MathDB

1

Guessing game with all but one of the numbers equal

A mysterious machine contains a secret combination of

2016

integer numbers

x_1,x_2,\ldots,x_{2016}

. It is known that all the numbers in the combination are equal but one. One may ask questions to the machine by giving to it a sequence of

2016

integer numbers

y_1,\ldots,y_{2016}

, and the machine answers by telling the value of the sum

x_1y_1+\dots+x_{2016}y_{2016}.

After answering the first question, the machine accepts a second question and then a third one, and so on.
Determine how many questions are necessary to determine the combination: (a) knowing that the number which is different from the others is equal to zero; (b) not knowing what the number different from the others is.

5

1

Sequence of rational numbers

Let

x_0,x_1,x_2,\ldots

be a sequence of rational numbers defined recursively as follows:

x_0

can be any rational number and, for

n\ge 0

,

x_{n+1}=\begin{cases} \left|\frac{x_n}2-1\right| & \text{if the numerator of }x_n\text{ is even}, \\ \left|\frac1{x_n}-1\right| & \text{if the numerator of }x_n\text{ is odd},\end{cases}

where by numerator of a rational number we mean the numerator of the fraction in its lowest terms. Prove that for any value of

x_0

: (a) the sequence contains only finitely many distinct terms; (b) the sequence contains exactly one of the numbers

0

and

2/3

(namely, either there exists an index

k

such that

x_k=0

, or there exists an index

m

such that

x_m=2/3

, but not both).

4

1

(a+1)^n+a-1 is a power of 2

Determine all pairs of positive integers

(a,n)

with

a\ge n\ge 2

for which

(a+1)^n+a-1

is a power of

2

.

3

1

Excircle-related intersection on Thales circle of BC

Let

\Gamma

be the excircle of triangle

ABC

opposite to the vertex

A

(namely, the circle tangent to

BC

and to the prolongations of the sides

AB

and

AC

from the part

B

and

C

). Let

D

be the center of

\Gamma

and

E

,

F

, respectively, the points in which

\Gamma

touches the prolongations of

AB

and

AC

. Let

J

be the intersection between the segments

BD

and

EF

.
Prove that

\angle CJB

is a right angle.

2

1

Almost completely different scores on IMO day 1

A mathematical contest had

3

problems, each of which was given a score between

0

and

7

(

0

and

7

included). It is known that, for any two contestants, there exists at most one problem in which they have obtained the same score (for example, there are no two contestants whose ordered scores are

7,1,2

and

7,1,5

, but there might be two contestants whose ordered scores are

7,1,2

and

7,2,1

). Find the maximum number of contestants.

1

Projection of vertex onto bisectors

Let

ABC

be a triangle, and let

D

and

E

be the orthogonal projections of

A

onto the internal bisectors from

B

and

C

. Prove that

DE

is parallel to

BC

.

2016 ITAMO

Subcontests

Guessing game with all but one of the numbers equal

Sequence of rational numbers

(a+1)^n+a-1 is a power of 2

Excircle-related intersection on Thales circle of BC

Almost completely different scores on IMO day 1

Projection of vertex onto bisectors