https://artofproblemsolving.com/wiki/index.php?title=1991_IMO_Problems&feed=atom&action=history 1991 IMO Problems - Revision history 2022-08-16T16:15:38Z Revision history for this page on the wiki MediaWiki 1.31.1 https://artofproblemsolving.com/wiki/index.php?title=1991_IMO_Problems&diff=144097&oldid=prev Hamstpan38825: Created page with "Problems of the 1991 IMO. ==Day I== ===Problem 1=== Given a triangle $\,ABC,\,$ let $\,I\,$ be the center of its inscribed circle. The internal bi..." 2021-01-30T21:35:56Z <p>Created page with &quot;Problems of the 1991 <a href="/wiki/index.php/IMO" class="mw-redirect" title="IMO">IMO</a>. ==Day I== ===Problem 1=== Given a triangle &lt;math&gt; \,ABC,\,&lt;/math&gt; let &lt;math&gt; \,I\,&lt;/math&gt; be the center of its inscribed circle. The internal bi...&quot;</p> <p><b>New page</b></p><div>Problems of the 1991 [[IMO]].<br /> <br /> ==Day I==<br /> ===Problem 1===<br /> Given a triangle &lt;math&gt; \,ABC,\,&lt;/math&gt; let &lt;math&gt; \,I\,&lt;/math&gt; be the center of its inscribed circle. The internal bisectors of the angles &lt;math&gt; \,A,B,C\,&lt;/math&gt; meet the opposite sides in &lt;math&gt; \,A^{\prime },B^{\prime },C^{\prime }\,&lt;/math&gt; respectively. Prove that<br /> &lt;cmath&gt; \frac {1}{4} &lt; \frac {AI\cdot BI\cdot CI}{AA^{\prime }\cdot BB^{\prime }\cdot CC^{\prime }} \leq \frac {8}{27}. &lt;/cmath&gt;<br /> <br /> [[1991 IMO Problems/Problem 1|Solution]]<br /> <br /> ===Problem 2===<br /> Let &lt;math&gt; \,n &gt; 6\,&lt;/math&gt; be an integer and &lt;math&gt; \,a_{1},a_{2},\cdots ,a_{k}\,&lt;/math&gt; be all the natural numbers less than &lt;math&gt; n&lt;/math&gt; and relatively prime to &lt;math&gt; n&lt;/math&gt;. If<br /> &lt;cmath&gt; a_{2} - a_{1} = a_{3} - a_{2} = \cdots = a_{k} - a_{k - 1} &gt; 0, &lt;/cmath&gt;<br /> prove that &lt;math&gt; \,n\,&lt;/math&gt; must be either a prime number or a power of &lt;math&gt; \,2&lt;/math&gt;.<br /> <br /> [[1991 IMO Problems/Problem 2|Solution]]<br /> <br /> ===Problem 3===<br /> Let &lt;math&gt; S = \{1,2,3,\cdots ,280\}&lt;/math&gt;. Find the smallest integer &lt;math&gt; n&lt;/math&gt; such that each &lt;math&gt; n&lt;/math&gt;-element subset of &lt;math&gt; S&lt;/math&gt; contains five numbers which are pairwise relatively prime. <br /> <br /> [[1991 IMO Problems/Problem 3|Solution]]<br /> <br /> ==Day II==<br /> ===Problem 4===<br /> Suppose &lt;math&gt; \,G\,&lt;/math&gt; is a connected graph with &lt;math&gt; \,k\,&lt;/math&gt; edges. Prove that it is possible to label the edges &lt;math&gt; 1,2,\ldots ,k\,&lt;/math&gt; 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 /> <br /> [[1991 IMO Problems/Problem 4|Solution]]<br /> <br /> ===Problem 5===<br /> Let &lt;math&gt; \,ABC\,&lt;/math&gt; be a triangle and &lt;math&gt; \,P\,&lt;/math&gt; an interior point of &lt;math&gt; \,ABC\,&lt;/math&gt;. Show that at least one of the angles &lt;math&gt; \,\angle PAB,\;\angle PBC,\;\angle PCA\,&lt;/math&gt; is less than or equal to &lt;math&gt; 30^{\circ }&lt;/math&gt;.<br /> <br /> [[1991 IMO Problems/Problem 5|Solution]]<br /> <br /> ===Problem 6===<br /> An infinite sequence &lt;math&gt; \,x_{0},x_{1},x_{2},\ldots \,&lt;/math&gt; of real numbers is said to be bounded if there is a constant &lt;math&gt; \,C\,&lt;/math&gt; such that &lt;math&gt; \, \vert x_{i} \vert \leq C\,&lt;/math&gt; for every &lt;math&gt; \,i\geq 0&lt;/math&gt;. Given any real number &lt;math&gt; \,a &gt; 1,\,&lt;/math&gt; construct a bounded infinite sequence &lt;math&gt; x_{0},x_{1},x_{2},\ldots \,&lt;/math&gt; such that<br /> &lt;cmath&gt; \vert x_{i} - x_{j} \vert \vert i - j \vert^{a}\geq 1 &lt;/cmath&gt;<br /> for every pair of distinct nonnegative integers &lt;math&gt; i, j&lt;/math&gt;.<br /> <br /> [[1991 IMO Problems/Problem 6|Solution]]<br /> <br /> * [[1991 IMO]] <br /> * [http://www.artofproblemsolving.com/Forum/resources.php?c=1&amp;cid=16&amp;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