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• 15:51, 14 December 2023 (diff | hist) . . (+444)‎ . . N 2007 OIM Problems/Problem 5 ‎ (Created page with "== Problem == A natural number <math>n</math> is "''daring''"<math> if the set of its divisors, from 1 to </math>n$ inclusive, can be divided into three subsets such that the...")
• 15:49, 14 December 2023 (diff | hist) . . (+1,015)‎ . . N 2007 OIM Problems/Problem 4 ‎ (Created page with "== Problem == On a <math>19 \times 19</math> grid board, a piece called a "''dragon''" jumps from side to side in the following way: it moves 4 squares in a direction parallel...") (current)
• 15:46, 14 December 2023 (diff | hist) . . (+11)‎ . . 2007 OIM Problems/Problem 3 ‎ (current)
• 15:46, 14 December 2023 (diff | hist) . . (+1,245)‎ . . N 2007 OIM Problems/Problem 3 ‎ (Created page with "== Problem == Two teams, <math>A4 and </math>B<math>, dispute the territory limited by a circle. </math>A<math> has </math>n<math> blue flags and </math>B<math> has </math>n<m...")
• 15:43, 14 December 2023 (diff | hist) . . (+994)‎ . . N 2007 OIM Problems/Problem 2 ‎ (Created page with "== Problem == Let <math>ABC</math> be a triangle with incenter <math>I</math> and <math>\Gamma</math> be a circle with center <math>I</math>, of radius greater than that of th...") (current)
• 15:40, 14 December 2023 (diff | hist) . . (+1)‎ . . 2007 OIM Problems/Problem 1 ‎ (current)
• 15:40, 14 December 2023 (diff | hist) . . (+462)‎ . . N 2007 OIM Problems/Problem 1 ‎ (Created page with "== Problem == Given a positive integer <math>m</math>, the sequence {<math>a_n</math>} is defined as follows: <math></math>a_1=\frac{m}{2},\;a_{n+1}=a_n\left\lceil a_n \right...")
• 15:37, 14 December 2023 (diff | hist) . . (+565)‎ . . N 2008 OIM Problems/Problem 6 ‎ (Created page with "== Problem == In a "''biribol''" match, two teams of four people each face each other. A ''biribol'' tournament is organized in which <math>n</math> people participate, who f...") (current)
• 15:35, 14 December 2023 (diff | hist) . . (+717)‎ . . N 2008 OIM Problems/Problem 5 ‎ (Created page with "== Problem == Let <math>ABC</math> be a triangle, and <math>X, Y, Z</math> be points on the sides <math>BC</math>, <math>AC</math>, <math>AB</math> respectively. Let <math>A'<...") (current)
• 15:32, 14 December 2023 (diff | hist) . . (0)‎ . . 2008 OIM Problems/Problem 4 ‎ (current)
• 15:32, 14 December 2023 (diff | hist) . . (+274)‎ . . N 2008 OIM Problems/Problem 4 ‎ (Created page with "== Problem == Prove that there are no positive integers <math>x</math> and <math>y</math> such that <cmath>s^{2008}+2008!=21^y</cmath> ~translated into English by Tomas Diaz...")
• 15:31, 14 December 2023 (diff | hist) . . (-2)‎ . . 2008 OIM Problems/Problem 3 ‎ (current)
• 15:31, 14 December 2023 (diff | hist) . . (+453)‎ . . N 2008 OIM Problems/Problem 3 ‎ (Created page with "== Problem == Let <math>m</math> and <math>n</math> be integers such that the polynomial <math>P(x) = x^3 + mx + n</math> has the following property: if <math>x</math> and <ma...")
• 15:30, 14 December 2023 (diff | hist) . . (+678)‎ . . N 2008 OIM Problems/Problem 2 ‎ (Created page with "== Problem == Let <math>ABC</math> be a scalene triangle and <math>r</math> be the external bisector of the angle <math>\angle ABC</math>. <math>P</math> and <math>Q</math> ar...") (current)
• 15:28, 14 December 2023 (diff | hist) . . (+546)‎ . . N 2008 OIM Problems/Problem 1 ‎ (Created page with "== Problem == Numbers <math>1, 2, 3, \cdots , 2008^2</math> were distributed on a <math>2008 \times 2008</math> board, such that in each box there is a different number. For e...") (current)
• 15:25, 14 December 2023 (diff | hist) . . (+495)‎ . . N 2009 OIM Problems/Problem 6 ‎ (Created page with "== Problem == Around a circle, 6000 points are marked and each one is colored with one of 10 given colors, such that among any 100 consecutive points the 10 colors always appe...") (current)
• 15:24, 14 December 2023 (diff | hist) . . (+12)‎ . . 2009 OIM Problems/Problem 5 ‎ (current)
• 15:24, 14 December 2023 (diff | hist) . . (+390)‎ . . N 2009 OIM Problems/Problem 5 ‎ (Created page with "== Problem == The sequence <math>a_n</math> is defined by <math>a_1 = 1, a_{2k} = 1 + a_k</math>, and <math>a_{2k+1} = \frac{1}{a_{2k}}, for all integer </math>k \ge 1$. Prov...")
• 15:22, 14 December 2023 (diff | hist) . . (+677)‎ . . N 2009 OIM Problems/Problem 4 ‎ (Created page with "== Problem == Let <math>ABC</math> be a triangle with <math>AB \ne AC</math>. Let <math>I</math> be the incenter of <math>ABC</math> and <math>P</math> the other point of inte...")
• 15:21, 14 December 2023 (diff | hist) . . (0)‎ . . 2009 OIM Problems/Problem 3 ‎ (current)
• 15:21, 14 December 2023 (diff | hist) . . (-3)‎ . . 2009 OIM Problems/Problem 3 ‎
• 15:20, 14 December 2023 (diff | hist) . . (+962)‎ . . N 2009 OIM Problems/Problem 3 ‎ (Created page with "== Problem == Let <math>C_1</math> and <math>C_2</math> be two circles with centers <math>O_1</math> and <math>O_2</math> with the same radius, which intersect at <math>A</mat...")
• 15:17, 14 December 2023 (diff | hist) . . (+391)‎ . . N 2009 OIM Problems/Problem 2 ‎ (Created page with "== Problem == For each positive integer <math>n</math> we define <math>a_n = n + m</math> where <math>m</math> is the largest integer such that <math>2^{2^m} \le n2^n</math>....") (current)
• 15:16, 14 December 2023 (diff | hist) . . (+558)‎ . . N 2009 OIM Problems/Problem 1 ‎ (Created page with "== Problem == Let <math>n</math> be a natural number greater than 2. Suppose that <math>n</math> islands are located in a circle and that between each two neighboring islands...") (current)
• 15:14, 14 December 2023 (diff | hist) . . (+367)‎ . . N 2010 OIM Problems/Problem 6 ‎ (Created page with "== Problem == 12 people sit around a circular table on which there are 28 vases. Two people can see each other if and only if there is no vase aligned with them. Prove that th...") (current)
• 15:13, 14 December 2023 (diff | hist) . . (+697)‎ . . N 2010 OIM Problems/Problem 5 ‎ (Created page with "== Problem == Let <math>ABCD</math> be a cyclic quadrilateral whose diagonals <math>AC</math> and <math>BD</math> are perpendicular. Let <math>O</math> be the circumcenter of...") (current)
• 15:12, 14 December 2023 (diff | hist) . . (+12)‎ . . 2010 OIM Problems/Problem 4 ‎ (current)
• 15:11, 14 December 2023 (diff | hist) . . (+523)‎ . . N 2010 OIM Problems/Problem 4 ‎ (Created page with "== Problem == The arithmetic, geometric and harmonic means of two different positive integers are integer numbers. Find the smallest possible value for the arithmetic mean. '...")
• 15:09, 14 December 2023 (diff | hist) . . (+1,088)‎ . . N 2010 OIM Problems/Problem 3 ‎ (Created page with "== Problem == The circle <math>\Gamma</math> inscribed in the scalene triangle <math>ABC</math> is tangent to the sides <math>BC</math>, <math>CA</math> and <math>AB</math> at...") (current)
• 15:07, 14 December 2023 (diff | hist) . . (+379)‎ . . N 2010 OIM Problems/Problem 2 ‎ (Created page with "== Problem == Find if there are positive integers <math>a</math> and <math>b</math> such that all terms of the sequence defined by <cmath>x_1 = 2010,\; x_2 = 2011,\;x_{n+2}=...") (current)
• 15:04, 14 December 2023 (diff | hist) . . (+525)‎ . . N 2010 OIM Problems/Problem 1 ‎ (Created page with "== Problem == There are ten indistinguishable coins placed in a line. It is known that two of them are false and occupy consecutive positions on the line. For each set of posi...") (current)
• 15:02, 14 December 2023 (diff | hist) . . (+721)‎ . . N 2011 OIM Problems/Problem 6 ‎ (Created page with "== Problem == Let <math>k</math> and <math>n</math> be positive integers, with <math>k \ge 2</math>. In a straight line there are <math>kn</math> stones of <math>k</math> diff...") (current)
• 15:00, 14 December 2023 (diff | hist) . . (0)‎ . . 2011 OIM Problems/Problem 5 ‎ (current)
• 14:58, 14 December 2023 (diff | hist) . . (+1)‎ . . 2011 OIM Problems/Problem 5 ‎
• 14:57, 14 December 2023 (diff | hist) . . (+372)‎ . . N 2011 OIM Problems/Problem 5 ‎ (Created page with "== Problem == Let <math>x_1, \cdots , x_n</math> be positive real numbers. Prove that there exist <math>a1, \cdots , a_n \in \left\{−1, 1\right\}</math> such that <cmath>a_...")
• 14:55, 14 December 2023 (diff | hist) . . (+439)‎ . . N 2011 OIM Problems/Problem 4 ‎ (Created page with "== Problem == Let <math>ABC</math> be an acute triangle, with <math>AC \ne BC</math>, and let <math>O</math> be its circumcenter. Let <math>P</math> and <math>Q</math> points...") (current)
• 14:54, 14 December 2023 (diff | hist) . . (+1,081)‎ . . N 2011 OIM Problems/Problem 3 ‎ (Created page with "== Problem == Let <math>ABC</math> be a triangle and let <math>X, Y, Z</math> be the points of tangency of its circumference inscribed with sides <math>BC</math>, <math>CA</ma...") (current)
• 14:51, 14 December 2023 (diff | hist) . . (+1)‎ . . 2011 OIM Problems/Problem 2 ‎ (current)
• 14:51, 14 December 2023 (diff | hist) . . (+342)‎ . . N 2011 OIM Problems/Problem 2 ‎ (Created page with "== Problem == Find all positive integers <math>n</math> for which there are three non-zero integers <math>x, y, z</math> such that <cmath>x+y+z=0,\;\frac{1}{x}+\frac{1}{y}\fr...")
• 14:50, 14 December 2023 (diff | hist) . . (+555)‎ . . N 2011 OIM Problems/Problem 1 ‎ (Created page with "== Problem == The number 2 is written on the board. Ana and Bruno play alternately, starting with Ana. Each one in turn replaces the written number with the one obtained by ap...") (current)
• 14:48, 14 December 2023 (diff | hist) . . (+324)‎ . . N 2012 OIM Problems/Problem 6 ‎ (Created page with "== Problem == Prove that for every positive integer <math>n</math> there are <math>n</math> consecutive positive integers such that none of them is divisible by the sum of the...") (current)
• 14:48, 14 December 2023 (diff | hist) . . (+647)‎ . . N 2012 OIM Problems/Problem 5 ‎ (Created page with "== Problem == Let <math>ABC</math> be a triangle and let <math>P</math> and <math>Q</math> be the points of intersection of the parallel to <math>BC</math> by <math>A</math> w...") (current)
• 14:46, 14 December 2023 (diff | hist) . . (+415)‎ . . N 2012 OIM Problems/Problem 4 ‎ (Created page with "== Problem == Let <math>a, b, c, d</math> be positive integers such that <math>a - b + c - d</math> is odd and divides <math>a^2 - b^2 + c^2 - d^2</math>. Show that <math>a -...") (current)
• 14:44, 14 December 2023 (diff | hist) . . (+701)‎ . . N 2012 OIM Problems/Problem 3 ‎ (Created page with "== Problem == Let <math>n</math> be a positive integer. Given a set <math>\left\{a_1, a_2, \cdots , a_n\right\}</math> of integers between 0 and <math>2^n-1</math> inclusive,...") (current)
• 14:41, 14 December 2023 (diff | hist) . . (+472)‎ . . N 2012 OIM Problems/Problem 2 ‎ (Created page with "== Problem == A positive integer is ''bisummed'' if it can be written as the sum of two integers positive numbers that have the same sum of their digits. For example, 2012 is...") (current)
• 14:39, 14 December 2023 (diff | hist) . . (+511)‎ . . N 2012 OIM Problems/Problem 1 ‎ (Created page with "== Problem == Equilateral triangles <math>BCX</math> and <math>DCY</math> are drawn on a rectangle <math>ABCD</math> so that each one shares points with the interior of the re...") (current)
• 14:37, 14 December 2023 (diff | hist) . . (+549)‎ . . N 2013 OIM Problems/Problem 6 ‎ (Created page with "== Problem == A ''configuration'' is a finite set <math>S</math> of points in the plane between which there are no three collinear and each point is assigned some color, so if...") (current)
• 14:36, 14 December 2023 (diff | hist) . . (+526)‎ . . N 2013 OIM Problems/Problem 5 ‎ (Created page with "== Problem == Let <math>A</math> and <math>B</math> be two sets such that: 1. <math>A \cup B</math> is the set of positive integers. 2. <math>A \cap B</math> is empty. 3. I...") (current)
• 14:34, 14 December 2023 (diff | hist) . . (+881)‎ . . N 2013 OIM Problems/Problem 4 ‎ (Created page with "== Problem == Let <math>\Gamma</math> be a circle with center <math>O</math>, <math>AE</math> be a diameter of <math>\Gamma</math> and <math>B</math> be the midpoint of one of...") (current)
• 14:32, 14 December 2023 (diff | hist) . . (+531)‎ . . N 2013 OIM Problems/Problem 3 ‎ (Created page with "== Problem == Let <math>A = \left\{1, 2, 3, \cdots , n\right\}</math> with <math>n > 5</math>. Prove that there exists a finite set <math>B</math> of distinct positive integer...") (current)

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