP5CRK8ppSYp\npVYA2At+InNOJ4D/BtAOwMOpJ4caIhrjoM6jvhKlvsM/6u24L1FKvQlgHoBaAH8hokWu6QTwCwBP\nZJT+/RcRbYtSp1LqGQAJAO2VUmsA3AM2YM5cQ150woFryKPOTKwN5PJw3H1fQzIwTRAEQQBgv8pI\nEARBcAQxCIIgCAIAMQiCIAhCCjEIgiAIAgAxCIIgCEIKMQiCIAgCADEIgiAIQgoxCIIgCAIA4P8D\n+4jBoPP8HN8AAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x57e3b00>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# ------------------------------------------------------------\n",
    "# Задача 11\n",
    "from numpy import *\n",
    "x = arange(0, pi/2, pi/500)\n",
    "y = 100*cos(10*x**2) - x**4 - 1\n",
    "pl.plot(x, y); pl.grid()\n",
    "# Графиката пресича 8 пъти абсцисната ос --> 8 корена"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 7.72525184])"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# ------------------------------------------------------------\n",
    "# Задача 12\n",
    "# За избягване на безкрайните нараствания на tan(x)\n",
    "# преобразуваме уравнението, като умножим двете му\n",
    "# части на cos(x) --> x*cos(x) - sin(x) = 0\n",
    "from scipy.optimize import fsolve # Зареждаме функцията fsolve\n",
    "f = lambda x: x*cos(x) - sin(x)\n",
    "fsolve(f, 8)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "3\n",
      "7\n",
      "11\n",
      "13\n",
      "21\n",
      "33\n",
      "37\n",
      "39\n",
      "77\n",
      "91\n",
      "111\n",
      "143\n",
      "231\n",
      "259\n",
      "273\n",
      "407\n",
      "429\n",
      "481\n",
      "777\n",
      "1001\n",
      "1221\n",
      "1443\n",
      "2849\n",
      "3003\n",
      "3367\n",
      "5291\n",
      "8547\n",
      "10101\n",
      "15873\n",
      "37037\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "30"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# -------------------------------------------------------------\n",
    "# Задача 13\n",
    "n  = 0   # брояч на делителите\n",
    "nn = 2   # числото, проверявано за делител\n",
    "# N = 1691146334800   # Защо се задават такива астрономически числа???\n",
    "                      # Компютърът работи над 3 часа безрезултатно!\n",
    "N = 111111\n",
    "while nn <= N/2:\n",
    "    if N % nn == 0:\n",
    "        print(nn)\n",
    "        n = n + 1\n",
    "    nn = nn + 1\n",
    "n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "3\n",
      "7\n",
      "11\n",
      "13\n",
      "21\n",
      "33\n",
      "37\n",
      "39\n",
      "77\n",
      "91\n",
      "111\n",
      "143\n",
      "231\n",
      "259\n",
      "273\n",
      "407\n",
      "429\n",
      "481\n",
      "777\n",
      "1001\n",
      "1221\n",
      "1443\n",
      "2849\n",
      "3003\n",
      "3367\n",
      "5291\n",
      "8547\n",
      "10101\n",
      "15873\n",
      "37037\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "30"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Вариант с for-цикъл\n",
    "n  = 0   # брояч на делителите\n",
    "N = 111111\n",
    "N2 = N//2 + 1 # Горна граница на делителите\n",
    "for nn in range(2, N2):\n",
    "    if N % nn == 0:\n",
    "        print(nn)\n",
    "        n = n + 1\n",
    "n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "97"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# -------------------------------------------------------------\n",
    "# Задача 14\n",
    "n_prime = 0  # брояч на простите числа\n",
    "for n in range(1,2016):\n",
    "    if sy.isprime(3*n**5 + 11):\n",
    "        n_prime += 1\n",
    "n_prime"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[183, 213, 208, 189, 197, 207, 202, 199, 203, 214]\n"
     ]
    }
   ],
   "source": [
    "# -------------------------------------------------------------\n",
    "# Задача 15\n",
    "Pi = int(sy.N(sy.pi,2020)*(10**2019))//10**5 \n",
    "nn = [0,0,0,0,0,0,0,0,0,0] # Списък с начални стойности на броячите\n",
    "for i in range(2015):\n",
    "    n = Pi % 10 # извличаме последната цифра\n",
    "    nn[n] += 1\n",
    "    Pi = Pi//10 # премахвме последната цифра\n",
    "print(nn)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[183, 212, 207, 189, 197, 206, 202, 199, 203, 213]\n"
     ]
    }
   ],
   "source": [
    "# Вариант 2 ***???***\n",
    "Pi2020 = str(sy.N(sy.pi,2014))\n",
    "nn = [0,0,0,1,0,0,0,0,0,0] # Списък с начални стойности на броячите\n",
    "for i in range(2,2012):\n",
    "    nn[int(Pi2020[i])-int('0')] += 1\n",
    "print(nn)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "x**6 - 9*x**5 + 33*x**4 - 65*x**3 + 74*x**2 - 46*x + 12"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# -------------------------------------------------------------\n",
    "# Задача 16\n",
    "x = sy.Symbol('x')\n",
    "sy.expand((x-2)*(x-3)*(x-1-sy.I)*(x-1+sy.I)*(x-1)**2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "matrix([[0, 0, 0],\n",
       "        [0, 0, 0],\n",
       "        [0, 0, 0]], dtype=object)"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# -------------------------------------------------------------\n",
    "# Задача 17\n",
    "A = np.matrix([[2,0,0],[5,-3,-15],[-1,1,5]]); \n",
    "A**2015 - 2**71*A**1944"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-a**3*b/(a**3 + b**3) + a**3*c/(a**3 + c**3) + a*b**3/(a**3 + b**3) - a*c**3/(a**3 + c**3) - b**3*c/(b**3 + c**3) + b*c**3/(b**3 + c**3)"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# -------------------------------------------------------------\n",
    "# Задача 18\n",
    "a, b, c = sy.symbols('a,b,c')\n",
    "A = sy.Matrix([[a, b, c],[a**3, b**3, c**3],[1/(b**3 + c**3), \\\n",
    "            1/(a**3 + c**3), 1/(a**3 + b**3)]]); A\n",
    "sy.det(A)\n",
    "# Невъзможност за опростяване до израз, позволяващ  \n",
    "# бързо и лесно определяне на знака му ;=("
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{x: 3/(a - 1), y: -6/(a - 1)}"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# -------------------------------------------------------------\n",
    "# Задача 19\n",
    "x, y, a = sy.symbols('x,y,a')\n",
    "eq1 = 2*x + (2-a)*y - 6\n",
    "eq2 = (a+1)*x + y - 3\n",
    "sol = sy.solve([eq1, eq2], [x, y]); sol"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Проверка\n",
    "eq1.subs(x, sol[x]).subs(y, sol[y]).simplify() "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "eq2.subs(x, sol[x]).subs(y, sol[y]).simplify()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Вариант с еднократно викане на метода subs\n",
    "eq1.subs([(x,sol[x]),(y,sol[y])]).simplify() "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "eq2.subs([(x,sol[x]),(y,sol[y])]).simplify()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Едновременна проверка на двете уравнения\n",
    "a, b = sy.symbols('a b') \n",
    "(a*eq1 + b*eq2).subs([(x,sol[x]),(y,sol[y])]).simplify()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# Полученото решение е в сила само за а различно от 0 и 1 !\n",
    "#\n",
    "# При а = 0 получяваме системата:\n",
    "# 2*x + 2*y = 6\n",
    "# x + y = 3\n",
    "# Уравненията са линейно зависими.\n",
    "# От второте уравнение получаваме безброй много решения:\n",
    "# x = t, y = 3 - t\n",
    "# където t е произволен параметър.\n",
    "#\n",
    "# При а = 1 получаваме системата:\n",
    "# 2*x + y = 6\n",
    "# 2*x + y = 3,\n",
    "# която няма решение - уравненията са несъвместими!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[(1, -2, -1, -7), (1, -2, -1, 7)]"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# -------------------------------------------------------------\n",
    "# Задача 20\n",
    "x,y,z,xc,yc,zc,r = sy.symbols('x,y,z,xc,yc,zc,r')\n",
    "sf = (x - xc)**2 + (y - yc)**2 + (z - zc)**2 - r**2\n",
    "eq1 = sf.subs([(x, 3),(y, 1),(z, 5)])\n",
    "eq2 = sf.subs([(x, 4),(y,-8),(z, 1)])\n",
    "eq3 = sf.subs([(x,-5),(y, 1),(z,-3)])\n",
    "eq4 = 2*xc + yc - zc - 1\n",
    "sol = sy.solve([eq1,eq2,eq3,eq4],[xc,yc,zc,r]); sol"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(x - 1)**2 + (y + 2)**2 + (z + 1)**2 - 49  = 0\n"
     ]
    }
   ],
   "source": [
    "print (sf.subs([(xc, 1),(yc,-2),(zc,-1),(r,7)]),' = 0')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "2\n",
      "8\n",
      "188\n",
      "106028\n",
      "33725810348\n"
     ]
    }
   ],
   "source": [
    "# -------------------------------------------------------------\n",
    "# Задача 21\n",
    "f = 2\n",
    "print(f)\n",
    "for i in [1,2,3,4]:\n",
    "    f = 3*f**2 - 4\n",
    "    print(f)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "2  8  188  106028  33725810348  "
     ]
    }
   ],
   "source": [
    "# Вариант с извеждане на числата на 1 ред\n",
    "f = 2\n",
    "print(f, end = '  ')\n",
    "for i in [1,2,3,4]:\n",
    "    f = 3*f**2 - 4\n",
    "    print(f, end = '  ')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-oo*sign(a**n - exp(n*log(a)))"
      ]
     },
     "execution_count": 47,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# -------------------------------------------------------------\n",
    "# Задача 22\n",
    "x = sy.Symbol('x')\n",
    "a = sy.Symbol('a', real = True)\n",
    "n = sy.Symbol('n', integer = True)\n",
    "n = sy.Symbol('n', positive = True)\n",
    "sy.limit((x**n - a**n - n*a**(n-1)*(x-a))/(x-a)**2, x, a)\n",
    "# Не може да я реши ;=(   \n",
    "# Maxima я решава, без никакви assumptions за a и за n."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Екстремална точка x =  2.00000000000000\n",
      "f_min = f(2) =  -4.76220315590460\n"
     ]
    },
    {
     "data": {
      "image/png": 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L+hRnxzbdFI46qrTbbGxsLO0GS0z5ZVvM+cWcWz4KKvLuvqTVYlegOff7oUC9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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x86a4390>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# -------------------------------------------------------------\n",
    "# Задача 23\n",
    "x = sy.Symbol('x')\n",
    "def f(x):\n",
    "    X = abs(x)\n",
    "    return sign(x)*X**(5./3) - 5*X**(2./3)\n",
    "def fp(x):\n",
    "    return sy.diff(x**(5./3) - 5*x**(2./3), x)\n",
    "sol = sy.solve(fp(x), x)\n",
    "print('Екстремална точка x = ', sol[0])\n",
    "xx = np.linspace(-5,5,200)\n",
    "y = f(xx)\n",
    "pl.plot(xx, y); pl.grid() \n",
    "f_min = f(sol[0])\n",
    "print ('f_min = f(2) = ', f_min)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# Расте в интервалите (-оо,0] и [2, oo). Намалява в инт. [0,2]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2001"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# -------------------------------------------------------------\n",
    "# Задача 24\n",
    "fmax = -1.e100\n",
    "for n in range(100000):\n",
    "    f = n**2/1.001**n\n",
    "    if f > fmax:\n",
    "        fmax = f\n",
    "        nmax = n\n",
    "nmax"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[0.0, 2000.99983341683]"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Вариант с нулиране на производната\n",
    "n = sy.Symbol('n')\n",
    "fp = sy.diff(n**2/1.001**n, n)\n",
    "sol = sy.solve(fp, n); sol"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "541882.383856011\n",
      "541882.519191272\n"
     ]
    }
   ],
   "source": [
    "# Проверка за съседните две цели числа:\n",
    "print((n**2/1.001**n).subs(n, 2000))\n",
    "print((n**2/1.001**n).subs(n, 2001))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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0FayIFC+djF2Du+6CnXfObpFXHzQ+ymW8lM/0ylSP/j//gd//PgyrFBGRIFOt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\nSUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x9d05978>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "f = lambda n: n**2/1.001**n\n",
    "x = arange(1990, 2010)\n",
    "pl.plot(x, f(x)); grid()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      " Стойност на производната \n",
      "-2.22044604925031e-16\n",
      "0\n",
      "0\n",
      "0\n",
      "0\n",
      "0\n",
      "-1.11022302462516e-16\n",
      "0\n",
      "2.22044604925031e-16\n",
      "0\n",
      "0\n",
      "-1.11022302462516e-16\n",
      "1.11022302462516e-16\n",
      "-1.11022302462516e-16\n",
      "-1.11022302462516e-16\n",
      "-2.22044604925031e-16\n",
      "-2.22044604925031e-16\n",
      "-1.11022302462516e-16\n",
      " Стойност на функцията \n",
      "0.785398163397448\n",
      "0.785398163397448\n",
      "0.785398163397448\n",
      "0.785398163397448\n",
      "0.785398163397448\n",
      "0.785398163397448\n",
      "0.785398163397448\n",
      "0.785398163397448\n",
      "0.785398163397448\n",
      "0.785398163397448\n",
      "0.785398163397448\n",
      "0.785398163397448\n",
      "0.785398163397448\n",
      "0.785398163397448\n",
      "0.785398163397448\n",
      "0.785398163397448\n",
      "0.785398163397448\n",
      "0.785398163397448\n",
      "0.785398163397448  =  pi/4\n"
     ]
    }
   ],
   "source": [
    "# -------------------------------------------------------------\n",
    "# Задача 25\n",
    "x = sy.Symbol('x')\n",
    "def f(x):\n",
    "    return sy.atan(sy.sqrt((1+x)/(1-x))) - 1./2*sy.asin(x)\n",
    "sy.diff(f(x), x).simplify() # Не може да го упрости до 0\n",
    "print (' Стойност на производната ')\n",
    "for n in np.arange(-0.9,0.9,0.1):\n",
    "    print (sy.diff(f(x),x).subs(x, n))\n",
    "print (' Стойност на функцията ')\n",
    "for n in np.arange(-0.9,0.9,0.1):\n",
    "    print (f(x).subs(x, n))\n",
    "print (sy.N(sy.pi/4), ' = ', sy.pi/4 )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "-2.22044604925031e-16\n",
      "0\n",
      "0\n",
      "0\n",
      "0\n",
      "0\n",
      "-1.11022302462516e-16\n",
      "0\n",
      "2.22044604925031e-16\n",
      "0\n",
      "0\n",
      "-1.11022302462516e-16\n",
      "1.11022302462516e-16\n",
      "-1.11022302462516e-16\n",
      "-1.11022302462516e-16\n",
      "-2.22044604925031e-16\n",
      "-2.22044604925031e-16\n",
      "-1.11022302462516e-16\n"
     ]
    }
   ],
   "source": [
    "for n in np.arange(-0.9,0.9,0.1):\n",
    "    print sy.diff(f(x),x).subs(x, n)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# Първата производна в интервала (-1,1) е нула => f = const"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 63,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.785398163397448\n",
      "0.785398163397448\n",
      "0.785398163397448\n",
      "0.785398163397448\n",
      "0.785398163397448\n",
      "0.785398163397448\n",
      "0.785398163397448\n",
      "0.785398163397448\n",
      "0.785398163397448\n",
      "0.785398163397448\n",
      "0.785398163397448\n",
      "0.785398163397448\n",
      "0.785398163397448\n",
      "0.785398163397448\n",
      "0.785398163397448\n",
      "0.785398163397448\n",
      "0.785398163397448\n",
      "0.785398163397448\n"
     ]
    }
   ],
   "source": [
    "# Пресмятаме функцията за точки в интервала (-1,1)\n",
    "for n in np.arange(-0.9,0.9,0.1):\n",
    "    print f(x).subs(x, n)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 64,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.785398163397448  =  pi/4\n"
     ]
    }
   ],
   "source": [
    "print sy.N(sy.pi/4), ' = ', sy.pi/4 "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# Наистина в указания интервал f = const = pi/4"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 66,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# -------------------------------------------------------------\n",
    "# Задача 26\n",
    "# x = sy.Symbol('x')\n",
    "# sy.integrate(x/(sy.sin(x) + 2015), (x, 0, 2015))\n",
    "# Невъзможност за аналитично решение. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1007.4996731118351\n",
      "0.004238078663433953\n"
     ]
    },
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "C:\\Users\\Yordan Yordanov\\Anaconda3\\lib\\site-packages\\scipy\\integrate\\quadpack.py:357: IntegrationWarning: The maximum number of subdivisions (50) has been achieved.\n",
      "  If increasing the limit yields no improvement it is advised to analyze \n",
      "  the integrand in order to determine the difficulties.  If the position of a \n",
      "  local difficulty can be determined (singularity, discontinuity) one will \n",
      "  probably gain from splitting up the interval and calling the integrator \n",
      "  on the subranges.  Perhaps a special-purpose integrator should be used.\n",
      "  warnings.warn(msg, IntegrationWarning)\n"
     ]
    }
   ],
   "source": [
    "# Числено решение:\n",
    "from scipy.integrate import quad\n",
    "f = lambda x: x/(sin(x) + 2015)\n",
    "val, abserr = quad(f, 0.00001, 2015)\n",
    "print (val)\n",
    "print (abserr)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# Не много точно решение."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(1.1269280110429725, 1.2511414249549407e-14)"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# -------------------------------------------------------------\n",
    "# Задача 27\n",
    "# x = sy.Symbol('x')\n",
    "# yp = sy.diff(sy.log((sy.exp(x) + 1)/(sy.exp(x)-1)), x).simplify(); yp\n",
    "# sy.integrate(sy.sqrt(1 + yp**2), (x,1,2))\n",
    "# Системата не може да реши интеграла аналитично ;=()\n",
    "# Прилагаме числено решение\n",
    "f = lambda x: sqrt(1 + (-2*exp(x)/(exp(2*x) - 1))**2)\n",
    "val, abserr = quad(f, 1, 2)\n",
    "val, abserr"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# Изключително точно решение!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "16.9266441708756"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# -------------------------------------------------------------\n",
    "# Задача 28\n",
    "# Уравнението на окръжността: (x - 6)**2 + y**2 = 9,\n",
    "# откъдето следва y = sqrt(9 - (x - 6)**2)\n",
    "# 2*sy.integrate(sy.sqrt(9 - sy.expand((x - 6)**2)), (x, 5, 8))\n",
    "# Системата не може да реши интеграла аналитично.\n",
    "# Следва числено решение:\n",
    "# Вариант 1\n",
    "2*sy.N(sy.integrate(sy.sqrt(9 - sy.expand((x - 6)**2)), (x, 5, 8)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(16.926644170875566, 3.9894480080224267e-13)"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Вариант 2\n",
    "f = lambda x: 2*sqrt(9 - (x - 6)**2)\n",
    "val, abserr = quad(f, 5, 8)\n",
    "val, abserr"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[-1]"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# -------------------------------------------------------------\n",
    "# Задача 29\n",
    "x = sy.Symbol('x')\n",
    "# Функцията log(x+2) пресича оста x в точка с координата xA:\n",
    "xA = sy.solve(sy.log(x + 2), x); xA"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[1]"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Функцията 2*log(x) пресича оста x в точка с координата xB:\n",
    "xB = sy.solve(2*sy.log(x), x); xB"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[2]"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Определяме координатата xC, в която се пресичат двете криви:\n",
    "xC = sy.solve(sy.log(x + 2) - 2*sy.log(x), x); xC"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "F =  -1 + log(16)  =  1.77258872223978\n"
     ]
    }
   ],
   "source": [
    "F = sy.integrate(sy.log(x+2), (x,xA,xC)) - \\\n",
    "    sy.integrate(2*sy.log(x), (x,xB,xC));\n",
    "print ('F = ', F.simplify(), ' = ', sy.N(F))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[-1.47168783648703, -0.0283121635129678]\n",
      "[-2.36374306091976, 0.863743060919757]\n"
     ]
    }
   ],
   "source": [
    "# -------------------------------------------------------------\n",
    "# Задача 30\n",
    "b, x = sy.symbols('b,x')\n",
    "fun = lambda x: (x + 2)*(0.5 - x)\n",
    "S = sy.integrate(fun(x), (x,-2,0.5)); S\n",
    "print (sy.solve(2.5*fun(b) - S, b))\n",
    "print (sy.solve(-2.5*fun(b) - S, b))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# Задачата има 4 реални решения."
   ]
  }
 ],
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