# 2011 IMO Shortlist was also a joint work with Jan Vonk (Belgium). These two recent problems were submitted by Belgium. However, the other 16 problems were entirely my work, and thus submitted by Republic of Korea (South Korea). In 2010 and 2012, I submitted no problems. 1. MY PROBLEMS ON THE IMO EXAMS I1.IMO 2009 Problem 4 Let ABC be a triangle

IMO Shortlist Official 1992-2000 EN with solutions, scanned.pdf. IMO Shortlist Official 1992-2000 EN with solutions, scanned.pdf. Sign In. Details

Each pair of points is connected with a segment. Each of these segments is painted with one of k colors, in such a way that for any k of the ten points, there are k segments each joining two of them and no two being painted with the same color. IMO Shortlist 1991 17 Find all positive integer solutions x,y,z of the equation 3x +4y = 5z. 18 Find the highest degree k of 1991 for which 1991k divides the number 199019911992 +199219911990. 19 Let α be a rational number with 0 < α < 1 and cos(3πα)+2cos(2πα) = 0. geometry problems from Chinese Mathematical Olympiads (CMO) with aops links in the names 1986 - 2019 1986 CMO problem 2 In $\\triang IMO Shortlist 1990 19 Let P be a point inside a regular tetrahedron T of unit volume.

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Show that |a| + |b| + |c| + |d| ≤ 7. 6. Show that there are polynomials p(x), q(x) with integer coefficients such that p(x) (x + 1)2n+ q(x) (x2n+ 1) = k, for some positive integer k. AoPS Community 1988 IMO Shortlist the trains have zero length.) A series of K freight trains must be driven from Signal 1 to Signal N:Each train travels at a distinct but constant spped at all times when it is not blocked by the 2011 IMO Shortlist was also a joint work with Jan Vonk (Belgium). These two recent problems were submitted by Belgium. However, the other 16 problems were entirely my work, and thus submitted by Republic of Korea (South Korea). In 2010 and 2012, I submitted no problems.

## Problems. Language versions of problems are not complete. Please send relevant PDF files to the webmaster: webmaster@imo-official.org.

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### The International Mathematical Olympiad (IMO) is a mathematical olympiad for pre-university students, and is the oldest of the International Science Olympiads. The first IMO was held in Romania in 1959. It has since been held annually, except in 1980. More …

70. 6 [hide =”Comment”]Alternative formulation, from IMO ShortList 1974, Jul 25, 2013 (IMO 1986, Day 1, Problem 1) Let d be any positive integer not equal to 2, (IMO Shortlist 1996, Number Theory Problem 1) Four integers are (IMO Shortlist 1986) Find the minimum value of the constant c such that for any x1 ,x2, ··· > 0 for which xk+1 ≥ x1 + x2 + ··· + xk for any k, the inequality. √ x1 +. √.

The angle bisectors of the triangle ABC meet the circumcircle again at A', B', C'. Show that area A'B'C' ≥ area ABC.
Bosnia & Herzegovina TST 1996 - 2018 (IMO - EGMO) 46p; British TST 1985 - 2015 (UK FST, NST) 62p; Bulgaria TST 2003-08, 2012-15, 2020 25p; Chile 1989 - 2020 levels 1-2 and TST 66p (uc) China TST 1986 - 2020 104p; China Hong Kong 1999 - 2020 (CHKMO) 20p (uc) Croatia TST 2001-20 (IMO …
Show that 16 (area A'B'C')3≥ 27 area ABC R4, where R is the circumradius of ABC. 5. Show that any two points P and Q inside a regular n-gon can be joined by two circular arcs PQ which lie inside the n-gon and meet at an angle at least (1 - 2/n)π. 6. 2001 IMO Problems/Problem 6; 2001 IMO Shortlist Problems/N1; 2001 IMO Shortlist Problems/N2; 2001 IMO Shortlist Problems/N3; 2001 IMO Shortlist Problems/N4; 2001 IMO Shortlist Problems/N5; 2001 IMO Shortlist Problems/N6; 2001 USA TST Problems/Problem 8; 2001 USA TST Problems/Problem 9; 2001 USAMO Problems/Problem 5; 2002 IMO Shortlist Problems/N1
IMO Shortlist 1996 Combinatorics 1 We are given a positive integer r and a rectangular board ABCD with dimensions AB = 20,BC = 12.

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Show that any two points P and Q inside a regular n-gon can be joined by two circular arcs PQ which lie inside the n-gon and meet at an angle at least (1 - 2/n)π.

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### N1.What is the smallest positive integer such that there exist integers withtx 1, x 2,…,x t x3 1 + x 3 2 + … + x 3 t = 2002 2002? Solution.The answer is .t = 4 We first show that is not a sum of three cubes by considering numbers modulo 9.

In the beginning, the IMO was a much smaller competition than it is today. I vote for Problem 6, IMO 1988. Let [math]a[/math] and [math]b[/math] be positive integers such that [math](1+ab) | (a^2+b^2)[/math].

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### To the current moment, there is only a single IMO problem that has two distinct proposing countries: The if-part of problem 1994/2 was proposed by Australia and its only-if part by Armenia. See also. IMO problems statistics (eternal) IMO problems statistics since 2000 (modern history) IMO problems on the Resources page; IMO Shortlist Problems

Let f(P) be the joint volume of those pieces that are neither a tetrahedron nor a parallelepiped (i.e., pieces adjacent to an edge but not to a vertex). 1986 INMO problem 3 IMO shortlist, TSTs and unofficial; created by: Takis Chronopoulos (parmenides51) from Greece. contact email: parmenides51 # gmail.com. Massachusetts Institute of Technology. map © OpenStreetMap contributors AoPS Community 1989 IMO Shortlist 16Q3 27r4P; where Qand Pare the areas of the triangles A0B0C0and ABCrespectively. 7 Show that any two points lying inside a regular n gon Ecan be joined by two circular arcs lying inside Eand meeting at an angle of at least 1 2 n ˇ: 8 Let Rbe a rectangle that is the union of a ﬁnite number of rectangles R IMO Shortlist 1994 Combinatorics 1 Two players play alternately on a 5 × 5 board.

## 2011 IMO Shortlist was also a joint work with Jan Vonk (Belgium). These two recent problems were submitted by Belgium. However, the other 16 problems were entirely my work, and thus submitted by Republic of Korea (South Korea). In 2010 and 2012, I submitted no problems. 1. MY PROBLEMS ON THE IMO EXAMS I1.IMO 2009 Problem 4 Let ABC be a triangle

Problems from the IMO Shortlists, by year: 1973; 1974; 1975; 1976; 1977; 1978; 1979; There was no IMO in 1980. 1981; 1982; 1983; 1984; 1985; 1986; 1987; 1988; 1989; 1990; 1991; 1992; 1993; 1994; 1995; 1996; 1997; 1998; 1999; 2000; 2001; 2002; 2003; 2004; 2005; 2006; 2007; 2008; 2009; 2010; 2011; 2012; 2013; Resources. IMO Shortlist Collection on AoPS; IMO; IMO Longlist Problems IMO Shortlist 1986 problem 6: 1986 shortlist tb.

Let be a point on the arc , and a point on the segment , such that . Web arhiva zadataka iz matematike.