

Algebra


A2. a, b, c are integers such that a ≥ 0, b ≥ 0, ab ≥ c^{2}. Show that for some n we can find integers x_{1}, x_{2}, ... , x_{n}, y_{1}, y_{2}, ... , y_{n} such that x_{1}^{2} + x_{2}^{2} + ... + x_{n}^{2} = a, y_{1}^{2} + y_{2}^{2} + ... + y_{n}^{2} = b, x_{1}y_{1} + x_{2}y_{2} + ... + x_{n}y_{n} = c.


A3. n > 2. x_{1}, x_{2}, ... , x_{n} are real numbers such that 2 ≤ x_{i} ≤ 3. Show that (x_{1}^{2} + x_{2}^{2}  x_{3}^{2})/(x_{1} + x_{2}  x_{3}) + (x_{2}^{2} + x_{3}^{2}  x_{4}^{2})/(x_{2} + x_{3}  x_{4}) + ... + (x_{n1}^{2} + x_{n}^{2}  x_{1}^{2})/(x_{n1} + x_{n}  x_{1}) + (x_{n}^{2} + x_{1}^{2}  x_{2}^{2})/(x_{n} + x_{1}  x_{2}) ≤ 2(x_{1} + x_{2} + ... + x_{n})  2n.


A4. a, b, c are fixed positive reals. Find all positive real solutions x, y, z to: x + y + z = a + b + c and 4xyz  (a^{2}x + b^{2}y + c^{2}z) = abc.


A5. Does there exist a realvalued function f on the reals such that f(x) is bounded, f(1) = 1 and f(x + 1/x^{2}) = f(x) + f(1/x)^{2} for all nonzero x?


A6. x_{1} < x_{2} < ... < x_{n} are real numbers, where n > 2. Show that n(n1)/2 ∑_{i<j} x_{i}x_{j} > ( (n1)x_{1} + (n2)x_{2} + ... + 2x_{n2} + x_{n1}) (x_{2} + 2x_{3} + ... + (n1)x_{n}).


Geometry


G2. ABC is a triangle. Show that there is a unique point P such that PA^{2} + PB^{2} + AB^{2} = PB^{2} + PC^{2} + BC^{2} = PC^{2} + PA^{2} + CA^{2}.


G3. ABC is a triangle. The incircle touches BC, CA, AB at D, E, F respectively. X is a point inside the triangle such that the incircle of XBC touches BC at D. It touches CX at Y and XB at Z. Show that EFZY is cyclic.


G4. ABC is an acuteangled triangle. There are points A_{1}, A_{2} on the side BC, B_{1} and B_{2} on the side CA, and C_{1}, C_{2} on the side AB such that the points are in the order: A, C_{1}, C_{2}, B; B, A_{1}, A_{2}, C; and C, B_{1}, B_{2}, A. Also ∠AA_{1}A_{2} = ∠AA_{2}A_{1} = ∠BB_{1}B_{2} = ∠BB_{2}B_{1} = ∠CC_{1}C_{2} = ∠CC_{2}C_{1}. The three lines AA_{1}, BB_{1} and CC_{1} meet in three points and the three lines AA_{2}, BB_{2}, CC_{2} meet in three points. Show that all six points lie on a circle.


G6. ABCD is a tetrahedron with centroid G. The line AG meets the circumsphere again at A'. The points B', C' and D' are defined similarly. Show that GA.GB.GC.GD ≤ GA'.GB'.GC'.GD' and 1/GA + 1/GB + 1/GC + 1/GC ≥ 1/GA' + 1/GB' + 1/GC' + 1/GD'.


G7. O is a point inside the convex quadrilateral ABCD. The line through O parallel to AB meets the side BC at L and the line through O parallel to BC meets the side AB at K. The line through O parallel to AD meets the side CD at M and the line through O parallel to CD meets the side DA at N. The area of ABCD is k the area of AKON is k_{1} and the area of LOMC is k_{2}. Show that k^{1/2} ≥ k_{1}^{1/2} + k_{2}^{1/2}.


G8. ABC is a triangle. A circle through B and C meets the side AB again at C' and meets the side AC again at B'. Let H be the orthocenter of ABC and H' the orthocenter of AB'C'. Show that the lines BB', CC' and HH' are concurrent.


Number theory and combinatorics


N1. k is a positive integer. Show that there are infinitely many squares of the form 2^{k}n  7.


N2. Show that for any integers a, b one can find an integer c such that there are no integers m, n with m^{2} + am + b = 2n^{2} + 2n + c.


N4. Find all positive integers m, n such that m + n^{2} + d^{3} = mnd, where d is the greatest common divisor of m and n.


N5. A graph has 12k points. Each point has 3k+6 edges. For any two points the number of points joined to both is the same. Find k.


N7. Does there exist n > 1 such that the set of positive integers may be partitioned into n nonempty subsets so that if we take an arbitrary element from every set but one then their sum belongs to the remaining set?


N8. For each odd prime p, find positive integers m, n such that m ≤ n and (2p)^{1/2}  m^{1/2}  n^{1/2} is nonnegative and as small as possible.


Sequences


S1. Find a sequence f(1), f(2), f(3), ... of nonnegative integers such that 0 occurs in the sequence, all positive integers occur in the sequence infinitely often, and f( f(n^{163}) ) = f( f(n) ) + f( f(361) ).


S3. For any integer n > 1, let p(n) be the smallest prime which does not divide n and let q(n) = the product of all primes less than p(n), or 1 if p(n) = 2. Define the sequence a_{0}, a_{1}, a_{2}, ... by a_{0} = 1 and a_{n+1} = a_{n}p(a_{n})/q(a_{n}). Find all n such that a_{n} = 1995.


S4. x is a positive real such that 1 + x + x^{2} + ... x^{n1} = x^{n}. Show that 2  1/2^{n1} ≤ x < 2  1/2^{n}.


S5. The function f(n) is defined on the positive integers as follows. f(1) = 1. f(n+1) is the largest positive integer m such that there is a strictly increasing arithmetic progression of m positive integers ending with n such that f(k) = f(n) for each k in the arithmetic progression. Show that there are positive integers a and b such that f(an + b) = n + 2 for all positive integers n.


S6. Show that there is a unique function f on the positive integers with positive integer values such that f(m + f(n) ) = n + f(m + 95) for all m, n. Find f(1) + f(2) + ... + f(19).


Note: problems A1, G1, G5, N3, N6, S2 were used in the Olympiad and are not shown here.

