MathDB

1

Maximum number of m-tastic numbers

Call a rational number short if it has finitely many digits in its decimal expansion. For a positive integer

m

, we say that a positive integer

t

is

m-

tastic if there exists a number

c\in \{1,2,3,\ldots ,2017\}

such that

\dfrac{10^t-1}{c\cdot m}

is short, and such that

\dfrac{10^k-1}{c\cdot m}

is not short for any

1\le k<t

. Let

S(m)

be the set of

m-

tastic numbers. Consider

S(m)

for

m=1,2,\ldots{}.

What is the maximum number of elements in

S(m)

?

A7

1

Inequality on sequence of integers

Let

a_0,a_1,a_2,\ldots

be a sequence of integers and

b_0,b_1,b_2,\ldots

be a sequence of positive integers such that

a_0=0,a_1=1

, and

a_{n+1} = \begin{cases} a_nb_n+a_{n-1} & \text{if $b_{n-1}=1$} \\ a_nb_n-a_{n-1} & \text{if $b_{n-1}>1$} \end{cases}\qquad\text{for }n=1,2,\ldots.

for

n=1,2,\ldots.

Prove that at least one of the two numbers

a_{2017}

and

a_{2018}

must be greater than or equal to

2017

.

A3

1

Show that f is bijective on T

Let

S

be a finite set, and let

\mathcal{A}

be the set of all functions from

S

to

S

. Let

f

be an element of

\mathcal{A}

, and let

T=f(S)

be the image of

S

under

f

. Suppose that

f\circ g\circ f\ne g\circ f\circ g

for every

g

in

\mathcal{A}

with

g\ne f

. Show that

f(T)=T

.

C7

1

Operation on smallest positive integers not in a set

For any finite sets

X

and

Y

of positive integers, denote by

f_X(k)

the

k^{\text{th}}

smallest positive integer not in

X

, and let

X*Y=X\cup \{ f_X(y):y\in Y\}.

Let

A

be a set of

a>0

positive integers and let

B

be a set of

b>0

positive integers. Prove that if

A*B=B*A

, then

\underbrace{A*(A*\cdots (A*(A*A))\cdots )}_{\text{ A appears $b$ times}}=\underbrace{B*(B*\cdots (B*(B*B))\cdots )}_{\text{ B appears $a$ times}}.

Proposed by Alex Zhai, United States

C3

1

Upper bound on number of moves

Sir Alex plays the following game on a row of 9 cells. Initially, all cells are empty. In each move, Sir Alex is allowed to perform exactly one of the following two operations:
[*] Choose any number of the form

2^j

, where

j

is a non-negative integer, and put it into an empty cell. [*] Choose two (not necessarily adjacent) cells with the same number in them; denote that number by

2^j

. Replace the number in one of the cells with

2^{j+1}

and erase the number in the other cell.
At the end of the game, one cell contains

2^n

, where

n

is a given positive integer, while the other cells are empty. Determine the maximum number of moves that Sir Alex could have made, in terms of

n

.
Proposed by Warut Suksompong, Thailand

G6

1

Two regular n-gons in a plane

Let

n\ge3

be an integer. Two regular

n

-gons

\mathcal{A}

and

\mathcal{B}

are given in the plane. Prove that the vertices of

\mathcal{A}

that lie inside

\mathcal{B}

or on its boundary are consecutive.
(That is, prove that there exists a line separating those vertices of

\mathcal{A}

that lie inside

\mathcal{B}

or on its boundary from the other vertices of

\mathcal{A}

.)

A2

1

Algebra form IMO Shortlist

Let

q

be a real number. Gugu has a napkin with ten distinct real numbers written on it, and he writes the following three lines of real numbers on the blackboard:
[*]In the first line, Gugu writes down every number of the form

a-b

, where

a

and

b

are two (not necessarily distinct) numbers on his napkin. [*]In the second line, Gugu writes down every number of the form

qab

, where

a

and

b

are two (not necessarily distinct) numbers from the first line. [*]In the third line, Gugu writes down every number of the form

a^2+b^2-c^2-d^2

, where

a, b, c, d

are four (not necessarily distinct) numbers from the first line.
Determine all values of

q

such that, regardless of the numbers on Gugu's napkin, every number in the second line is also a number in the third line.

C2

1

Swapping string consisting a,b,c

Let

n

be a positive integer. Define a chameleon to be any sequence of

3n

letters, with exactly

n

occurrences of each of the letters

a, b,

and

c

. Define a swap to be the transposition of two adjacent letters in a chameleon. Prove that for any chameleon

X

, there exists a chameleon

Y

such that

X

cannot be changed to

Y

using fewer than

3n^2/2

swaps.

G8

1

Shortlist 2017/G8

There are

2017

mutually external circles drawn on a blackboard, such that no two are tangent and no three share a common tangent. A tangent segment is a line segment that is a common tangent to two circles, starting at one tangent point and ending at the other one. Luciano is drawing tangent segments on the blackboard, one at a time, so that no tangent segment intersects any other circles or previously drawn tangent segments. Luciano keeps drawing tangent segments until no more can be drawn.
Find all possible numbers of tangent segments when Luciano stops drawing.

N5

1

Weird expression being integer.

Find all pairs

(p,q)

of prime numbers which

p>q

and

\frac{(p+q)^{p+q}(p-q)^{p-q}-1}{(p+q)^{p-q}(p-q)^{p+q}-1}

is an integer.

A8

1

Shortlist 2017/A8

A function

f:\mathbb{R} \to \mathbb{R}

has the following property:

\text{For every } x,y \in \mathbb{R} \text{ such that }(f(x)+y)(f(y)+x) > 0, \text{ we have } f(x)+y = f(y)+x.

Prove that

f(x)+y \leq f(y)+x

whenever

x>y

.

C1

1

Tiling rectangle with smaller rectangles.

A rectangle

\mathcal{R}

with odd integer side lengths is divided into small rectangles with integer side lengths. Prove that there is at least one among the small rectangles whose distances from the four sides of

\mathcal{R}

are either all odd or all even.
Proposed by Jeck Lim, Singapore

N8

1

Number of times to do Euclidean GCD.

Let

p

be an odd prime number and

\mathbb{Z}_{>0}

be the set of positive integers. Suppose that a function

f:\mathbb{Z}_{>0}\times\mathbb{Z}_{>0}\to\{0,1\}

satisfies the following properties:
[*]

f(1,1)=0

. [*]

f(a,b)+f(b,a)=1

for any pair of relatively prime positive integers

(a,b)

not both equal to 1; [*]

f(a+b,b)=f(a,b)

for any pair of relatively prime positive integers

(a,b)

.
Prove that

\sum_{n=1}^{p-1}f(n^2,p) \geqslant \sqrt{2p}-2.

N6

1

Shortlist 2017/N6

Find the smallest positive integer

n

or show no such

n

exists, with the following property: there are infinitely many distinct

n

-tuples of positive rational numbers

(a_1, a_2, \ldots, a_n)

such that both a_1+a_2+\dots +a_n \text{and} \frac{1}{a_1} + \frac{1}{a_2} + \dots + \frac{1}{a_n} are integers.

N3

1

Number theory sequences : sums divisible by n

Determine all integers

n\geq 2

having the following property: for any integers

a_1,a_2,\ldots, a_n

whose sum is not divisible by

n

, there exists an index

1 \leq i \leq n

such that none of the numbers

a_i,a_i+a_{i+1},\ldots,a_i+a_{i+1}+\ldots+a_{i+n-1}

is divisible by

n

. Here, we let

a_i=a_{i-n}

when

i >n

.
Proposed by Warut Suksompong, Thailand

C8

1

2017 ISL C8

Let

n

be a given positive integer. In the Cartesian plane, each lattice point with nonnegative coordinates initially contains a butterfly, and there are no other butterflies. The neighborhood of a lattice point

c

consists of all lattice points within the axis-aligned

(2n+1) \times (2n+1)

square entered at

c

, apart from

c

itself. We call a butterfly lonely, crowded, or comfortable, depending on whether the number of butterflies in its neighborhood

N

is respectively less than, greater than, or equal to half of the number of lattice points in

N

. Every minute, all lonely butterflies fly away simultaneously. This process goes on for as long as there are any lonely butterflies. Assuming that the process eventually stops, determine the number of comfortable butterflies at the final state.

C6

1

2017 Shortlist/C6

Let

n > 1

be a given integer. An

n \times n \times n

cube is composed of

n^3

unit cubes. Each unit cube is painted with one colour. For each

n \times n \times 1

box consisting of

n^2

unit cubes (in any of the three possible orientations), we consider the set of colours present in that box (each colour is listed only once). This way, we get

3n

sets of colours, split into three groups according to the orientation.
It happens that for every set in any group, the same set appears in both of the other groups. Determine, in terms of

n

, the maximal possible number of colours that are present.

A5

1

2017 Shortlist/A5

An integer

n \geq 3

is given. We call an

n

-tuple of real numbers

(x_1, x_2, \dots, x_n)

Shiny if for each permutation

y_1, y_2, \dots, y_n

of these numbers, we have

\sum \limits_{i=1}^{n-1} y_i y_{i+1} = y_1y_2 + y_2y_3 + y_3y_4 + \cdots + y_{n-1}y_n \geq -1.

Find the largest constant

K = K(n)

such that

\sum \limits_{1 \leq i < j \leq n} x_i x_j \geq K

holds for every Shiny

n

-tuple

(x_1, x_2, \dots, x_n)

.

A4

1

A sequence must be bounded

A sequence of real numbers

a_1,a_2,\ldots

satisfies the relation a_n=-\max_{i+j=n}(a_i+a_j)\qquad\text{for all} n>2017. Prove that the sequence is bounded, i.e., there is a constant

M

such that

|a_n|\leq M

for all positive integers

n

.

G5

1

Shortlist 2017/G5

Let

ABCC_1B_1A_1

be a convex hexagon such that

AB=BC

, and suppose that the line segments

AA_1, BB_1

, and

CC_1

have the same perpendicular bisector. Let the diagonals

AC_1

and

A_1C

meet at

D

, and denote by

\omega

the circle

ABC

. Let

\omega

intersect the circle

A_1BC_1

again at

E \neq B

. Prove that the lines

BB_1

and

DE

intersect on

\omega

.

N2

1

a game involving digits

Let

p \geq 2

be a prime number. Eduardo and Fernando play the following game making moves alternately: in each move, the current player chooses an index

i

in the set

\{0,1,2,\ldots, p-1 \}

that was not chosen before by either of the two players and then chooses an element

a_i

from the set

\{0,1,2,3,4,5,6,7,8,9\}

. Eduardo has the first move. The game ends after all the indices have been chosen .Then the following number is computed:

M=a_0+a_110+a_210^2+\cdots+a_{p-1}10^{p-1}= \sum_{i=0}^{p-1}a_i.10^i

. The goal of Eduardo is to make

M

divisible by

p

, and the goal of Fernando is to prevent this.
Prove that Eduardo has a winning strategy.
Proposed by Amine Natik, Morocco

G4

1

Shortlist 2017/G4

In triangle

ABC

, let

\omega

be the excircle opposite to

A

. Let

D, E

and

F

be the points where

\omega

is tangent to

BC, CA

, and

AB

, respectively. The circle

AEF

intersects line

BC

at

P

and

Q

. Let

M

be the midpoint of

AD

. Prove that the circle

MPQ

is tangent to

\omega

.

A1

1

IMO Shortlist 2017 A1

Let

a_1,a_2,\ldots a_n,k

, and

M

be positive integers such that \frac{1}{a_1}+\frac{1}{a_2}+\cdots+\frac{1}{a_n}=k \text{and} a_1a_2\cdots a_n=M. If

M>1

, prove that the polynomial

P(x)=M(x+1)^k-(x+a_1)(x+a_2)\cdots (x+a_n)

has no positive roots.

G3

1

Shortlist 2017/G3

Let

O

be the circumcenter of an acute triangle

ABC

. Line

OA

intersects the altitudes of

ABC

through

B

and

C

at

P

and

Q

, respectively. The altitudes meet at

H

. Prove that the circumcenter of triangle

PQH

lies on a median of triangle

ABC

.

G7

1

Shortlist 2017/G7

A convex quadrilateral

ABCD

has an inscribed circle with center

I

. Let

I_a, I_b, I_c

and

I_d

be the incenters of the triangles

DAB, ABC, BCD

and

CDA

, respectively. Suppose that the common external tangents of the circles

AI_bI_d

and

CI_bI_d

meet at

X

, and the common external tangents of the circles

BI_aI_c

and

DI_aI_c

meet at

Y

. Prove that

\angle{XIY}=90^{\circ}

.

G1

1

Shortlist 2017/G1

Let

ABCDE

be a convex pentagon such that

AB=BC=CD

,

\angle{EAB}=\angle{BCD}

, and

\angle{EDC}=\angle{CBA}

. Prove that the perpendicular line from

E

to

BC

and the line segments

AC

and

BD

are concurrent.

2017 IMO Shortlist

Subcontests

Maximum number of m-tastic numbers

Inequality on sequence of integers

Show that f is bijective on T

Operation on smallest positive integers not in a set

Upper bound on number of moves

Two regular n-gons in a plane

Algebra form IMO Shortlist

Swapping string consisting a,b,c

Shortlist 2017/G8

Weird expression being integer.

Shortlist 2017/A8

Tiling rectangle with smaller rectangles.

Number of times to do Euclidean GCD.

Shortlist 2017/N6

Number theory sequences : sums divisible by n

2017 ISL C8

2017 Shortlist/C6

2017 Shortlist/A5

A sequence must be bounded

Shortlist 2017/G5

a game involving digits

Shortlist 2017/G4

IMO Shortlist 2017 A1

Shortlist 2017/G3

Shortlist 2017/G7

Shortlist 2017/G1