User contributions
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- 13:26, 13 December 2023 (diff | hist) . . (+658) . . N 1994 OIM Problems/Problem 2 (Created page with "== Problem == Let there be a quadrilateral inscribed in a circle, whose vertices are denoted consecutively by <math>A</math>, <math>B</math>, <math>C</math> and <math>D</math>...") (current)
- 13:24, 13 December 2023 (diff | hist) . . (+536) . . N 1994 OIM Problems/Problem 1 (Created page with "== Problem == A natural number <math>n</math> is said to be "''sensible''" if there exists an integer <math>r</math>, with <math>1<r<n-1</math>, such that the representation o...")
- 13:22, 13 December 2023 (diff | hist) . . (+4) . . 1993 OIM Problems/Problem 6 (current)
- 13:22, 13 December 2023 (diff | hist) . . (-2) . . 1993 OIM Problems/Problem 6
- 13:21, 13 December 2023 (diff | hist) . . (+684) . . N 1993 OIM Problems/Problem 6 (Created page with "== Problem == Two non-negative integers <math>a</math> and <math>b</math> are ''cuates'' if the decimal expression <math>a+b</math> consists only of zeros and ones. Let <math>...")
- 13:19, 13 December 2023 (diff | hist) . . (+2) . . 1993 OIM Problems/Problem 5 (current)
- 13:19, 13 December 2023 (diff | hist) . . (+922) . . N 1993 OIM Problems/Problem 5 (Created page with "== Problem == Let <math>P</math> and <math>Q</math> be two different points on the plane. Let us denote by <math>m(PQ)</math> the bisector of the segment <math>PQ</math>. Let...")
- 13:16, 13 December 2023 (diff | hist) . . (+468) . . N 1993 OIM Problems/Problem 4 (Created page with "== Problem == Let <math>ABC</math> be an equilateral triangle and <math>Q</math> be its inscribed circle. If <math>D</math> and <math>E</math> are points on sides <math>AB</ma...") (current)
- 13:15, 13 December 2023 (diff | hist) . . (+4) . . 1993 OIM Problems/Problem 3 (current)
- 13:15, 13 December 2023 (diff | hist) . . (+417) . . N 1993 OIM Problems/Problem 3 (Created page with "== Problem == Let <math>N*={1,2,3,\cdots }</math>. Find all functions <math>f: N* \to N*</math> such that: i. If <math>x < y</math> then <math>f(x) < f(y)</math> ii. <math>f...")
- 13:12, 13 December 2023 (diff | hist) . . (+264) . . N 1993 OIM Problems/Problem 2 (Created page with "== Problem == Show that for any convex polygon of area 1 there exists a parallelogram of area 2 that contains it. ~translated into English by Tomas Diaz. ~orders@tomasdiaz.co...") (current)
- 13:12, 13 December 2023 (diff | hist) . . (+565) . . N 1993 OIM Problems/Problem 1 (Created page with "== Problem == A natural number is a palindrome if when written in decimal notation, it can be read equally from left to right and from right to left, for example: 8, 23432, 64...") (current)
- 13:08, 13 December 2023 (diff | hist) . . (0) . . 1992 OIM Problems/Problem 6
- 13:08, 13 December 2023 (diff | hist) . . (+27) . . 1992 OIM Problems/Problem 6
- 13:07, 13 December 2023 (diff | hist) . . (0) . . N File:Ibe7 2.gif (current)
- 13:06, 13 December 2023 (diff | hist) . . (+2) . . 1992 OIM Problems/Problem 6
- 13:06, 13 December 2023 (diff | hist) . . (+500) . . N 1992 OIM Problems/Problem 6 (Created page with "== Problem == From the triangle <math>T</math> with vertices <math>A</math>, <math>B</math> and <math>C</math>, the hexagon <math>H</math> with vertices <math>A_1</math>, <mat...")
- 13:04, 13 December 2023 (diff | hist) . . (+495) . . N 1992 OIM Problems/Problem 5 (Created page with "== Problem == The circumference <math>C</math> and the positive numbers <math>h</math> and <math>m</math> are given so that there is a trapezoid <math>ABCD</math> inscribed in...")
- 13:02, 13 December 2023 (diff | hist) . . (+694) . . N 1992 OIM Problems/Problem 4 (Created page with "== Problem == Let <math>(a_n)</math> and <math>(b_n)</math> be two sequences of integers that verify the following conditions: i. <math>a_0 = 0</math>, <math>b_0 = 8</math>...")
- 12:57, 13 December 2023 (diff | hist) . . (+707) . . N 1992 OIM Problems/Problem 3 (Created page with "== Problem == In an equilateral triangle <math>ABC</math> whose side has length 2, the circle <math>G</math> is inscribed. a. Show that for every point <math>P</math> of <mat...")
- 12:55, 13 December 2023 (diff | hist) . . (+480) . . N 1992 OIM Problems/Problem 2 (Created page with "== Problem == Given the collection of <math>n</math> positive real numbers <math>a_1 < a_2 < a_3 < \cdots < a_n</math> and the function: <cmath>f(x) = \frac{a_1}{x+a_1}+\frac...")
- 12:53, 13 December 2023 (diff | hist) . . (+11) . . 1992 OIM Problems/Problem 1
- 12:53, 13 December 2023 (diff | hist) . . (+339) . . N 1992 OIM Problems/Problem 1 (Created page with "== Problem == For each positive integer 4n<math>, let </math>a_n<math> be the last digit of the number. </math>1+2+3+\cdots +n<math>. Calculate </math>a_1 + a_2 + a_3 + \cdots...")
- 12:51, 13 December 2023 (diff | hist) . . (+436) . . N 1991 OIM Problems/Problem 6 (Created page with "== Problem == Given 3 non-aligned points <math>M</math>, <math>N</math> and <math>P</math>, we know that <math>M</math> and <math>N</math> are midpoints of two sides of a tria...")
- 12:50, 13 December 2023 (diff | hist) . . (+543) . . N 1991 OIM Problems/Problem 5 (Created page with "== Problem == Let <math>P(x,y) = 2x^2 - 6xy + 5y^2</math>. We will say that an integer <math>a</math> is a value of <math>P</math> if there exist integers <math>b</math> and <...")
- 12:49, 13 December 2023 (diff | hist) . . (+358) . . N 1991 OIM Problems/Problem 4 (Created page with "== Problem == Find a number <math>N</math> of five different and non-zero digits, which is equal to the sum of all the numbers of three different digits that can be formed wit...")
- 12:48, 13 December 2023 (diff | hist) . . (+418) . . N 1991 OIM Problems/Problem 3 (Created page with "== Problem == Let <math>f</math> be an increasing function defined for every real number <math>x</math>, <math>0 \le x \le 1</math>, such that: a. <math>f(0) = 0</math> b. <...")
- 12:46, 13 December 2023 (diff | hist) . . (+308) . . N 1991 OIM Problems/Problem 2 (Created page with "== Problem == Two perpendicular lines divide a square into four parts, three of which each have an area equal to 1. Show that the area of the square is four. ~translated into...")
- 12:46, 13 December 2023 (diff | hist) . . (+360) . . N 1991 OIM Problems/Problem 1 (Created page with "== Problem == Each vertex of a cube is assigned the value of +1 or -1, and each face is assigned the product of the values assigned to each vertex. What values can the sum of...")
- 12:45, 13 December 2023 (diff | hist) . . (+360) . . N 1991 OIM Problems/Problem 7 (Created page with "== Problem == Each vertex of a cube is assigned the value of +1 or -1, and each face is assigned the product of the values assigned to each vertex. What values can the sum of...") (current)
- 12:44, 13 December 2023 (diff | hist) . . (+359) . . N 1990 OIM Problems/Problem 6 (Created page with "== Problem == Let <math>f(x)</math> be a polynomial of degree 3 with rational coefficients. Prove that if the graph of <math>f</math> is tangent to the x-axis, then <math>f(x)...") (current)
- 12:44, 13 December 2023 (diff | hist) . . (+911) . . N 1990 OIM Problems/Problem 5 (Created page with "== Problem == Let <math>A</math> and <math>B</math> be opposite vertices of a gridded board of <math>n</math> by <math>n</math> squares <math>n \ge 1</math>, to each of which...") (current)
- 12:41, 13 December 2023 (diff | hist) . . (+862) . . N 1990 OIM Problems/Problem 4 (Created page with "== Problem == Let: <math>C_1</math> be a circle, <math>AB</math> be one of its diameters, <math>t</math> be its tangent at <math>B</math> and <math>M</math> be a point on <mat...") (current)
- 12:39, 13 December 2023 (diff | hist) . . (+823) . . N 1990 OIM Problems/Problem 3 (Created page with "== Problem == Let <math>f(x) = (x + b)^22-c</math>, be a polynomial with <math>b</math> and <math>c</math> as integers. a. If <math>p</math> is a prime number such that <math...")
- 12:36, 13 December 2023 (diff | hist) . . (+699) . . N 1990 OIM Problems/Problem 2 (Created page with "== Problem == In a triangle <math>ABC</math>, let <math>I</math> be the center of the inscribed circle and <math>D</math>, <math>E</math> and <math>F</math> be its points of t...") (current)
- 12:34, 13 December 2023 (diff | hist) . . (+670) . . N 1990 OIM Problems/Problem 1 (Created page with "== Problem == Let <math>f</math> be a function defined in the set of integers greater or equal to zero such that: (i) If <math>n=2^j-1</math>, for all <math>n=0, 1, 2, \cdots...")
- 12:30, 13 December 2023 (diff | hist) . . (+59) . . 1989 OIM Problems/Problem 4 (current)
- 12:30, 13 December 2023 (diff | hist) . . (+59) . . 1989 OIM Problems/Problem 3 (current)
- 12:30, 13 December 2023 (diff | hist) . . (-63) . . 1989 OIM Problems/Problem 2 (current)
- 12:29, 13 December 2023 (diff | hist) . . (+122) . . 1989 OIM Problems/Problem 2
- 12:29, 13 December 2023 (diff | hist) . . (+121) . . 1989 OIM Problems/Problem 1 (current)
- 12:29, 13 December 2023 (diff | hist) . . (+122) . . 1988 OIM Problems/Problem 6 (current)
- 12:28, 13 December 2023 (diff | hist) . . (+122) . . 1988 OIM Problems/Problem 5 (current)
- 12:28, 13 December 2023 (diff | hist) . . (+122) . . 1988 OIM Problems/Problem 4 (current)
- 12:28, 13 December 2023 (diff | hist) . . (+122) . . 1988 OIM Problems/Problem 3 (current)
- 12:28, 13 December 2023 (diff | hist) . . (+122) . . 1988 OIM Problems/Problem 2 (current)
- 12:28, 13 December 2023 (diff | hist) . . (+122) . . 1988 OIM Problems/Problem 1 (current)
- 12:27, 13 December 2023 (diff | hist) . . (+122) . . 1987 OIM Problems/Problem 6 (current)
- 12:27, 13 December 2023 (diff | hist) . . (+119) . . 1987 OIM Problems/Problem 5 (current)
- 12:27, 13 December 2023 (diff | hist) . . (+122) . . 1987 OIM Problems/Problem 4 (current)
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