https://artofproblemsolving.com/wiki/api.php?action=feedcontributions&user=Muhaboug&feedformat=atomAoPS Wiki - User contributions [en]2024-03-28T19:03:57ZUser contributionsMediaWiki 1.31.1https://artofproblemsolving.com/wiki/index.php?title=2006_IMO_Problems&diff=1334632006 IMO Problems2020-09-10T13:26:57Z<p>Muhaboug: </p>
<hr />
<div>==Problem 1==<br />
Let <math>ABC</math> be a triangle with incentre <math>I.</math> A point <math>P</math> in the interior of the triangle satisfies <math>\angle PBA + \angle PCA = \angle PBC + \angle PCB</math>.<br />
Show that <math>AP \ge AI,</math> and that equality holds if and only if <math>P = I.</math><br />
<br />
==Problem 2==<br />
Let <math>P</math> be a regular 2006-gon. A diagonal of <math>P</math> is called good if its endpoints divide the boundary of <math>P</math> into two parts, each composed of an odd number of sides of <math>P</math>. The sides of <math>P</math> are also called good. Suppose <math>P</math> has been dissected into triangles by 2003 diagonals, no two of which have a common point in the interior of <math>P</math>. Find the maximum number of isosceles triangles having two good sides that could appear in such a configuration.<br />
<br />
==Problem 3==<br />
Determine the least real number <math>M</math> such that the inequality <cmath> \left| ab\left(a^{2}-b^{2}\right)+bc\left(b^{2}-c^{2}\right)+ca\left(c^{2}-a^{2}\right)\right|\leq M\left(a^{2}+b^{2}+c^{2}\right)^{2} </cmath> holds for all real numbers <math>a,b</math> and <math>c</math><br />
<br />
==Problem 4==<br />
Determine all pairs <math>(x, y)</math> of integers such that <cmath>1+2^{x}+2^{2x+1}= y^{2}.</cmath><br />
<br />
==Problem 5==<br />
Let <math>P(x)</math> be a polynomial of degree <math>n>1</math> with integer coefficients, and let <math>k</math> be a positive integer. Consider the polynomial <math>Q(x) = P( P ( \ldots P(P(x)) \ldots ))</math>, where <math>P</math> occurs <math>k</math> times. Prove that there are at most <math>n</math> integers <math>t</math> such that <math>Q(t)=t</math>.<br />
<br />
==Problem 6==<br />
Assign to each side b of a convex polygon P the maximum area of a triangle that has b as a side and is contained in P. Show that the sum of the areas assigned to the sides of P is at least twice the area of P.<br />
<br />
==See Also==<br />
* [[2006 IMO]]<br />
* [[IMO Problems and Solutions]]<br />
* [[IMO]]<br />
<br />
{{IMO box|year=2006|before=[[2005 IMO Problems]]|after=[[2007 IMO Problems]]}}</div>Muhaboughttps://artofproblemsolving.com/wiki/index.php?title=2006_IMO_Problems&diff=1334622006 IMO Problems2020-09-10T13:24:53Z<p>Muhaboug: </p>
<hr />
<div>==Problem 1==<br />
Let <math>ABC</math> be a triangle with incentre <math>I.</math> A point <math>P</math> in the interior of the triangle satisfies <math>\angle PBA + \angle PCA = \angle PBC + \angle PCB</math>.<br />
Show that <math>AP \ge AI,</math> and that equality holds if and only if <math>P = I.</math><br />
<br />
==Problem 2==<br />
Let <math>P</math> be a regular 2006-gon. A diagonal of <math>P</math> is called good if its endpoints divide the boundary of <math>P</math> into two parts, each composed of an odd number of sides of <math>P</math>. The sides of <math>P</math> are also called good. Suppose <math>P</math> has been dissected into triangles by 2003 diagonals, no two of which have a common point in the interior of <math>P</math>. Find the maximum number of isosceles triangles having two good sides that could appear in such a configuration.<br />
<br />
==Problem 3==<br />
<br />
==Problem 4==<br />
<br />
==Problem 5==<br />
<br />
==Problem 6==<br />
Assign to each side b of a convex polygon P the maximum area of a triangle that has b as a side and is contained in P. Show that the sum of the areas assigned to the sides of P is at least twice the area of P.<br />
<br />
==See Also==<br />
* [[2006 IMO]]<br />
* [[IMO Problems and Solutions]]<br />
* [[IMO]]<br />
<br />
{{IMO box|year=2006|before=[[2005 IMO Problems]]|after=[[2007 IMO Problems]]}}</div>Muhaboughttps://artofproblemsolving.com/wiki/index.php?title=2007_IMO_Problems&diff=1334612007 IMO Problems2020-09-10T13:24:25Z<p>Muhaboug: </p>
<hr />
<div>==Problem 1==<br />
<hr><br />
Real numbers <math>a_1, a_2, \dots , a_n</math> are given. For each <math>i</math> (<math>1\le i\le n</math>) define<br />
<cmath>d_i=\max\{a_j:1\le j\le i\}-\min\{a_j:i\le j\le n\}</cmath><br />
and let<br />
<cmath>d=\max\{d_i:1\le i\le n\}.</cmath><br />
<br />
(a) Prove that, for any real numbers <math>x_1\le x_2\le \cdots\le x_n</math>,<br />
<cmath>\max\{|x_i-a_i|:1\le i\le n\}\ge \dfrac{d}{2} \qquad (*)</cmath> <br />
<br />
(b) Show that there are real numbers <math>x_1\le x_2\le x_n</math> such that equality holds in (*)<br />
<br />
<br />
[[2007 IMO Problems/Problem 1 | Solution]]<br />
<br />
==Problem 2==<br />
Consider five points <math>A,B,C,D</math>, and <math>E</math> such that <math>ABCD</math> is a parallelogram and <math>BCED</math> is a cyclic quadrilateral.<br />
Let <math>\ell</math> be a line passing through <math>A</math>. Suppose that <math>\ell</math> intersects the interior of the segment <math>DC</math> at <math>F</math> and intersects <br />
line <math>BC</math> at <math>G</math>. Suppose also that <math>EF=EG=EC</math>. Prove that <math>\ell</math> is the bisector of <math>\angle DAB</math>.<br />
<br />
[[2007 IMO Problems/Problem 2 | Solution]]<br />
<br />
==Problem 3==<br />
<br />
In a mathematical competition some competitors are friends. Friendship is always mutual. Call a group of competitors a clique if each two of them are friends. (In particular, any group of fewer than two competitors is a clique.) The number of members of a clique is called its size. Given that, in this competition, the largest size of a clique is even, prove that the competitors can be arranged in two rooms such that the largest size of a clique contained in one room is the same as the largest size of a clique contained in the other room.<br />
<br />
[[2007 IMO Problems/Problem 3 | Solution]]<br />
<br />
==Problem 4==<br />
<br />
In <math>\triangle ABC</math> the bisector of <math>\angle{BCA}</math> intersects the circumcircle again at <math>R</math>, the perpendicular bisector of <math>BC</math> at <math>P</math>, and the perpendicular bisector of <math>AC</math> at <math>Q</math>. The midpoint of <math>BC</math> is <math>K</math> and the midpoint of <math>AC</math> is <math>L</math>. Prove that the triangles <math>RPK</math> and <math>RQL</math> have the same area.<br />
<br />
[[2007 IMO Problems/Problem 4 | Solution]]<br />
<br />
==Problem 5==<br />
<br />
(''Kevin Buzzard and Edward Crane, United Kingdom'')<br />
Let <math>a</math> and <math>b</math> be positive integers. Show that if <math>4ab-1</math> divides <math>(4a^2-1)^2</math>, then <math>a=b</math>.<br />
<br />
[[2007 IMO Problems/Problem 5 | Solution]]<br />
<br />
==Problem 6==<br />
<br />
Let <math>n</math> be a positive integer. Consider<br />
<cmath>S=\{(x,y,z)~:~x,y,z\in \{0,1,\ldots,n \},~x+y+z>0\}</cmath><br />
as a set of <math>(n+1)^3-1</math> points in three-dimensional space. <br />
Determine the smallest possible number of planes, the union of which contain <math>S</math> but does not include <math>(0,0,0)</math>.<br />
<br />
[[2007 IMO Problems/Problem 6 | Solution]]<br />
<br />
{{IMO box|year=2007|before=[[2006 IMO Problems]]|after=[[2008 IMO Problems]]}}</div>Muhaboughttps://artofproblemsolving.com/wiki/index.php?title=2008_IMO_Problems&diff=1334602008 IMO Problems2020-09-10T13:23:58Z<p>Muhaboug: </p>
<hr />
<div>Problems of the 49th [[IMO]] 2008 Spain.<br />
<br />
== Day I ==<br />
<br />
=== Problem 1 ===<br />
Let <math>H</math> be the orthocenter of an acute-angled triangle <math>ABC</math>. The circle <math>\Gamma_{A}</math> centered at the midpoint of <math>BC</math> and passing through <math>H</math> intersects line <math>BC</math> at points <math>A_{1}</math> and <math>A_{2}</math>. Similarly, define the points <math>B_{1}</math>, <math>B_{2}</math>, <math>C_{1}</math> and <math>C_{2}</math>.<br />
<br />
Prove that six points <math>A_{1}</math> , <math>A_{2}</math>, <math>B_{1}</math>, <math>B_{2}</math>, <math>C_{1}</math> and <math>C_{2}</math> are concyclic.<br />
<br />
[[2008 IMO Problems/Problem 1 | Solution]]<br />
<br />
=== Problem 2 ===<br />
'''(i)''' If <math>x</math>, <math>y</math> and <math>z</math> are three real numbers, all different from <math>1</math>, such that <math>xyz = 1</math>, then prove that<br />
<math>\frac {x^{2}}{\left(x - 1\right)^{2}} + \frac {y^{2}}{\left(y - 1\right)^{2}} + \frac {z^{2}}{\left(z - 1\right)^{2}} \geq 1</math>.<br />
(With the <math>\sum</math> sign for cyclic summation, this inequality could be rewritten as <math>\sum \frac {x^{2}}{\left(x - 1\right)^{2}} \geq 1</math>.)<br />
<br />
'''(ii)''' Prove that equality is achieved for infinitely many triples of rational numbers <math>x</math>, <math>y</math> and <math>z</math>.<br />
<br />
[[2008 IMO Problems/Problem 2 | Solution]]<br />
<br />
=== Problem 3 ===<br />
Prove that there are infinitely many positive integers <math>n</math> such that <math>n^{2} + 1</math> has a prime divisor greater than <math>2n + \sqrt {2n}</math>.<br />
<br />
[[2008 IMO Problems/Problem 3 | Solution]]<br />
<br />
== Day II ==<br />
<br />
=== Problem 4 ===<br />
Find all functions <math>f: (0, \infty) \mapsto (0, \infty)</math> (so <math>f</math> is a function from the positive real numbers) such that<br />
<center><br />
<math>\frac {\left( f(w) \right)^2 + \left( f(x) \right)^2}{f(y^2) + f(z^2) } = \frac {w^2 + x^2}{y^2 + z^2}</math><br />
</center><br />
for all positive real numbes <math>w,x,y,z,</math> satisfying <math>wx = yz.</math><br />
<br />
[[2008 IMO Problems/Problem 4 | Solution]]<br />
<br />
=== Problem 5 ===<br />
Let <math>n</math> and <math>k</math> be positive integers with <math>k \geq n</math> and <math>k - n</math> an even number. Let <math>2n</math> lamps labelled <math>1</math>, <math>2</math>, ..., <math>2n</math> be given, each of which can be either ''on'' or ''off''. Initially all the lamps are off. We consider sequences of steps: at each step one of the lamps is switched (from on to off or from off to on).<br />
<br />
Let <math>N</math> be the number of such sequences consisting of <math>k</math> steps and resulting in the state where lamps <math>1</math> through <math>n</math> are all on, and lamps <math>n + 1</math> through <math>2n</math> are all off.<br />
<br />
Let <math>M</math> be number of such sequences consisting of <math>k</math> steps, resulting in the state where lamps <math>1</math> through <math>n</math> are all on, and lamps <math>n + 1</math> through <math>2n</math> are all off, but where none of the lamps <math>n + 1</math> through <math>2n</math> is ever switched on.<br />
<br />
Determine <math>\frac {N}{M}</math>.<br />
<br />
[[2008 IMO Problems/Problem 5 | Solution]]<br />
<br />
=== Problem 6 ===<br />
Let <math>ABCD</math> be a convex quadrilateral with <math>BA</math> different from <math>BC</math>. Denote the incircles of triangles <math>ABC</math> and <math>ADC</math> by <math>k_{1}</math> and <math>k_{2}</math> respectively. Suppose that there exists a circle <math>k</math> tangent to ray <math>BA</math> beyond <math>A</math> and to the ray <math>BC</math> beyond <math>C</math>, which is also tangent to the lines <math>AD</math> and <math>CD</math>. <br />
<br />
Prove that the common external tangents to <math>k_{1}</math> and <math>k_{2}</math> intersect on <math>k</math>.<br />
<br />
[[2008 IMO Problems/Problem 6 | Solution]]<br />
<br />
== Resources ==<br />
<br />
* [[2008 IMO]]<br />
* [http://www.artofproblemsolving.com/Forum/resources.php?c=1&cid=16&year=2008 IMO 2008 Problems on the Resources page]<br />
<br />
{{IMO box|year=2008|before=[[2007 IMO Problems]]|after=[[2009 IMO Problems]]}}</div>Muhaboughttps://artofproblemsolving.com/wiki/index.php?title=2009_IMO_Problems&diff=1334592009 IMO Problems2020-09-10T13:23:26Z<p>Muhaboug: </p>
<hr />
<div>Problems of the 50th [[IMO]] 2009 in Bremen, Germany. <br />
<br />
== Day I == <br />
<br />
=== Problem 1. === <br />
Let <math>n</math> be a positive integer and let <math>a_1,\ldots,a_k (k\ge2)</math> be distinct integers in the set <math>\{1,\ldots,n\}</math> such that <math>n</math> divides <math>a_i(a_{i+1}-1)</math> for <math>i=1,\ldots,k-1</math>. Prove that <math>n</math> doesn't divide <math>a_k(a_1-1)</math>.<br />
<br />
''Author: Ross Atkins, Australia''<br />
<br />
=== Problem 2. ===<br />
<br />
Let <math>ABC</math> be a triangle with circumcentre <math>O</math>. The points <math>P</math> and <math>Q</math> are interior points of the sides <math>CA</math> and <math>AB</math> respectively. Let <math>K,L</math> and <math>M</math> be the midpoints of the segments <math>BP,CQ</math> and <math>PQ</math>, respectively, and let <math>\Gamma</math> be the circle passing through <math>K,L</math> and <math>M</math>. Suppose that the line <math>PQ</math> is tangent to the circle <math>\Gamma</math>. Prove that <math>OP=OQ</math>. <br />
<br />
''Author: Sergei Berlov, Russia''<br />
<br />
=== Problem 3. ===<br />
<br />
Suppose that <math>s_1,s_2,s_3,\ldots</math> is a strictly increasing sequence of positive integers such that the subsequences <br />
<br />
<center> <math>s_{s_1},s_{s_2},s_{s_3},\ldots</math> and <math>s_{s_1+1},s_{s_2+1},s_{s_3+1},\ldots</math> </center><br />
<br />
are both arithmetic progressions. Prove that the sequence <math>s_1,s_2,s_3,\ldots</math> is itself an arithmetic progression.<br />
<br />
''Author: Gabriel Carroll, USA''<br />
<br />
== Day 2 ==<br />
<br />
=== Problem 4. ===<br />
<br />
Let <math>ABC</math> be a triangle with <math>AB=AC</math>. The angle bisectors of <math>\angle CAB</math> and <math>\angle ABC</math> meet the sides <math>BC</math> and <math>CA</math> at <math>D</math> and <math>E</math>, respectively. Let <math>K</math> be the incentre of triangle <math>ADC</math>. Suppose that <math>\angle BEK=45^\circ</math>. Find all possible values of <math>\angle CAB</math>. <br />
<br />
''Authors: Jan Vonk and Peter Vandendriessche, Belgium, and Hojoo Lee, South Korea''<br />
<br />
=== Problem 5. ===<br />
<br />
Determine all functions <math>f</math> from the set of positive integers to the set of positive integers such that, for all positive integers <math>a</math> and <math>b</math>, there exists a non-degenerate triangle with sides of lengths <br />
<br />
<center> <math>a,f(b)</math> and <math>f(b+f(a)-1)</math>. </center><br />
<br />
(A triangle is ''non-degenerate'' if its vertices are not collinear.) <br />
<br />
''Author: Bruno Le Floch, France''<br />
<br />
=== Problem 6. ===<br />
<br />
Let <math>a_1,a_2,\ldots,a_n</math> be distinct positive integers and let <math>M</math> be a set of <math>n-1</math> positive integers not containing <math>s=a_1+a_2+\ldots+a_n</math>. A grasshopper is to jump along the real axis, starting at the point <math>0</math> and making <math>n</math> jumps to the right with lengths <math>a_1,a_2,\ldots,a_n</math> in some order. Prove that the order can be chosen in such a way that the grasshopper never lands on any point in <math>M</math>.<br />
<br />
''Author: Dmitry Khramtsov, Russia''<br />
<br />
{{IMO box|year=2009|before=[[2008 IMO Problems]]|after=[[2010 IMO Problems]]}}</div>Muhaboughttps://artofproblemsolving.com/wiki/index.php?title=2010_IMO_Problems&diff=1334582010 IMO Problems2020-09-10T13:22:47Z<p>Muhaboug: </p>
<hr />
<div>Problems of the 51st [[IMO]] 2010 in Astana, Kazakhstan. <br />
<br />
== Day 1 == <br />
<br />
=== Problem 1. === <br />
Find all functions <math>f:\mathbb{R}\rightarrow\mathbb{R}</math> such that for all <math>x,y\in\mathbb{R}</math> the following equality holds<br />
<br />
<cmath>f(\left\lfloor x\right\rfloor y)=f(x)\left\lfloor f(y)\right\rfloor</cmath><br />
<br />
where <math>\left\lfloor a\right\rfloor</math> is greatest integer not greater than <math>a.</math><br />
<br />
''Author: Pierre Bornsztein, France ''<br />
<br />
[[2010 IMO Problems/Problem 1 | Solution]]<br />
=== Problem 2. ===<br />
<br />
Given a triangle <math>ABC</math>, with <math>I</math> as its incenter and <math>\Gamma</math> as its circumcircle, <math>AI</math> intersects <math>\Gamma</math> again at <math>D</math>. Let <math>E</math> be a point on arc <math>BDC</math>, and <math>F</math> a point on the segment <math>BC</math>, such that <math>\angle BAF=\angle CAE< \frac12\angle BAC</math>. If <math>G</math> is the midpoint of <math>IF</math>, prove that the intersection of lines <math>EI</math> and <math>DG</math> lies on <math>\Gamma</math>.<br />
<br />
''Authors: Tai Wai Ming and Wang Chongli, Hong Kong''<br />
<br />
[[2010 IMO Problems/Problem 2 | Solution]]<br />
=== Problem 3. ===<br />
<br />
Find all functions <math>g:\mathbb{N}\rightarrow\mathbb{N}</math> such that <math>\left(g(m)+n\right)\left(g(n)+m\right)</math> is a perfect square for all <math>m,n\in\mathbb{N}.</math><br />
<br />
''Author: Gabriel Carroll, USA''<br />
<br />
[[2010 IMO Problems/Problem 3 | Solution]]<br />
== Day 2 ==<br />
<br />
=== Problem 4. ===<br />
<br />
Let <math>P</math> be a point interior to triangle <math>ABC</math> (with <math>CA \neq CB</math>). The lines <math>AP</math>, <math>BP</math> and <math>CP</math> meet again its circumcircle <math>\Gamma</math> at <math>K</math>, <math>L</math>, respectively <math>M</math>. The tangent line at <math>C</math> to <math>\Gamma</math> meets the line <math>AB</math> at <math>S</math>. Show that from <math>SC = SP</math> follows <math>MK = ML</math>.<br />
<br />
''Author: Unknown currently''<br />
<br />
[[2010 IMO Problems/Problem 4 | Solution]]<br />
=== Problem 5. ===<br />
<br />
Each of the six boxes <math>B_1</math>, <math>B_2</math>, <math>B_3</math>, <math>B_4</math>, <math>B_5</math>, <math>B_6</math> initially contains one coin. The following operations are allowed<br />
<br />
Type 1) Choose a non-empty box <math>B_j</math>, <math>1\leq j \leq 5</math>, remove one coin from <math>B_j</math> and add two coins to <math>B_{j+1}</math>; <br />
<br />
Type 2) Choose a non-empty box <math>B_k</math>, <math>1\leq k \leq 4</math>, remove one coin from <math>B_k</math> and swap the contents (maybe empty) of the boxes <math>B_{k+1}</math> and <math>B_{k+2}</math>.<br />
<br />
Determine if there exists a finite sequence of operations of the allowed types, such that the five boxes <math>B_1</math>, <math>B_2</math>, <math>B_3</math>, <math>B_4</math>, <math>B_5</math> become empty, while box <math>B_6</math> contains exactly <math>2010^{2010^{2010}}</math> coins.<br />
<br />
''Author: Hans Zantema, Netherlands''<br />
<br />
[[2010 IMO Problems/Problem 5 | Solution]]<br />
=== Problem 6. ===<br />
<br />
Let <math>a_1, a_2, a_3, \ldots</math> be a sequence of positive real numbers, and <math>s</math> be a positive integer, such that<br />
<cmath>a_n = \max \{ a_k + a_{n-k} \mid 1 \leq k \leq n-1 \} \ \textrm{ for all } \ n > s.</cmath><br />
Prove there exist positive integers <math>\ell \leq s</math> and <math>N</math>, such that <br />
<cmath>a_n = a_{\ell} + a_{n - \ell} \ \textrm{ for all } \ n \geq N.</cmath><br />
<br />
''Author: Morteza Saghafiyan, Iran''<br />
<br />
[[2010 IMO Problems/Problem 6 | Solution]]<br />
== Resources ==<br />
* [[2010 IMO]]<br />
* [http://www.artofproblemsolving.com/Forum/resources.php?c=1&cid=16&year=2010&sid=d01bf5fde3957e46434bfbcddbb9a0cb 2010 IMO Problems on the Resources page]<br />
<br />
{{IMO box|year=2010|before=[[2009 IMO Problems]]|after=[[2012 IMO Problems]]}}</div>Muhaboughttps://artofproblemsolving.com/wiki/index.php?title=2011_IMO_Problems&diff=1334572011 IMO Problems2020-09-10T13:21:59Z<p>Muhaboug: </p>
<hr />
<div>Problems of the 52st [[IMO]] 2011 in Amsterdam, Netherlands. <br />
<br />
== Day 1 ==<br />
=== Problem 1. ===<br />
Given any set <math>A = \{a_1, a_2, a_3, a_4\}</math> of four distinct positive integers, we denote the sum <math>a_1+a_2+a_3+a_4</math> by <math>s(A)</math>. Let <math>n(A)</math> denote the number of pairs <math>(i,j)</math> with <math>1 \le i < j \le 4</math> for which <math>a_i+a_j</math> divides <math>s(A)</math>. Find all sets <math>A</math> of four distinct positive integers which achieve the largest possible value of <math>n(A)</math>.<br />
<br />
''Author: Fernando Campos, Mexico''<br />
<br />
[[2011 IMO Problems/Problem 1 | Solution]]<br />
<br />
=== Problem 2. === <br />
Let <math>S</math> be a finite set of at least two points in the plane. Assume that no three points of <math>S</math> are collinear. A windmill is a process that starts with a line <math>l</math> going through a single point <math>P \in S</math>. The line rotates clockwise about the pivot <math>P</math> until the first time that the line meets some other point belonging to <math>S</math>. This point, <math>Q</math>, takes over as the new pivot, and the line now rotates clockwise about <math>Q</math>, until it next meets a point of <math>S</math>. This process continues indefinitely.<br />
Show that we can choose a point <math>P</math> in <math>S</math> and a line <math>l</math> going through <math>P</math> such that the resulting windmill uses each point of <math>S</math> as a pivot infinitely many times.<br />
<br />
''Author: Geoffrey Smith, United Kingdom''<br />
<br />
[[2011 IMO Problems/Problem 2 | Solution]]<br />
<br />
=== Problem 3. ===<br />
Let <math>f : R \rightarrow R</math> be a real-valued function defined on the set of real numbers that satisfies<br />
<math>f(x + y) \le yf(x) + f(f(x))</math> for all real numbers <math>x</math> and <math>y</math>. Prove that <math>f(x)=0</math> for all <math>x \le 0</math>.<br />
<br />
''Author: Igor Voronovich, Belarus''<br />
<br />
[[2011 IMO Problems/Problem 3 | Solution]]<br />
<br />
<br />
== Day 2 ==<br />
=== Problem 4. ===<br />
Let <math>n > 0</math> be an integer. We are given a balance and <math>n</math> weights of weight <math>2^0, 2^1,\ldots, 2^{n-1}</math> . We are to place each of the <math>n</math> weights on the balance, one after another, in such a way that the right pan is never heavier than the left pan. At each step we choose one of the weights that has not yet been placed on the balance, and place it on either the left pan or the right pan, until all of the weights have been placed.<br />
Determine the number of ways in which this can be done.<br />
<br />
''Author: Morteza Saghafian, Iran''<br />
<br />
[[2011 IMO Problems/Problem 4 | Solution]]<br />
<br />
=== Problem 5. ===<br />
Let <math>f</math> be a function from the set of integers to the set of positive integers. Suppose that, for any two integers <math>m</math> and <math>n</math>, the difference <math>f(m) - f(n)</math> is divisible by <math>f(m - n)</math>. Prove that, for all integers <math>m</math> and <math>n</math> with <math>f(m) \le f(n)</math>, the number <math>f(n)</math> is divisible by <math>f(m)</math>.<br />
<br />
''Author: Mahyar Sefidgaran, Iran''<br />
<br />
[[2011 IMO Problems/Problem 5 | Solution]]<br />
<br />
=== Problem 6. ===<br />
Let <math>ABC</math> be an acute triangle with circumcircle <math>\Gamma</math>. Let <math>l</math> be a tangent line to <math>\Gamma</math>, and let <math>l_a</math>, <math>l_b</math> and <math>l_c</math> be the lines obtained by reflecting <math>l</math> in the lines <math>BC</math>, <math>CA</math> and <math>AB</math>, respectively. Show that the circumcircle of the triangle determined by the lines <math>l_a</math>, <math>l_b</math> and <math>l_c</math> is tangent to the circle <math>\Gamma</math>.<br />
<br />
''Author: Japan''<br />
<br />
[[2011 IMO Problems/Problem 6 | Solution]]<br />
<br />
<br />
== Resources ==<br />
* [[2011 IMO]]<br />
* [http://www.artofproblemsolving.com/Forum/resources.php?c=1&cid=16&year=2011&sid=8fa5e6d20f9aad1a4dd4efa73519b417 2011 IMO Problems on the Resources page]<br />
<br />
{{IMO box|year=2011|before=[[2010 IMO Problems]]|after=[[2012 IMO Problems]]}}</div>Muhaboughttps://artofproblemsolving.com/wiki/index.php?title=2013_IMO_Problems&diff=1334562013 IMO Problems2020-09-10T13:10:40Z<p>Muhaboug: Created page with "==Problem 1== Prove that for any pair of positive integers <math>k</math> and <math>n</math>, there exist <math>k</math> positive integers <math>m_1,m_2,...,m_k</math> (not ne..."</p>
<hr />
<div>==Problem 1==<br />
Prove that for any pair of positive integers <math>k</math> and <math>n</math>, there exist <math>k</math> positive integers <math>m_1,m_2,...,m_k</math> (not necessarily different) such that<br />
<br />
<math>1+\frac{2^k-1}{n}=(1+\frac{1}{m_1})(1+\frac{1}{m_2})...(1+\frac{1}{m_k})</math>.<br />
<br />
[[2013 IMO Problems/Problem 1|Solution]]<br />
<br />
==Problem 2==<br />
A configuration of <math>4027</math> points in the plane is called ''Colombian'' if it consists of <math>2013</math> red points and <math>2014</math> blue points, and no three of the points of the configuration are collinear. By drawing some lines, the plane is divided into several regions. An arrangement of lines is ''good'' for a Colombian<br />
configuration if the following two conditions are satisfied:<br />
*no line passes through any point of the configuration;<br />
*no region contains points of both colours.<br />
Find the least value of <math>k</math> such that for any Colombian configuration of <math>4027</math> points, there is a good<br />
arrangement of <math>k</math> lines.<br />
<br />
[[2013 IMO Problems/Problem 2|Solution]]<br />
<br />
==Problem 3==<br />
Let the excircle of triangle <math>ABC</math> opposite the vertex <math>A</math> be tangent to the side <math>BC</math> at the point <math>A_1</math>. Define the points <math>B_1</math> on <math>CA</math> and <math>C_1</math> on <math>AB</math> analogously, using the excircles opposite <math>B</math> and <math>C</math>, respectively. Suppose that the circumcentre of triangle <math>A_1B_1C_1</math> lies on the circumcircle of triangle <math>ABC</math>. Prove that triangle <math>ABC</math> is right-angled.<br />
<br />
[[2013 IMO Problems/Problem 3|Solution]]<br />
<br />
==Problem 4==<br />
Let <math>ABC</math> be an acute triangle with orthocenter <math>H</math>, and let <math>W</math> be a point on the side <math>BC</math>, lying strictly between <math>B</math> and <math>C</math>. The points <math>M</math> and <math>N</math> are the feet of the altitudes from <math>B</math> and <math>C</math>, respectively. Denote by <math>\omega_1</math> is [sic] the circumcircle of <math>BWN</math>, and let <math>X</math> be the point on <math>\omega_1</math> such that <math>WX</math> is a diameter of <math>\omega_1</math>. Analogously, denote by <math>\omega_2</math> the circumcircle of triangle <math>CWM</math>, and let <math>Y</math> be the point such that <math>WY</math> is a diameter of <math>\omega_2</math>. Prove that <math>X, Y</math> and <math>H</math> are collinear.<br />
<br />
[[2013 IMO Problems/Problem 4|Solution]]<br />
<br />
==Problem 5==<br />
Let <math>\mathbb Q_{>0}</math> be the set of all positive rational numbers. Let <math>f:\mathbb Q_{>0}\to\mathbb R</math> be a function satisfying the following three conditions:<br />
<br />
(i) for all <math>x,y\in\mathbb Q_{>0}</math>, we have <math>f(x)f(y)\geq f(xy)</math>;<br />
(ii) for all <math>x,y\in\mathbb Q_{>0}</math>, we have <math>f(x+y)\geq f(x)+f(y)</math>;<br />
(iii) there exists a rational number <math>a>1</math> such that <math>f(a)=a</math>.<br />
<br />
Prove that <math>f(x)=x</math> for all <math>x\in\mathbb Q_{>0}</math>.<br />
<br />
[[2013 IMO Problems/Problem 5|Solution]]<br />
<br />
==Problem 6==<br />
Let <math>n \ge 3</math> be an integer, and consider a circle with <math>n + 1</math> equally spaced points marked on it. Consider all labellings of these points with the numbers <math>0, 1, ... , n</math> such that each label is used exactly once; two such labellings are considered to be the same if one can be obtained from the other by a rotation of the circle. A labelling is called beautiful if, for any four labels <math>a < b < c < d</math> with <math>a + d = b + c</math>, the chord joining the points labelled <math>a</math> and <math>d</math> does not intersect the chord joining the points labelled <math>b</math> and <math>c</math>.<br />
<br />
Let <math>M</math> be the number of beautiful labelings, and let N be the number of ordered pairs <math>(x, y)</math> of positive integers such that <math>x + y \le n</math> and <math>\gcd(x, y) = 1</math>. Prove that <cmath>M = N + 1.</cmath><br />
<br />
[[2013 IMO Problems/Problem 6|Solution]]<br />
<br />
{{IMO box|year=2013|before=[[2012 IMO Problems]]|after=[[2014 IMO Problems]]}}</div>Muhaboughttps://artofproblemsolving.com/wiki/index.php?title=2012_IMO_Problems&diff=1334552012 IMO Problems2020-09-10T13:04:19Z<p>Muhaboug: </p>
<hr />
<div>Problems of the 53st [[IMO]] 2012 in Mar del Plata, Argentina. <br />
<br />
== Day 1 ==<br />
=== Problem 1. ===<br />
Given triangle ABC the point <math>J</math> is the centre of the excircle opposite the vertex <math>A</math>.<br />
This excircle is tangent to the side <math>BC</math> at <math>M</math>, and to the lines <math>AB</math> and <math>AC</math> at <math>K</math> and <math>L</math>, respectively.<br />
The lines <math>LM</math> and <math>BJ</math> meet at <math>F</math>, and the lines <math>KM</math> and <math>CJ</math> meet at <math>G</math>. Let <math>S</math> be the point of<br />
intersection of the lines <math>AF</math> and <math>BC</math>, and let <math>T</math> be the point of intersection of the lines <math>AG</math> and <math>BC</math>.<br />
Prove that <math>M</math> is the midpoint of <math>ST</math>.<br />
(The excircle of <math>ABC</math> opposite the vertex <math>A</math> is the circle that is tangent to the line segment <math>BC</math>,<br />
to the ray <math>AB</math> beyond <math>B</math>, and to the ray <math>AC</math> beyond <math>C</math>.)<br />
<br />
''Author: Evangelos Psychas, Greece''<br />
<br />
[[2012 IMO Problems/Problem 1 | Solution]]<br />
<br />
=== Problem 2. === <br />
Let <math>{{a}_{2}}, {{a}_{3}}, \cdots, {{a}_{n}}</math> be positive real numbers that satisfy <math>{{a}_{2}}\cdot {{a}_{3}}\cdots {{a}_{n}}=1</math> . Prove that<br />
<cmath> \left(a_2+1\right)^2\cdot \left(a_3+1\right)^3\cdots \left(a_n+1\right)^n\gneq n^n</cmath><br />
<br />
''Author: Angelo di Pasquale, Australia''<br />
<br />
[[2012 IMO Problems/Problem 2 | Solution]]<br />
<br />
=== Problem 3. ===<br />
The ''liar’s guessing game'' is a game played between two players <math>A</math> and <math>B</math>. The rules of the game depend on two positive integers <math>k</math> and <math>n</math> which are known to both players. At the start of the game A chooses integers <math>x</math> and <math>N</math> with <math>1\le x\le N</math>. Player <math>A</math> keeps <math>x</math> secret, and truthfully tells <math>N</math> to player <math>B</math>. Player <math>B</math> now tries to obtain information about <math>x</math> by asking player <math>A</math> questions as follows: each question consists of <math>B</math> specifying an arbitrary set <math>S</math> of positive integers (possibly one specified in some previous question), and asking <math>A</math> whether <math>x</math> belongs to <math>S</math>. Player <math>B</math> may ask as many such questions as he wishes. After each question, player <math>A</math> must immediately answer it with yes or no, but is allowed to lie as many times as she wants; the only restriction is<br />
that, among any <math>k + 1</math> consecutive answers, at least one answer must be truthful. After <math>B</math> has asked as many questions as he wants, he must specify a set <math>X</math> of at most <math>n</math> positive integers. If <math>x</math> belongs to <math>X</math>, then <math>B</math> wins; otherwise, he loses. Prove that:<br />
# If <math>n\ge {{2}^{k}}</math>, then <math>B</math> can guarantee a win.<br />
# For all sufficiently large <math>k</math>, there exists an integer <math>n\ge {1.99^k}</math> such that <math>B</math> cannot guarantee a win.<br />
<br />
''Author: David Arthur, Canada ''<br />
<br />
[[2012 IMO Problems/Problem 3 | Solution]]<br />
<br />
<br />
== Day 2 ==<br />
=== Problem 4. ===<br />
Find all functions <math>f:\mathbb{Z}\to \mathbb{Z}</math> such that, for all integers <math>a</math>, <math>b</math>, <math>c</math> that satisfy <math>a+b+c = 0</math>, the following equality holds:<br />
<cmath>f(a)^2 + f(b)^2 + f(c)^2 = 2f(a)f(b) + 2f(b)f(c) + 2f(c)f(a).</cmath><br />
(Here <math>\mathbb{Z}</math> denotes the set of integers.)<br />
<br />
''Author: Liam Baker, South Africa ''<br />
<br />
[[2012 IMO Problems/Problem 4 | Solution]]<br />
<br />
=== Problem 5. ===<br />
Let <math>ABC</math> be a triangle with <math>\angle BCA=90{}^\circ </math>, and let <math>D</math> be the foot of the altitude from <math>C</math>. Let <math>X</math> be a point in the interior of the segment <math>CD</math>. Let K be the point on the segment <math>AX</math> such that <math>BK = BC</math>. Similarly, let <math>L</math> be the point on the segment <math>BX</math> such that <math>AL = AC</math>. Let <math>M</math> be the point of intersection of <math>AL</math> and <math>BK</math>.<br />
Show that <math>MK = ML</math>.<br />
<br />
''Author: Josef Tkadlec, Czech Republic''<br />
<br />
[[2012 IMO Problems/Problem 5 | Solution]]<br />
<br />
=== Problem 6. ===<br />
Find all positive integers n for which there exist non-negative integers <math>a_1</math>, <math>a_2</math>, <math>\ldots</math> , <math>a_n</math> such that<br />
<math>\frac{1}{{{2}^{{{a}_{1}}}}}+\frac{1Problems}{{{2}^{{{a}_{2}}}}}+\cdots +\frac{1}{{{2}^{{{a}_{n}}}}}=\frac{1}{{{3}^{{{a}_{1}}}}}+\frac{2}{{{3}^{{{a}_{2}}}}}+\cdots +\frac{n}{{{3}^{{{a}_{n}}}}}=1</math><br />
<br />
''Author: Dušan Djukić, Serbia''<br />
<br />
[[2012 IMO Problems/Problem 6 | Solution]]<br />
<br />
<br />
== Resources ==<br />
* [[2012 IMO]]<br />
* [http://www.artofproblemsolving.com/Forum/resources.php?c=1&cid=16&year=2012&sid=0b159fdf0134b865758167cc5cf255dd 2012 IMO Problems on the Resources page]<br />
<br />
{{IMO box|year=2012|before=[[2011 IMO Problems]]|after=[[2013 IMO Problems]]}}</div>Muhaboughttps://artofproblemsolving.com/wiki/index.php?title=2014_IMO_Problems&diff=1334542014 IMO Problems2020-09-10T13:02:54Z<p>Muhaboug: </p>
<hr />
<div>==Problem 1==<br />
Let <math>a_0<a_1<a_2<\cdots \quad </math> be an infinite sequence of positive integers, Prove that there exists a unique integer <math>n\ge1</math> such that <br />
<cmath>a_n<\frac{a_0+a_1+\cdots + a_n}{n}\le a_{n+1}.</cmath><br />
<br />
[[2014 IMO Problems/Problem 1|Solution]]<br />
<br />
==Problem 2==<br />
Let <math>n\ge2</math> be an integer. Consider an <math>n\times n</math> chessboard consisting of <math>n^2</math> unit squares. A configuration of <math>n</math> rooks on this board is <math>peaceful</math> if every row and every column contains exactly one rook. Find the greatest positive integer <math>k</math> such that, for each peaceful configuration of <math>n</math> rooks, there is a <math>k\times k</math> square which does not contain a rook on any of its <math>k^2</math> squares.<br />
<br />
[[2014 IMO Problems/Problem 2|Solution]]<br />
<br />
==Problem 3==<br />
Convex quadrilateral <math>ABCD</math> has <math>\angle{ABC}=\angle{CDA}=90^{\circ}</math>. Point <math>H</math> is the foot of the perpendicular from <math>A</math> to <math>BD</math>. Points <math>S</math> and <math>T</math> lie on sides <math>AB</math> and <math>AD</math>, respectively, such that <math>H</math> lies inside <math>\triangle{SCT}</math> and<br />
<cmath>\angle{CHS}-\angle{CSB}=90^{\circ},\quad \angle{THC}-\angle{DTC} = 90^{\circ}.</cmath><br />
<br />
Prove that line <math>BD</math> is tangent to the circumcircle of <math>\triangle{TSH}.</math><br />
<br />
[[2014 IMO Problems/Problem 3|Solution]]<br />
<br />
==Problem 4==<br />
Points <math>P</math> and <math>Q</math> lie on side <math>BC</math> of acute-angled <math>\triangle{ABC}</math> so that <math>\angle{PAB}=\angle{BCA}</math> and <math>\angle{CAQ}=\angle{ABC}</math>. Points <math>M</math> and <math>N</math> lie on lines <math>AP</math> and <math>AQ</math>, respectively, such that <math>P</math> is the midpoint of <math>AM</math>, and <math>Q</math> is the midpoint of <math>AN</math>. Prove that lines <math>BM</math> and <math>CN</math> intersect on the circumcircle of <math>\triangle{ABC}</math>.<br />
<br />
[[2014 IMO Problems/Problem 4|Solution]]<br />
<br />
==Problem 5==<br />
For each positive integer <math>n</math>, the Bank of Cape Town issues coins of denomination <math>\tfrac{1}{n}</math>. Given a finite collection of such coins (of not necessarily different denominations) with total value at most <math>99+\tfrac{1}{2}</math>, prove that it is possible to split this collection into <math>100</math> or fewer groups, such that each group has total value at most <math>1</math>.<br />
<br />
[[2014 IMO Problems/Problem 5|Solution]]<br />
<br />
==Problem 6==<br />
A set of lines in the plane is in <math>\textit{general position}</math> if no two are parallel and no three pass through the same point. A set of lines in general position cuts the plane into regions, some of which have finite are; we call these its <math>\textit{finite regions}</math>. Prove that for all sufficiently large <math>n</math>, in any set of <math>n</math> lines in general position it is possible to colour at least <math>\sqrt{n}</math> of the lines blue in such a way that none of its finite regions has a completely blue boundary.<br />
<br />
[[2014 IMO Problems/Problem 6|Solution]]<br />
<br />
==Problem 1==<br />
''Let <math>\mathbb{Z}</math> be the set of integers. Determine all functions <math>f : \mathbb{Z} \to \mathbb{Z}</math> such that, for all<br />
''integers <math>a</math> and <math>b</math>, <cmath>f(2a) + 2f(b) = f(f(a + b)).</cmath>''<br />
<br />
[[2019 IMO Problems/Problem 1|Solution]]<br />
<br />
==Problem 2==<br />
In triangle <math>ABC</math>, point <math>A_1</math> lies on side <math>BC</math> and point <math>B_1</math> lies on side <math>AC</math>. Let <math>P</math> and <math>Q</math> be points on segments <math>AA_1</math> and <math>BB_1</math>, respectively, such that <math>PQ</math> is parallel to <math>AB</math>. Let <math>P_1</math> be a point on line <math>PB_1</math>, such that <math>B_1</math> lies strictly between <math>P</math> and <math>P_1</math>, and <math>\angle PP_1C=\angle BAC</math>. Similarly, let <math>Q_1</math> be the point on line <math>QA_1</math>, such that <math>A_1</math> lies strictly between <math>Q</math> and <math>Q_1</math>, and <math>\angle CQ_1Q=\angle CBA</math>.<br />
<br />
Prove that points <math>P,Q,P_1</math>, and <math>Q_1</math> are concyclic.<br />
<br />
[[2019 IMO Problems/Problem 2|Solution]]<br />
<br />
==Problem 3==<br />
A social network has <math>2019</math> users, some pairs of whom are friends. Whenever user <math>A</math> is friends with user <math>B</math>, user <math>B</math> is also friends with user <math>A</math>. Events of the following kind may happen repeatedly, one at a time:<br />
Three users <math>A</math>, <math>B</math>, and <math>C</math> such that <math>A</math> is friends with both <math>B</math> and <math>C</math>, but <math>B</math> and <math>C</math> are not friends, change their friendship statuses such that <math>B</math> and <math>C</math> are now friends, but <math>A</math> is no longer friends with <math>B</math>, and no longer friends with <math>C</math>. All other friendship statuses are unchanged.<br />
Initially, <math>1010</math> users have <math>1009</math> friends each, and <math>1009</math> users have <math>1010</math> friends each. Prove that there exists a sequence of such events after which each user is friends with at most one other user.<br />
<br />
[[2019 IMO Problems/Problem 3|Solution]]<br />
<br />
==Problem 4==<br />
Find all pairs <math>(k,n)</math> of positive integers such that <br />
<br />
<cmath>k!=(2^n-1)(2^n-2)(2^n-4)\dots(2^n-2^{n-1}).</cmath><br />
<br />
[[2019 IMO Problems/Problem 4|Solution]]<br />
<br />
==Problem 5==<br />
The Bank of Bath issues coins with an <math>H</math> on one side and a <math>T</math> on the other. Harry has <math>n</math> of these coins arranged in a line from left to right. He repeatedly performs the following operation:<br />
<br />
If there are exactly <math>k > 0</math> coins showing <math>H</math>, then he turns over the <math>k^{th}</math> coin from the left; otherwise, all coins show <math>T</math> and he stops. For example, if <math>n = 3</math> the process starting with the configuration <math>THT</math> would be <math>THT \rightarrow HHT \rightarrow HTT \rightarrow TTT</math>, which stops after three operations.<br />
<br />
(a) Show that, for each initial configuration, Harry stops after a finite number of operations.<br />
<br />
(b) For each initial configuration <math>C</math>, let <math>L(C)</math> be the number of operations before Harry stops. For<br />
example, <math>L(THT) = 3</math> and <math>L(TTT) = 0</math>. Determine the average value of <math>L(C)</math> over all <math>2^n</math><br />
possible initial configurations <math>C</math>.<br />
<br />
[[2019 IMO Problems/Problem 5|Solution]]<br />
<br />
==Problem 6==<br />
Let <math>I</math> be the incenter of acute triangle <math>ABC</math> with <math>AB \neq AC</math>. The incircle <math>\omega</math> of <math>ABC</math> is tangent to sides <math>BC</math>, <math>CA</math>, and <math>AB</math> at <math>D</math>, <math>E</math>, and <math>F</math>, respectively. The line through <math>D</math> perpendicular to <math>EF</math> meets ω again at <math>R</math>. Line <math>AR</math> meets ω again at <math>P</math>. The circumcircles of triangles <math>PCE</math> and <math>PBF</math> meet again at <math>Q</math>.<br />
Prove that lines <math>DI</math> and <math>PQ</math> meet on the line through <math>A</math> perpendicular to <math>AI</math>.<br />
<br />
[[2019 IMO Problems/Problem 6|Solution]]<br />
<br />
{{IMO box|year=2014|before=[[2013 IMO Problems]]|after=[[2015 IMO Problems]]}}</div>Muhaboughttps://artofproblemsolving.com/wiki/index.php?title=2015_IMO_Problems&diff=1334532015 IMO Problems2020-09-10T13:02:15Z<p>Muhaboug: </p>
<hr />
<div>==Problem 1==<br />
We say that a finite set <math>\mathcal{S}</math> in the plane is <i> balanced </i><br />
if, for any two different points <math>A</math>, <math>B</math> in <math>\mathcal{S}</math>, there is<br />
a point <math>C</math> in <math>\mathcal{S}</math> such that <math>AC=BC</math>. We say that<br />
<math>\mathcal{S}</math> is <i>centre-free</i> if for any three points <math>A</math>, <math>B</math>, <math>C</math> in<br />
<math>\mathcal{S}</math>, there is no point <math>P</math> in <math>\mathcal{S}</math> such that<br />
<math>PA=PB=PC</math>.<br />
<ol style="list-style-type: lower-latin;"><br />
<li> Show that for all integers <math>n\geq 3</math>, there exists a balanced set consisting of <math>n</math> points. </li><br />
<li> Determine all integers <math>n\geq 3</math> for which there exists a balanced centre-free set consisting of <math>n</math> points. </li> <br />
</ol><br />
<br />
[[2015 IMO Problems/Problem 1|Solution]]<br />
<br />
==Problem 2==<br />
Determine all triples of positive integers <math>(a,b,c)</math> such that each of the numbers <br />
<cmath> ab-c,\; bc-a,\; ca-b </cmath><br />
is a power of 2. <br />
<br />
(<i>A power of 2 is an integer of the form <math>2^n</math> where <math>n</math> is a non-negative integer </i>).<br />
<br />
[[2015 IMO Problems/Problem 2|Solution]]<br />
<br />
==Problem 3==<br />
Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.<br />
<br />
Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.<br />
<br />
[[2015 IMO Problems/Problem 3|Solution]]<br />
<br />
==Problem 4==<br />
Triangle <math>ABC</math> has circumcircle <math>\Omega</math> and circumcenter <math>O</math>. A circle <math>\Gamma</math> with center <math>A</math> intersects the segment <math>BC</math> at points <math>D</math> and <math>E</math>, such that <math>B</math>, <math>D</math>, <math>E</math>, and <math>C</math> are all different and lie on line <math>BC</math> in this order. Let <math>F</math> and <math>G</math> be the points of intersection of <math>\Gamma</math> and <math>\Omega</math>, such that <math>A</math>, <math>F</math>, <math>B</math>, <math>C</math>, and <math>G</math> lie on <math>\Omega</math> in this order. Let <math>K</math> be the second point of intersection of the circumcircle of triangle <math>BDF</math> and the segment <math>AB</math>. Let <math>L</math> be the second point of intersection of the circumcircle of triangle <math>CGE</math> and the segment <math>CA</math>.<br /><br /><br />
Suppose that the lines <math>FK</math> and <math>GL</math> are different and intersect at the point <math>X</math>. Prove that <math>X</math> lies on the line <math>AO</math>.<br /><br /><br />
<br />
[[2015 IMO Problems/Problem 4|Solution]]<br />
<br />
==Problem 5==<br />
Let <math>\mathbb{R}</math> be the set of real numbers. Determine all functions <math>f</math>:<math>\mathbb{R}\rightarrow\mathbb{R}</math> satisfying the equation<br />
<br />
<math>f(x+f(x+y))+f(xy) = x+f(x+y)+yf(x)</math><br />
<br />
for all real numbers <math>x</math> and <math>y</math>.<br />
<br />
[[2015 IMO Problems/Problem 5|Solution]]<br />
<br />
==Problem 6==<br />
The sequence <math>a_1,a_2,\dots</math> of integers satisfies the conditions:<br /><br /><br />
(i) <math>1\le a_j\le2015</math> for all <math>j\ge1</math>,<br /><br />
(ii) <math>k+a_k\neq \ell+a_\ell</math> for all <math>1\le k<\ell</math>.<br /><br /><br />
Prove that there exist two positive integers <math>b</math> and <math>N</math> for which<cmath>\left\vert\sum_{j=m+1}^n(a_j-b)\right\vert\le1007^2</cmath>for all integers <math>m</math> and <math>n</math> such that <math>n>m\ge N</math>.<br /><br /><br />
<br />
[[2015 IMO Problems/Problem 6|Solution]]<br />
<br />
==Problem 1==<br />
''Let <math>\mathbb{Z}</math> be the set of integers. Determine all functions <math>f : \mathbb{Z} \to \mathbb{Z}</math> such that, for all<br />
''integers <math>a</math> and <math>b</math>, <cmath>f(2a) + 2f(b) = f(f(a + b)).</cmath>''<br />
<br />
[[2019 IMO Problems/Problem 1|Solution]]<br />
<br />
==Problem 2==<br />
In triangle <math>ABC</math>, point <math>A_1</math> lies on side <math>BC</math> and point <math>B_1</math> lies on side <math>AC</math>. Let <math>P</math> and <math>Q</math> be points on segments <math>AA_1</math> and <math>BB_1</math>, respectively, such that <math>PQ</math> is parallel to <math>AB</math>. Let <math>P_1</math> be a point on line <math>PB_1</math>, such that <math>B_1</math> lies strictly between <math>P</math> and <math>P_1</math>, and <math>\angle PP_1C=\angle BAC</math>. Similarly, let <math>Q_1</math> be the point on line <math>QA_1</math>, such that <math>A_1</math> lies strictly between <math>Q</math> and <math>Q_1</math>, and <math>\angle CQ_1Q=\angle CBA</math>.<br />
<br />
Prove that points <math>P,Q,P_1</math>, and <math>Q_1</math> are concyclic.<br />
<br />
[[2019 IMO Problems/Problem 2|Solution]]<br />
<br />
==Problem 3==<br />
A social network has <math>2019</math> users, some pairs of whom are friends. Whenever user <math>A</math> is friends with user <math>B</math>, user <math>B</math> is also friends with user <math>A</math>. Events of the following kind may happen repeatedly, one at a time:<br />
Three users <math>A</math>, <math>B</math>, and <math>C</math> such that <math>A</math> is friends with both <math>B</math> and <math>C</math>, but <math>B</math> and <math>C</math> are not friends, change their friendship statuses such that <math>B</math> and <math>C</math> are now friends, but <math>A</math> is no longer friends with <math>B</math>, and no longer friends with <math>C</math>. All other friendship statuses are unchanged.<br />
Initially, <math>1010</math> users have <math>1009</math> friends each, and <math>1009</math> users have <math>1010</math> friends each. Prove that there exists a sequence of such events after which each user is friends with at most one other user.<br />
<br />
[[2019 IMO Problems/Problem 3|Solution]]<br />
<br />
==Problem 4==<br />
Find all pairs <math>(k,n)</math> of positive integers such that <br />
<br />
<cmath>k!=(2^n-1)(2^n-2)(2^n-4)\dots(2^n-2^{n-1}).</cmath><br />
<br />
[[2019 IMO Problems/Problem 4|Solution]]<br />
<br />
==Problem 5==<br />
The Bank of Bath issues coins with an <math>H</math> on one side and a <math>T</math> on the other. Harry has <math>n</math> of these coins arranged in a line from left to right. He repeatedly performs the following operation:<br />
<br />
If there are exactly <math>k > 0</math> coins showing <math>H</math>, then he turns over the <math>k^{th}</math> coin from the left; otherwise, all coins show <math>T</math> and he stops. For example, if <math>n = 3</math> the process starting with the configuration <math>THT</math> would be <math>THT \rightarrow HHT \rightarrow HTT \rightarrow TTT</math>, which stops after three operations.<br />
<br />
(a) Show that, for each initial configuration, Harry stops after a finite number of operations.<br />
<br />
(b) For each initial configuration <math>C</math>, let <math>L(C)</math> be the number of operations before Harry stops. For<br />
example, <math>L(THT) = 3</math> and <math>L(TTT) = 0</math>. Determine the average value of <math>L(C)</math> over all <math>2^n</math><br />
possible initial configurations <math>C</math>.<br />
<br />
[[2019 IMO Problems/Problem 5|Solution]]<br />
<br />
==Problem 6==<br />
Let <math>I</math> be the incenter of acute triangle <math>ABC</math> with <math>AB \neq AC</math>. The incircle <math>\omega</math> of <math>ABC</math> is tangent to sides <math>BC</math>, <math>CA</math>, and <math>AB</math> at <math>D</math>, <math>E</math>, and <math>F</math>, respectively. The line through <math>D</math> perpendicular to <math>EF</math> meets ω again at <math>R</math>. Line <math>AR</math> meets ω again at <math>P</math>. The circumcircles of triangles <math>PCE</math> and <math>PBF</math> meet again at <math>Q</math>.<br />
Prove that lines <math>DI</math> and <math>PQ</math> meet on the line through <math>A</math> perpendicular to <math>AI</math>.<br />
<br />
[[2019 IMO Problems/Problem 6|Solution]]<br />
<br />
{{IMO box|year=2015|before=[[2014 IMO Problems]]|after=[[2016 IMO Problems]]}}</div>Muhaboughttps://artofproblemsolving.com/wiki/index.php?title=2019_IMO&diff=1334522019 IMO2020-09-10T13:00:42Z<p>Muhaboug: Undo revision 133445 by Muhaboug (talk)</p>
<hr />
<div>mathlete6453 was here</div>Muhaboughttps://artofproblemsolving.com/wiki/index.php?title=2019_IMO&diff=1334512019 IMO2020-09-10T13:00:25Z<p>Muhaboug: Undo revision 133448 by Muhaboug (talk)</p>
<hr />
<div>mathlete6453 was here<br />
<br />
==Problem 1==<br />
''Let <math>\mathbb{Z}</math> be the set of integers. Determine all functions <math>f : \mathbb{Z} \to \mathbb{Z}</math> such that, for all<br />
''integers <math>a</math> and <math>b</math>, <cmath>f(2a) + 2f(b) = f(f(a + b)).</cmath>''<br />
<br />
[[2019 IMO Problems/Problem 1|Solution]]<br />
<br />
==Problem 2==<br />
In triangle <math>ABC</math>, point <math>A_1</math> lies on side <math>BC</math> and point <math>B_1</math> lies on side <math>AC</math>. Let <math>P</math> and <math>Q</math> be points on segments <math>AA_1</math> and <math>BB_1</math>, respectively, such that <math>PQ</math> is parallel to <math>AB</math>. Let <math>P_1</math> be a point on line <math>PB_1</math>, such that <math>B_1</math> lies strictly between <math>P</math> and <math>P_1</math>, and <math>\angle PP_1C=\angle BAC</math>. Similarly, let <math>Q_1</math> be the point on line <math>QA_1</math>, such that <math>A_1</math> lies strictly between <math>Q</math> and <math>Q_1</math>, and <math>\angle CQ_1Q=\angle CBA</math>.<br />
<br />
Prove that points <math>P,Q,P_1</math>, and <math>Q_1</math> are concyclic.<br />
<br />
[[2019 IMO Problems/Problem 2|Solution]]<br />
<br />
==Problem 3==<br />
A social network has <math>2019</math> users, some pairs of whom are friends. Whenever user <math>A</math> is friends with user <math>B</math>, user <math>B</math> is also friends with user <math>A</math>. Events of the following kind may happen repeatedly, one at a time:<br />
Three users <math>A</math>, <math>B</math>, and <math>C</math> such that <math>A</math> is friends with both <math>B</math> and <math>C</math>, but <math>B</math> and <math>C</math> are not friends, change their friendship statuses such that <math>B</math> and <math>C</math> are now friends, but <math>A</math> is no longer friends with <math>B</math>, and no longer friends with <math>C</math>. All other friendship statuses are unchanged.<br />
Initially, <math>1010</math> users have <math>1009</math> friends each, and <math>1009</math> users have <math>1010</math> friends each. Prove that there exists a sequence of such events after which each user is friends with at most one other user.<br />
<br />
[[2019 IMO Problems/Problem 3|Solution]]<br />
<br />
==Problem 4==<br />
Find all pairs <math>(k,n)</math> of positive integers such that <br />
<br />
<cmath>k!=(2^n-1)(2^n-2)(2^n-4)\dots(2^n-2^{n-1}).</cmath><br />
<br />
[[2019 IMO Problems/Problem 4|Solution]]<br />
<br />
==Problem 5==<br />
The Bank of Bath issues coins with an <math>H</math> on one side and a <math>T</math> on the other. Harry has <math>n</math> of these coins arranged in a line from left to right. He repeatedly performs the following operation:<br />
<br />
If there are exactly <math>k > 0</math> coins showing <math>H</math>, then he turns over the <math>k^{th}</math> coin from the left; otherwise, all coins show <math>T</math> and he stops. For example, if <math>n = 3</math> the process starting with the configuration <math>THT</math> would be <math>THT \rightarrow HHT \rightarrow HTT \rightarrow TTT</math>, which stops after three operations.<br />
<br />
(a) Show that, for each initial configuration, Harry stops after a finite number of operations.<br />
<br />
(b) For each initial configuration <math>C</math>, let <math>L(C)</math> be the number of operations before Harry stops. For<br />
example, <math>L(THT) = 3</math> and <math>L(TTT) = 0</math>. Determine the average value of <math>L(C)</math> over all <math>2^n</math><br />
possible initial configurations <math>C</math>.<br />
<br />
[[2019 IMO Problems/Problem 5|Solution]]<br />
<br />
==Problem 6==<br />
Let <math>I</math> be the incenter of acute triangle <math>ABC</math> with <math>AB \neq AC</math>. The incircle <math>\omega</math> of <math>ABC</math> is tangent to sides <math>BC</math>, <math>CA</math>, and <math>AB</math> at <math>D</math>, <math>E</math>, and <math>F</math>, respectively. The line through <math>D</math> perpendicular to <math>EF</math> meets ω again at <math>R</math>. Line <math>AR</math> meets ω again at <math>P</math>. The circumcircles of triangles <math>PCE</math> and <math>PBF</math> meet again at <math>Q</math>.<br />
Prove that lines <math>DI</math> and <math>PQ</math> meet on the line through <math>A</math> perpendicular to <math>AI</math>.<br />
<br />
[[2019 IMO Problems/Problem 6|Solution]]</div>Muhaboughttps://artofproblemsolving.com/wiki/index.php?title=2019_IMO_Problems&diff=1334502019 IMO Problems2020-09-10T12:59:12Z<p>Muhaboug: Created page with "==Problem 1== ''Let <math>\mathbb{Z}</math> be the set of integers. Determine all functions <math>f : \mathbb{Z} \to \mathbb{Z}</math> such that, for all ''integers <math>a</m..."</p>
<hr />
<div>==Problem 1==<br />
''Let <math>\mathbb{Z}</math> be the set of integers. Determine all functions <math>f : \mathbb{Z} \to \mathbb{Z}</math> such that, for all<br />
''integers <math>a</math> and <math>b</math>, <cmath>f(2a) + 2f(b) = f(f(a + b)).</cmath>''<br />
<br />
[[2019 IMO Problems/Problem 1|Solution]]<br />
<br />
==Problem 2==<br />
In triangle <math>ABC</math>, point <math>A_1</math> lies on side <math>BC</math> and point <math>B_1</math> lies on side <math>AC</math>. Let <math>P</math> and <math>Q</math> be points on segments <math>AA_1</math> and <math>BB_1</math>, respectively, such that <math>PQ</math> is parallel to <math>AB</math>. Let <math>P_1</math> be a point on line <math>PB_1</math>, such that <math>B_1</math> lies strictly between <math>P</math> and <math>P_1</math>, and <math>\angle PP_1C=\angle BAC</math>. Similarly, let <math>Q_1</math> be the point on line <math>QA_1</math>, such that <math>A_1</math> lies strictly between <math>Q</math> and <math>Q_1</math>, and <math>\angle CQ_1Q=\angle CBA</math>.<br />
<br />
Prove that points <math>P,Q,P_1</math>, and <math>Q_1</math> are concyclic.<br />
<br />
[[2019 IMO Problems/Problem 2|Solution]]<br />
<br />
==Problem 3==<br />
A social network has <math>2019</math> users, some pairs of whom are friends. Whenever user <math>A</math> is friends with user <math>B</math>, user <math>B</math> is also friends with user <math>A</math>. Events of the following kind may happen repeatedly, one at a time:<br />
Three users <math>A</math>, <math>B</math>, and <math>C</math> such that <math>A</math> is friends with both <math>B</math> and <math>C</math>, but <math>B</math> and <math>C</math> are not friends, change their friendship statuses such that <math>B</math> and <math>C</math> are now friends, but <math>A</math> is no longer friends with <math>B</math>, and no longer friends with <math>C</math>. All other friendship statuses are unchanged.<br />
Initially, <math>1010</math> users have <math>1009</math> friends each, and <math>1009</math> users have <math>1010</math> friends each. Prove that there exists a sequence of such events after which each user is friends with at most one other user.<br />
<br />
[[2019 IMO Problems/Problem 3|Solution]]<br />
<br />
==Problem 4==<br />
Find all pairs <math>(k,n)</math> of positive integers such that <br />
<br />
<cmath>k!=(2^n-1)(2^n-2)(2^n-4)\dots(2^n-2^{n-1}).</cmath><br />
<br />
[[2019 IMO Problems/Problem 4|Solution]]<br />
<br />
==Problem 5==<br />
The Bank of Bath issues coins with an <math>H</math> on one side and a <math>T</math> on the other. Harry has <math>n</math> of these coins arranged in a line from left to right. He repeatedly performs the following operation:<br />
<br />
If there are exactly <math>k > 0</math> coins showing <math>H</math>, then he turns over the <math>k^{th}</math> coin from the left; otherwise, all coins show <math>T</math> and he stops. For example, if <math>n = 3</math> the process starting with the configuration <math>THT</math> would be <math>THT \rightarrow HHT \rightarrow HTT \rightarrow TTT</math>, which stops after three operations.<br />
<br />
(a) Show that, for each initial configuration, Harry stops after a finite number of operations.<br />
<br />
(b) For each initial configuration <math>C</math>, let <math>L(C)</math> be the number of operations before Harry stops. For<br />
example, <math>L(THT) = 3</math> and <math>L(TTT) = 0</math>. Determine the average value of <math>L(C)</math> over all <math>2^n</math><br />
possible initial configurations <math>C</math>.<br />
<br />
[[2019 IMO Problems/Problem 5|Solution]]<br />
<br />
==Problem 6==<br />
Let <math>I</math> be the incenter of acute triangle <math>ABC</math> with <math>AB \neq AC</math>. The incircle <math>\omega</math> of <math>ABC</math> is tangent to sides <math>BC</math>, <math>CA</math>, and <math>AB</math> at <math>D</math>, <math>E</math>, and <math>F</math>, respectively. The line through <math>D</math> perpendicular to <math>EF</math> meets ω again at <math>R</math>. Line <math>AR</math> meets ω again at <math>P</math>. The circumcircles of triangles <math>PCE</math> and <math>PBF</math> meet again at <math>Q</math>.<br />
Prove that lines <math>DI</math> and <math>PQ</math> meet on the line through <math>A</math> perpendicular to <math>AI</math>.<br />
<br />
[[2019 IMO Problems/Problem 6|Solution]]<br />
<br />
{{IMO box|year=2019|before=[[2018 IMO Problems]]|after=[[2020 IMO Problems]]}}</div>Muhaboughttps://artofproblemsolving.com/wiki/index.php?title=2018_IMO_Problems&diff=1334492018 IMO Problems2020-09-10T12:57:27Z<p>Muhaboug: </p>
<hr />
<div>==Problem 1==<br />
Let <math>\Gamma</math> be the circumcircle of acute triangle <math>ABC</math>. Points <math>D</math> and <math>E</math> are on segments <math>AB</math> and <math>AC</math> respectively such that <math>AD = AE</math>. The perpendicular bisectors of <math>BD</math> and <math>CE</math> intersect minor arcs <math>AB</math> and <math>AC</math> of <math>\Gamma</math> at points <math>F</math> and <math>G</math> respectively. Prove that lines <math>DE</math> and <math>FG</math> are either parallel or they are the same line.<br />
<br />
[[2018 IMO Problems/Problem 1|Solution]]<br />
<br />
==Problem 2==<br />
Find all numbers <math>n \ge 3</math> for which there exists real numbers <math>a_1, a_2, ..., a_{n+2}</math> satisfying <math>a_{n+1} = a_1, a_{n+2} = a_2</math> and <br />
<cmath>a_{i}a_{i+1} + 1 = a_{i+2}</cmath><br />
for <math>i = 1, 2, ..., n.</math><br />
<br />
[[2018 IMO Problems/Problem 2|Solution]]<br />
<br />
==Problem 3==<br />
An anti-Pascal triangle is an equilateral triangular array of numbers such that, except for the numbers in the bottom row, each number is the absolute value of the difference of the two numbers immediately below it. For example, the following is an anti-Pascal triangle with four rows which contains every integer from <math>1</math> to <math>10</math><br />
<br />
<cmath>4</cmath><br />
<cmath>2\quad 6</cmath><br />
<cmath>5\quad 7 \quad 1</cmath><br />
<cmath>8\quad 3 \quad 10 \quad 9</cmath><br />
<br />
Does there exist an anti-Pascal triangle with <math>2018</math> rows which contains every integer from <math>1</math> to <math>1 + 2 + 3 + \dots + 2018</math>?<br />
<br />
[[2018 IMO Problems/Problem 3|Solution]]<br />
<br />
==Problem 4==<br />
A site is any point <math>(x, y)</math> in the plane such that <math>x</math> and <math>y</math> are both positive integers less<br />
than or equal to 20.<br />
Initially, each of the 400 sites is unoccupied. Amy and Ben take turns placing stones with Amy<br />
going first. On her turn, Amy places a new red stone on an unoccupied site such that the distance<br />
between any two sites occupied by red stones is not equal to <math>\sqrt{5}</math>. On his turn, Ben places a new blue<br />
stone on any unoccupied site. (A site occupied by a blue stone is allowed to be at any distance from<br />
any other occupied site.) They stop as soon as a player cannot place a stone.<br />
Find the greatest <math>K</math> such that Amy can ensure that she places at least <math>K</math> red stones, no matter<br />
how Ben places his blue stones.<br />
<br />
[[2018 IMO Problems/Problem 4|Solution]]<br />
<br />
==Problem 5==<br />
Let <math>a_1, a_2, \dots</math> be an infinite sequence of positive integers. Suppose that there is an integer<math> N > 1</math> such that, for each <math>n \geq N</math>, the number <math>\frac{a_1}{a_2}+\frac{a_2}{a_3}+\dots +\frac{a_{n-1}}{a_n}+\frac{a_n}{a_1}</math> is an integer. Prove that there is a positive integer <math>M</math> such that <math>a_m = a_{m+1}</math> for all <math>m \geq M.</math><br />
<br />
[[2018 IMO Problems/Problem 5|Solution]]<br />
<br />
==Problem 6==<br />
A convex quadrilateral <math>ABCD</math> satisfies <math>AB\cdot CD=BC \cdot DA.</math> Point <math>X</math> lies inside<br />
<math>ABCD</math> so that<br />
<math>\angle XAB = \angle XCD</math> and <math>\angle XBC = \angle XDA.</math><br />
Prove that <math>\angle BXA + \angle DXC = 180^{\circ}</math>.<br />
<br />
[[2018 IMO Problems/Problem 6|Solution]]<br />
<br />
{{IMO box|year=2018|before=[[2017 IMO Problems]]|after=[[2019 IMO Problems]]}}</div>Muhaboughttps://artofproblemsolving.com/wiki/index.php?title=2019_IMO&diff=1334482019 IMO2020-09-10T12:56:49Z<p>Muhaboug: </p>
<hr />
<div>mathlete6453 was here<br />
<br />
==Problem 1==<br />
''Let <math>\mathbb{Z}</math> be the set of integers. Determine all functions <math>f : \mathbb{Z} \to \mathbb{Z}</math> such that, for all<br />
''integers <math>a</math> and <math>b</math>, <cmath>f(2a) + 2f(b) = f(f(a + b)).</cmath>''<br />
<br />
[[2019 IMO Problems/Problem 1|Solution]]<br />
<br />
==Problem 2==<br />
In triangle <math>ABC</math>, point <math>A_1</math> lies on side <math>BC</math> and point <math>B_1</math> lies on side <math>AC</math>. Let <math>P</math> and <math>Q</math> be points on segments <math>AA_1</math> and <math>BB_1</math>, respectively, such that <math>PQ</math> is parallel to <math>AB</math>. Let <math>P_1</math> be a point on line <math>PB_1</math>, such that <math>B_1</math> lies strictly between <math>P</math> and <math>P_1</math>, and <math>\angle PP_1C=\angle BAC</math>. Similarly, let <math>Q_1</math> be the point on line <math>QA_1</math>, such that <math>A_1</math> lies strictly between <math>Q</math> and <math>Q_1</math>, and <math>\angle CQ_1Q=\angle CBA</math>.<br />
<br />
Prove that points <math>P,Q,P_1</math>, and <math>Q_1</math> are concyclic.<br />
<br />
[[2019 IMO Problems/Problem 2|Solution]]<br />
<br />
==Problem 3==<br />
A social network has <math>2019</math> users, some pairs of whom are friends. Whenever user <math>A</math> is friends with user <math>B</math>, user <math>B</math> is also friends with user <math>A</math>. Events of the following kind may happen repeatedly, one at a time:<br />
Three users <math>A</math>, <math>B</math>, and <math>C</math> such that <math>A</math> is friends with both <math>B</math> and <math>C</math>, but <math>B</math> and <math>C</math> are not friends, change their friendship statuses such that <math>B</math> and <math>C</math> are now friends, but <math>A</math> is no longer friends with <math>B</math>, and no longer friends with <math>C</math>. All other friendship statuses are unchanged.<br />
Initially, <math>1010</math> users have <math>1009</math> friends each, and <math>1009</math> users have <math>1010</math> friends each. Prove that there exists a sequence of such events after which each user is friends with at most one other user.<br />
<br />
[[2019 IMO Problems/Problem 3|Solution]]<br />
<br />
==Problem 4==<br />
Find all pairs <math>(k,n)</math> of positive integers such that <br />
<br />
<cmath>k!=(2^n-1)(2^n-2)(2^n-4)\dots(2^n-2^{n-1}).</cmath><br />
<br />
[[2019 IMO Problems/Problem 4|Solution]]<br />
<br />
==Problem 5==<br />
The Bank of Bath issues coins with an <math>H</math> on one side and a <math>T</math> on the other. Harry has <math>n</math> of these coins arranged in a line from left to right. He repeatedly performs the following operation:<br />
<br />
If there are exactly <math>k > 0</math> coins showing <math>H</math>, then he turns over the <math>k^{th}</math> coin from the left; otherwise, all coins show <math>T</math> and he stops. For example, if <math>n = 3</math> the process starting with the configuration <math>THT</math> would be <math>THT \rightarrow HHT \rightarrow HTT \rightarrow TTT</math>, which stops after three operations.<br />
<br />
(a) Show that, for each initial configuration, Harry stops after a finite number of operations.<br />
<br />
(b) For each initial configuration <math>C</math>, let <math>L(C)</math> be the number of operations before Harry stops. For<br />
example, <math>L(THT) = 3</math> and <math>L(TTT) = 0</math>. Determine the average value of <math>L(C)</math> over all <math>2^n</math><br />
possible initial configurations <math>C</math>.<br />
<br />
[[2019 IMO Problems/Problem 5|Solution]]<br />
<br />
==Problem 6==<br />
Let <math>I</math> be the incenter of acute triangle <math>ABC</math> with <math>AB \neq AC</math>. The incircle <math>\omega</math> of <math>ABC</math> is tangent to sides <math>BC</math>, <math>CA</math>, and <math>AB</math> at <math>D</math>, <math>E</math>, and <math>F</math>, respectively. The line through <math>D</math> perpendicular to <math>EF</math> meets ω again at <math>R</math>. Line <math>AR</math> meets ω again at <math>P</math>. The circumcircles of triangles <math>PCE</math> and <math>PBF</math> meet again at <math>Q</math>.<br />
Prove that lines <math>DI</math> and <math>PQ</math> meet on the line through <math>A</math> perpendicular to <math>AI</math>.<br />
<br />
[[2019 IMO Problems/Problem 6|Solution]]<br />
<br />
{{IMO box|year=2019|before=[[2018 IMO Problems]]|after=[[2020 IMO Problems]]}}</div>Muhaboughttps://artofproblemsolving.com/wiki/index.php?title=2016_IMO_Problems&diff=1334472016 IMO Problems2020-09-10T12:56:25Z<p>Muhaboug: </p>
<hr />
<div>==Problem 1==<br />
Triangle <math>BCF</math> has a right angle at <math>B</math>. Let <math>A</math> be the point on line <math>CF</math> such that <math>FA=FB</math> and <math>F</math> lies between <math>A</math> and <math>C</math>. Point <math>D</math> is chosen so that <math>DA=DC</math> and <math>AC</math> is the bisector of <math>\angle{DAB}</math>. Point <math>E</math> is chosen so that <math>EA=ED</math> and <math>AD</math> is the bisector of <math>\angle{EAC}</math>. Let <math>M</math> be the midpoint of <math>CF</math>. Let <math>X</math> be the point such that <math>AMXE</math> is a parallelogram. Prove that <math>BD,FX</math> and <math>ME</math> are concurrent.<br />
<br />
[[2016 IMO Problems/Problem 1|Solution]]<br />
<br />
==Problem 2==<br />
Find all integers <math>n</math> for which each cell of <math>n \times n</math> table can be filled with one of the letters <math>I,M</math> and <math>O</math> in such a way that:<br />
{| border="0" cellpadding="5"<br />
| valign="top"|<br />
* in each row and each column, one third of the entries are <math>I</math>, one third are <math>M</math> and one third are <math>O</math>; and<br />
* in any diagonal, if the number of entries on the diagonal is a multiple of three, then one third of the entries are <math>I</math>, one third are <math>M</math> and one third are <math>O</math>.<br />
|}<br />
'''Note.''' The rows and columns of an <math>n \times n</math> table are each labelled <math>1</math> to <math>n</math> in a natural order. Thus each cell corresponds to a pair of positive integer <math>(i,j)</math> with <math>1 \le i,j \le n</math>. For <math>n>1</math>, the table has <math>4n-2</math> diagonals of two types. A diagonal of first type consists all cells <math>(i,j)</math> for which <math>i+j</math> is a constant, and the diagonal of this second type consists all cells <math>(i,j)</math> for which <math>i-j</math> is constant.<br />
<br />
[[2016 IMO Problems/Problem 2|Solution]]<br />
<br />
==Problem 3==<br />
Let <math>P = A_1A_2 \cdots A_k</math> be a convex polygon in the plane. The vertices <math>A_1,A_2,\dots, A_k</math> have integral coordinates and lie on a circle. Let <math>S</math> be the area of <math>P</math>. An odd positive integer <math>n</math> is given such that the squares of the side lengths of <math>P</math> are integers divisible by <math>n</math>. Prove that <math>2S</math> is an integer divisible by <math>n</math>.<br />
<br />
[[2016 IMO Problems/Problem 3|Solution]]<br />
<br />
==Problem 4==<br />
A set of positive integers is called ''fragrant'' if it contains at least two elements and each of its elements has a prime factor in common with at least one of the other elements. Let <math>P(n)=n^2+n+1</math>. What is the least possible positive integer value of <math>b</math> such that there exists a non-negative integer <math>a</math> for which the set <math>\{P(a+1),P(a+2),\ldots,P(a+b)\}</math> is fragrant?<br />
<br />
[[2016 IMO Problems/Problem 4|Solution]]<br />
<br />
==Problem 5==<br />
The equation<br />
<center><math>(x-1)(x-2)\cdots(x-2016)=(x-1)(x-2)\cdots (x-2016)</math></center><br />
is written on the board, with <math>2016</math> linear factors on each side. What is the least possible value of <math>k</math> for which it is possible to erase exactly <math>k</math> of these <math>4032</math> linear factors so that at least one factor remains on each side and the resulting equation has no real solutions?<br />
<br />
[[2016 IMO Problems/Problem 5|Solution]]<br />
<br />
==Problem 6==<br />
There are <math>n\ge 2</math> line segments in the plane such that every two segments cross and no three segments meet at a point. Geoff has to choose an endpoint of each segment and place a frog on it facing the other endpoint. Then he will clap his hands <math>n-1</math> times. Every time he claps,each frog will immediately jump forward to the next intersection point on its segment. Frogs never change the direction of their jumps. Geoff wishes to place the frogs in such a way that no two of them will ever occupy the same intersection point at the same time.<br />
<br />
(a) Prove that Geoff can always fulfill his wish if <math>n</math> is odd.<br />
<br />
(b) Prove that Geoff can never fulfill his wish if <math>n</math> is even.<br />
<br />
[[2016 IMO Problems/Problem 6|Solution]]<br />
<br />
{{IMO box|year=2016|before=[[2015 IMO Problems]]|after=[[2017 IMO Problems]]}}</div>Muhaboughttps://artofproblemsolving.com/wiki/index.php?title=2017_IMO_Problems&diff=1334462017 IMO Problems2020-09-10T12:55:49Z<p>Muhaboug: </p>
<hr />
<div>==Problem 1==<br />
For each integer <math>a_0 > 1</math>, define the sequence <math>a_0, a_1, a_2, \ldots</math> for <math>n \geq 0</math> as<br />
<cmath>a_{n+1} = <br />
\begin{cases}<br />
\sqrt{a_n} & \text{if } \sqrt{a_n} \text{ is an integer,} \\<br />
a_n + 3 & \text{otherwise.}<br />
\end{cases}<br />
</cmath>Determine all values of <math>a_0</math> such that there exists a number <math>A</math> such that <math>a_n = A</math> for infinitely many values of <math>n</math>.<br />
<br />
[[2017 IMO Problems/Problem 1|Solution]]<br />
<br />
==Problem 2==<br />
Let <math>\mathbb{R}</math> be the set of real numbers , determine all functions <br />
<math>f:\mathbb{R}\rightarrow\mathbb{R}</math> such that for any real numbers <math>x</math> and <math>y</math> <math>{f(f(x)f(y)) + f(x+y)}</math> =<math>f(xy)</math><br />
<br />
[[2017 IMO Problems/Problem 2|Solution]]<br />
<br />
==Problem 3==<br />
A hunter and an invisible rabbit play a game in the Euclidean plane. The rabbit's starting point, <math>A_0</math>, and the hunter's starting point, <math>B_0</math>, are the same. After <math>n-1</math> rounds of the game, the rabbit is at point <math>A_{n-1}</math> and the hunter is at point <math>B_{n-1}</math>. In the nth round of the game, three things occur in order.<br />
<br />
(i) The rabbit moves invisibly to a point <math>A_n</math> such that the distance between <math>A_{n-1}</math> and <math>A_n</math> is exactly 1.<br />
<br />
(ii) A tracking device reports a point <math>P_n</math> to the hunter. The only guarantee provided by the tracking device is that the distance between <math>P_n</math> and <math>A_n</math> is at most 1.<br />
<br />
(iii) The hunter moves visibly to a point <math>B_n</math> such that the distance between <math>B_{n-1}</math> and <math>B_n</math> is exactly 1.<br />
<br />
Is it always possible, no matter how the rabbit moves, and no matter what points are reported by the tracking device, for the hunter to choose her moves so that after <math>10^9</math> rounds she can ensure that the distance between her and the rabbit is at most 100?<br />
<br />
[[2017 IMO Problems/Problem 3|Solution]]<br />
<br />
==Problem 4==<br />
Let <math>R</math> and <math>S</math> be different points on a circle <math>\Omega</math> such that <math>RS</math> is not a diameter. Let <math>\ell</math> be the tangent line to <math>\Omega</math> at <math>R</math>. Point <math>T</math> is such that <math>S</math> is the midpoint of the line segment <math>RT</math>. Point <math>J</math> is chosen on the shorter arc <math>RS</math> of <math>\Omega</math> so that the circumcircle <math>\Gamma</math> of triangle <math>JST</math> intersects <math>\ell</math> at two distinct points. Let <math>A</math> be the common point of <math>\Gamma</math> and <math>\ell</math> that is closer to <math>R</math>. Line <math>AJ</math> meets <math>\Omega</math> again at <math>K</math>. Prove that the line <math>KT</math> is tangent to <math>\Gamma</math>.<br />
<br />
[[2017 IMO Problems/Problem 4|Solution]]<br />
<br />
==Problem 5==<br />
An integer <math>N \ge 2</math> is given. A collection of <math>N(N + 1)</math> soccer players, no two of whom are of the same height, stand in a row. Sir Alex wants to remove <math>N(N - 1)</math> players from this row leaving a new row of <math>2N</math> players in which the following <math>N</math> conditions hold:<br />
<br />
(<math>1</math>) no one stands between the two tallest players,<br />
<br />
(<math>2</math>) no one stands between the third and fourth tallest players,<br />
<br />
<math>\;\;\vdots</math><br />
<br />
(<math>N</math>) no one stands between the two shortest players.<br />
<br />
Show that this is always possible.<br />
<br />
[[2017 IMO Problems/Problem 5|Solution]]<br />
<br />
==Problem 6==<br />
An ordered pair <math>(x, y)</math> of integers is a primitive point if the greatest common divisor of <math>x</math> and <math>y</math> is <math>1</math>. Given a finite set <math>S</math> of primitive points, prove that there exist a positive integer <math>n</math> and integers <math>a_0, a_1, \ldots , a_n</math> such that, for each <math>(x, y)</math> in <math>S</math>, we have:<br />
<cmath>a_0x^n + a_1x^{n-1} y + a_2x^{n-2}y^2 + \cdots + a_{n-1}xy^{n-1} + a_ny^n = 1.</cmath><br />
<br />
[[2017 IMO Problems/Problem 6|Solution]]<br />
<br />
{{IMO box|year=2017|before=[[2016 IMO Problems]]|after=[[2018 IMO Problems]]}}</div>Muhaboughttps://artofproblemsolving.com/wiki/index.php?title=2019_IMO&diff=1334452019 IMO2020-09-10T12:51:54Z<p>Muhaboug: </p>
<hr />
<div>mathlete6453 was here<br />
<br />
==Problem 1==<br />
''Let <math>\mathbb{Z}</math> be the set of integers. Determine all functions <math>f : \mathbb{Z} \to \mathbb{Z}</math> such that, for all<br />
''integers <math>a</math> and <math>b</math>, <cmath>f(2a) + 2f(b) = f(f(a + b)).</cmath>''<br />
<br />
[[2019 IMO Problems/Problem 1|Solution]]<br />
<br />
==Problem 2==<br />
In triangle <math>ABC</math>, point <math>A_1</math> lies on side <math>BC</math> and point <math>B_1</math> lies on side <math>AC</math>. Let <math>P</math> and <math>Q</math> be points on segments <math>AA_1</math> and <math>BB_1</math>, respectively, such that <math>PQ</math> is parallel to <math>AB</math>. Let <math>P_1</math> be a point on line <math>PB_1</math>, such that <math>B_1</math> lies strictly between <math>P</math> and <math>P_1</math>, and <math>\angle PP_1C=\angle BAC</math>. Similarly, let <math>Q_1</math> be the point on line <math>QA_1</math>, such that <math>A_1</math> lies strictly between <math>Q</math> and <math>Q_1</math>, and <math>\angle CQ_1Q=\angle CBA</math>.<br />
<br />
Prove that points <math>P,Q,P_1</math>, and <math>Q_1</math> are concyclic.<br />
<br />
[[2019 IMO Problems/Problem 2|Solution]]<br />
<br />
==Problem 3==<br />
A social network has <math>2019</math> users, some pairs of whom are friends. Whenever user <math>A</math> is friends with user <math>B</math>, user <math>B</math> is also friends with user <math>A</math>. Events of the following kind may happen repeatedly, one at a time:<br />
Three users <math>A</math>, <math>B</math>, and <math>C</math> such that <math>A</math> is friends with both <math>B</math> and <math>C</math>, but <math>B</math> and <math>C</math> are not friends, change their friendship statuses such that <math>B</math> and <math>C</math> are now friends, but <math>A</math> is no longer friends with <math>B</math>, and no longer friends with <math>C</math>. All other friendship statuses are unchanged.<br />
Initially, <math>1010</math> users have <math>1009</math> friends each, and <math>1009</math> users have <math>1010</math> friends each. Prove that there exists a sequence of such events after which each user is friends with at most one other user.<br />
<br />
[[2019 IMO Problems/Problem 3|Solution]]<br />
<br />
==Problem 4==<br />
Find all pairs <math>(k,n)</math> of positive integers such that <br />
<br />
<cmath>k!=(2^n-1)(2^n-2)(2^n-4)\dots(2^n-2^{n-1}).</cmath><br />
<br />
[[2019 IMO Problems/Problem 4|Solution]]<br />
<br />
==Problem 5==<br />
The Bank of Bath issues coins with an <math>H</math> on one side and a <math>T</math> on the other. Harry has <math>n</math> of these coins arranged in a line from left to right. He repeatedly performs the following operation:<br />
<br />
If there are exactly <math>k > 0</math> coins showing <math>H</math>, then he turns over the <math>k^{th}</math> coin from the left; otherwise, all coins show <math>T</math> and he stops. For example, if <math>n = 3</math> the process starting with the configuration <math>THT</math> would be <math>THT \rightarrow HHT \rightarrow HTT \rightarrow TTT</math>, which stops after three operations.<br />
<br />
(a) Show that, for each initial configuration, Harry stops after a finite number of operations.<br />
<br />
(b) For each initial configuration <math>C</math>, let <math>L(C)</math> be the number of operations before Harry stops. For<br />
example, <math>L(THT) = 3</math> and <math>L(TTT) = 0</math>. Determine the average value of <math>L(C)</math> over all <math>2^n</math><br />
possible initial configurations <math>C</math>.<br />
<br />
[[2019 IMO Problems/Problem 5|Solution]]<br />
<br />
==Problem 6==<br />
Let <math>I</math> be the incenter of acute triangle <math>ABC</math> with <math>AB \neq AC</math>. The incircle <math>\omega</math> of <math>ABC</math> is tangent to sides <math>BC</math>, <math>CA</math>, and <math>AB</math> at <math>D</math>, <math>E</math>, and <math>F</math>, respectively. The line through <math>D</math> perpendicular to <math>EF</math> meets ω again at <math>R</math>. Line <math>AR</math> meets ω again at <math>P</math>. The circumcircles of triangles <math>PCE</math> and <math>PBF</math> meet again at <math>Q</math>.<br />
Prove that lines <math>DI</math> and <math>PQ</math> meet on the line through <math>A</math> perpendicular to <math>AI</math>.<br />
<br />
[[2019 IMO Problems/Problem 6|Solution]]</div>Muhaboughttps://artofproblemsolving.com/wiki/index.php?title=2018_IMO_Problems&diff=1334442018 IMO Problems2020-09-10T12:47:53Z<p>Muhaboug: Created page with "==Problem 1== Let <math>\Gamma</math> be the circumcircle of acute triangle <math>ABC</math>. Points <math>D</math> and <math>E</math> are on segments <math>AB</math> and <mat..."</p>
<hr />
<div>==Problem 1==<br />
Let <math>\Gamma</math> be the circumcircle of acute triangle <math>ABC</math>. Points <math>D</math> and <math>E</math> are on segments <math>AB</math> and <math>AC</math> respectively such that <math>AD = AE</math>. The perpendicular bisectors of <math>BD</math> and <math>CE</math> intersect minor arcs <math>AB</math> and <math>AC</math> of <math>\Gamma</math> at points <math>F</math> and <math>G</math> respectively. Prove that lines <math>DE</math> and <math>FG</math> are either parallel or they are the same line.<br />
<br />
[[2018 IMO Problems/Problem 1|Solution]]<br />
<br />
==Problem 2==<br />
Find all numbers <math>n \ge 3</math> for which there exists real numbers <math>a_1, a_2, ..., a_{n+2}</math> satisfying <math>a_{n+1} = a_1, a_{n+2} = a_2</math> and <br />
<cmath>a_{i}a_{i+1} + 1 = a_{i+2}</cmath><br />
for <math>i = 1, 2, ..., n.</math><br />
<br />
[[2018 IMO Problems/Problem 2|Solution]]<br />
<br />
==Problem 3==<br />
An anti-Pascal triangle is an equilateral triangular array of numbers such that, except for the numbers in the bottom row, each number is the absolute value of the difference of the two numbers immediately below it. For example, the following is an anti-Pascal triangle with four rows which contains every integer from <math>1</math> to <math>10</math><br />
<br />
<cmath>4</cmath><br />
<cmath>2\quad 6</cmath><br />
<cmath>5\quad 7 \quad 1</cmath><br />
<cmath>8\quad 3 \quad 10 \quad 9</cmath><br />
<br />
Does there exist an anti-Pascal triangle with <math>2018</math> rows which contains every integer from <math>1</math> to <math>1 + 2 + 3 + \dots + 2018</math>?<br />
<br />
[[2018 IMO Problems/Problem 3|Solution]]<br />
<br />
==Problem 4==<br />
A site is any point <math>(x, y)</math> in the plane such that <math>x</math> and <math>y</math> are both positive integers less<br />
than or equal to 20.<br />
Initially, each of the 400 sites is unoccupied. Amy and Ben take turns placing stones with Amy<br />
going first. On her turn, Amy places a new red stone on an unoccupied site such that the distance<br />
between any two sites occupied by red stones is not equal to <math>\sqrt{5}</math>. On his turn, Ben places a new blue<br />
stone on any unoccupied site. (A site occupied by a blue stone is allowed to be at any distance from<br />
any other occupied site.) They stop as soon as a player cannot place a stone.<br />
Find the greatest <math>K</math> such that Amy can ensure that she places at least <math>K</math> red stones, no matter<br />
how Ben places his blue stones.<br />
<br />
[[2018 IMO Problems/Problem 4|Solution]]<br />
<br />
==Problem 5==<br />
Let <math>a_1, a_2, \dots</math> be an infinite sequence of positive integers. Suppose that there is an integer<math> N > 1</math> such that, for each <math>n \geq N</math>, the number <math>\frac{a_1}{a_2}+\frac{a_2}{a_3}+\dots +\frac{a_{n-1}}{a_n}+\frac{a_n}{a_1}</math> is an integer. Prove that there is a positive integer <math>M</math> such that <math>a_m = a_{m+1}</math> for all <math>m \geq M.</math><br />
<br />
[[2018 IMO Problems/Problem 5|Solution]]<br />
<br />
==Problem 6==<br />
A convex quadrilateral <math>ABCD</math> satisfies <math>AB\cdot CD=BC \cdot DA.</math> Point <math>X</math> lies inside<br />
<math>ABCD</math> so that<br />
<math>\angle XAB = \angle XCD</math> and <math>\angle XBC = \angle XDA.</math><br />
Prove that <math>\angle BXA + \angle DXC = 180^{\circ}</math>.<br />
<br />
[[2018 IMO Problems/Problem 6|Solution]]</div>Muhaboughttps://artofproblemsolving.com/wiki/index.php?title=2017_IMO_Problems&diff=1334432017 IMO Problems2020-09-10T12:44:00Z<p>Muhaboug: Created page with "==Problem 1== For each integer <math>a_0 > 1</math>, define the sequence <math>a_0, a_1, a_2, \ldots</math> for <math>n \geq 0</math> as <cmath>a_{n+1} = \begin{cases} \sqrt{..."</p>
<hr />
<div>==Problem 1==<br />
For each integer <math>a_0 > 1</math>, define the sequence <math>a_0, a_1, a_2, \ldots</math> for <math>n \geq 0</math> as<br />
<cmath>a_{n+1} = <br />
\begin{cases}<br />
\sqrt{a_n} & \text{if } \sqrt{a_n} \text{ is an integer,} \\<br />
a_n + 3 & \text{otherwise.}<br />
\end{cases}<br />
</cmath>Determine all values of <math>a_0</math> such that there exists a number <math>A</math> such that <math>a_n = A</math> for infinitely many values of <math>n</math>.<br />
<br />
[[2017 IMO Problems/Problem 1|Solution]]<br />
<br />
==Problem 2==<br />
Let <math>\mathbb{R}</math> be the set of real numbers , determine all functions <br />
<math>f:\mathbb{R}\rightarrow\mathbb{R}</math> such that for any real numbers <math>x</math> and <math>y</math> <math>{f(f(x)f(y)) + f(x+y)}</math> =<math>f(xy)</math><br />
<br />
[[2017 IMO Problems/Problem 2|Solution]]<br />
<br />
==Problem 3==<br />
A hunter and an invisible rabbit play a game in the Euclidean plane. The rabbit's starting point, <math>A_0</math>, and the hunter's starting point, <math>B_0</math>, are the same. After <math>n-1</math> rounds of the game, the rabbit is at point <math>A_{n-1}</math> and the hunter is at point <math>B_{n-1}</math>. In the nth round of the game, three things occur in order.<br />
<br />
(i) The rabbit moves invisibly to a point <math>A_n</math> such that the distance between <math>A_{n-1}</math> and <math>A_n</math> is exactly 1.<br />
<br />
(ii) A tracking device reports a point <math>P_n</math> to the hunter. The only guarantee provided by the tracking device is that the distance between <math>P_n</math> and <math>A_n</math> is at most 1.<br />
<br />
(iii) The hunter moves visibly to a point <math>B_n</math> such that the distance between <math>B_{n-1}</math> and <math>B_n</math> is exactly 1.<br />
<br />
Is it always possible, no matter how the rabbit moves, and no matter what points are reported by the tracking device, for the hunter to choose her moves so that after <math>10^9</math> rounds she can ensure that the distance between her and the rabbit is at most 100?<br />
<br />
[[2017 IMO Problems/Problem 3|Solution]]<br />
<br />
==Problem 4==<br />
Let <math>R</math> and <math>S</math> be different points on a circle <math>\Omega</math> such that <math>RS</math> is not a diameter. Let <math>\ell</math> be the tangent line to <math>\Omega</math> at <math>R</math>. Point <math>T</math> is such that <math>S</math> is the midpoint of the line segment <math>RT</math>. Point <math>J</math> is chosen on the shorter arc <math>RS</math> of <math>\Omega</math> so that the circumcircle <math>\Gamma</math> of triangle <math>JST</math> intersects <math>\ell</math> at two distinct points. Let <math>A</math> be the common point of <math>\Gamma</math> and <math>\ell</math> that is closer to <math>R</math>. Line <math>AJ</math> meets <math>\Omega</math> again at <math>K</math>. Prove that the line <math>KT</math> is tangent to <math>\Gamma</math>.<br />
<br />
[[2017 IMO Problems/Problem 4|Solution]]<br />
<br />
==Problem 5==<br />
An integer <math>N \ge 2</math> is given. A collection of <math>N(N + 1)</math> soccer players, no two of whom are of the same height, stand in a row. Sir Alex wants to remove <math>N(N - 1)</math> players from this row leaving a new row of <math>2N</math> players in which the following <math>N</math> conditions hold:<br />
<br />
(<math>1</math>) no one stands between the two tallest players,<br />
<br />
(<math>2</math>) no one stands between the third and fourth tallest players,<br />
<br />
<math>\;\;\vdots</math><br />
<br />
(<math>N</math>) no one stands between the two shortest players.<br />
<br />
Show that this is always possible.<br />
<br />
[[2017 IMO Problems/Problem 5|Solution]]<br />
<br />
==Problem 6==<br />
An ordered pair <math>(x, y)</math> of integers is a primitive point if the greatest common divisor of <math>x</math> and <math>y</math> is <math>1</math>. Given a finite set <math>S</math> of primitive points, prove that there exist a positive integer <math>n</math> and integers <math>a_0, a_1, \ldots , a_n</math> such that, for each <math>(x, y)</math> in <math>S</math>, we have:<br />
<cmath>a_0x^n + a_1x^{n-1} y + a_2x^{n-2}y^2 + \cdots + a_{n-1}xy^{n-1} + a_ny^n = 1.</cmath><br />
<br />
[[2017 IMO Problems/Problem 6|Solution]]</div>Muhaboughttps://artofproblemsolving.com/wiki/index.php?title=2014_AIME_II_Problems/Problem_1&diff=624082014 AIME II Problems/Problem 12014-06-28T19:40:58Z<p>Muhaboug: /* Solution */</p>
<hr />
<div>==Problem==<br />
<br />
Abe can paint the room in 15 hours, Bea can paint 50 percent faster than Abe, and Coe can paint twice as fast as Abe. Abe begins to paint the room and works alone for the first hour and a half. Then Bea joins Abe, and they work together until half the room is painted. Then Coe joins Abe and Bea, and they work together until the entire room is painted. Find the number of minutes after Abe begins for the three of them to finish painting the room.<br />
<br />
==Solution==<br />
<br />
From the given information, we can see that Abe can paint <math>\frac{1}{15}</math> of the room in an hour, Bea can paint <math>\frac{1}{15}\times\frac{3}{2} = \frac{1}{10}</math> of the room in an hour, and Coe can paint the room in <math>\frac{1}{15}\times 2 = \frac{2}{15}</math> of the room in an hour. After <math>90</math> minutes, Abe has painted <math>\frac{1}{15}\times\frac{3}{2}=\frac{1}{10}</math> of the room. Working together, Abe and Bea can paint <math>\frac{1}{15}+\frac{1}{10}=\frac{1}{6}</math> of the room in an hour, so it takes then <math>\frac{2}{5}\div \frac{1}{6}= \frac{12}{5}</math> hours to finish the first half of the room. All three working together can paint <math>\frac{1}{6}+\frac{2}{15}=\frac{3}{10}</math> of the room in an hour, and it takes them <math>\frac{1}{2}\div \frac{3}{10}=\frac{5}{3}</math> hours to finish the room. The total amount of time they take is<br />
<cmath>90+\frac{12}{5}\times 60+\frac{5}{3}\times 60 = 90+ 144 + 100 = \boxed{334} \text{\ minutes.}</cmath><br />
<br />
== See also ==<br />
{{AIME box|year=2014|n=II|before=First Question|num-a=2}}<br />
{{MAA Notice}}</div>Muhaboughttps://artofproblemsolving.com/wiki/index.php?title=Asymptote:_3D_graphics&diff=51905Asymptote: 3D graphics2013-03-24T18:35:59Z<p>Muhaboug: /* Projection */</p>
<hr />
<div>{{Asymptote}}<br />
<br />
<br />
==Three==<br />
Three is a module in Asymptote that allows the user to create three dimensional graphics. Usually all you must do is import three,<br />
<code><br />
import three;<br />
</code><br />
<br />
then change from using doubles eg. (x,y) to using triples eg. (x,y,z) as coordinates. Some functions do not work when three is active. For example: In order to fill a surface one must define a surface and draw that. instead of using <tt>[[asymptote: Filling|filldraw]]</tt>. This is also described <url>http://www.artofproblemsolving.com/Forum/viewtopic.php?f=519&t=399845 here</url>.<br />
<br />
===Data types===<br />
three defines the data types:<br />
* path3, (3D version of path)<br />
* guide3, (3D version of guide)<br />
* and surface (a surface bounded by a path(3))<br />
and other, less important ones.<br />
<br />
===Definitions===<br />
three defines the surfaces:<br />
* unitcube<br />
* unitsphere<br />
* unitdisk <br />
* unitplane <br />
* unitcylinder<br />
* unitcone<br />
* unitsolidcone<br />
* and unithemisphere.<br />
These can be drawn like you would normally draw an object in 2D<br />
<pre><nowiki><br />
draw(unitcube,green);<br />
</nowiki></pre><br />
Transforms also work<br />
<pre><nowiki><br />
draw(shift(2,3,4)*scale(5,20,7)*unitcone,paleblue);<br />
</nowiki></pre><br />
<asy><br />
import three;<br />
draw(shift(2,3,4)*scale(5,20,7)*unitcone,paleblue);</asy><br />
<br />
==Projection==<br />
You can use<br />
<code><br />
currentprojection=orthographic(x,y,z);<br />
</code><br />
To change current the view.<br />
<code><br />
currentprojection=perspective(x,y,z);<br />
</code><br />
Does the same thing, but it distorts the picture to imitate actual perspective.<br />
<br />
'''Example:'''<br />
<br />
base code:<br />
<pre><nowiki><br />
import three;<br />
/* perspective line /*<br />
draw(unitcube,palegrey);<br />
</nowiki></pre><br />
Using<br />
<code><br />
currentprojection=orthographic(1,1/2,1/2);<br />
</code><br />
We get a unit cube as:<br />
<asy><br />
import three;<br />
currentprojection=orthographic(1,1/2,1/2);<br />
draw(unitcube,palegrey);<br />
</asy><br />
Using<br />
<code><br />
currentprojection=perspective(1,1/2,1/2);<br />
</code><br />
We get a unit cube as:<br />
<asy><br />
import three;<br />
currentprojection=perspective(1,1/2,1/2);<br />
draw(unitcube,palegrey);<br />
</asy><br />
<br />
'''Note:''' When current projection is not given, <tt>three</tt> tries to find the "best" view.<br />
<br />
==Interactive Projection==<br />
When using Asymptote on your computer (not on AoPS), you can add some code that lets you rotate/pan/zoom with the mouse.<br />
<pre><nowiki><br />
import settings;<br />
leftbutton=new string[] {"rotate","zoom","shift","pan"};<br />
middlebutton=new string[] {"menu"};<br />
rightbutton=new string[] {"zoom/menu","rotateX","rotateY","rotateZ"};<br />
wheelup=new string[] {"zoomin"};<br />
wheeldown=new string[] {"zoomout"};<br />
</nowiki></pre><br />
When compiling to PDF, it will allow you to rotate/pan/zoom with the mouse.<br />
==Arrows and bars==<br />
Arrows and bars in 3D are the same as in 2D except you add a 3 to the end of the name.<br />
Example.<br />
<pre><nowiki><br />
import three;<br />
draw((0,0,0)--(1,1,1),green,Arrows3);<br />
draw((0,1,0)--(1,0,1),blue,Bars3);<br />
</nowiki></pre><br />
<asy><br />
import three;<br />
draw((0,0,0)--(1,1,1),green,Arrows3);<br />
draw((0,1,0)--(1,0,1),blue,Bars3);<br />
</asy><br />
==Examples==<br />
<pre><nowiki><br />
import three;<br />
unitsize(1cm);<br />
size(200);<br />
currentprojection=perspective(1/3,-1,1/2);<br />
draw((0,0,0)--(1,0,0)--(1,1,0)--(0,1,0)--cycle,red);<br />
draw((0,0,0)--(0,0,1),red);<br />
draw((0,1,0)--(0,1,1),red);<br />
draw((1,1,0)--(1,1,1),red);<br />
draw((1,0,0)--(1,0,1),red);<br />
draw((0,0,1)--(1,0,1)--(1,1,1)--(0,1,1)--cycle,red);<br />
draw((0,0,0)--(1,0,0)--(1,1,0)--cycle,red);<br />
draw((0,0,0)--(1,1,0)--(1,1,1)--cycle,blue);<br />
label("$o$",(0,0,0),NW);<br />
label("$x=1$",(0.5,0,0),S);<br />
label("$y=1$",(1,1,0.5),E);<br />
label("$z=1$",(1,0.5,0),SE);<br />
label("$c$",(0.5,0.5,0.5),N);[/asy]<br />
</nowiki></pre><br />
Which renders to<br />
<asy><br />
import three;<br />
unitsize(1cm);<br />
size(200);<br />
currentprojection=orthographic(1/3,-1,1/2);<br />
draw((0,0,0)--(1,0,0)--(1,1,0)--(0,1,0)--cycle,red);<br />
draw((0,0,0)--(0,0,1),red);<br />
draw((0,1,0)--(0,1,1),red);<br />
draw((1,1,0)--(1,1,1),red);<br />
draw((1,0,0)--(1,0,1),red);<br />
draw((0,0,1)--(1,0,1)--(1,1,1)--(0,1,1)--cycle,red);<br />
draw((0,0,0)--(1,0,0)--(1,1,0)--cycle,red);<br />
draw((0,0,0)--(1,1,0)--(1,1,1)--cycle,blue);<br />
label("$o$",(0,0,0),NW);<br />
label("$x=1$",(0.5,0,0),S);<br />
label("$y=1$",(1,1,0.5),E);<br />
label("$z=1$",(1,0.5,0),SE);<br />
label("$c$",(0.5,0.5,0.5),N);</asy><br />
<br />
For other examples, see [[Platonic solids]] and [[2000 AMC 12 Problems/Problem 25]].<br />
<br />
==Other 3D Modules==<br />
Other modules in Asymptote that are for 3D are:<br />
* graph3<br />
* grid3<br />
* contour3<br />
* and solids.</div>Muhaboug