Korrespondenskurs 2010/2011
Problemblad 4
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.
1.
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
dierent for the two dice). Do there exist unfair dice such that P2 = P3 = · · · = P12 ?
2.
Given a triangle ABC , prove that if there exist two dierent interior points M
and N , such that ∠M AC = ∠N BC, ∠M CA = ∠N CB, AN = BM , then ABC
is isosceles.
3.
4.
Prove that for all positive real numbers a, b, c, the inequality holds
1 + a(b + c) 1 + b(c + a) 1 + c(a + b)
+
+
≥ 1.
(1 + b + c)2
(1 + c + a)2 (1 + a + b)2
When does equality occur?
Denote by O, I, H the circumcentre, incentre, orthocentre, respectively, of an
acute scalene triangle ABC . Prove that ∠OIH > 135◦ .
5.
Let S and T be trees with no vertices of degree 2. For each edge of S and T
there is a xed 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 .