Point $B$ lies on line segment $\overline{AC}$ with $AB=16$ and $BC=4$. Points $D$ and $E$ lie on the same side of line $AC$ forming equilateral triangles $\triangle ABD$ and $\triangle BCE$. Let $M$ be the midpoint of $\overline{AE}$, and $N$ be the midpoint of $\overline{CD}$. The area of $\triangle BMN$ is $x$. Find $x^2$.
Point $A,B,C,D,$ and $E$ are equally spaced on a minor arc of a cirle. Points $E,F,G,H,I$ and $A$ are equally spaced on a minor arc of a second circle with center $C$ as shown in the figure below. The angle $\angle ABD$ exceeds $\angle AHG$ by $12^\circ$. Find the degree measure of $\angle BAG$.
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAQIAAACfCAIAAACdokAuAAAAAXNSR0IArs4c6QAAAARnQU1BAACxjwv8YQUAAAAJcEhZcwAADsMAAA7DAcdvqGQAACKVSURBVHhe7Z13XBTX+salWFCCNdEY4SJgsICgxqBiD4otEUWvooLGQhSigl2RKlYU7Cb2inLVqLFeIKKiUWMXFBQQUQRREAvmJv/9Hvcdzm+yC0jbndnlfP/Yz5kzs7tnznme97xnd3a22t8cTpWH24DD4TbgcLgNOBzAbcDhcBtwONwGHA7gNuBwuA1kwF9//fXnn38KGwr+97//4RH1tMlRN9wGEkOKp0JGRkZKSkpubi7VgIKCAqHEUSfcBlJC8T4tLW3u3LnDhg3z9PT08fGZOHGin59ffn7+woULDx48yA7jqA9uA4lZvny5jY3N0qVLYQaqef36dUREhJWVVb169R4/fkyVHLXCbSANFODHjBljZmaWkJBAlYAF/t69e3fq1InKHHXDbSABpHVvb+9atWqlp6ej/P79e8WeD6CMA0JDQ2fNmoVNpdUzRx1wG2gaWhMfP368WrVqW7ZsQVnsAUAHbNq06dSpUyjwhYEG4DaQAAR4rAcsLCwKCgqKU3lOTo6SPTjqg9tAo5Do4+PjMRXMnz8f5SJzHj4DaBhdswEyCgRRVaA2OWiLRL9q1Sp9ff1ffvkFZfa9gfxB44Xe/CdFOlm7qFqzgeSag2jwOG/ePD09vdu3b6OsZE7avHfv3tOnT6lGW9DqGUx3bEDDgKVnQEBAUFCQn59fiIJFixb5+/uvWLEiNTWVHSYVFDhXr16N2eDOnTtojFJ7yKhottxsgHZiQR+oAP2JXqXuRW/v2rVL2ycEXZsNEEcPHz6MzNvR0TE2Nvb06dN4RM20adOaNm2KAcMxSsrTJPTWSUlJaOHOnTtRfvfu3QcriC4r2rNnz5EjR1Cgg2UC/Hnt2jUEF7QcUSYuLg59Gx0dfeDAga+++srBwSErK0s4VAvRKRuQbh4+fFi9evWoqCiqZOzbtw9DiPFDWcLsiBo5a9asTz/9FE2lSqKgoGD79u3r1q1DWVYeAOTSDRs2GBkZKcX+R48eoWOR6aGspdOCTtkAYwD1hIeHY1QyMjJQhtzpkYanVatW/fr1Q0FakdG7z549u3Xr1suXLz958iSmLKQciLUynAcIalKvXr26d++OAnUsehWrnby8PENDQw8PD6r/cLS2oYOzgYuLS4sWLaiGQcM2ZsyYTz75hNap0kJNTUhIgGmRY4SFhR07duzly5e0V568fv26Vq1aCxcuRAe+ffuWBf7Fixcj7hw9ehRlbgNZgLGpX7/+lClTUBYPCcluxIgRGLD8/HyqlJYiFSNPGVGrMGWh9+7evUuVAH6AgVGJRbNQpZ3ojg1I6PTNFJYBKKtKys7OrnHjxnKYDQi0mfIKPMrTAAQF/kWLFunp6S1btgzSByEhIV27dnV2dr506RL2Uv9rKbpjAxoqpNoYqidPnlClmKysrBo1avTv31/YlhSIBrqnR4DGAyrgEfVAOFQ29OzZE2uD5OTk+/fvJyUloXDmzJnRo0cjnROO0Fp0bTZAcGrbti3VMKAtPO7duxcTxbZt21CG1BR7Kh80g9SMGK+md8Fb0Oszzwg71AktDDAVCNuFUK/+/vvvKMvQuqVEp9YGWLdhBezr64sySZ+gcvfu3Zs1a4bhpMoKIpa7+L2KgyRSUFCABoBHjx7duXMHS2QE1F9//RXry6ioqBMnTqB87dq1e/fuJSYmvnr1Cke+efOG3otepwRwjDq8QW9NC4MLFy6wGrwF3gvTgoGBwYIFC1CDd8ejNqIjNqCBOXv2LIZK6Vod+jnv/v37sQuaQ5ntKhM06kDYLgpIFrI4ffr05s2b58+fj4QB2XOXLl06duzYvHlzS0vLevXqoRllwtDQEM/917/+ZWtri1dzcHBAXjdt2rTw8PBDhw5dv379+fPnJZwRtbkiriBxBwUFGRkZiX8bTfUHDx5EI2mOLblz5IyO2ICGJzQ0FAuDZ8+eKYXPixcvYkJfsWIFymUSRHHSR01OTg7C9uHDh1euXDl16tR+/frZ2NjUrFmTtMvA/AMRt2jRAmaAgpGzeXh4uLm5/fjjjwsXLvTz81u3bt3WrVu3b98eGRkJ82zZsiU4OBj1c+fOHTNmjLu7u6ura+fOnWEA2ABmsLCwgByFVy/E1NQUB4wbNw5i3b17d1xcXHp6+rt374TmFoLTgXbRM2XqBOpJvH6PHj2oY8VPb9OmDVqVl5cnbGsnOpUUderUqX379sKGgszMTKjKysoKIsNmCVGToGGGVlSFkpGRceTIEWjXyckJ4hYEWIixsTFkOmjQoBkzZkRERGDVeOvWrdzcXBimTJorGZIgcr+UlJSYmBjEYH9/f5jK0dGxadOmQlMKwcxjb2/v6ekJd929e1fVFQqDl6p5L168QHxZsmSJsK3g9u3b8Ia5uTl9hFqJp6l5dMEGEMfy5cuRgdStW7dDhw4+Pj7e3t5IGyZOnPjvf/973rx59MP2EjyAXRCEsFEIdIMBhoZ++OEHRHpBXNWqmZiYQHYQH0IvojjkCFFCmsLTSgRawXvR28FshEKN/49QqwjboJTywhOzsrIuX76MNUZYWNiUKVP69u2LiUhodLVqn3766XfffYeOQpYPWQtPK4SapPpeT58+nT17NuY6nPXAgQNhcvTt9OnT0du9e/eeNWsWUjIcptUeADoyG0DoWHQi+iIjgihTU1PxiBqWy2KYqSCGxl7YUAB9QCXQChQD3QgKqlYNWQ0Gfs2aNdCZalhlQA2kYLxs6RVcDvDK1Hhyi1CrAnahHzATIihgqqxevTqdDgrYROW+fftwgLgT6GVZy9GBOAAzIfqWbqPE+hZvTceU0ABtQaeSoiIRDypBIy1sKCR15cqVwMBApOBioWDT19cXGsKoqwoaL6IBuZcJNINOjRom1IoozuRYb2DmxDQivpoDr1bki4jBATI59wqi+zZgQCJA2FB80HH+/PkFCxZYW1uTGqAMFxeXZcuWnTt3TvXyHjy35NArT0jNqoJmKd/kyZNbtWpFPYAVDia9PXv2iK+axivQieuG4otE923wQfsi7WL4ERGRD5iZmdHYo4BNVFZkEakV4ETQFdC00hlhMzk5OTw8vFu3btQnmAyHDh26Y8cOpe/jdak3xOisDTBaGDNhQ/El6PHjxydNmvTZZ5/RSGMSwFSACQGyEA6qGpGPQV0k7iWADHDTpk3Ozs7USwBl1ND9lAhFYNH69YAYHbSBeJAw0idPnnR3d69fvz4Nqr29fXBwMBYDYqFDCjo2rmWFLCFsKMAiGCkT5gS2XurevfvatWszMzOFIxT9JpS0HJ2ygXhUsrOz161b16ZNGxpCR0fHkJAQ+hU8g6tfFVU/YJ2wc+dOrBmwckBP1q5de8qUKVevXhV260SmpAs2UBo5jBDGicYMq96ZM2eKL5EHXP2lgXpVrO/c3NytW7c6ODgoAsuHZOk///kPSyk/TMFa26vabQNx12M8MCosqcVoYcxevXpFewFXf/kgP4i77uzZsyNHjtTX10c/W1hYLFmyBBmUsO+fc7K2oK02wKiwQIUxwEhgPMgAI0aMwDjRLqAU0jjlhvwgbChufYA809zcHH1uYGCArImuPyW0ywzaZwMMBotMFy9eHDt2LK3hTE1NFy1a9ODBA9oFtDEsaQXiGFRQULB///5vvvlGEYKqderUaceOHSxT0pYh0CYbiA1w6dKloUOHUtf37NkTI8E+9RcfxlEfSpMDlmSenp516tTBiFhbW2/bto2NgvyHQztsIFb2jRs33N3dyQDDhw/XsY8stBGxGTIzM8PCwho0aIDRadeuHf1pFSFnM2iBDVgvJycne3l5kQEGDhxIvwQntGXy1WGgciZ0mCEgIKBGjRoYqW7dutGvnYBsJ2pZ24CJ++nTpwsWLKA1QK9eveTfrVUWcaaEsDVt2jRF1KrWp08fFrYwZHKbtGVqA9aVeXl5ixcvpu+Av/76azbJcgPIGbEZbt68yZJYLOeuX79O9bIaPtnZAD0IUHj//v2GDRtMTU3RfS1btty+fTsdALgBtAKMIxspqJ99pAFX3Llzh+plMpTysgELISdOnGjdujW6zMzMDGbQug/gOAyxGc6fP4/sCMOqr6/v7+9Ptw/EXgp8EiIXG3yYAhR9gWXApEmT0FN0Vxz2NTA3gFYj1jqWdshvMcS2trYxMTFUKe34ysIGrAv2799PPy0fPHgw+7dgbgCdgU0LKCDGGRoaYqy9vb2fPXuGShYKNY/ENmBB4v79+y4uLuiUJk2a0P9f0F4qcHQJNqxYPdM1YBYWFnRDbCBJ1JPSBqw71qxZQ/ex+v777+nnHRIGBo5mYKO/fv16+iTQzc0tLS0NNRh6DUdAaWwgjge9e/dGF1haWkobDziaBzKgYIdcwNXVFTJo1KgR/WU67aWCBpDABnR60HpoaChlh1OnTs3JyaFdfBKoarCod+DAAfqBeP/+/ekT1Q8pgUb0UCobUGugUZKpGKoUjisFdPCVK1fo1xtt27ZlF0XzSaDKQlpCgX1OaGRktGrVKtpbJoGVD83NBuxkNm/eXKtWLZxqcHAw3U6LuoBTxWFx8PTp01ZWVlCIu7s7SxNoV0Vg925TpVQ22Lp1q5eXF9av48eP/1EEKocPH+7p6ZmbmyscWgx0hq9fv6Zr47ASOHfuHO3SgNc52gICIunh+fPnHh4ekErr1q3p+oty64SeCA1PmDABhSJjbqls8OTJkwsXLqBNUHxKSkpycjIewY0bN5DWY/4S/9ZRFWpHYmIiJUKDBg3KyMhADc+COEXChBGm+GO12rVrI4OgmvKZAWEai+/OnTsL2yp83Ab0xqdOnTIwMBBf28yAExDmhY1/wpx36NChhg0b4pQCAwOpptzm5lQFoBwST2xsLP28dtq0aZTVlEk5dPBPP/2EF7G3t6dKVT5uA1gTDfLx8UHUJ7mzdqD+5cuXERER7JofMewwf39/nEbjxo3Zn2TRGXI4JUPTQlpa2sCBAyEhR0dH5BSoKaUTSGa//fZbZGTknDlzjI2Ni0tASrtEbtu2be/evfEqdAfz33//nW5GkJ2dvX//fsUh/4CdwIABA8pxAhwOUZFgigPevXs3c+ZMhGn682a6Db0qpbJBVlaWnp4eZhbaxKvPmDFDfPeyIomOjq7IdMbhEEzuR48epdQ6ICCAakqAxLZly5aoqCgUNm3aBA3funULZVX/fMQG9Fr0x2Fjx44NCQmBqxDa3d3dUV+cHVG/ZMkSfX19LG7Y7wS4BzgVgfSDnAIrXaixX79+KSkpqClBhPfv3/f19UX+guf+8ssveNbx48exS1WKpbKBl5eXiYkJcqwYBcOGDVuzZg3qi8u08vPz8ZYNGjSo4EddHI4Y0hsWqKNGjYLA6M/sSliXBgcHs9/33L59G0/Zu3cvymW2AWFtbe3k5CRs/P33nj174uPjqZyXl6f0rQS5E2uaOnXqPH78GG9ZnF85nLJCThgxYkTNmjWTk5NRVlUXqfzUqVPffvvtpUuXTp8+jdgNAyApCgsLwy7V8P1xG2RkZOD5SIfw6phf8K5YH9O/YODlVq1apfRHWvQeSONgPixrWA2HU0FI8ciLIC2kJKxGCQgV+YiHhwdEGxoaijkBhaCgIDwLa1ocUDYbkKsiIyPx/Li4OFYD6O0vXLiwfv16FFg9Ayt0rI/NzMxU/zuDwykfJLN58+ZBkCdOnGA1YqgmPDxcfIsk4pNPPhk5ciQKquYpyQaUdXl7e2OlC3vhyXgPPLJXwZrh4cOHVBZDTVm6dCmaS+t0PiFwKoU3b9588cUXlpaWJSwJkpKSKA1Bug7h4UgU8GhnZ+fo6Ih6JmDGx5Mic3Pzvn37ChuFZGVlubm5jRs3DmVVR9LbpKam1qhRw9nZmdVwOOWGZIYUH7GVUnxV4QGk6C4uLvQllRJfffUV/CNs/JNibYBkxs/Pb/To0ZhK8HwfHx+6nG7q1KkwQIsWLbACpg+CipQ4VSKBw7pCB/4+miMTevToYWRkRP/Ipqoo5EIODg5IXtzd3WnJSmlIbGzs2LFjGzdu/Nlnn40fPx6bqBS7qFgb4D0Qzh+p/NkwSE9Pz87OLu46IoLeAyt0eNfX15fVcDjlgBR/69YtyGnUqFEoFymnjIyMzMzMvLw8SFScNb18+RJKxiPEnJaWpvo3px9PiioC2mpvb9+wYUPYRqjicMoO2YDufkcf1lduVFWjDaihhw8fRtPpwlI+IXDKAXkgISHB0NCwV69e2KSaSkS9swG1uF27dg0aNOATAqd8kOjHjBmDeKr0wX1lofakCI98QuCUG/IApgIDA4PBgwezmspFvTYAH6YD0YSgjnPg6DAkmNGjRyOSXrt2DWV1RFK120A8IdCXGnxC4JQS8kBiYqJapwKgdhsANB20b9/e2Ng4PT2dNoV9HE7xUMSk60nVNxUATdiAmk4XfHt6erIaDqcEKFbGx8dDNkOGDGE16kATNgCKCeCvnj176unpXb58GTXcCZySgWDev3/v4OBQvXp1dhM72lXpaMgGJPqbN28aGhp27doV58NtwCkBkse6deswFcydO5fVqAkN2QDQadB/wv3888+shsNR4kPm8NdfmZmZjRo1Mjc3z8vLw6awTz1ozgYAJ5OTk2Nqavr5558/f/5c3efG0VIoPo4fPx4RMzIyktWoD43agE5m9+7dOL3JkyezGg6HQZLAAhIi6devH6tRKxq1AaBT4mtlTnFAD8DR0VEDK2OGNDbAWhkn2aFDB/pxM+3icEgedOtSDayMGZq2AaATCw0Nxan6+PiwGk4VhwLilStXDA0NbWxs3rx5o7EQKYENAHT/559/duvWDU44ffo01dAuTpUFon/16lWbNm2QKcAMVEO71I00NqDTu3fvXv369Zs3b073gdTYOXNkCP1a0tvbG5GxhJ8aqwlpbADoJHfs2IHTdnNzQ5nfvaLKQmI4cuQIxNC3b18ERA1nB5LZANCp0oVT/Au1KgtlAcgImjZt2qhRI7rlj4ZTAyltAHC22dnZFhYWdevWpb/F506oatCIf/vtt4iGxd1jVN1IbAM64ejoaD09PayNsrKyqJ5TRaBMeP78+fCAhFcfS2wDQKcdERGBjhg4cCA2JekIjuYhD2zfvh1D36NHj9evX0s19NLbAIg/JaCbrXIn6Dw06OfPnzcyMrK0tHz8+DE2pfq0UBY2oE8G0C/Ozs5wwsaNG1FZ5E0qOboByT01NdXMzMzY2Fjyy2pkYQNA/ZKZmYnAYGBgcObMGWxSwODoGBhrkJ+fT/9bs3v3blRKO//LxQaAnHDz5s26des2adKE/2WgrkJjSvcdCgoKQlnyeCcjGwDqjsOHD+vp6VlbW9OfbXIn6BI0mnPnzoUHPDw8WI20yMsGgJzw888/o5tsbW25E3QJsQewDsTyjxIk2ishsrMBoM6iq21tbGzS09NZJUd7EXugT58+ubm52JSDB4AcbQDETujSpYusuoxTDsQD6uTkJLcBlakNAHUc/bk5Cx58TtBGxB6wt7envxeQVVCTrw0AdR+bRqn7uBO0CxqvlStXYhCx2EtLS2OV8kHWNgBiJzg4OFAn8u8TtAIW7wMCAjB8WObJ9gMPudsAUG/SjzZbtmxJ/6TGnSBzmAe8vLwwcN26dZPzh35aYANAfbpx40Z0aLNmzei/HrgTZAtpHcu5YcOGYchcXFzy8vJYvQzRDhsAEn1UVFTt2rWNjY0PHz6MTdl2a1WGRio9Pb1Xr17wAGYDimJsfpAhsrABOqgEhIMK+/fs2bNNmjRB/9IP1gA3g3ygsUhMTMRKAGPk5+dH9eJxlCFaMxsQ5IQ7d+5YWVmhl2fMmEH9zp0gOSxmHTt2rHHjxhidlStXYlMrhkZ6G7x69SomJuby5cu3b9+OjY2NLgSV8fHxqCwoKBAOVUBOSEtLo6uye/bsmZSUhBruBAlhnR8YGIhBadiw4Uez1rdv36akpGRmZmIJkZGRgfKbN2+EfRpHehtAxCEhIXZ2doaGhjNnzgwuxN/fv3fv3qg8efIkDhPPqtS5qKEf7yFHOnHiBNWID+NoBgpMkPLgwYMxHA4ODqW5OjghIcHX19fR0dHExKR///6Y2CmcSYJckiJ0RLNmzYSNQqDpAQMGnD9/nspUSbBNLJoRe9D78BLV0KhwNACETgOBedvS0hKjgAXx69evUfPRUaAn4nh9ff0XL15QpVRIbwN0B6bF2rVrjxkzBt367t279+/foxNRwN61a9fev3+fjlSF4g1iT8eOHTEGLi4u7Do8JdtwKh0mdPqG2MjIaPPmzVRT8jwgpmXLlkhrUSj9U9SBxDagk4+Li0M/btu2DWXqXIiYdLx+/fqnT5+iUBx0fF5enqenJ17k888/379/v3gXp9JhUebu3bu0QoOa6XaLZVIz8ig8d9GiRShL+5tbiW1ASkU+g+54+PAhde7jx49Zro9Ir7REVoV1fWRkJH2W6ubmRpddfDATnxYqFdbbq1evrlOnDnobAejZs2fiXR+FjsR44elq+uP7MiGLtYGTkxPCibChmGRjYmJQKJOCqR8fPXoED6BzGzVqtGXLFtrFp4VKgSn15s2bNAlYWVnRnZhBmXRMB2NhgGSY1hLSIr0NXr161aBBA3Soj4JBgwZZW1vTwgCUyQlM7siLmjZtinHCaoF9/sDNUG6gWjYQYWFhhoaG6Nsff/wxJydHaW+ZwEB/8803woakSGkDCgmYE/X09FatWpWSkpKamnrgwIGxY8eivnw9i2fRE588eTJp0iSMVt26dfHiLOSUKWhx0JmsxxD4u3Tpgi61tbVlk0A5ggsNEIYb404LA8kjlJQ2oJNHR6BnWcxGVlPx2/qybj1y5AhCDl7/yy+/3LdvH1WKh5ZTAqwbr1+/7urqim4E8+bNy8/PR2W5+5BedteuXXg1OSwMgPRJUd++fU1NTSlCAMyz9HcHFYRpHfMAZgPkXeh0R0dHcS4ree/LE3QdMwCiEl0pDYYPH37jxg2qZweUA+r2CRMmmJiYvH37liqlRUoboLsLCgqQtGBRq6YIzUYrIyNj+vTpNJwIbAhvVK+m99VSxAbAmm358uX16tVDj/Xq1evs2bNUX/HuwrsAS0tLREDapHoJkXg2QHQxMDA4evSosK0e2NDi7RDSyAzu7u5Xr16lelCR8KYDQNxM38+ePQsPDzczM0MvtWzZMioqiupJvlSuINnZ2Vhnb9q0CeWK+6riSGaDhw8fxsTEuLi4oK+RtCDYqPWu7hg/1t143z59+ii8UO27775jaRKAGSprpLUF8SknJyfPnTuXZgDYYO3atexLm8oSa2pqKvp/4sSJeIuFCxfGx8d/9HshDSCBDajTofuQkBB0hL+/f0BAABbKDx48oAPUB96aDfnFixdHjBih8EK1zp07R0ZGsi8ycYzOTw6QtfgcMTF6eHhQb9jZ2W3dupXWwaCyuoJ6nsZ9wYIFGPTAwEBEwCpqA8kRB7abN296e3tXr14dw//ll18iGaCfzBI4UnywDqDkcKxQjx07NmTIEDKAk5PTwYMHhX1VKVGUzAaQF6KvGBanNYNY4pB+UFAQfeMGhg4diskBa0TaCyAIdrA2QuoX9/Dly5dnzpzJTnnYsGEXLlwQ9qnTAOhGYbwVyMRpVXE2EKPwgqDvvLy8Xbt2YbVAymjUqNHkyZPj4uLYAQBlJT3JFpK+ks5SUlKQh7Rv357OsVWrVkhR7t27J+yuSjOAmKpuA4IUI2wolu/IjphWrK2tkcvGxsYqXf2i0Ji8LIHGwKiIskqtSkxMxHp3wIABdEaqDqcnUrkKwm3wD5RkTZkDfXQI6tevP3LkyI0bNyYnJwtHFIJn0RSvpD91Q/LFW6uKOCcnB3m/r68vQj61X09Pr8h8T8NtliHcBkUAWUAcwobiUvhLly4FBwd37dqV9ARat249Z84cLCjT0tJwgHCoCLwCGQMCrRSdkeLpZUGRrwnpnz9/fuXKlQj8NWrUoKZaWFgg9qOp4o+k8VJA2KjycBuUBMlO2FDw5MmTPXv2TJo0ydTUlEQGmjVrhhUFkuxDhw5hta30FFVIgoAETTB9A2F3Kfzz4sWLc+fORURETJgwwdbWFvGemgQP9O/ff8WKFTdu3MDrCEerOJxDcBuUCihJST0Q6x9//LFhw4YpU6Z06NChZs2apD8Ah/Tp0wdWwQSyY8eOM2fOIDVHHlKROeHt27ePHz++cOHCgQMHwsLCkOoMGTLExsaG6R5YWVkNGzZs2bJlJ0+epEugGWg8qEgDdBtug7JB0VQcX4n8/HwkTnAF0o+OHTvSF7FKNG/evFOnTsisRo0a5e7uPnr0aCw8Fi5c6Ofnh3C+detWf39/lMH48eOx18PDw9nZuXPnznZ2dvQ7LzGGhoYtW7Z0dXVdvnz58ePHMzIylFSOTZpYhG1O8XAblB+aIoCwLeLdu3dIxK9cuYI0ae3atbNmzRoxYgQEbWlp+cUXXwhCLgWwEzJ7rEP69evn5eW1dOnSvXv3xsbGYo1e5I+2mPR54C8T3AZlhtQPteGRYi2bIj54osTMm46EggEmkKSkpLt37yJliomJ+fXXX2/dunXv3r2EhITnz5/TMQUFBSULWlX3KKOGQS0EwraCkhtZBeE2KANFqqc4maKejEHKY3IsH3g1eik8ikXPqRS4DUoLKQ8SPHHixIoVKxYtWoTH6OhoVCJmYx1cDqHjNQkyDOmbEI4oO5GRkQEKFixYEBQUdOrUKVSmpqaiwax+8eLFdC8JDsFtUAZ27tzZo0ePSZMmbd68GWnMkSNHJk6cOGfOHBcXF29vbxxQ5HShYW7fvv3TTz9hXdG3b9+TJ0/ShRI5OTlxcXFYYKAee8+dOyfhDUNlCLdBqcCSd8iQIe3bt4+PjxeqCqFL5+EKlMsxIVQu1IDffvvN0NCQbnopxtnZ2dbWVtjgiOA2+AiUrjg6OrZq1YquQUANQN6CTB2PWOCam5vT3xNKDpqKJk2fPr1mzZovXrz4kF0V5leYEKpXr+7l5YXGo+VUySG4DUqCNPTDDz/UqFEjIyMDZdW058GDB25ubsKGPLCzs+vevTvJHdBtYY8ePYpZi+5siV10JIfgNigW0sqlS5egnhDF7bKLTP1TUlJoGcrirrRkZ2fr6emtXr1a2C4EaxicyJMnT4Rtjghug2IhWbu6uiLBoO9omdBRoEALWKXkkG+xcIfcPTw8YN2goCA8gtDQ0ObNm3/99dd0JEcJboOSePPmjbGxMRYGwra8ocnKx8fHxMQEq2Ss2qOjo//73/+iEBUVBW/Mnj2bHcYRw21QEgkJCVAPlpUoU6yl2H/r1i2E2MDAQPoYfv369fLJtq2trZ2cnISNQuAEfX19TBQo84WBKtwGJZGUlAQbQOuUBQm1ilvPx8bG+vn5Ye/48eOvXr0qkxCL5A0LA1gUWqcLMfCIsq+vL2yAZYNwHOefcBuUBKTfsGHD77//HmVVoe/atQvaojtZSL5CoBhf3D8GtG/fvl27dsIGRwVug2IhGWGVWadOHfbnXGK529vbW1lZ4TDJPQBosvL29q5duza7xRAaBnJycgwMDOjPc/nCoEi4DUoCGoJuXFxcbGxs/vjjD2yiEjWJiYnh4eEWFhbjxo1DjTjuSou5ubnqwuDYsWOYIvCIsnyaKiu4DUoFRD9gwAAPD4/p06fPnz8fy2KsBx48eFDkv3RqHiwAsF4fPXq0iYkJkh808tq1a6jHynjatGkdOnSoV6+eq6trRESE5E2VJ9wGZQAL0JSUlOfPnwvbsgHiTktLwyolLy8vKysLjaS8KDc3F2XUoIBlPX0RzlGF26BUKKXUkB1DqOJoM9wGHA63AYfDbcDh/P333/8HPzbp8nwjWQsAAAAASUVORK5CYII=)
Triangle $ABC$ has positive integer side lengths with $AB=AC$. Let $I$ be the intersection of the bisectors of $\angle B$ and $\angle C$. Suppose $BI=8$. Find the smallest possible perimeter of $\triangle ABC$.
Triangle $ABC$ has side lengths $AB = 12$, $BC = 25$, and $CA = 17$. Rectangle $PQRS$ has vertex $P$ on $\overline{AB}$, vertex $Q$ on $\overline{AC}$, and vertices $R$ and $S$ on $\overline{BC}$. In terms of the side length $PQ = w$, the area of $PQRS$ can be expressed as the quadratic polynomial Area($PQRS$) = $\alpha w - \beta \cdot w^2$.
Then the coefficient $\beta = \frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
The circumcircle of acute $\triangle ABC$ has center $O$. The line passing through point $O$ perpendicular to $\overline{OB}$ intersects lines $AB$ and $BC$ and $P$ and $Q$, respectively. Also $AB=5$, $BC=4$, $BQ=4.5$, and $BP=\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
Circles $\mathcal{P}$ and $\mathcal{Q}$ have radii $1$ and $4$, respectively, and are externally tangent at point $A$. Point $B$ is on $mathcal{P}$ and point $C$ is on $\mathcal{Q}$ so that line $BC$ is a common external tangent of the two circles. A line $\ell$ through $A$ intersects $\mathcal{P}$ again at $D$ and intersects $\mathcal{Q}$ again at $E$. Points $B$ and $C$ lie on the same side of $\ell$, and the areas of $\triangle DBA$ and $\triangle ACE$ are equal. This common area is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
![](data:image/png;base64,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)
A disk with radius $1$ is externally tangent to a disk with radius $5$. Let $A$ be the point where the disks are tangent, $C$ be the center of the smaller disk, and $E$ be the center of the larger disk. While the larger disk remains fixed, the smaller disk is allowed to roll along the outside of the larger disk until the smaller disk has turned through an angle of $360^\circ$. That is, if the center of the smaller disk has moved to the point $D$, and the point on the smaller disk that began at $A$ has now moved to point $B$, then $\overline{AC}$ is parallel to $\overline{BD}$. Then $sin^2(\angle BEA)=\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
On square $ABCD$, points $E,F,G$, and $H$ lie on sides $\overline{AB},\overline{BC},\overline{CD},$ and $\overline{DA},$ respectively, so that $\overline{EG} \perp \overline{FH}$ and $EG=FH = 34$. Segments $\overline{EG}$ and $\overline{FH}$ intersect at a point $P$, and the areas of the quadrilaterals $AEPH, BFPE, CGPF,$ and $DHPG$ are in the ratio $269:275:405:411.$ Find the area of square $ABCD$.
![](data:image/png;base64,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)
A rectangle has sides of length $a$ and 36. A hinge is installed at each vertex of the rectangle, and at the midpoint of each side of length 36. The sides of length $a$ can be pressed toward each other keeping those two sides parallel so the rectangle becomes a convex hexagon as shown. When the figure is a hexagon with the sides of length $a$ parallel and separated by a distance of 24, the hexagon has the same area as the original rectangle. Find $a^2$.
![](data:image/png;base64,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)
Circle $C$ with radius 2 has diameter $\overline{AB}$. Circle D is internally tangent to circle $C$ at $A$. Circle $E$ is internally tangent to circle $C$, externally tangent to circle $D$, and tangent to $\overline{AB}$. The radius of circle $D$ is three times the radius of circle $E$, and can be written in the form $\sqrt{m}-n$, where $m$ and $n$ are positive integers. Find $m+n$.
In $\triangle RED$, $\angle DRE=75^{\circ}$ and $\angle RED=45^{\circ}$. $|RD|=1$. Let $M$ be the midpoint of segment $\overline{RD}$. Point $C$ lies on side $\overline{ED}$ such that $\overline{RC}\perp\overline{EM}$. Extend segment $\overline{DE}$ through $E$ to point $A$ such that $CA=AR$. Then $AE=\frac{a-\sqrt{b}}{c}$, where $a$ and $c$ are relatively prime positive integers, and $b$ is a positive integer. Find $a+b+c$.
In $\triangle{ABC}, AB=10, \angle{A}=30^{\circ}$, and $\angle{C=45^{\circ}}$. Let $H, D,$ and $M$ be points on the line $BC$ such that $AH\perp{BC}$, $\angle{BAD}=\angle{CAD}$, and $BM=CM$. Point $N$ is the midpoint of the segment $HM$, and point $P$ is on ray $AD$ such that $PN\perp{BC}$. Then $AP^2=\dfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
Let $ABCD$ be a square, and let $E$ and $F$ be points on $\overline{AB}$ and $\overline{BC},$ respectively. The line through $E$ parallel to $\overline{BC}$ and the line through $F$ parallel to $\overline{AB}$ divide $ABCD$ into two squares and two nonsquare rectangles. The sum of the areas of the two squares is $\frac{9}{10}$ of the area of square $ABCD.$ Find $\frac{AE}{EB} + \frac{EB}{AE}.$
A paper equilateral triangle $ABC$ has side length $12$. The paper triangle is folded so that vertex $A$ touches a point on side $\overline{BC}$ a distance $9$ from point $B$. The length of the line segment along which the triangle is folded can be written as $\frac{m\sqrt{p}}{n}$, where $m$, $n$, and $p$ are positive integers, $m$ and $n$ are relatively prime, and $p$ is not divisible by the square of any prime. Find $m+n+p$.
![](data:image/png;base64,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)
Let $\bigtriangleup PQR$ be a triangle with $\angle P = 75^\circ$ and $\angle Q = 60^\circ$. A regular hexagon $ABCDEF$ with side length 1 is drawn inside $\triangle PQR$ so that side $\overline{AB}$ lies on $\overline{PQ}$, side $\overline{CD}$ lies on $\overline{QR}$, and one of the remaining vertices lies on $\overline{RP}$. There are positive integers $a, b, c,$ and $d$ such that the area of $\triangle PQR$ can be expressed in the form $\frac{a+b\sqrt{c}}{d}$, where $a$ and $d$ are relatively prime, and c is not divisible by the square of any prime. Find $a+b+c+d$.
Triangle $AB_0C_0$ has side lengths $AB_0 = 12$, $B_0C_0 = 17$, and $C_0A = 25$. For each positive integer $n$, points $B_n$ and $C_n$ are located on $\overline{AB_{n-1}}$ and $\overline{AC_{n-1}}$, respectively, creating three similar triangles $\triangle AB_nC_n \sim \triangle B_{n-1}C_nC_{n-1} \sim \triangle AB_{n-1}C_{n-1}$. The area of the union of all triangles $B_{n-1}C_nB_n$ for $n\geq1$ can be expressed as $\tfrac pq$, where $p$ and $q$ are relatively prime positive integers. Find $q$.
In equilateral $\triangle ABC$ let points $D$ and $E$ trisect $\overline{BC}$. Then $\sin(\angle DAE)$ can be expressed in the form $\frac{a\sqrt{b}}{c}$, where $a$ and $c$ are relatively prime positive integers, and $b$ is an integer that is not divisible by the square of any prime. Find $a+b+c$.
A hexagon that is inscribed in a circle has side lengths $22$, $22$, $20$, $22$, $22$, and $20$ in that order. The radius of the circle can be written as $p+\sqrt{q}$, where $p$ and $q$ are positive integers. Find $p+q$.
Given a circle of radius $\sqrt{13}$, let $A$ be a point at a distance $4 + \sqrt{13}$ from the center $O$ of the circle. Let $B$ be the point on the circle nearest to point $A$. A line passing through the point $A$ intersects the circle at points $K$ and $L$. The maximum possible area for $\triangle BKL$ can be written in the form $\frac{a - b\sqrt{c}}{d}$, where $a$, $b$, $c$, and $d$ are positive integers, $a$ and $d$ are relatively prime, and $c$ is not divisible by the square of any prime. Find $a+b+c+d$.
In $\triangle ABC$, $AC = BC$, and point $D$ is on $\overline{BC}$ so that $CD = 3\cdot BD$. Let $E$ be the midpoint of $\overline{AD}$. Given that $CE = \sqrt{7}$ and $BE = 3$, the area of $\triangle ABC$ can be expressed in the form $m\sqrt{n}$, where $m$ and $n$ are positive integers and $n$ is not divisible by the square of any prime. Find $m+n$.
Let $\triangle ABC$ be a right triangle with right angle at $C.$ Let $D$ and $E$ be points on $\overline{AB}$ with $D$ between $A$ and $E$ such that $\overline{CD}$ and $\overline{CE}$ trisect $\angle C.$ If $\frac{DE}{BE} = \frac{8}{15},$ then $\tan B$ can be written as $\frac{m \sqrt{p}}{n},$ where $m$ and $n$ are relatively prime positive integers, and $p$ is a positive integer not divisible by the square of any prime. Find $m+n+p.$
Three concentric circles have radii $3,$ $4,$ and $5.$ An equilateral triangle with one vertex on each circle has side length $s.$ The largest possible area of the triangle can be written as $a + \tfrac{b}{c} \sqrt{d},$ where $a,$ $b,$ $c,$ and $d$ are positive integers, $b$ and $c$ are relatively prime, and $d$ is not divisible by the square of any prime. Find $a+b+c+d.$
Equilateral $\triangle ABC$ has side length $\sqrt{111}$. There are four distinct triangles $AD_1E_1$, $AD_1E_2$, $AD_2E_3$, and $AD_2E_4$, each congruent to $\triangle ABC$, with $BD_1 = BD_2 = \sqrt{11}$. Find $\displaystyle\sum_{k=1}^4(CE_k)^2$.
Triangle $ABC$ is inscribed in circle $\omega$ with $AB=5$, $BC=7$, and $AC=3$. The bisector of angle $A$ meets side $\overline{BC}$ at $D$ and circle $\omega$ at a second point $E$. Let $\gamma$ be the circle with diameter $\overline{DE}$. Circles $\omega$ and $\gamma$ meet at $E$ and a second point $F$. Then $AF^2 = \frac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
In rectangle $ABCD$, $AB=12$ and $BC=10$. Points $E$ and $F$ lie inside rectangle $ABCD$ so that $BE=9$,$DF=8$,$\overline{BE}||\overline{DF}$,$\overline{EF}||\overline{AB}$, and line $BE$ intersects segment $\overline{AD}$. The length $EF$ can be expressed in the form $m\sqrt{n}-p$, where $m$,$n$, and $p$ are positive integers and $n$ is not divisible by the square of any prime. Find $m+n+p$.