MathDB

1

IMO Shortlist 2013, Number Theory #7

\nu

be an irrational positive number, and let

m

be a positive integer. A pair of

(a,b)

of positive integers is called good if

a \left \lceil b\nu \right \rceil - b \left \lfloor a \nu \right \rfloor = m.

A good pair

(a,b)

is called excellent if neither of the pair

(a-b,b)

and

(a,b-a)

is good.
Prove that the number of excellent pairs is equal to the sum of the positive divisors of

m

.

N6

1

IMO Shortlist 2013, Number Theory #6

Determine all functions

f: \mathbb{Q} \rightarrow \mathbb{Z}

satisfying

f \left( \frac{f(x)+a} {b}\right) = f \left( \frac{x+a}{b} \right)

for all

x \in \mathbb{Q}

,

a \in \mathbb{Z}

, and

b \in \mathbb{Z}_{>0}

. (Here,

\mathbb{Z}_{>0}

denotes the set of positive integers.)

N4

1

IMO Shortlist 2013, Number Theory #4

Determine whether there exists an infinite sequence of nonzero digits

a_1 , a_2 , a_3 , \cdots

and a positive integer

N

such that for every integer

k > N

, the number

\overline{a_k a_{k-1}\cdots a_1 }

is a perfect square.

N3

1

IMO Shortlist 2013, Number Theory #3

Prove that there exist infinitely many positive integers

n

such that the largest prime divisor of

n^4 + n^2 + 1

is equal to the largest prime divisor of

(n+1)^4 + (n+1)^2 +1

.

N1

1

IMO Shortlist 2013, Number Theory #1

Let

\mathbb{Z} _{>0}

be the set of positive integers. Find all functions

f: \mathbb{Z} _{>0}\rightarrow \mathbb{Z} _{>0}

such that

m^2 + f(n) \mid mf(m) +n

for all positive integers

m

and

n

.

G5

1

IMO Shortlist 2013, Geometry #5

Let

ABCDEF

be a convex hexagon with

AB=DE

,

BC=EF

,

CD=FA

, and

\angle A-\angle D = \angle C -\angle F = \angle E -\angle B

. Prove that the diagonals

AD

,

BE

, and

CF

are concurrent.

G4

1

IMO Shortlist 2013, Geometry #4

Let

ABC

be a triangle with

\angle B > \angle C

. Let

P

and

Q

be two different points on line

AC

such that

\angle PBA = \angle QBA = \angle ACB

and

A

is located between

P

and

C

. Suppose that there exists an interior point

D

of segment

BQ

for which

PD=PB

. Let the ray

AD

intersect the circle

ABC

at

R \neq A

. Prove that

QB = QR

.

G3

1

IMO Shortlist 2013, Geometry #3

In a triangle

ABC

, let

D

and

E

be the feet of the angle bisectors of angles

A

and

B

, respectively. A rhombus is inscribed into the quadrilateral

AEDB

(all vertices of the rhombus lie on different sides of

AEDB

). Let

\varphi

be the non-obtuse angle of the rhombus. Prove that

\varphi \le \max \{ \angle BAC, \angle ABC \}

.

G2

1

IMO Shortlist 2013, Geometry #2

Let

\omega

be the circumcircle of a triangle

ABC

. Denote by

M

and

N

the midpoints of the sides

AB

and

AC

, respectively, and denote by

T

the midpoint of the arc

BC

of

\omega

not containing

A

. The circumcircles of the triangles

AMT

and

ANT

intersect the perpendicular bisectors of

AC

and

AB

at points

X

and

Y

, respectively; assume that

X

and

Y

lie inside the triangle

ABC

. The lines

MN

and

XY

intersect at

K

. Prove that

KA=KT

.

C8

1

IMO Shortlist 2013, Combinatorics #8

Players

A

and

B

play a "paintful" game on the real line. Player

A

has a pot of paint with four units of black ink. A quantity

p

of this ink suffices to blacken a (closed) real interval of length

p

. In every round, player

A

picks some positive integer

m

and provides

1/2^m

units of ink from the pot. Player

B

then picks an integer

k

and blackens the interval from

k/2^m

to

(k+1)/2^m

(some parts of this interval may have been blackened before). The goal of player

A

is to reach a situation where the pot is empty and the interval

[0,1]

is not completely blackened. Decide whether there exists a strategy for player

A

to win in a finite number of moves.

C6

1

IMO Shortlist 2013, Combinatorics #6

In some country several pairs of cities are connected by direct two-way flights. It is possible to go from any city to any other by a sequence of flights. The distance between two cities is defined to be the least possible numbers of flights required to go from one of them to the other. It is known that for any city there are at most

100

cities at distance exactly three from it. Prove that there is no city such that more than

2550

other cities have distance exactly four from it.

C5

1

IMO Shortlist 2013, Combinatorics #5

Let

r

be a positive integer, and let

a_0 , a_1 , \cdots

be an infinite sequence of real numbers. Assume that for all nonnegative integers

m

and

s

there exists a positive integer

n \in [m+1, m+r]

such that

a_m + a_{m+1} +\cdots +a_{m+s} = a_n + a_{n+1} +\cdots +a_{n+s}

Prove that the sequence is periodic, i.e. there exists some

p \ge 1

such that

a_{n+p} =a_n

for all

n \ge 0

.

C4

1

IMO Shortlist 2013, Combinatorics #4

Let

n

be a positive integer, and let

A

be a subset of

\{ 1,\cdots ,n\}

. An

A

-partition of

n

into

k

parts is a representation of n as a sum

n = a_1 + \cdots + a_k

, where the parts

a_1 , \cdots , a_k

belong to

A

and are not necessarily distinct. The number of different parts in such a partition is the number of (distinct) elements in the set

\{ a_1 , a_2 , \cdots , a_k \}

. We say that an

A

-partition of

n

into

k

parts is optimal if there is no

A

-partition of

n

into

r

parts with

r<k

. Prove that any optimal

A

-partition of

n

contains at most

\sqrt[3]{6n}

different parts.

C3

1

IMO Shortlist 2013, Combinatorics #3

A crazy physicist discovered a new kind of particle wich he called an imon, after some of them mysteriously appeared in his lab. Some pairs of imons in the lab can be entangled, and each imon can participate in many entanglement relations. The physicist has found a way to perform the following two kinds of operations with these particles, one operation at a time. (i) If some imon is entangled with an odd number of other imons in the lab, then the physicist can destroy it. (ii) At any moment, he may double the whole family of imons in the lab by creating a copy

I'

of each imon

I

. During this procedure, the two copies

I'

and

J'

become entangled if and only if the original imons

I

and

J

are entangled, and each copy

I'

becomes entangled with its original imon

I

; no other entanglements occur or disappear at this moment.
Prove that the physicist may apply a sequence of such operations resulting in a family of imons, no two of which are entangled.

C1

1

IMO Shortlist 2013, Combinatorics #1

Let

n

be an positive integer. Find the smallest integer

k

with the following property; Given any real numbers

a_1 , \cdots , a_d

such that

a_1 + a_2 + \cdots + a_d = n

and

0 \le a_i \le 1

for

i=1,2,\cdots ,d

, it is possible to partition these numbers into

k

groups (some of which may be empty) such that the sum of the numbers in each group is at most

1

.

A6

1

IMO Shortlist 2013, Algebra 6

Let

m \neq 0

be an integer. Find all polynomials

P(x)

with real coefficients such that

(x^3 - mx^2 +1 ) P(x+1) + (x^3+mx^2+1) P(x-1) =2(x^3 - mx +1 ) P(x)

for all real number

x

.

A5

1

IMO Shortlist 2013, Algebra #5

Let

\mathbb{Z}_{\ge 0}

be the set of all nonnegative integers. Find all the functions

f: \mathbb{Z}_{\ge 0} \rightarrow \mathbb{Z}_{\ge 0}

satisfying the relation

f(f(f(n))) = f(n+1 ) +1

for all

n\in \mathbb{Z}_{\ge 0}

.

A4

1

IMO Shortlist 2013, Algebra #4

Let

n

be a positive integer, and consider a sequence

a_1 , a_2 , \dotsc , a_n

of positive integers. Extend it periodically to an infinite sequence

a_1 , a_2 , \dotsc

by defining

a_{n+i} = a_i

for all

i \ge 1

. If

a_1 \le a_2 \le \dots \le a_n \le a_1 +n

and a_{a_i } \le n+i-1 \text{for} i=1,2,\dotsc, n, prove that

a_1 + \dots +a_n \le n^2.

A2

1

IMO Shortlist 2013, Algebra #2

Prove that in any set of

2000

distinct real numbers there exist two pairs

a>b

and

c>d

with

a \neq c

or

b \neq d

, such that

\left| \frac{a-b}{c-d} - 1 \right|< \frac{1}{100000}.

A1

1

IMO Shortlist 2013, Algebra #1

Let

n

be a positive integer and let

a_1, \ldots, a_{n-1}

be arbitrary real numbers. Define the sequences

u_0, \ldots, u_n

and

v_0, \ldots, v_n

inductively by

u_0 = u_1 = v_0 = v_1 = 1

, and

u_{k+1} = u_k + a_k u_{k-1}

,

v_{k+1} = v_k + a_{n-k} v_{k-1}

for

k=1, \ldots, n-1.

Prove that

u_n = v_n.

2013 IMO Shortlist

Subcontests

IMO Shortlist 2013, Number Theory #7

IMO Shortlist 2013, Number Theory #6

IMO Shortlist 2013, Number Theory #4

IMO Shortlist 2013, Number Theory #3

IMO Shortlist 2013, Number Theory #1

IMO Shortlist 2013, Geometry #5

IMO Shortlist 2013, Geometry #4

IMO Shortlist 2013, Geometry #3

IMO Shortlist 2013, Geometry #2

IMO Shortlist 2013, Combinatorics #8

IMO Shortlist 2013, Combinatorics #6

IMO Shortlist 2013, Combinatorics #5

IMO Shortlist 2013, Combinatorics #4

IMO Shortlist 2013, Combinatorics #3

IMO Shortlist 2013, Combinatorics #1

IMO Shortlist 2013, Algebra 6

IMO Shortlist 2013, Algebra #5

IMO Shortlist 2013, Algebra #4

IMO Shortlist 2013, Algebra #2

IMO Shortlist 2013, Algebra #1