00.15
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16th Balkan Mathematical Olympiad Problems 1999A1. O is the circumcenter of the triangle ABC. XY is the diameter of the circumcircle perpendicular to BC. It meets BC at M. X is closer to M than Y. Z is the point on MY such that MZ = MX. W is the midpoint of AZ. Show that W lies on the circle through the midpoints of the sides of ABC. Show that MW is perpendicular to
00.13
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15th Balkan Mathematical Olympiad Problems 1998A1. How many different integers can be written as [n2/1998] for n = 1, 2, ... , 1997? A2. xi are distinct positive reals satisfying x1 < x2 < ... < x2n+1. Show that x1 - x2 + x3 - x4 + ... - x2n + x2n+1 < (x1n - x2n + ... - x2nn + x2n+1n)1/n.
14th Balkan Mathematical Olympiad Problems 1997A1. ABCD is a convex quadrilateral. X is a point inside it. XA2 + XB2 + XC2 + XD2 is twice the area of the quadrilateral. Show that it is a square and that X is its center.
13th Balkan Mathematical Olympiad Problems 1996A1. Let d be the distance between the circumcenter and the centroid of a triangle. Let R be its circumradius and r the radius of its inscribed circle. Show that d2 ≤ R(R - 2r).
12th Balkan Mathematical Olympiad Problems 1995A1. Define an by a3 = (2 + 3)/(1 + 6), an = (an-1 + n)/(1 + n an-1). Find a1995. A2. Two circles centers O and O' meet at A and B, so that OA is perpendicular to O'A. OO' meets the circles at C, E, D, F, so that the points C, O, E, D, O', F lie on the line in that order. BE meets the circle again at K and
19.10
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11th Balkan Mathematical Olympiad Problems 1994A1. Given a point P inside an acute angle XAY, show how to construct a line through P meeting the line AX at B and the line AY at C such that the area of the triangle ABC is AP2.
10th Balkan Mathematical Olympiad Problems 1993A1. Given reals a1 ≤ a2 ≤ a3 ≤ a4 ≤ a5 ≤ a6 satisfying a1 + a2 + a3 + a4 + a5 + a6 = 10 and (a1 - 1)2 + (a2 - 1)2 + (a3 - 1)2 + (a4 - 1)2 + (a5 - 1)2 + (a6 - 1)2 = 6, what is the largest possible a6?
9th Balkan Mathematical Olympiad Problems 1992A1. Let a(n) = 34n. For which n is (ma(n)+6 - ma(n)+4 - m5 + m3) always divisible by 1992? A2. Prove that (2n2 + 3n + 1)n ≥ 6nn! n! for all positive integers.
8th Balkan Mathematical Olympiad Problems 1991A1. The circumcircle of the acute-angled triangle ABC has center O. M lies on the minor arc AB. The line through M perpendicular to OA cuts AB at K and AC at L. The line through M perpendicular to OB cuts AB at N and BC at P. MN = KL. Find angle MLP in terms of angles A, B and C.
7th Balkan Mathematical Olympiad Problems 1990A1. The sequence un is defined by u1 = 1, u2 = 3, un = (n+1) un-1 - n un-2. Which members of the sequence which are divisible by 11? A2. Expand (x + 2x2 + 3x3 + ... + nxn)2 and add the coefficients of xn+1 through x2n. Show that the result is n(n+1)(5n2 + 5n + 2)/24.
6th Balkan Mathematical Olympiad Problems 1989A1. Find all integers which are the sum of the squares of their four smallest positive divisors. A2. A prime p has decimal digits pnpn-1...p0 with pn > 1. Show that the polynomial pnxn + pn-1xn-1 + ... + p1x + p0 has no factors which are polynomials with integer coefficients and degree strictly between
03.19
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5th Balkan Mathematical Olympiad Problems 1988A1. ABC is a triangle area 1. AH is an altitude, M is the midpoint of BC and K is the point where the angle bisector at A meets the segment BC. The area of the triangle AHM is 1/4 and the area of AKM is 1 - (√3)/2. Find the angles of the triangle.
4th Balkan Mathematical Olympiad Problems 1987A1. f is a real valued function on the reals satisfying (1) f(0) = 1/2, (2) for some real a we have f(x+y) = f(x) f(a-y) + f(y) f(a-x) for all x, y. Prove that f is constant.
3rd Balkan Mathematical Olympiad Problems 1986A1. A line through the incenter of a triangle meets the circumcircle and incircle in the points A, B, C, D (in that order). Show that AB·CD ≥ BC2/4. When do you have equality?
2nd Balkan Mathematical Olympiad Problems 1985A1. ABC is a triangle. O is the circumcenter, D is the midpoint of AB, and E is the centroid of ACD. Prove that OE is perpendicular to CD iff AB = AC. A2. The reals w, x
03.10
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1st Balkan Mathematical Olympiad Problems 1984A1. Let x1, x2, ... , xn be positive reals with sum 1. Prove that x1/(2 - x1) + x2/(2 - x2) + ... + xn/(2 - xn) ≥ n/(2n - 1). A2. ABCD is a cyclic quadrilateral.
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