# Площадь 2

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&auml;&icirc; 04.03.200 .
&Iuml;&euml;&icirc;&ugrave;&agrave;&auml;&uuml; II .
Problem 1 (1 &aacute;&agrave;&euml;&euml;). &Ntilde;&aring;&eth;&aring;&auml;&egrave;&iacute;&ucirc; &ntilde;&ograve;&icirc;&eth;&icirc;&iacute; &acirc;&ucirc;&iuml;&oacute;&ecirc;&euml;&icirc;&atilde;&icirc; &divide;&aring;&ograve;&ucirc;&eth;&frac14;&otilde;&oacute;&atilde;&icirc;&euml;&uuml;&iacute;&egrave;&ecirc;&agrave; &iuml;&icirc;&ntilde;&euml;&aring;&auml;&icirc;&acirc;&agrave;&ograve;&aring;&euml;&uuml;&iacute;&icirc; &ntilde;&icirc;&aring;&auml;&egrave;&iacute;&aring;&iacute;&ucirc;
&icirc;&ograve;&eth;&aring;&ccedil;&ecirc;&agrave;&igrave;&egrave;. &Auml;&icirc;&ecirc;&agrave;&aelig;&egrave;&ograve;&aring;, &divide;&ograve;&icirc; &iuml;&euml;&icirc;&ugrave;&agrave;&auml;&uuml; &iuml;&icirc;&euml;&oacute;&divide;&aring;&iacute;&iacute;&icirc;&atilde;&icirc; &divide;&aring;&ograve;&ucirc;&eth;&frac14;&otilde;&oacute;&atilde;&icirc;&euml;&uuml;&iacute;&egrave;&ecirc;&agrave; &acirc;&auml;&icirc;&aring; &igrave;&aring;&iacute;&uuml;&oslash;&aring; &iuml;&euml;&icirc;&ugrave;&agrave;&auml;&egrave; &egrave;&ntilde;&otilde;&icirc;&auml;&iacute;&icirc;&atilde;&icirc;.
Problem 2 (1 &aacute;&agrave;&euml;&euml;). &Auml;&icirc;&ecirc;&agrave;&aelig;&egrave;&ograve;&aring;, &divide;&ograve;&icirc; &aring;&ntilde;&euml;&egrave; &auml;&egrave;&agrave;&atilde;&icirc;&iacute;&agrave;&euml;&uuml; &ecirc;&agrave;&ecirc;&icirc;&atilde;&icirc;-&iacute;&egrave;&aacute;&oacute;&auml;&uuml; &divide;&aring;&ograve;&ucirc;&eth;&frac14;&otilde;&oacute;&atilde;&icirc;&euml;&uuml;&iacute;&egrave;&ecirc;&agrave; &auml;&aring;&euml;&egrave;&ograve; &auml;&eth;&oacute;&atilde;&oacute;&thorn;
&auml;&egrave;&agrave;&atilde;&icirc;&iacute;&agrave;&euml;&uuml; &iuml;&icirc;&iuml;&icirc;&euml;&agrave;&igrave;, &ograve;&icirc; &icirc;&iacute;&agrave; &eth;&agrave;&ccedil;&aacute;&egrave;&acirc;&agrave;&aring;&ograve; &yacute;&ograve;&icirc;&ograve; &divide;&aring;&ograve;&ucirc;&eth;&frac14;&otilde;&oacute;&atilde;&icirc;&euml;&uuml;&iacute;&egrave;&ecirc; &iacute;&agrave; &auml;&acirc;&aring; &eth;&agrave;&acirc;&iacute;&icirc;&acirc;&aring;&euml;&egrave;&ecirc;&egrave;&aring; &divide;&agrave;&ntilde;&ograve;&egrave;.
Problem 3 (1 &aacute;&agrave;&euml;&euml;). &Iacute;&agrave; &iuml;&eth;&icirc;&auml;&icirc;&euml;&aelig;&aring;&iacute;&egrave;&yuml;&otilde; &ntilde;&ograve;&icirc;&eth;&icirc;&iacute; AB, BC, CD &egrave; DA &acirc;&ucirc;&iuml;&oacute;&ecirc;&euml;&icirc;&atilde;&icirc; &divide;&aring;&ograve;&ucirc;&eth;&frac14;&otilde;&oacute;&atilde;&icirc;&euml;&uuml;&iacute;&egrave;&ecirc;&agrave;
ABCD &ntilde;&icirc;&icirc;&ograve;&acirc;&aring;&ograve;&ntilde;&ograve;&acirc;&aring;&iacute;&iacute;&icirc; &ccedil;&agrave; &ograve;&icirc;&divide;&ecirc;&egrave; B, C, D, A &icirc;&ograve;&euml;&icirc;&aelig;&aring;&iacute;&ucirc; &icirc;&ograve;&eth;&aring;&ccedil;&ecirc;&egrave; BB1 , CC1 , DD1 &egrave; AA1 , &eth;&agrave;&acirc;&iacute;&ucirc;&aring; &yacute;&ograve;&egrave;&igrave;
&ntilde;&ograve;&icirc;&eth;&icirc;&iacute;&agrave;&igrave;. &Iacute;&agrave;&eacute;&auml;&egrave;&ograve;&aring; &iuml;&euml;&icirc;&ugrave;&agrave;&auml;&uuml; &divide;&aring;&ograve;&ucirc;&eth;&frac14;&otilde;&oacute;&atilde;&icirc;&euml;&uuml;&iacute;&egrave;&ecirc;&agrave; A1 B1 C1 D1 , &aring;&ntilde;&euml;&egrave; &iuml;&euml;&icirc;&ugrave;&agrave;&auml;&uuml; &divide;&aring;&ograve;&ucirc;&eth;&frac14;&otilde;&oacute;&atilde;&icirc;&euml;&uuml;&iacute;&egrave;&ecirc;&agrave; ABCD
&eth;&agrave;&acirc;&iacute;&agrave; s.
Problem 4 (1 &aacute;&agrave;&euml;&euml;). &Auml;&icirc;&ecirc;&agrave;&aelig;&egrave;&ograve;&aring;, &divide;&ograve;&icirc; &ntilde;&oacute;&igrave;&igrave;&agrave; &eth;&agrave;&ntilde;&ntilde;&ograve;&icirc;&yuml;&iacute;&egrave;&eacute; &icirc;&ograve; &iuml;&eth;&icirc;&egrave;&ccedil;&acirc;&icirc;&euml;&uuml;&iacute;&icirc;&eacute; &ograve;&icirc;&divide;&ecirc;&egrave; &iacute;&agrave; &icirc;&ntilde;&iacute;&icirc;&acirc;&agrave;&iacute;&egrave;&egrave; &eth;&agrave;&acirc;&iacute;&icirc;-
&aacute;&aring;&auml;&eth;&aring;&iacute;&iacute;&icirc;&atilde;&icirc; &ograve;&eth;&aring;&oacute;&atilde;&icirc;&euml;&uuml;&iacute;&egrave;&ecirc;&agrave; &auml;&icirc; &aring;&atilde;&icirc; &aacute;&icirc;&ecirc;&icirc;&acirc;&ucirc;&otilde; &ntilde;&ograve;&icirc;&eth;&icirc;&iacute; &acirc;&ntilde;&aring;&atilde;&auml;&agrave; &icirc;&auml;&iacute;&agrave; &egrave; &ograve;&agrave; &aelig;&aring;, &ograve;.&aring;. &iacute;&aring; &ccedil;&agrave;&acirc;&egrave;&ntilde;&egrave;&ograve; &icirc;&ograve; &iuml;&icirc;&euml;&icirc;&aelig;&aring;&iacute;&egrave;&yuml;
&ograve;&icirc;&divide;&ecirc;&egrave;.
Problem 5 (1 &aacute;&agrave;&euml;&euml;). &Ntilde;&ograve;&icirc;&eth;&icirc;&iacute;&ucirc; AB &egrave; AC &ograve;&eth;&aring;&oacute;&atilde;&icirc;&uuml;&iacute;&egrave;&ecirc;&agrave; ABC &eth;&agrave;&acirc;&iacute;&ucirc; &ntilde;&icirc;&icirc;&ograve;&acirc;&aring;&ograve;&ntilde;&ograve;&acirc;&aring;&iacute;&iacute;&icirc; a &egrave; b. &Iacute;&agrave; &igrave;&aring;&auml;&egrave;&agrave;&iacute;&aring;,
&iuml;&eth;&icirc;&acirc;&aring;&auml;&frac14;&iacute;&iacute;&icirc;&eacute; &ecirc; &ntilde;&ograve;&icirc;&eth;&icirc;&iacute;&aring; BC , &acirc;&ccedil;&yuml;&ograve;&agrave; &ograve;&icirc;&divide;&ecirc;&agrave; M . &Ntilde;&oacute;&igrave;&igrave;&agrave; &eth;&agrave;&ntilde;&ntilde;&ograve;&icirc;&yuml;&iacute;&egrave;&eacute; &icirc;&ograve; &yacute;&ograve;&icirc;&eacute; &ograve;&icirc;&divide;&ecirc;&egrave; &auml;&icirc; &iuml;&eth;&yuml;&igrave;&ucirc;&otilde; AB &egrave; AC
&eth;&agrave;&acirc;&iacute;&agrave; c. &Iacute;&agrave;&eacute;&auml;&egrave;&ograve;&aring; &yacute;&ograve;&egrave; &eth;&agrave;&ntilde;&ntilde;&ograve;&icirc;&yuml;&iacute;&egrave;&yuml;.
Problem 6 (2 &aacute;&agrave;&euml;&euml;&agrave;). &Iacute;&agrave;&eacute;&auml;&egrave;&ograve;&aring; &iuml;&euml;&icirc;&ugrave;&agrave;&auml;&uuml; &iuml;&eth;&yuml;&igrave;&icirc;&oacute;&atilde;&icirc;&euml;&uuml;&iacute;&icirc;&atilde;&icirc; &ograve;&eth;&aring;&oacute;&atilde;&icirc;&euml;&uuml;&iacute;&egrave;&ecirc;&agrave; &ntilde; &atilde;&egrave;&iuml;&icirc;&ograve;&aring;&iacute;&oacute;&ccedil;&icirc;&eacute;, &eth;&agrave;&acirc;&iacute;&icirc;&eacute; c, &egrave;
&icirc;&ntilde;&ograve;&eth;&ucirc;&igrave; &oacute;&atilde;&euml;&icirc;&igrave; 15◦ .
Problem 7 (1 &aacute;&agrave;&euml;&euml;). &Ograve;&icirc;&divide;&ecirc;&egrave; K, L, M &egrave; N &ntilde;&aring;&eth;&aring;&auml;&egrave;&iacute;&ucirc; &ntilde;&ograve;&icirc;&eth;&icirc;&iacute; &ntilde;&icirc;&icirc;&ograve;&acirc;&aring;&ograve;&ntilde;&ograve;&acirc;&aring;&iacute;&iacute;&icirc; AB, BC, CD &egrave; AD &iuml;&agrave;&eth;&agrave;&euml;&euml;&aring;&euml;&icirc;&atilde;&eth;&agrave;&igrave;&igrave;&agrave; ABCD, &iuml;&euml;&icirc;&ugrave;&agrave;&auml;&uuml; &ecirc;&icirc;&ograve;&icirc;&eth;&icirc;&atilde;&icirc; &eth;&agrave;&acirc;&iacute;&agrave; 1. &Iacute;&agrave;&eacute;&auml;&egrave;&ograve;&aring; &iuml;&euml;&icirc;&ugrave;&agrave;&auml;&uuml; &iuml;&agrave;&eth;&agrave;&euml;&euml;&aring;&euml;&icirc;&atilde;&eth;&agrave;&igrave;&igrave;&agrave;, &icirc;&aacute;&eth;&agrave;&ccedil;&icirc;&acirc;&agrave;&iacute;&iacute;&icirc;&atilde;&icirc;
&iuml;&aring;&eth;&aring;&ntilde;&aring;&divide;&aring;&iacute;&egrave;&yuml;&igrave;&egrave; &iuml;&eth;&yuml;&igrave;&ucirc;&otilde; AL, BM, CN &egrave; DK .
Problem 8 (3 &aacute;&agrave;&euml;&euml;&agrave;). &Auml;&agrave;&iacute; &ograve;&eth;&aring;&oacute;&atilde;&icirc;&euml;&uuml;&iacute;&egrave; ABC . &Iacute;&agrave;&eacute;&auml;&egrave;&ograve;&aring; &atilde;&aring;&icirc;&igrave;&aring;&ograve;&eth;&egrave;&divide;&aring;&ntilde;&ecirc;&icirc;&aring; &igrave;&aring;&ntilde;&ograve;&icirc; &ograve;&icirc;&ecirc;&egrave;&otilde; &ograve;&icirc;&divide;&aring;&ecirc; M , &auml;&euml;&yuml;
&ecirc;&icirc;&ograve;&icirc;&eth;&ucirc;&otilde;:
&agrave;) &ograve;&eth;&aring;&oacute;&atilde;&icirc;&euml;&uuml;&iacute;&egrave;&ecirc;&egrave; AM B &egrave; ABC &eth;&agrave;&acirc;&iacute;&icirc;&acirc;&aring;&euml;&egrave;&ecirc;&egrave;;
&aacute;) &ograve;&eth;&aring;&oacute;&atilde;&icirc;&euml;&uuml;&iacute;&egrave;&ecirc;&egrave; AM B &egrave; AM C &eth;&agrave;&acirc;&iacute;&icirc;&acirc;&aring;&euml;&egrave;&ecirc;&egrave;;
c) &ograve;&eth;&aring;&oacute;&atilde;&icirc;&euml;&uuml;&iacute;&egrave;&ecirc;&egrave; AM B, AM C &egrave; BM C &eth;&agrave;&acirc;&iacute;&icirc;&acirc;&aring;&euml;&egrave;&ecirc;&egrave;.
Problem 9 (1 &aacute;&agrave;&euml;&euml;). &Icirc;&ograve;&eth;&aring;&ccedil;&icirc;&ecirc;, &ntilde;&icirc;&aring;&auml;&egrave;&iacute;&yuml;&thorn;&ugrave;&egrave;&eacute; &ntilde;&aring;&eth;&aring;&auml;&egrave;&iacute;&ucirc; &auml;&acirc;&oacute;&otilde; &iuml;&eth;&icirc;&ograve;&egrave;&acirc;&icirc;&iuml;&icirc;&euml;&icirc;&aelig;&iacute;&ucirc;&otilde; &ntilde;&ograve;&icirc;&eth;&icirc;&iacute; &acirc;&ucirc;&iuml;&oacute;&ecirc;&euml;&icirc;&atilde;&icirc;
&divide;&aring;&ograve;&ucirc;&eth;&frac14;&otilde;&oacute;&atilde;&icirc;&euml;&uuml;&iacute;&egrave;&ecirc;&agrave;, &eth;&agrave;&ccedil;&auml;&aring;&euml;&egrave;&euml; &aring;&atilde;&icirc; &iacute;&agrave; &auml;&acirc;&agrave; &divide;&aring;&ograve;&ucirc;&eth;&frac14;&otilde;&oacute;&atilde;&icirc;&euml;&uuml;&iacute;&egrave;&ecirc;&agrave;, &egrave;&igrave;&aring;&thorn;&ugrave;&egrave;&otilde; &eth;&agrave;&acirc;&iacute;&ucirc;&aring; &iuml;&euml;&icirc;&ugrave;&agrave;&auml;&egrave;. &Auml;&icirc;&ecirc;&agrave;&aelig;&egrave;&ograve;&aring;, &divide;&ograve;&icirc;
&yacute;&ograve;&egrave; &ntilde;&ograve;&icirc;&eth;&icirc;&iacute;&ucirc; &iuml;&agrave;&eth;&agrave;&euml;&euml;&aring;&euml;&uuml;&iacute;&ucirc;.
Problem 10 (2 &aacute;&agrave;&euml;&euml;&agrave;). &Auml;&agrave;&iacute; &oacute;&atilde;&icirc;&euml; XAY &egrave; &ograve;&icirc;&divide;&ecirc;&agrave; O &acirc;&iacute;&oacute;&ograve;&eth;&egrave; &iacute;&aring;&atilde;&icirc;. &Iuml;&eth;&icirc;&acirc;&aring;&auml;&egrave;&ograve;&aring; &divide;&aring;&eth;&aring;&ccedil; &ograve;&icirc;&divide;&ecirc;&oacute; O &iuml;&eth;&yuml;&igrave;&oacute;&thorn;,
&icirc;&ograve;&ntilde;&aring;&ecirc;&agrave;&thorn;&ugrave;&oacute;&thorn; &icirc;&ograve; &auml;&agrave;&iacute;&iacute;&icirc;&atilde;&icirc; &oacute;&atilde;&euml;&agrave; &ograve;&eth;&aring;&oacute;&atilde;&icirc;&euml;&uuml;&iacute;&egrave;&ecirc; &iacute;&agrave;&egrave;&igrave;&aring;&iacute;&uuml;&oslash;&aring;&eacute; &iuml;&euml;&icirc;&ugrave;&agrave;&auml;&egrave;.
Problem 11 (2 &aacute;&agrave;&euml;&euml;&agrave;). &Iuml;&oacute;&ntilde;&ograve;&uuml; M &egrave; N &ntilde;&aring;&eth;&aring;&auml;&egrave;&iacute;&ucirc; &iuml;&eth;&icirc;&ograve;&egrave;&acirc;&icirc;&iuml;&icirc;&euml;&icirc;&aelig;&iacute;&ucirc;&otilde; &ntilde;&ograve;&icirc;&eth;&icirc;&iacute; BC &egrave; AD &acirc;&ucirc;&iuml;&oacute;&ecirc;&euml;&icirc;&atilde;&icirc;
&divide;&aring;&ograve;&ucirc;&eth;&frac14;&otilde;&oacute;&atilde;&icirc;&euml;&uuml;&iacute;&egrave;&ecirc;&agrave; ABCD, &icirc;&ograve;&eth;&aring;&ccedil;&ecirc;&egrave; AM &egrave; BN &iuml;&aring;&eth;&aring;&ntilde;&aring;&ecirc;&agrave;&thorn;&ograve;&ntilde;&yuml; &acirc; &ograve;&icirc;&divide;&ecirc;&aring; P , &agrave; &icirc;&ograve;&eth;&aring;&ccedil;&ecirc;&egrave; DM &egrave; CN &acirc; &ograve;&icirc;&divide;&ecirc;&aring;
Q. &Auml;&icirc;&ecirc;&agrave;&aelig;&egrave;&ograve;&aring;, &divide;&ograve;&icirc; &ntilde;&oacute;&igrave;&igrave;&agrave; &iuml;&euml;&icirc;&ugrave;&agrave;&auml;&aring;&eacute; &ograve;&eth;&aring;&oacute;&atilde;&icirc;&euml;&uuml;&iacute;&egrave;&ecirc;&icirc;&acirc; AP B &egrave; CQD &eth;&agrave;&acirc;&iacute;&agrave; &iuml;&euml;&icirc;&ugrave;&agrave;&auml;&egrave; &divide;&aring;&ograve;&ucirc;&eth;&frac14;&otilde;&oacute;&atilde;&icirc;&euml;&uuml;&iacute;&egrave;&ecirc;&agrave;
M P N Q.
Problem 12 (2 &aacute;&agrave;&euml;&euml;&agrave;). &Egrave;&ccedil; &ntilde;&aring;&eth;&aring;&auml;&egrave;&iacute;&ucirc; &ecirc;&agrave;&aelig;&auml;&icirc;&eacute; &ntilde;&ograve;&icirc;&eth;&icirc;&iacute;&ucirc; &icirc;&ntilde;&ograve;&eth;&icirc;&oacute;&atilde;&icirc;&euml;&uuml;&iacute;&icirc;&atilde;&icirc; &ograve;&eth;&aring;&oacute;&atilde;&icirc;&euml;&uuml;&iacute;&egrave;&ecirc;&agrave; &icirc;&iuml;&oacute;&ugrave;&aring;&iacute;&ucirc; &iuml;&aring;&eth;&iuml;&aring;&iacute;-
&auml;&egrave;&ecirc;&oacute;&euml;&yuml;&eth;&ucirc; &iacute;&agrave; &auml;&acirc;&aring; &auml;&eth;&oacute;&atilde;&egrave;&aring; &ntilde;&ograve;&icirc;&eth;&icirc;&iacute;&ucirc;. &Auml;&icirc;&ecirc;&agrave;&aelig;&egrave;&ograve;&aring;, &divide;&ograve;&icirc; &iuml;&euml;&icirc;&ugrave;&agrave;&auml;&uuml; &icirc;&atilde;&eth;&agrave;&iacute;&egrave;&divide;&aring;&iacute;&iacute;&icirc;&atilde;&icirc; &egrave;&igrave;&egrave; &oslash;&aring;&ntilde;&ograve;&egrave;&oacute;&atilde;&icirc;&euml;&uuml;&iacute;&egrave;&ecirc;&agrave; &eth;&agrave;&acirc;&iacute;&agrave;
&iuml;&icirc;&euml;&icirc;&acirc;&egrave;&iacute;&aring; &iuml;&euml;&icirc;&ugrave;&agrave;&auml;&egrave; &ograve;&eth;&aring;&oacute;&atilde;&icirc;&euml;&uuml;&iacute;&egrave;&ecirc;&agrave;.
&Iacute;&agrave;&aacute;&eth;&agrave;&euml;&egrave; ≥ 7 &aacute;&agrave;&euml;&euml;&icirc;&acirc; &iuml;&icirc;&euml;&oacute;&divide;&agrave;&aring;&ograve;&aring; 3;
&Iacute;&agrave;&aacute;&eth;&agrave;&euml;&egrave; ≥ 12 &aacute;&agrave;&euml;&euml;&icirc;&acirc; &iuml;&icirc;&euml;&oacute;&divide;&agrave;&aring;&ograve;&aring; 4;
&Iacute;&agrave;&aacute;&eth;&agrave;&euml;&egrave; ≥ 15 &aacute;&agrave;&euml;&euml;&icirc;&acirc; &iuml;&icirc;&euml;&oacute;&divide;&agrave;&aring;&ograve;&aring; 5;
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