Problems for the Team Competition Baltic Way 2001

1. A set of 8 problems was prepared for an examination. Each student was given 3 of them. No
two students received more than one common problem. What is the largest possible number
of students?

2. Let n ≥ 2 be a positive integer. Find whether there exist n pairwise nonintersecting nonempty
subsets of {1, 2, 3, . . . } such that each positive integer can be expressed in a unique way as a
sum of at most n integers, all from different subsets.

3. The numbers 1, 2, . . . , 49 are placed in a 7 × 7 array, and the sum of the numbers in each row
and in each column is computed. Some of these 14 sums are odd while others are even. Let
A denote the sum of all the odd sums and B the sum of all even sums. Is it possible that the
numbers were placed in the array in such a way that A = B?

4. Let p and q be two different primes. Prove that

       
p 2p 3p (q − 1)p 1
+ + + ··· + = (p − 1)(q − 1) .
q q q q 2
(Here bxc denotes the largest integer not greater than x.)

5. Let 2001 given points on a circle be colored either red or green. In one step all points are
recolored simultaneously in the following way: If both direct neighbors of a point P have the
same color as P , then the color of P remains unchanged, otherwise P obtains the other color.
Starting with the first coloring F1 , we obtain the colorings F2 , F3 , . . . after several recoloring
steps. Prove that there is a number n0 ≤ 1000 such that Fn0 = Fn0 +2 . Is the assertion also
true if 1000 is replaced by 999?

6. The points A, B, C, D, E lie on the circle c in this order and satisfy AB k EC and AC k ED.
The line tangent to the circle c at E meets the line AB at P . The lines BD and EC meet at
Q. Prove that |AC| = |P Q|.

7. Given a parallelogram ABCD. A circle passing through A meets the line segments AB, AC
and AD at inner points M , K, N , respectively. Prove that
|AB| · |AM | + |AD| · |AN | = |AK| · |AC|.
8. Let ABCD be a convex quadrilateral, and let N be the midpoint of BC. Suppose further
1
that ∠AN D = 135◦ . Prove that |AB| + |CD| + √ · |BC| ≥ |AD|.
2
9. Given a rhombus ABCD, find the locus of the points P lying inside the rhombus and satisfying
∠AP D + ∠BP C = 180◦ .

10. In a triangle ABC, the bisector of ∠BAC meets the side BC at the point D. Knowing that
|BD| · |CD| = |AD|2 and ∠ADB = 45◦ , determine the angles of triangle ABC.

11. The real-valued function f is defined for all positive integers. For any integers a > 1, b > 1
with d = gcd(a, b), we have
    
a b
f (ab) = f (d) f +f .
d d

1
 Determine all possible values of f (2001).
n
X n
X
12. Let a1 , a2 , ..., an be positive real numbers such that a3i = 3 and a5i = 5. Prove that
i=1 i=1
n
X
ai > 3/2.
i=1

13. Let a0 , a1 , a2 , . . . be a sequence of real numbers satisfying a0 = 1 and an = ab7n/9c + abn/9c for
k
n = 1, 2, . . . . Prove that there exists a positive integer k with ak < .
2001!
(Here bxc denotes the largest integer not greater than x.)

14. There are 2n cards. On each card some real number x, 1 ≤ x ≤ 2, is written (there can be
different numbers on different cards). Prove that the cards can be divided into two heaps
n s1
with sums s1 and s2 so that ≤ ≤ 1.
n+1 s2
15. Let a0 , a1 , a2 , . . . be a sequence of positive real numbers satisfying i · a2i ≥ (i + 1) · ai−1 ai+1 for
i = 1, 2, . . . Furthermore, let x and y be positive reals, and let bi = xai +yai−1 for i = 1, 2, . . . .
Prove that the inequality i · b2i > (i + 1) · bi−1 bi+1 holds for all integers i ≥ 2.

16. Let f be a real-valued function defined on the positive integers satisfying the following condi-
tion: For all n > 1 there exists a prime divisor p of n such that f (n) = f (n/p) − f (p). Given
that f (2001) = 1, what is the value of f (2002)?

17. Let n be a positive integer. Prove that at least 2n−1 + n numbers can be chosen from the set
{1, 2, 3, . . . , 2n } such that for any two different chosen numbers x and y, x + y is not a divisor
of x · y.
n n m m
18. Let a be an odd integer. Prove that a2 +22 and a2 +22 are relatively prime for all positive
integers n and m with n 6= m.

19. What is the smallest positive odd integer having the same number of positive divisors as 360?

20. From a sequence of integers (a, b, c, d) each of the sequences

(c, d, a, b), (b, a, d, c), (a + nc, b + nd, c, d), (a + nb, b, c + nd, d)

for arbitrary integer n can be obtained by one step. Is it possible to obtain (3, 4, 5, 7) from
(1, 2, 3, 4) through a sequence of such steps?