2011 ITAMO

Part of ITAMO

Subcontests

(6)

1

All 3 element subsets are coloured

X = \{1, 2, 3, 4, 5, 6, 7, 8\}

. We want to color, using

k

colors, all subsets of

3

X

in such a way that, two disjoint subsets have distinct colors. Prove that: (a)

4

colors are sufficient; (b)

3

colors are not sufficient.

1

all solutions of (p,n)

Determine all solutions

(p,n)

of the equation

n^3=p^2-p-1

p

is a prime number and

n

1

P is the intersection of external bisectors

ABCD

is a convex quadrilateral.

P

is the intersection of external bisectors of

\angle DAC

\angle DBC

\angle APD = \angle BPC

AD+AC=BC+BD

1

Integers written on a board

Integers between

1

7

are written on a blackboard. It is possible that not all the numbers from 1 to 7 are present, and it is also possible that one, some or all of the numbers are repeated, one or more times. A move consists of choosing one or more numbers on the blackboard, where all distinct, delete them and write different numbers in their place, such that the written numbers together with those deleted form the whole set

\{1, 2, 3, 4, 5, 6 , 7\}

For example, moves allowed are: • delete a

4

5

, and write in their place the numbers

1, 2, 3, 6

7

; • deleting a

1

2

3

4

5

6

7

and write nothing in their place.
Prove that, if it is possible to find a sequence of moves, starting from the initial situation, leading to have on board a single number (written once), then this number does not depend on the sequence of moves used.

1

strings of n consecutive integers

A sequence of positive integers

a_1, a_2,\ldots, a_n

is called ladder of length

n

if it consists of

n

consecutive integers in ascending order. (a) Prove that for every positive integer

n

there exist two ladders of length

n

, with no elements in common,

a_1, a_2,\ldots, a_n

b_1, b_2,\ldots, b_n

, such that for all

i

1

n

, the greatest common divisor of

a_i

b_i

1

. (b) Prove that for every positive integer

n

there exist two ladders of length

n

, with no elements in common,

a_1, a_2,\ldots, a_n

b_1, b_2,\ldots, b_n

, such that for all

i

1

n

, the greatest common divisor of

a_i

b_i

is greater than

1

1

A trapezium divide into m+n trapeziums

A trapezium is given with parallel bases having lengths

1

4

. Split it into two trapeziums by a cut, parallel to the bases, of length

3

. We now want to divide the two new trapeziums, always by means of cuts parallel to the bases, in

m

n

trapeziums, respectively, so that all the

m + n

trapezoids obtained have the same area. Determine the minimum possible value for

m + n

and the lengths of the cuts to be made to achieve this minimum value.