1. A rectangular grid is colored in checkerboard fashion, and each cell contains an integer. It is given that the sum of the numbers in each row and the sum of the numbers in each column is even. Prove that the sum of all numbers in black cells is even.
2. Find all solutions in real numbers to the following system of equa-tions:
x1+ x2+· · · + x1997 = 1997
x41+ x42+· · · + x19974 = x31+ x32+· · · + x31997.
3. Let d(n) denote the greatest odd divisor of the natural number n. We define the function f : N→ N such that f(2n − 1) = 2nand f (2n) = n +d(n)2n for all n∈ N. Find all k such that f(f(· · · f(1) · · ·)) = 1997, where f is iterated k times.
4. Two regular pentagons ABCDE and AEKP L are situated in space so that ∠DAK = 60◦. Prove that the planes ACK and BAL are perpendicular.
5. The equation ax3+ bx2+ cx + d = 0 is known to have three distinct real roots. How many real roots are there of the equation
4(ax3+ bx2+ cx + d)(3ax + b) = (3ax2+ 2bx + c)2? 6. Let Q+ denote the set of positive rational numbers. Find all
func-tions f : Q+→ Q+ such that for all x∈ Q+: (a) f (x + 1) = f (x) + 1;
(b) f (x2) = f (x)2.
7. Find the smallest integer n such that among any n integers, there exist 18 integers whose sum is divisible by 18.
8. Points K, L, M, N lie on the edges AB, BC, CD, DA of a (not nec-essarily right) parallelepiped ABCDA1B1C1D1. Prove that the centers of the circumscribed spheres of the tetrahedra A1AKN , B1BKL, C1CLM , D1DM N are the vertices of a parallelogram.
3.24 United Kingdom
1. (a) Let M and N be two 9-digit positive integers with the property that if any one digit of M is replaced by the digit of N in the corresponding place, the resulting integer is a multiple of 7.
Prove that any number obtained by replacing a digit of N with the corresponding digit of M is also a multiple of 7.
(b) Find an integer d > 9 such that the above result remains true when M and N are two d-digit positive integers.
2. In acute triangle ABC, CF is an altitude, with F on AB, and BM is a median, with M on CA. Given that BM = CF and ∠M BC =
∠F CA, prove that the triangle ABC is equilateral.
3. Find the number of polynomials of degree 5 with distinct coefficients from the set{1, 2, . . . , 9} that are divisible by x2− x + 1.
4. The set S = {1/r : r = 1, 2, 3, . . .} of reciprocals of the positive integers contains arithmetic progressions of various lengths. For in-stance, 1/20, 1/8, 1/5 is such a progression, of length 3 and common difference 3/40. Moreover, this is a maximal progression in S of length 3 since it cannot be extended to the left or right within S (−1/40 and 11/40 not being members of S).
(a) Find a maximal progression in S of length 1996.
(b) Is there a maximal progression in S of length 1997?
3.25 United States of America
1. Let p1, p2, p3, . . . be the prime numbers listed in increasing order, and let x0 be a real number between 0 and 1. For positive integer k, define
xk = 0 if xk−1= 0,
pk xk−1
if xk−16= 0,
where {x} = x − bxc denotes the fractional part of x. Find, with proof, all x0satisfying 0 < x0< 1 for which the sequence x0, x1, x2, . . . eventually becomes 0.
2. Let ABC be a triangle, and draw isosceles triangles BCD, CAE, ABF externally to ABC, with BC, CA, AB as their respective bases. Prove that the lines through A, B, C perpendicular to the lines EF, F D, DE, respectively, are concurrent.
3. Prove that for any integer n, there exists a unique polynomial Q with coefficients in{0, 1, . . . , 9} such that Q(−2) = Q(−5) = n.
4. To clip a convex n-gon means to choose a pair of consecutive sides AB, BC and to replace them by the three segments AM, M N , and N C, where M is the midpoint of AB and N is the midpoint of BC.
In other words, one cuts off the triangle M BN to obtain a convex (n + 1)-gon. A regular hexagonP6 of area 1 is clipped to obtain a heptagonP7. ThenP7is clipped (in one of the seven possible ways) to obtain an octagonP8, and so on. Prove that no matter how the clippings are done, the area ofPn is greater than 1/3, for all n≥ 6.
5. Prove that, for all positive real numbers a, b, c,
(a3+ b3+ abc)−1+ (b3+ c3+ abc)−1+ (c3+ a3+ abc)−1≤ (abc)−1. 6. Suppose the sequence of nonnegative integers a1, a2, . . . , a1997
satis-fies
ai+ aj≤ ai+j ≤ ai+ aj+ 1
for all i, j ≥ 1 with i + j ≤ 1997. Show that there exists a real number x such that an=bnxc for all 1 ≤ n ≤ 1997.
3.26 Vietnam
1. Determine the smallest integer k for which there exists a graph on 25 vertices such that every vertex is adjacent to exactly k others, and any two nonadjacent vertices are both adjacent to some third vertex.
2. Find the largest real number α for which there exists an infinite sequence a1, a2, . . . of positive integers satisfying the following prop-erties.
(a) For each n∈ N, an > 1997n.
(b) For every n ≥ 2, aαn does not exceed the greatest common divisor of the set{ai+ aj: i + j = n}.
3. Let f : N→ Z be the function defined by
f (0) = 2, f (1) = 503, f (n + 2) = 503f (n + 1)− 1996f(n).
For k∈ N, let s1, . . . , sk be integers not less than k, and let pi be a prime divisor of f (2si) for i = 1, . . . , k. Prove that for t = 1, . . . , k,
k
X
i=1
pi| 2t if and only if k| 2t.
4. Find all pairs (a, b) of positive reals such that for every n∈ N and every real number x satisfying
4n2x = log2(2n2x + 1), we have ax+ bx≥ 2 + 3x.
5. Let n, k, p be positive integers such that k ≥ 2 and k(p + 1) ≤ n.
Determine the number of ways to color n labeled points on a circle in blue or red, so that exactly k points are colored blue, and any arc whose endpoints are blue but contains no blue points in its interior contains exactly p red points.
4 1997 Regional Contests:
where the denominators contain partial sums of the sequence of re-ciprocals of triangular numbers. Prove that S > 1001.2. Find an integer n with 100≤ n ≤ 1997 such that n divides 2n+ 2.
3. Let ABC be a triangle and let la= ma
where ma, mb, mcare the lengths of the internal angle bisectors and Ma, Mb, Mc are the lengths of the extensions of the internal angle bisectors to the circumcircle. Prove that
la
sin2A+ lb
sin2B + lc
sin2C ≥ 3, with equality if and only if ABC is equilateral.
4. The triangle A1A2A3 has a right angle at A3. For n ≥ 3, let An+1
be the foot of the perpendicular from An to An−1An−2.
(a) Show that there is a unique point P in the plane interior to the triangles An−2An−1An for all n≥ 3.
(b) For fixed A1 and A3, determine the locus of P as A2 varies.
5. Persons A1, . . . , An (n≥ 3) are seated in a circle in that order, and each person Ai holds a number ai of objects, such that (a1+· · · + an)/n is an integer. It is desired to redistribute the objects so that each person holds the same number; objects may only be passed from one person to either of her two neighbors. How should the redistribution take place so as to minimize the number of passes?