Korrespondenskurs 2010/2011
Problemblad 4

1.Find the smallest and the largest among all 10-digit numbers, divisible by 99,
and in which each of the digits 0, 1, 2, . . . , 9 occurs exactly once.

2. We shall call a die unfair if the probability to throw k, 1 ≤ k ≤ 6, is pk (≥ 0),
with p1 + p2 + p3 + p4 + p5 + p6 = 1. Given that two dice are thrown simultaneously,
denote by Pi the probability that the sum of the two numbers shown is equal to i,
i = 2, 3, . . . , 12 (the probabilities to throw a certain number k, 1 ≤ k ≤ 6, may be
different for the two dice). Do there exist unfair dice such that P2 = P3 = · · · = P12 ?

3. Given a triangle ABC , prove that if there exist two different interior points M
and N , such that ∠MAC = ∠NBC, ∠MCA = ∠NCB, AN = BM , then ABC is isosceles.

4. Prove that for all positive real numbers a, b, c, the inequality holds
(1 + a(b + c))/(1 + b + c)^2 + (1 + b(c + a))/(1 + c + a)^2 + (1 + c(a + b))/ (1 + a + b)^2>=1.
When does equality occur?

5. Denote by O, I, H the circumcentre, incentre, orthocentre, respectively, of an
acute scalene triangle ABC . Prove that ∠OIH > 135◦ .

6. Let S and T be trees with no vertices of degree 2. For each edge of S and T
there is a fixed positive number, called length of the edge. The distance between two
vertices in a tree is dened as the sum of the lengths of the edges connecting them.
The vertices of degree 1 are called leaves of the tree. Let f be a bijection between the
leaves of S and the leaves of T , preserving distances, i.e. the distance between two
leaves u, v ∈ S is equal to the distance between the two leaves f (u), f (v) ∈ T . Prove
that there exists a bijection g between the vertices of S and the vertices of T such
that the distance between any two vertices u, v ∈ S is equal to the distance between
the corresponding vertices f (u), f (v) ∈ T .