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• 14:29, 14 December 2023 (diff | hist) . . (+845)‎ . . N 2013 OIM Problems/Problem 2 ‎ (Created page with "== Problem == Let <math>X</math>, <math>Y</math> be the ends of a diameter of a circle <math>\Gamma</math> and <math>N</math> the the midpoint of one of the <math>XY</math> ar...")
• 14:27, 14 December 2023 (diff | hist) . . (+855)‎ . . N 2013 OIM Problems/Problem 1 ‎ (Created page with "== Problem == A set <math>S</math> of positive integers is called "''canalero''" if for any three numbers <math>a, b, c \in S</math>, all different, it holds that <math>a</mat...") (current)
• 14:23, 14 December 2023 (diff | hist) . . (+3)‎ . . 2014 OIM Problems/Problem 6 ‎ (current)
• 14:23, 14 December 2023 (diff | hist) . . (-21)‎ . . 2014 OIM Problems/Problem 6 ‎
• 14:22, 14 December 2023 (diff | hist) . . (+140)‎ . . 2014 OIM Problems/Problem 6 ‎
• 14:21, 14 December 2023 (diff | hist) . . (+259)‎ . . 2014 OIM Problems/Problem 6 ‎
• 14:19, 14 December 2023 (diff | hist) . . (+40)‎ . . 2014 OIM Problems/Problem 6 ‎
• 14:18, 14 December 2023 (diff | hist) . . (+12)‎ . . 2014 OIM Problems/Problem 6 ‎
• 14:17, 14 December 2023 (diff | hist) . . (+784)‎ . . N 2014 OIM Problems/Problem 6 ‎ (Created page with "== Problem == Given a set <math>X</math> and a function <math>f: X \to X</math>, we say that for each <math>x \in X</math>, <math>f^1(x)=f(x)</math>, and for each <math>j \ge...")
• 14:13, 14 December 2023 (diff | hist) . . (+750)‎ . . N 2014 OIM Problems/Problem 5 ‎ (Created page with "== Problem == Let <math>ABC</math> be an acute triangle and <math>H</math> be the point of intersection of the heights. The height from <math>A</math> cuts <math>BC</math> in...") (current)
• 14:11, 14 December 2023 (diff | hist) . . (+735)‎ . . N 2014 OIM Problems/Problem 4 ‎ (Created page with "== Problem == There are <math>N</math> coins, of which <math>N-1</math> are authentic of equal weight and one is false, of different weight than the others. The objective is,...") (current)
• 14:09, 14 December 2023 (diff | hist) . . (+516)‎ . . N 2014 OIM Problems/Problem 3 ‎ (Created page with "== Problem == 2014 points are marked on a circle. On each of the segments whose ends are two of the 2014 points, a non-negative real number is written. It's known that For any...") (current)
• 14:08, 14 December 2023 (diff | hist) . . (+348)‎ . . N 2014 OIM Problems/Problem 2 ‎ (Created page with "== Problem == Find all the polynomials <math>P(x)</math> with real coefficients such that <math>P(2014) = 1</math> and, for some integer <math>c</math>, it holds that <cmath>...") (current)
• 14:07, 14 December 2023 (diff | hist) . . (+392)‎ . . N 2014 OIM Problems/Problem 1 ‎ (Created page with "== Problem == For each positive integer <math>n</math>, define <math>s(n)</math> as the sum of the digits of <math>n</math>. Find the smallest positive integer <math>k</math>...") (current)
• 14:06, 14 December 2023 (diff | hist) . . (+1)‎ . . 2015 OIM Problems/Problem 6 ‎ (current)
• 14:06, 14 December 2023 (diff | hist) . . (+31)‎ . . 2015 OIM Problems/Problem 6 ‎ (→‎See also)
• 14:06, 14 December 2023 (diff | hist) . . (+31)‎ . . 2015 OIM Problems/Problem 5 ‎ (→‎See also) (current)
• 14:05, 14 December 2023 (diff | hist) . . (+31)‎ . . 2015 OIM Problems/Problem 4 ‎ (→‎See also) (current)
• 14:05, 14 December 2023 (diff | hist) . . (+31)‎ . . 2015 OIM Problems/Problem 3 ‎ (→‎See also) (current)
• 14:05, 14 December 2023 (diff | hist) . . (+31)‎ . . 2015 OIM Problems/Problem 2 ‎ (→‎See also) (current)
• 14:05, 14 December 2023 (diff | hist) . . (+31)‎ . . 2015 OIM Problems/Problem 1 ‎ (current)
• 14:04, 14 December 2023 (diff | hist) . . (+488)‎ . . N 2016 OIM Problems/Problem 6 ‎ (Created page with "== Problem == Let <math>k</math> be a positive integer and <math>a_1, a_2, \cdots a_k</math> be digits. Prove that a positive integer <math>n</math> exists such that the last...") (current)
• 14:01, 14 December 2023 (diff | hist) . . (-2)‎ . . 2016 OIM Problems/Problem 5 ‎ (current)
• 14:01, 14 December 2023 (diff | hist) . . (+11)‎ . . 2016 OIM Problems/Problem 5 ‎
• 14:01, 14 December 2023 (diff | hist) . . (+861)‎ . . N 2016 OIM Problems/Problem 5 ‎ (Created page with "== Problem == The circles <math>\Gamma_1</math> and <math>\Gamma_2</math> intersect at two different points <math>A</math> and <math>K</math>. The tangent common to <math>\Gam...")
• 13:58, 14 December 2023 (diff | hist) . . (+540)‎ . . N 2016 OIM Problems/Problem 4 ‎ (Created page with "== Problem == Find the greatest number of bishops that can be placed on an <math>8 \times 8</math> chess board, so that no two bishops are on the same square and each bishop i...") (current)
• 13:57, 14 December 2023 (diff | hist) . . (+1,052)‎ . . N 2016 OIM Problems/Problem 3 ‎ (Created page with "== Problem == Let <math>ABC</math> be an acute triangle whose circumcircle is <math>\Gamma</math>. The tangents to <math>\Gamma</math> through <math>B</math> and <math>C</math...") (current)
• 13:53, 14 December 2023 (diff | hist) . . (+2)‎ . . 2016 OIM Problems/Problem 2 ‎ (current)
• 13:53, 14 December 2023 (diff | hist) . . (+292)‎ . . N 2016 OIM Problems/Problem 2 ‎ (Created page with "== Problem == Find all positive real solutions of the system of equations: <cmath>x=\frac{1}{y^2+y-1},\\y=\frac{1}{z^2+z-1},\\z=\frac{1}{x^2+x-1}</cmath> ~translated into Eng...")
• 13:51, 14 December 2023 (diff | hist) . . (+257)‎ . . N 2016 OIM Problems/Problem 1 ‎ (Created page with "== Problem == Find all positive prime numbers <math>p, q, r, k</math> such that <cmath>pq + qr + rp = 12k + 1</cmath> ~translated into English by Tomas Diaz. ~orders@tomasdi...") (current)
• 13:50, 14 December 2023 (diff | hist) . . (+1)‎ . . 2017 OIM Problems/Problem 6 ‎ (current)
• 13:50, 14 December 2023 (diff | hist) . . (+695)‎ . . N 2017 OIM Problems/Problem 6 ‎ (Created page with "== Problem == Let <math>n > 2</math> be an even positive integer and <math>a_1 < a_2 < \cdots < a_n</math> real numbers such that <math>a_{k+1} -a_k \le 1</math> for all <math...")
• 13:46, 14 December 2023 (diff | hist) . . (+746)‎ . . N 2017 OIM Problems/Problem 5 ‎ (Created page with "== Problem == Given a positive integer <math>n</math>, all its positive integer divisors are written in a blackboard. Ana and Beto play the following game: By turns, each one...") (current)
• 13:44, 14 December 2023 (diff | hist) . . (+868)‎ . . N 2017 OIM Problems/Problem 4 ‎ (Created page with "== Problem == Let <math>ABC</math> be an acute triangle with <math>AC > AB</math> and <math>O</math> its circumcenter. Let <math>D</math> be a point on the segment <math>BC</m...") (current)
• 13:42, 14 December 2023 (diff | hist) . . (+536)‎ . . N 2017 OIM Problems/Problem 3 ‎ (Created page with "== Problem == We consider configurations of integers: <math>a_{1,1},\\ a_{2,1}, a_{2,2},\\ a_{3,1}, a_{3,2}, a_{3,3},\\ \cdots\\ a_{2017,1}, a_{2017,2}, a_{2017,3},\cdots,a_{...") (current)
• 13:38, 14 December 2023 (diff | hist) . . (+11)‎ . . 2017 OIM Problems/Problem 2 ‎ (current)
• 13:38, 14 December 2023 (diff | hist) . . (+814)‎ . . N 2017 OIM Problems/Problem 2 ‎ (Created page with "== Problem == Let <math>ABC</math> be a right triangle and <math>\Gamma</math> its circumcircle. Let <math>D</math> be a point on the segment <math>BC</math>, distinct from <m...")
• 13:36, 14 December 2023 (diff | hist) . . (+11)‎ . . 2017 OIM Problems/Problem 1 ‎ (current)
• 13:36, 14 December 2023 (diff | hist) . . (+686)‎ . . N 2017 OIM Problems/Problem 1 ‎ (Created page with "== Problem == For each positive integer <math>n</math>, let <math>S(n)</math> be the sum of its digits. We say that <math>n</math> has the property <math>E</math> if the terms...")
• 13:33, 14 December 2023 (diff | hist) . . (+765)‎ . . N 2018 OIM Problems/Problem 6 ‎ (Created page with "== Problem == Let <math>ABC</math> be an acute triangle with <math>AC > AB > BC</math>. The perpendicular bisectors of <math>AC</math> and <math>AB</math> cut the line <math>B...") (current)
• 13:30, 14 December 2023 (diff | hist) . . (+656)‎ . . N 2018 OIM Problems/Problem 5 ‎ (Created page with "== Problem == Let <math>n</math> be a positive integer. For a permutation <math>a_1, a_2, \cdots , a_n,</math> of the numbers <math>1, 2, \cdots , n,</math> we define <cmath>...") (current)
• 13:26, 14 December 2023 (diff | hist) . . (+557)‎ . . N 2018 OIM Problems/Problem 4 ‎ (Created page with "== Problem == A set "X" of positive integers is "''Iberian''" if <math>X</math> is a subset of <math>{2,3,4,\cdots,2018}</math>, and when <math>m</math> and <math>n</math> are...") (current)
• 13:22, 14 December 2023 (diff | hist) . . (+1,003)‎ . . N 2018 OIM Problems/Problem 3 ‎ (Created page with "== Problem == In a plane we have <math>n</math> lines without two being parallel, nor two perpendicular, nor three concurrent. A system of Cartesian axes is chosen with one o...") (current)
• 13:19, 14 December 2023 (diff | hist) . . (+633)‎ . . N 2018 OIM Problems/Problem 2 ‎ (Created page with "== Problem == Let <math>ABC</math> be a triangle such that <math>\angle BAC = 90^{\circ}</math> and <math>BA = CA</math>. Let <math>M</math> be the midpoint of <math>BC</math>...") (current)
• 13:17, 14 December 2023 (diff | hist) . . (+473)‎ . . N 2018 OIM Problems/Problem 1 ‎ (Created page with "== Problem == For each natural number <math>n \ge 2</math>, find the integer solutions to the following system of equations: <cmath>x_1 = (x_2 + x_3 + x_4 + \cdots + x_n)^{20...") (current)
• 13:15, 14 December 2023 (diff | hist) . . (+4)‎ . . 2019 OIM Problems/Problem 6 ‎ (current)
• 13:14, 14 December 2023 (diff | hist) . . (+439)‎ . . N 2019 OIM Problems/Problem 6 ‎ (Created page with "== Problem == Let <math>a_1, a_2, \cdots , a_{2019}</math> be positive integers and <math>P</math> be a polynomial with integer coefficients such that, for every positive inte...")
• 13:12, 14 December 2023 (diff | hist) . . (+813)‎ . . N 2019 OIM Problems/Problem 5 ‎ (Created page with "== Problem == Don Miguel places a game piece on one of the <math>(n + 1)^2</math> vertices defined by a <math>n \times n</math> board. A ''move'' consists of moving the piece...") (current)
• 13:07, 14 December 2023 (diff | hist) . . (+759)‎ . . N 2019 OIM Problems/Problem 4 ‎ (Created page with "== Problem == Let <math>ABCD</math> be a trapezoid with <math>AB \parallel CD</math> and inscribed in the circle <math>\Gamma</math>. Let <math>P</math> and <math>Q</math> be...") (current)
• 13:05, 14 December 2023 (diff | hist) . . (+1,217)‎ . . N 2019 OIM Problems/Problem 3 ‎ (Created page with "== Problem == Let <math>\Gamma</math> be the circumcircle of triangle <math>ABC</math>. The parallel to <math>AC</math> that passes through <math>B</math> cuts <math>\Gamma</m...") (current)

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