https://artofproblemsolving.com/wiki/index.php?title=2003_JBMO_Problems&feed=atom&action=history2003 JBMO Problems - Revision history2024-03-28T19:17:08ZRevision history for this page on the wikiMediaWiki 1.31.1https://artofproblemsolving.com/wiki/index.php?title=2003_JBMO_Problems&diff=97407&oldid=prevRockmanex3: 2003 JBMO Problems are up!2018-08-25T17:59:04Z<p>2003 JBMO Problems are up!</p>
<p><b>New page</b></p><div>==Problem 1==<br />
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Let <math>n</math> be a positive integer. A number <math>A</math> consists of <math>2n</math> digits, each of which is 4; and a number <math>B</math> consists of <math>n</math> digits, each of which is 8. Prove that <math>A+2B+4</math> is a perfect square.<br />
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[[2003 JBMO Problems/Problem 1|Solution]]<br />
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==Problem 2==<br />
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Suppose there are <math>n</math> points in a plane no three of which are collinear with the property that if we label these points as <math>A_1,A_2,\ldots,A_n</math> in any way whatsoever, the broken line <math>A_1A_2\ldots A_n</math> does not intersect itself. Find the maximum value of <math>n</math>.<br />
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[[2003 JBMO Problems/Problem 2|Solution]]<br />
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==Problem 3==<br />
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Let <math>D</math>, <math>E</math>, <math>F</math> be the midpoints of the arcs <math>BC</math>, <math>CA</math>, <math>AB</math> on the circumcircle of a triangle <math>ABC</math> not containing the points <math>A</math>, <math>B</math>, <math>C</math>, respectively. Let the line <math>DE</math> meets <math>BC</math> and <math>CA</math> at <math>G</math> and <math>H</math>, and let <math>M</math> be the midpoint of the segment <math>GH</math>. Let the line <math>FD</math> meet <math>BC</math> and <math>AB</math> at <math>K</math> and <math>J</math>, and let <math>N</math> be the midpoint of the segment <math>KJ</math>.<br />
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a) Find the angles of triangle <math>DMN</math>;<br />
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b) Prove that if <math>P</math> is the point of intersection of the lines <math>AD</math> and <math>EF</math>, then the circumcenter of triangle <math>DMN</math> lies on the circumcircle of triangle <math>PMN</math>.<br />
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[[2003 JBMO Problems/Problem 3|Solution]]<br />
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==Problem 4==<br />
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Let <math>x, y, z > -1</math>. Prove that <cmath> \frac{1+x^2}{1+y+z^2} + \frac{1+y^2}{1+z+x^2} + \frac{1+z^2}{1+x+y^2} \geq 2. </cmath><br />
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[[2003 JBMO Problems/Problem 4|Solution]]<br />
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==See Also==<br />
{{JBMO box|year=2003|before=[[2002 JBMO]]|after=[[2004 JBMO]]|five=}}</div>Rockmanex3