7

Part of 2007 All-Russian Olympiad

Problems(4)

product of all primes, which are less than $n$

Source: All-Russian 2007

n>3

n?

the product of all primes less than

n

. Solve the equation

n?=2n+16

cover entries of matrix 10x10 by 50 rectangles $1\times 2$

Source: All-Russian 2007

\{a_{ij}\}_{i,j=0}^{9}

a_{ij}=10i+j+1

. Andrei is going to cover its entries by

50

1\times 2

(each such rectangle contains two adjacent entries) so that the sum of

50

products in these rectangles is minimal possible. Help him. A. Badzyan

linear algebramatrixgeometryrectanglecombinatorics proposedcombinatorics

colouring of edges of a convex polyhedron

Source: All-Russian 2007

Given a convex polyhedron

F

A

5

, other vertices have degree

3

. A colouring of edges of

F

is called nice, if for any vertex except

A

all three edges from it have different colours. It appears that the number of nice colourings is not divisible by

5

. Prove that there is a nice colouring, in which some three consecutive edges from

A

are coloured the same way. D. Karpov

combinatorics unsolvedcombinatorics

tetrahedron covered by balls diameters are two edges

Source: All-Russian 2007

Given a tetrahedron

T

. Valentin wants to find two its edges

a,b

with no common vertices so that

T

is covered by balls with diameters

a,b

. Can he always find such a pair? A. Zaslavsky

geometry3D geometrytetrahedroninequalitiesgeometry proposed