1.
IMO 1996
37th IMO 1996. A1. We are given a positive integer r and a rectangular board divided into 20 x 12 unit squares. The following moves are permitted on the ...
2.
Balkan 89/3 solution
Then the distance of M from the line XY is the average of the distances of B and C, ... passes through G. AB'C' is similar to ABC with sides 2/3 as long, so area AB'C' = 4/9. Thus we wish to show that area AXG + area AYG ≥ area AB'C'. ...
3.
15th Vietnam 1977
Show that there are 1977 non-similar triangles such that the angles A, B, C satisfy (sin A + sin B + sin C)/(cos A + cos B + cos C) = 12/7 and sin A sin B ...
4.
24th Vietnam 1986
P is a pyramid whose base is a regular 1986-gon, and whose sloping sides are all equal. Its inradius is r and its circumradius is R. Show that R/r ≥ 1 + ...
5.
IMO 1994
35th IMO 1994. A1. Let m and n be positive integers. Let a1, a2, ... , am be distinct elements of {1, 2, ... , n} such that whenever ai + aj ≤ n for some i ...
6.
IMO 1997
38th IMO 1997 ... (a) there is no silver matrix for n = 1997; (b) silver matrices exist for infinitely many values of n. B2. Find all pairs (a, ...
7.
26th Russian 2000
26th Russian 2000 problems. 1. The equations x2 + ax + 1 = 0 and x2 + bx + c = 0 have a common real root, and the equations x2 + x + a = 0 and x2 + cx + b ...
8.
17th Balkan 2000
Lần thứ 17: Balkan 2000. A1. Find all real-valued functions on the reals which satisfy f( xf(x) + f(y) ) = f(x)2 + y for all x, y. ...
9.
Kiếm tiền trên mạng
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10.
13th USAMO 1984
13th USAMO 1984 ... Each pupil attempted 7 questions. Each pair of questions was attempted by just two pupils. Show that one pupil attempted either nil or ...
