https://artofproblemsolving.com/wiki/index.php?title=2014_Indonesia_MO_Problems&feed=atom&action=history2014 Indonesia MO Problems - Revision history2024-03-28T13:23:44ZRevision history for this page on the wikiMediaWiki 1.31.1https://artofproblemsolving.com/wiki/index.php?title=2014_Indonesia_MO_Problems&diff=97420&oldid=prevRockmanex3: 2014 Indonesia MO Problems are up!2018-08-27T03:40:37Z<p>2014 Indonesia MO Problems are up!</p>
<p><b>New page</b></p><div>==Day 1==<br />
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===Problem 1===<br />
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Is it possible to fill a <math>3 \times 3</math> grid with each of the numbers <math>1,2,\ldots,9</math> once each such that the sum of any two numbers sharing a side is prime?<br />
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[[2014 Indonesia MO Problems/Problem 1|Solution]]<br />
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===Problem 2===<br />
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For some positive integers <math>m,n</math>, the system <math>x+y^2 = m</math> and <math>x^2+y = n</math> has exactly one integral solution <math>(x,y)</math>. Determine all possible values of <math>m-n</math>.<br />
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[[2014 Indonesia MO Problems/Problem 2|Solution]]<br />
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===Problem 3===<br />
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Let <math>ABCD</math> be a trapezoid (quadrilateral with one pair of parallel sides) such that <math>AB < CD</math>. Suppose that <math>AC</math> and <math>BD</math> meet at <math>E</math> and <math>AD</math> and <math>BC</math> meet at <math>F</math>. Construct the parallelograms <math>AEDK</math> and <math>BECL</math>. Prove that <math>EF</math> passes through the midpoint of the segment <math>KL</math>.<br />
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[[2014 Indonesia MO Problems/Problem 3|Solution]]<br />
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===Problem 4===<br />
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Determine all polynomials with integral coefficients <math>P(x)</math> such that if <math>a,b,c</math> are the sides of a right-angled triangle, then <math>P(a), P(b), P(c)</math> are also the sides of a right-angled triangle. (Sides of a triangle are necessarily positive. Note that it's not necessary for the order of sides to be preserved; if <math>c</math> is the hypotenuse of the first triangle, it's not necessary that <math>P(c)</math> is the hypotenuse of the second triangle, and similar with the others.)<br />
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[[2014 Indonesia MO Problems/Problem 4|Solution]]<br />
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==Day 2==<br />
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===Problem 5===<br />
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A sequence of positive integers <math>a_1, a_2, \ldots</math> satisfies <math>a_k + a_l = a_m + a_n</math> for all positive integers <math>k,l,m,n</math> satisfying <math>kl = mn</math>. Prove that if <math>p</math> divides <math>q</math> then <math>a_p \le a_q</math>.<br />
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[[2014 Indonesia MO Problems/Problem 5|Solution]]<br />
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===Problem 6===<br />
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Let <math>ABC</math> be a triangle. Suppose <math>D</math> is on <math>BC</math> such that <math>AD</math> bisects <math>\angle BAC</math>. Suppose <math>M</math> is on <math>AB</math> such that <math>\angle MDA = \angle ABC</math>, and <math>N</math> is on <math>AC</math> such that <math>\angle NDA = \angle ACB</math>. If <math>AD</math> and <math>MN</math> intersect on <math>P</math>, prove that <math>AD^3 = AB \cdot AC \cdot AP</math>.<br />
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[[2014 Indonesia MO Problems/Problem 6|Solution]]<br />
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===Problem 7===<br />
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Suppose that <math>k,m,n</math> are positive integers with <math>k \le n</math>. Prove that:<br />
<cmath>\sum_{r=0}^m \dfrac{k \binom{m}{r} \binom{n}{k}}{(r+k) \binom{m+n}{r+k}} = 1</cmath><br />
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[[2014 Indonesia MO Problems/Problem 7|Solution]]<br />
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===Problem 8===<br />
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A positive integer is called beautiful if it can be represented in the form <math>\dfrac{x^2+y^2}{x+y}</math> for two distinct positive integers <math>x,y</math>. A positive integer that is not beautiful is ugly.<br />
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a) Prove that <math>2014</math> is a product of a beautiful number and an ugly number.<br />
b) Prove that the product of two ugly numbers is also ugly.<br />
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[[2014 Indonesia MO Problems/Problem 8|Solution]]<br />
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==See Also==<br />
{{Indonesia MO box|year=2014|before=[[2013 Indonesia MO]]|after=[[2015 Indonesia MO]]}}</div>Rockmanex3