2010 ITAMO

Part of ITAMO

Subcontests

(6)

1

reduce the string of binary numbers

In the land of Cockaigne, people play the following solitaire. It starts from a finite string of zeros and ones, and are granted the following moves: (i) cancel each two consecutive ones; (ii) delete three consecutive zeros; (iii) if the substring within the string is

01

, one may replace this by substring

100

. The moves (i), (ii) and (iii) must be made one at a time. You win if you can reduce the string to a string formed by two digits or less. (For example, starting from

0101

, one can win using move (iii) first in the last two digits, resulting in

01100

, then playing the move (i) on two 'ones', and finally the move (ii) on the three zeros, one will get the empty string.) Among all the

1024

possible strings of ten-digit binary numbers, how many are there from which it is not possible to win the solitary?

1

find the number of participants

In a mathematics test number of participants is

N < 40

. The passmark is fixed at

65

. The test results are the following: The average of all participants is

66

, that of the promoted

71

and that of the repeaters

56

. However, due to an error in the wording of a question, all scores are increased by

5

. At this point the average of the promoted participants becomes

75

and that of the non-promoted

59

. (a) Find all possible values of

N

. (b) Find all possible values of

N

in the case where, after the increase, the average of the promoted had become

79

and that of non-promoted

47

1

white and red numbers

Every non-negative integer is coloured white or red, so that: • there are at least a white number and a red number; • the sum of a white number and a red number is white; • the product of a white number and a red number is red. Prove that the product of two red numbers is always a red number, and the sum of two red numbers is always a red number.

1

two triangles are divided into 4 disjoint triangles

ABCD

AB

CD

are parallel and the angles

\angle ABC

\angle BAD

are acute. Show that it is possible to divide the triangle

ABC

into 4 disjoint triangle

X_1. . . , X_4

and the triangle

ABD

into 4 disjoint triangles

Y_1,. . . , Y_4

such that the triangles

X_i

Y_i

are congruent for all

i

1

angle BCM=angle DBA in a convex quad

ABCD

be a convex quadrilateral. such that

\angle CAB = \angle CDA

\angle BCA = \angle ACD

M

be the midpoint of

AB

\angle BCM = \angle DBA

1

infinitely many such primes

Prove that there are infinitely many prime numbers that divide at least one integer of the form

2^{n^3+1}-3^{n^2+1}+5^{n+1}

n

is a positive integer.