https://artofproblemsolving.com/wiki/index.php?title=1991_IMO_Problems&feed=atom&action=history1991 IMO Problems - Revision history2024-03-29T12:36:16ZRevision history for this page on the wikiMediaWiki 1.31.1https://artofproblemsolving.com/wiki/index.php?title=1991_IMO_Problems&diff=144097&oldid=prevHamstpan38825: Created page with "Problems of the 1991 IMO. ==Day I== ===Problem 1=== Given a triangle <math> \,ABC,\,</math> let <math> \,I\,</math> be the center of its inscribed circle. The internal bi..."2021-01-30T21:35:56Z<p>Created page with "Problems of the 1991 <a href="/wiki/index.php/IMO" class="mw-redirect" title="IMO">IMO</a>. ==Day I== ===Problem 1=== Given a triangle <math> \,ABC,\,</math> let <math> \,I\,</math> be the center of its inscribed circle. The internal bi..."</p>
<p><b>New page</b></p><div>Problems of the 1991 [[IMO]].<br />
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==Day I==<br />
===Problem 1===<br />
Given a triangle <math> \,ABC,\,</math> let <math> \,I\,</math> be the center of its inscribed circle. The internal bisectors of the angles <math> \,A,B,C\,</math> meet the opposite sides in <math> \,A^{\prime },B^{\prime },C^{\prime }\,</math> respectively. Prove that<br />
<cmath> \frac {1}{4} < \frac {AI\cdot BI\cdot CI}{AA^{\prime }\cdot BB^{\prime }\cdot CC^{\prime }} \leq \frac {8}{27}. </cmath><br />
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[[1991 IMO Problems/Problem 1|Solution]]<br />
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===Problem 2===<br />
Let <math> \,n > 6\,</math> be an integer and <math> \,a_{1},a_{2},\cdots ,a_{k}\,</math> be all the natural numbers less than <math> n</math> and relatively prime to <math> n</math>. If<br />
<cmath> a_{2} - a_{1} = a_{3} - a_{2} = \cdots = a_{k} - a_{k - 1} > 0, </cmath><br />
prove that <math> \,n\,</math> must be either a prime number or a power of <math> \,2</math>.<br />
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[[1991 IMO Problems/Problem 2|Solution]]<br />
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===Problem 3===<br />
Let <math> S = \{1,2,3,\cdots ,280\}</math>. Find the smallest integer <math> n</math> such that each <math> n</math>-element subset of <math> S</math> contains five numbers which are pairwise relatively prime. <br />
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[[1991 IMO Problems/Problem 3|Solution]]<br />
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==Day II==<br />
===Problem 4===<br />
Suppose <math> \,G\,</math> is a connected graph with <math> \,k\,</math> edges. Prove that it is possible to label the edges <math> 1,2,\ldots ,k\,</math> in such a way that at each vertex which belongs to two or more edges, the greatest common divisor of the integers labeling those edges is equal to 1.<br />
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[[1991 IMO Problems/Problem 4|Solution]]<br />
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===Problem 5===<br />
Let <math> \,ABC\,</math> be a triangle and <math> \,P\,</math> an interior point of <math> \,ABC\,</math>. Show that at least one of the angles <math> \,\angle PAB,\;\angle PBC,\;\angle PCA\,</math> is less than or equal to <math> 30^{\circ }</math>.<br />
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[[1991 IMO Problems/Problem 5|Solution]]<br />
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===Problem 6===<br />
An infinite sequence <math> \,x_{0},x_{1},x_{2},\ldots \,</math> of real numbers is said to be bounded if there is a constant <math> \,C\,</math> such that <math> \, \vert x_{i} \vert \leq C\,</math> for every <math> \,i\geq 0</math>. Given any real number <math> \,a > 1,\,</math> construct a bounded infinite sequence <math> x_{0},x_{1},x_{2},\ldots \,</math> such that<br />
<cmath> \vert x_{i} - x_{j} \vert \vert i - j \vert^{a}\geq 1 </cmath><br />
for every pair of distinct nonnegative integers <math> i, j</math>.<br />
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[[1991 IMO Problems/Problem 6|Solution]]<br />
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* [[1991 IMO]] <br />
* [http://www.artofproblemsolving.com/Forum/resources.php?c=1&cid=16&year=1991 IMO 1991 Problems on the Resources page] <br />
* [[IMO Problems and Solutions, with authors]] <br />
* [[Mathematics competition resources]] {{IMO box|year=1991|before=[[1990 IMO]]|after=[[1992 IMO]]}}</div>Hamstpan38825