MathDB

1

IMO Shortlist 2014 A2

f:(0,1)\to (0,1)

by

\displaystyle f(x) = \left\{ \begin{array}{lr} x+\frac 12 & \text{if}\ \ x < \frac 12\\ x^2 & \text{if}\ \ x \ge \frac 12 \end{array} \right.

Let

a

and

b

be two real numbers such that

0 < a < b < 1

. We define the sequences

a_n

and

b_n

by

a_0 = a, b_0 = b

, and

a_n = f( a_{n -1})

,

b_n = f (b_{n -1} )

for

n > 0

. Show that there exists a positive integer

n

such that

(a_n - a_{n-1})(b_n-b_{n-1})<0.

Proposed by Denmark

N8

1

IMO Shortlist 2014 N8

For every real number

x

, let

||x||

denote the distance between

x

and the nearest integer. Prove that for every pair

(a, b)

of positive integers there exist an odd prime

p

and a positive integer

k

satisfying

\displaystyle\left|\left|\frac{a}{p^k}\right|\right|+\left|\left|\frac{b}{p^k}\right|\right|+\left|\left|\frac{a+b}{p^k}\right|\right|=1.

Proposed by Geza Kos, Hungary

N7

1

IMO Shortlist 2014 N7

Let

c \ge 1

be an integer. Define a sequence of positive integers by

a_1 = c

and

a_{n+1}=a_n^3-4c\cdot a_n^2+5c^2\cdot a_n+c

for all

n\ge 1

. Prove that for each integer

n \ge 2

there exists a prime number

p

dividing

a_n

but none of the numbers

a_1 , \ldots , a_{n -1}

.
Proposed by Austria

N6

1

IMO Shortlist 2014 N6

Let

a_1 < a_2 < \cdots <a_n

be pairwise coprime positive integers with

a_1

being prime and

a_1 \ge n + 2

. On the segment

I = [0, a_1 a_2 \cdots a_n ]

of the real line, mark all integers that are divisible by at least one of the numbers

a_1 , \ldots , a_n

. These points split

I

into a number of smaller segments. Prove that the sum of the squares of the lengths of these segments is divisible by

a_1

.
Proposed by Serbia

N5

1

IMO Shortlist 2014 N5

Find all triples

(p, x, y)

consisting of a prime number

p

and two positive integers

x

and

y

such that

x^{p -1} + y

and

x + y^ {p -1}

are both powers of

p

.
Proposed by Belgium

N4

1

IMO Shortlist 2014 N4

Let

n > 1

be a given integer. Prove that infinitely many terms of the sequence

(a_k )_{k\ge 1}

, defined by

a_k=\left\lfloor\frac{n^k}{k}\right\rfloor,

are odd. (For a real number

x

,

\lfloor x\rfloor

denotes the largest integer not exceeding

x

.)
Proposed by Hong Kong

N2

1

IMO Shortlist 2014 N2

Determine all pairs

(x, y)

of positive integers such that

\sqrt[3]{7x^2-13xy+7y^2}=|x-y|+1.

Proposed by Titu Andreescu, USA

N1

1

IMO Shortlist 2014 N1

Let

n \ge 2

be an integer, and let

A_n

be the set

A_n = \{2^n - 2^k\mid k \in \mathbb{Z},\, 0 \le k < n\}.

Determine the largest positive integer that cannot be written as the sum of one or more (not necessarily distinct) elements of

A_n

.
Proposed by Serbia

G7

1

IMO Shortlist 2014 G7

Let

ABC

be a triangle with circumcircle

\Omega

and incentre

I

. Let the line passing through

I

and perpendicular to

CI

intersect the segment

BC

and the arc

BC

(not containing

A

) of

\Omega

at points

U

and

V

, respectively. Let the line passing through

U

and parallel to

AI

intersect

AV

at

X

, and let the line passing through

V

and parallel to

AI

intersect

AB

at

Y

. Let

W

and

Z

be the midpoints of

AX

and

BC

, respectively. Prove that if the points

I, X,

and

Y

are collinear, then the points

I, W ,

and

Z

are also collinear.
Proposed by David B. Rush, USA

G6

1

IMO Shortlist 2014 G6

Let

ABC

be a fixed acute-angled triangle. Consider some points

E

and

F

lying on the sides

AC

and

AB

, respectively, and let

M

be the midpoint of

EF

. Let the perpendicular bisector of

EF

intersect the line

BC

at

K

, and let the perpendicular bisector of

MK

intersect the lines

AC

and

AB

at

S

and

T

, respectively. We call the pair

(E, F )

<span class='latex-italic'>interesting</span>

, if the quadrilateral

KSAT

is cyclic. Suppose that the pairs

(E_1 , F_1 )

and

(E_2 , F_2 )

are interesting. Prove that

\displaystyle\frac{E_1 E_2}{AB}=\frac{F_1 F_2}{AC}

Proposed by Ali Zamani, Iran

G4

1

IMO Shortlist 2014 G4

Consider a fixed circle

\Gamma

with three fixed points

A, B,

and

C

on it. Also, let us fix a real number

\lambda \in(0,1)

. For a variable point

P \not\in\{A, B, C\}

on

\Gamma

, let

M

be the point on the segment

CP

such that

CM =\lambda\cdot CP

. Let

Q

be the second point of intersection of the circumcircles of the triangles

AMP

and

BMC

. Prove that as

P

varies, the point

Q

lies on a fixed circle.
Proposed by Jack Edward Smith, UK

G3

1

IMO Shortlist 2014 G3

Let

\Omega

and

O

be the circumcircle and the circumcentre of an acute-angled triangle

ABC

with

AB > BC

. The angle bisector of

\angle ABC

intersects

\Omega

at

M \ne B

. Let

\Gamma

be the circle with diameter

BM

. The angle bisectors of

\angle AOB

and

\angle BOC

intersect

\Gamma

at points

P

and

Q,

respectively. The point

R

is chosen on the line

P Q

so that

BR = MR

. Prove that

BR\parallel AC

. (Here we always assume that an angle bisector is a ray.)
Proposed by Sergey Berlov, Russia

G2

1

IMO Shortlist 2014 G2

Let

ABC

be a triangle. The points

K, L,

and

M

lie on the segments

BC, CA,

and

AB,

respectively, such that the lines

AK, BL,

and

CM

intersect in a common point. Prove that it is possible to choose two of the triangles

ALM, BMK,

and

CKL

whose inradii sum up to at least the inradius of the triangle

ABC

.
Proposed by Estonia

C9

1

IMO Shortlist 2014 C9

There are

n

circles drawn on a piece of paper in such a way that any two circles intersect in two points, and no three circles pass through the same point. Turbo the snail slides along the circles in the following fashion. Initially he moves on one of the circles in clockwise direction. Turbo always keeps sliding along the current circle until he reaches an intersection with another circle. Then he continues his journey on this new circle and also changes the direction of moving, i.e. from clockwise to anticlockwise or

<span class='latex-italic'>vice versa</span>

. Suppose that Turbo’s path entirely covers all circles. Prove that

n

must be odd.
Proposed by Tejaswi Navilarekallu, India

C8

1

IMO Shortlist 2014 C8

A card deck consists of

1024

cards. On each card, a set of distinct decimal digits is written in such a way that no two of these sets coincide (thus, one of the cards is empty). Two players alternately take cards from the deck, one card per turn. After the deck is empty, each player checks if he can throw out one of his cards so that each of the ten digits occurs on an even number of his remaining cards. If one player can do this but the other one cannot, the one who can is the winner; otherwise a draw is declared. Determine all possible first moves of the first player after which he has a winning strategy.
Proposed by Ilya Bogdanov & Vladimir Bragin, Russia

C7

1

IMO Shortlist 2014 C7

Let

M

be a set of

n \ge 4

points in the plane, no three of which are collinear. Initially these points are connected with

n

segments so that each point in

M

is the endpoint of exactly two segments. Then, at each step, one may choose two segments

AB

and

CD

sharing a common interior point and replace them by the segments

AC

and

BD

if none of them is present at this moment. Prove that it is impossible to perform

n^3 /4

or more such moves.
Proposed by Vladislav Volkov, Russia

C6

1

IMO Shortlist 2014 C6

We are given an infinite deck of cards, each with a real number on it. For every real number

x

, there is exactly one card in the deck that has

x

written on it. Now two players draw disjoint sets

A

and

B

of

100

cards each from this deck. We would like to define a rule that declares one of them a winner. This rule should satisfy the following conditions: 1. The winner only depends on the relative order of the

200

cards: if the cards are laid down in increasing order face down and we are told which card belongs to which player, but not what numbers are written on them, we can still decide the winner. 2. If we write the elements of both sets in increasing order as

A =\{ a_1 , a_2 , \ldots, a_{100} \}

and

B= \{ b_1 , b_2 , \ldots , b_{100} \}

, and

a_i > b_i

for all

i

, then

A

beats

B

. 3. If three players draw three disjoint sets

A, B, C

from the deck,

A

beats

B

and

B

beats

C

then

A

also beats

C

. How many ways are there to define such a rule? Here, we consider two rules as different if there exist two sets

A

and

B

such that

A

beats

B

according to one rule, but

B

beats

A

according to the other.
Proposed by Ilya Bogdanov, Russia

C4

1

IMO Shortlist 2014 C4

Construct a tetromino by attaching two

2 \times 1

dominoes along their longer sides such that the midpoint of the longer side of one domino is a corner of the other domino. This construction yields two kinds of tetrominoes with opposite orientations. Let us call them

S

- and

Z

-tetrominoes, respectively. Assume that a lattice polygon

P

can be tiled with

S

-tetrominoes. Prove that no matter how we tile

P

using only

S

- and

Z

-tetrominoes, we always use an even number of

Z

-tetrominoes.
Proposed by Tamas Fleiner and Peter Pal Pach, Hungary

C2

1

IMO Shortlist 2014 C2

We have

2^m

sheets of paper, with the number

1

written on each of them. We perform the following operation. In every step we choose two distinct sheets; if the numbers on the two sheets are

a

and

b

, then we erase these numbers and write the number

a + b

on both sheets. Prove that after

m2^{m -1}

steps, the sum of the numbers on all the sheets is at least

4^m

.
Proposed by Abbas Mehrabian, Iran

C1

1

IMO Shortlist 2014 C1

Let

n

points be given inside a rectangle

R

such that no two of them lie on a line parallel to one of the sides of

R

. The rectangle

R

is to be dissected into smaller rectangles with sides parallel to the sides of

R

in such a way that none of these rectangles contains any of the given points in its interior. Prove that we have to dissect

R

into at least

n + 1

smaller rectangles.
Proposed by Serbia

A6

1

IMO Shortlist 2014 A6

Find all functions

f : \mathbb{Z} \to\mathbb{ Z}

such that

n^2+4f(n)=f(f(n))^2

for all

n\in \mathbb{Z}

.
Proposed by Sahl Khan, UK

A5

1

IMO Shortlist 2014 A5

Consider all polynomials

P(x)

with real coefficients that have the following property: for any two real numbers

x

and

y

one has |y^2-P(x)|\le 2|x| \text{if and only if} |x^2-P(y)|\le 2|y|. Determine all possible values of

P(0)

.
Proposed by Belgium

A3

1

IMO Shortlist 2014 A3

For a sequence

x_1,x_2,\ldots,x_n

of real numbers, we define its

<span class='latex-italic'>price</span>

as

\max_{1\le i\le n}|x_1+\cdots +x_i|.

Given

n

real numbers, Dave and George want to arrange them into a sequence with a low price. Diligent Dave checks all possible ways and finds the minimum possible price

D

. Greedy George, on the other hand, chooses

x_1

such that

|x_1 |

is as small as possible; among the remaining numbers, he chooses

x_2

such that

|x_1 + x_2 |

is as small as possible, and so on. Thus, in the

i

-th step he chooses

x_i

among the remaining numbers so as to minimise the value of

|x_1 + x_2 + \cdots x_i |

. In each step, if several numbers provide the same value, George chooses one at random. Finally he gets a sequence with price

G

.
Find the least possible constant

c

such that for every positive integer

n

, for every collection of

n

real numbers, and for every possible sequence that George might obtain, the resulting values satisfy the inequality

G\le cD

.
Proposed by Georgia

A4

1

IMO shortlist 2014 A4

Determine all functions

f: \mathbb{Z}\to\mathbb{Z}

satisfying

f\big(f(m)+n\big)+f(m)=f(n)+f(3m)+2014

for all integers

m

and

n

.
Proposed by Netherlands

2014 IMO Shortlist

Subcontests

IMO Shortlist 2014 A2

IMO Shortlist 2014 N8

IMO Shortlist 2014 N7

IMO Shortlist 2014 N6

IMO Shortlist 2014 N5

IMO Shortlist 2014 N4

IMO Shortlist 2014 N2

IMO Shortlist 2014 N1

IMO Shortlist 2014 G7

IMO Shortlist 2014 G6

IMO Shortlist 2014 G4

IMO Shortlist 2014 G3

IMO Shortlist 2014 G2

IMO Shortlist 2014 C9

IMO Shortlist 2014 C8

IMO Shortlist 2014 C7

IMO Shortlist 2014 C6

IMO Shortlist 2014 C4

IMO Shortlist 2014 C2

IMO Shortlist 2014 C1

IMO Shortlist 2014 A6

IMO Shortlist 2014 A5

IMO Shortlist 2014 A3

IMO shortlist 2014 A4