40th British Mathematical Olympiad 2004 Problems

40th British Mathematical Olympiad 2004 Problems1.  ABC is an equilateral triangle. D is a point on the side BC (not at the endpoints). A circle touches BC at D and meets the side AB at M and N, and the side AC at P and Q. Show that BD + AM + AN = CD + AP + AQ. 2.  D is a point on the side AB of the triangle ABC such that AB = 4·AD. P is a point on the

38th British Mathematical Olympiad 2002 Problems

38th British Mathematical Olympiad 2002 Problems1.  From the foot of an altitude in an acute-angled triangle perpendiculars are drawn to the other two sides. Show that the distance between their feet is independent of the choice of altitude. 2.

34th British Mathematical Olympiad 1998 Problems

34th British Mathematical Olympiad 1998 Problems1.  A station issues 3800 tickets covering 200 destinations. Show that there are at least 6 destinations for which the number of tickets sold is the same. Show that this is not necessarily true for 7. 2.  The triangle ABC has ∠A > ∠C. P lies inside the triangle so that ∠PAC = ∠C. Q is taken outside the triangle so

33rd British Mathematical Olympiad 1997 Problems

33rd British Mathematical Olympiad 1997 Problems1.  M and N are 9-digit numbers. If any digit of M is replaced by the corresponding digit of N (eg the 10s digit of M replaced by the 10s digit of N), then the resulting integer is a multiple of 7. Show that if any digit of N is replaced by the corresponding digit of M, then the resulting integer must be a multiple of 7.

32nd British Mathematical Olympiad 1996 Problems

32nd British Mathematical Olympiad 1996 Problems1.  Find all non-negative integer solutions to 2m + 3n = k2. 2.  The triangle ABC has sides a, b, c, and the triangle UVW has sides u, v, w such that a2 = u(v + w - u), b2 = v(w + u - v), c2 = w(u + v - w). Show that ABC must be acute angled and express the angles U, V, W in terms of the angles A, B, C.

31st British Mathematical Olympiad 1995 Problems

31st British Mathematical Olympiad 1995 Problems1.  Find all positive integers a ≥ b ≥ c such that (1 + 1/a)(1 + 1/b)(1 + 1/c) = 2. 2.  ABC is a triangle. D, E, F are the midpoints of BC, CA, AB. Show that ∠DAC = ∠ABE iff ∠AFC = ∠ADB. 2.  Show that 12/(w + x + y + z) ≤ 1/(w + x

27th British Mathematical Olympiad 1991 Problems

27th British Mathematical Olympiad 1991 Problems 1.  ABC is a triangle with ∠B = 90o and M the midpoint of AB. Show that sin ACM ≤ 1/3. 2.  Twelve dwarfs live in a forest. Some pairs of dwarfs are friends. Each has a

26th British Mathematical Olympiad 1990 Problems

26th British Mathematical Olympiad 1990 Problems1.  Show that if a polynomial with integer coefficients takes the value 1990 at four different integers, then it cannot take the value 1997 at any integer. 2.  The fractional part { x } of a real number is defined as x - [x]. Find a positive real x such that { x } + { 1/x } = 1 (*). Is there a rational x satisfying

25th British Mathematical Olympiad 1989 Problems

25th British Mathematical Olympiad 1989 Problems1.  Find the smallest positive integer a such that ax2 - bx + c = 0 has two distinct roots in the interval 0 < x < 1 for some integers b, c. 2.  Find the number of different

24th British Mathematical Olympiad 1988 Problems

24th British Mathematical Olympiad 1988 Problems1.  ABC is an equilateral triangle. S is the circle diameter AB. P is a point on AC such that the circle center P radius PC touches S at T. Show that AP/AC = 4/5. Find AT/AC.