https://artofproblemsolving.com/wiki/index.php?title=1966_IMO_Problems&feed=atom&action=history1966 IMO Problems - Revision history2024-03-29T06:28:46ZRevision history for this page on the wikiMediaWiki 1.31.1https://artofproblemsolving.com/wiki/index.php?title=1966_IMO_Problems&diff=143748&oldid=prevHamstpan38825 at 16:39, 29 January 20212021-01-29T16:39:53Z<p></p>
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</table>Hamstpan38825https://artofproblemsolving.com/wiki/index.php?title=1966_IMO_Problems&diff=57285&oldid=prevDANCH: /* Problem 1 */2013-10-14T08:34:46Z<p><span dir="auto"><span class="autocomment">Problem 1</span></span></p>
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<tr><td class='diff-marker'>−</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>In a mathematical contest, three problems, <math>A,B,C</math> were posed. Among the participants <del class="diffchange diffchange-inline">ther </del>were 25 students who solved at least one problem each. Of all the contestants who did not solve problem <math>A</math>, the number who solved <math>B</math> was twice the number who solved <math>C</math>. The number of students who solved only problem <math>A</math> was one more than the number of students who solved <math>A</math> and at least one other problem. Of all students who solved just one problem, half did not solve problem <math>A</math>. How many students solved only problem <math>B</math>?</div></td><td class='diff-marker'>+</td><td style="color: #222; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>In a mathematical contest, three problems, <math>A,B,C</math> were posed. Among the participants <ins class="diffchange diffchange-inline">there </ins>were 25 students who solved at least one problem each. Of all the contestants who did not solve problem <math>A</math>, the number who solved <math>B</math> was twice the number who solved <math>C</math>. The number of students who solved only problem <math>A</math> was one more than the number of students who solved <math>A</math> and at least one other problem. Of all students who solved just one problem, half did not solve problem <math>A</math>. How many students solved only problem <math>B</math>?</div></td></tr>
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</table>DANCHhttps://artofproblemsolving.com/wiki/index.php?title=1966_IMO_Problems&diff=57284&oldid=prevDANCH at 08:33, 14 October 20132013-10-14T08:33:31Z<p></p>
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</table>DANCHhttps://artofproblemsolving.com/wiki/index.php?title=1966_IMO_Problems&diff=35129&oldid=prev1=2: Created page with '==Problem 1== In a mathematical contest, three problems, <math>A,B,C</math> were posed. Among the participants ther were 25 students who solved at least one problem each. Of all …'2010-07-08T12:10:50Z<p>Created page with '==Problem 1== In a mathematical contest, three problems, <math>A,B,C</math> were posed. Among the participants ther were 25 students who solved at least one problem each. Of all …'</p>
<p><b>New page</b></p><div>==Problem 1==<br />
In a mathematical contest, three problems, <math>A,B,C</math> were posed. Among the participants ther were 25 students who solved at least one problem each. Of all the contestants who did not solve problem <math>A</math>, the number who solved <math>B</math> was twice the number who solved <math>C</math>. The number of students who solved only problem <math>A</math> was one more than the number of students who solved <math>A</math> and at least one other problem. Of all students who solved just one problem, half did not solve problem <math>A</math>. How many students solved only problem <math>B</math>?<br />
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[[1966 IMO Problems/Problem 1|Solution]]<br />
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==Problem 2==<br />
Let <math>a,b,c</math> be the lengths of the sides of a triangle, and <math>\alpha, \beta, \gamma</math> respectively, the angles opposite these sides. Prove that if <cmath> a+b=\tan{\frac{\gamma}{2}}(a\tan{\alpha}+b\tan{\beta}) </cmath> the triangle is isosceles.<br />
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[[1966 IMO Problems/Problem 2|Solution]]<br />
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==Problem 3==<br />
Prove that the sum of the distances of the vertices of a regular tetrahedron from the center of its circumscribed sphere is less than the sum of the distances of these vertices from any other point in space. <br />
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[[1966 IMO Problems/Problem 3|Solution]]<br />
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==Problem 4==<br />
Prove that for every natural number <math>n</math>, and for every real number <math>x \neq \frac{k\pi}{2^t}</math> (<math>t=0,1, \dots, n</math>; <math>k</math> any integer)<br />
<cmath> \frac{1}{\sin{2x}}+\frac{1}{\sin{4x}}+\dots+\frac{1}{\sin{2^nx}}=\cot{x}-\cot{2^nx} </cmath><br />
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[[1966 IMO Problems/Problem 4|Solution]]<br />
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==Problem 5==<br />
Solve the system of equations<br />
<cmath> |a_1-a_2|x_2+|a_1-a_3|x_3+|a_1-a_4|x_4=1 </cmath><br />
<cmath> |a_2-a_1|x_1+|a_2-a_3|x_3+|a_2-a_4|x_4=1 </cmath><br />
<cmath> |a_3-a_1|x_1+|a_3-a_2|x_2+|a_3-a_4|x_4=1 </cmath><br />
<cmath> |a_4-a_1|x_1+|a_4-a_2|x_2+|a_4-a_3|x_3=1 </cmath><br />
where <math>a_1, a_2, a_3, a_4</math> are four different real numbers.<br />
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[[1966 IMO Problems/Problem 5|Solution]]<br />
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==Problem 6==<br />
Let <math> ABC</math> be a triangle, and let <math> P</math>, <math> Q</math>, <math> R</math> be three points in the interiors of the sides <math> BC</math>, <math> CA</math>, <math> AB</math> of this triangle. Prove that the area of at least one of the three triangles <math> AQR</math>, <math> BRP</math>, <math> CPQ</math> is less than or equal to one quarter of the area of triangle <math> ABC</math>.<br />
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''Alternative formulation:'' Let <math> ABC</math> be a triangle, and let <math> P</math>, <math> Q</math>, <math> R</math> be three points on the segments <math> BC</math>, <math> CA</math>, <math> AB</math>, respectively. Prove that<br />
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<math> \min\left\{\left|AQR\right|,\left|BRP\right|,\left|CPQ\right|\right\}\leq\frac14\cdot\left|ABC\right|</math>,<br />
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where the abbreviation <math> \left|P_1P_2P_3\right|</math> denotes the (non-directed) area of an arbitrary triangle <math> P_1P_2P_3</math>.<br />
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[[1966 IMO Problems/Problem 6|Solution]]</div>1=2