https://artofproblemsolving.com/wiki/index.php?title=1989_IMO_Problems&feed=atom&action=history 1989 IMO Problems - Revision history 2022-08-16T16:33:01Z Revision history for this page on the wiki MediaWiki 1.31.1 https://artofproblemsolving.com/wiki/index.php?title=1989_IMO_Problems&diff=144061&oldid=prev Hamstpan38825: Created page with "Problems of the 1989 IMO. ==Day I== ===Problem 1=== Prove that in the set $\{1,2, \ldots, 1989\}$ can be expressed as the disjoint union of subsets [itex] A_i..." 2021-01-30T15:47:30Z <p>Created page with &quot;Problems of the 1989 <a href="/wiki/index.php/IMO" class="mw-redirect" title="IMO">IMO</a>. ==Day I== ===Problem 1=== Prove that in the set &lt;math&gt; \{1,2, \ldots, 1989\}&lt;/math&gt; can be expressed as the disjoint union of subsets &lt;math&gt; A_i...&quot;</p> <p><b>New page</b></p><div>Problems of the 1989 [[IMO]].<br /> <br /> ==Day I==<br /> ===Problem 1===<br /> Prove that in the set &lt;math&gt; \{1,2, \ldots, 1989\}&lt;/math&gt; can be expressed as the disjoint union of subsets &lt;math&gt; A_i, \{i = 1,2, \ldots, 117\}&lt;/math&gt; such that<br /> <br /> i.) each &lt;math&gt; A_i&lt;/math&gt; contains 17 elements<br /> <br /> ii.) the sum of all the elements in each &lt;math&gt; A_i&lt;/math&gt; is the same.<br /> <br /> [[1989 IMO Problems/Problem 1|Solution]]<br /> <br /> ===Problem 2===<br /> &lt;math&gt; ABC&lt;/math&gt; is a triangle, the bisector of angle &lt;math&gt; A&lt;/math&gt; meets the circumcircle of triangle &lt;math&gt; ABC&lt;/math&gt; in &lt;math&gt; A_1&lt;/math&gt;, points &lt;math&gt; B_1&lt;/math&gt; and &lt;math&gt; C_1&lt;/math&gt; are defined similarly. Let &lt;math&gt; AA_1&lt;/math&gt; meet the lines that bisect the two external angles at &lt;math&gt; B&lt;/math&gt; and &lt;math&gt; C&lt;/math&gt; in &lt;math&gt; A_0&lt;/math&gt;. Define &lt;math&gt; B_0&lt;/math&gt; and &lt;math&gt; C_0&lt;/math&gt; similarly. Prove that the area of triangle &lt;math&gt; A_0B_0C_0 = 2 \cdot&lt;/math&gt; area of hexagon &lt;math&gt; AC_1BA_1CB_1 \geq 4 \cdot&lt;/math&gt; area of triangle &lt;math&gt; ABC&lt;/math&gt;.<br /> <br /> [[1989 IMO Problems/Problem 2|Solution]]<br /> <br /> ===Problem 3===<br /> Let &lt;math&gt; n&lt;/math&gt; and &lt;math&gt; k&lt;/math&gt; be positive integers and let &lt;math&gt; S&lt;/math&gt; be a set of &lt;math&gt; n&lt;/math&gt; points in the plane such that<br /> <br /> i.) no three points of &lt;math&gt; S&lt;/math&gt; are collinear, and<br /> <br /> ii.) for every point &lt;math&gt; P&lt;/math&gt; of &lt;math&gt; S&lt;/math&gt; there are at least &lt;math&gt; k&lt;/math&gt; points of &lt;math&gt; S&lt;/math&gt; equidistant from &lt;math&gt; P.&lt;/math&gt;<br /> <br /> Prove that:<br /> &lt;cmath&gt; k &lt; \frac {1}{2} + \sqrt {2 \cdot n} &lt;/cmath&gt;<br /> <br /> [[1989 IMO Problems/Problem 3|Solution]]<br /> <br /> ==Day II==<br /> ===Problem 4===<br /> Let &lt;math&gt; ABCD&lt;/math&gt; be a convex quadrilateral such that the sides &lt;math&gt; AB, AD, BC&lt;/math&gt; satisfy &lt;math&gt; AB = AD + BC.&lt;/math&gt; There exists a point &lt;math&gt; P&lt;/math&gt; inside the quadrilateral at a distance &lt;math&gt; h&lt;/math&gt; from the line &lt;math&gt; CD&lt;/math&gt; such that &lt;math&gt; AP = h + AD&lt;/math&gt; and &lt;math&gt; BP = h + BC.&lt;/math&gt; Show that:<br /> &lt;cmath&gt; \frac {1}{\sqrt {h}} \geq \frac {1}{\sqrt {AD}} + \frac {1}{\sqrt {BC}} &lt;/cmath&gt;<br /> <br /> [[1989 IMO Problems/Problem 4|Solution]]<br /> <br /> ===Problem 5===<br /> Prove that for each positive integer &lt;math&gt; n&lt;/math&gt; there exist &lt;math&gt; n&lt;/math&gt; consecutive positive integers none of which is an integral power of a prime number.<br /> <br /> [[1989 IMO Problems/Problem 5|Solution]]<br /> <br /> ===Problem 6===<br /> A permutation &lt;math&gt; \{x_1, x_2, \ldots, x_{2n}\}&lt;/math&gt; of the set &lt;math&gt; \{1,2, \ldots, 2n\}&lt;/math&gt; where &lt;math&gt; n&lt;/math&gt; is a positive integer, is said to have property &lt;math&gt; T&lt;/math&gt; if &lt;math&gt; |x_i - x_{i + 1}| = n&lt;/math&gt; for at least one &lt;math&gt; i&lt;/math&gt; in &lt;math&gt; \{1,2, \ldots, 2n - 1\}.&lt;/math&gt; Show that, for each &lt;math&gt; n&lt;/math&gt;, there are more permutations with property &lt;math&gt; T&lt;/math&gt; than without.<br /> <br /> [[1989 IMO Problems/Problem 6|Solution]]<br /> <br /> * [[1989 IMO]] <br /> * [http://www.artofproblemsolving.com/Forum/resources.php?c=1&amp;cid=16&amp;year=1989 IMO 1989 Problems on the Resources page] <br /> * [[IMO Problems and Solutions, with authors]] <br /> * [[Mathematics competition resources]] {{IMO box|year=1979|before=[[1988 IMO]]|after=[[1990 IMO]]}}</div> Hamstpan38825