28th Spanish Mathematical Olympiad Problems 1992

A1.  Find the smallest positive integer N which is a multiple of 83 and is such that N² has exactly 63 positive divisors.

A2.  Given two circles (neither inside the other) with different radii, a line L, and k > 0, show how to construct a line L' parallel to L so that L intersects the two circles in chords with total length k.

A3.  a, b, c, d are positive integers such that (a+b)² + 2a + b = (c+d)² + 2c + d. Show that a = c and b = d. Show that the same is true if a, b, c, d satisfy (a+b)² + 3a + b = (c+d)² + 3c + d. But show that there exist a, b, c, d such that (a+b)² + 4a + b = (c+d)² + 4c + d, but a ≠ c and b ≠ d.

B1.  Show that there are infinitely many primes in the arithmetic progression 3, 7, 11, 15, ... .

B2.  Given the triangle ABC, show how to find geometrically the point P such that ∠PAB = ∠PBC = ∠PCA. Express this angle in terms of ∠A, ∠B, ∠C using trigonometric functions.

B3.  For each positive integer n let S(n) be the set of complex numbers z such that |z| = 1 and (z + 1/z)ⁿ = 2^n-1(zⁿ + 1/zⁿ). Find S(2), S(3), S(4). Find an upper bound for |S(n)| for n ≥ 5.

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27th Spanish Mathematical Olympiad Problems 1991

A1.  Let S be the set of all points in the plane with integer coordinates. Let T be the set of all segments AB, where A, B ∈ S and AB has integer length. Prove that we cannot find two elements of T making an angle 45^o. Is the same true in three dimensions?

A2.  a, b are distinct elements of {0,1,-1}. A is the matrix:

a+b    a+b2    a+b3   ...    a+bm
a2+b   a2+b2   a2+b3   ...   a2+bm
a3+b   a3+b2   a3+b3   ...   a3+bm
 ...
an+b   an+b2   an+b3   ...   an+bm

Find the smallest possible number of columns of A such that any other column is a linear combination of these columns with integer coefficients.

A3.  What condition must be satisfied by the coefficients u, v, w if the roots of the polynomial x³ - ux² + vx - w can be the sides of a triangle?

B1.  The incircle of ABC touches BC, CA, AB at A', B', C' respectively. The line A'C' meets the angle bisector of A at D. Find ∠ADC.

B2.  Let s(n) be the sum of the binary digits of n. Find s(1) + s(2) + s(3) + ... + s(2^k) for each positive integer k.

B3.  Find the integral part of 1/√1 + 1/√2 + 1/√3 + ... + 1/√1000.

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26th Spanish Mathematical Olympiad Problems 1990

A1.  Show that √x + √y + √(xy) = √x + √(y + xy + 2y√x). Hence show that √3 + √(10 + 2√3) = √(5 + √22) + √(8 - √22 + 2√(15 - 3√22)).

A2.  Every point of the plane is painted with one of three colors. Can we always find two points a distance 1 apart which are the same color?

A3.  Show that [(4 + √11)ⁿ] is odd for any positive integer n.

B1.  Show that ((a+1)/2 + ((a+3)/6)√((4a+3)/3) )^1/3 + ((a+1)/2 - ((a+3)/6)√((4a+3)/3) )^1/3 is independent of a for a ≥ 3/4 and find it.

B2.  ABC is a triangle with area S. Points A', B', C' are taken on the sides BC, CA, AB, so that AC'/AB = BA'/BC = CB'/CA = k, where 0 < k < 1. Find the area of A'B'C' in terms of S and k. Find the value of k which minimises the area. The line through A' parallel to AB and the line through C' parallel to AC meet at P. Find the locus of P as k varies.

B3.  There are n points in the plane so that no two pairs are the same distance apart. Each point is connected to the nearest point by a line. Show that no point is connected to more than 5 points.

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