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.

1.  Find all integers 0 < a < b < c such that b - a = c - b and none of a, b, c have a prime factor greater than 3. 2.  D is a point on the side AB of the triangle ABC such that AB = 4·AD. P is a point on the

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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.

37th British Mathematical Olympiad 2001 Problems1.  A has a marbles and B has b < a marbles. Starting with A each gives the other enough marbles to double the number he has. After 2n such transfers A has b marbles. Find a/b in terms of n. 2.  Find all integer solutions to m2n + 1 = m2 + 2mn + 2m + n.

36th British Mathematical Olympiad 2000 Problems1.  Two circles meet at A and B and touch a common tangent at C and D. Show that triangles ABC and ABD have the same area. 2.  Find the smallest value of x2 + 4xy + 4y2 + 2z2 for positive reals x, y, z with product 32.

35th British Mathematical Olympiad 1999 Problems1.  Let Xn = {1, 2, 3, ... , n}. For which n can we partition Xn into two parts with the same sum? For which n can we partition Xn into three parts with the same sum? 2.

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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

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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.

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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.

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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.

30th British Mathematical Olympiad 1994 Problems1.  Find the smallest integer n > 1 such that (12 + 22 + 32 + ... + n2)/n is a square. 2.  How many incongruent triangles have integer sides and perimeter 1994?

29th British Mathematical Olympiad 1993 Problems1.  The angles in the diagram below are measured in some unknown unit, so that a, b, ... , k, l are all distinct positive integers. Find the smallest possible value of a + b + c and give the corresponding values of a, b, ... , k, l. 2.  p > 3 is prime. m = (4p - 1)/3. Show that 2m-1 = 1 mod m. 3.  P is

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28th British Mathematical Olympiad 1992 Problems

28th British Mathematical Olympiad 1992 Problems1.  p is an odd prime. Show that there are unique positive integers m, n such that m2 = n(n + p). Find m and n in terms of p. 2.  Show that 12/(w + x + y + z) ≤ 1/(w + x

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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

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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

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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

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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.

22th British Mathematical Olympiad 1986 Problems - Further International Selection Test 1.  A rational point is a point both of whose coordinates are rationals. Let A, B, C, D be rational points such that AB and CD are not both equal and parallel. Show that there is just one point P such that the triangle PCD can be obtained from the triangle PAB by enlargement and

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21th British Mathematical Olympiad 1985 Problems1.  Prove that ∑1n ∑1n | xi - xj | ≤ n2 for all real xi such that 0 ≤ xi ≤ 2. When does equality hold? 2.  (1) The incircle of the triangle ABC touches BC at L. LM is a diameter of the incircle. The ray AM meets BC at N. Show that | NL | = | AB - AC |.

1.  In the triangle ABC, ∠C = 90o. Find all points D such that AD·BC = AC·BD = AB·CD/√2. 2.  ABCD is a tetrahedron such that DA = DB = DC = d and AB = BC = CA = e. M and N are the midpoints of AB and CD. A variable plane through MN meets AD at P and BC at Q. Show that AP/AD = BQ/BC. Find the value of this ratio in terms of d and e which minimises the area of MQNP

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