MathDB

1

2^k divides a_n = 2*a_(n-1) + a_(n-2)

An integer sequence is defined by { a_n = 2 a_{n-1} + a_{n-2}}, (n > 1), a_0 = 0, a_1 = 1. Prove that

2^k

divides

a_n

if and only if

2^k

divides

n

.

31

1

After a break they sit again around the circular table

Around a circular table an even number of persons have a discussion. After a break they sit again around the circular table in a different order. Prove that there are at least two people such that the number of participants sitting between them before and after a break is the same.

30

1

USS cot triangle equality [equal-inradius cevian length]

A point

M

is chosen on the side

AC

of the triangle

ABC

in such a way that the radii of the circles inscribed in the triangles

ABM

and

BMC

are equal. Prove that BM^{2} \equal{} X \cot \left( \frac {B}{2}\right) where X is the area of triangle

ABC.

29

1

American one-way railroad track

A number of signal lights are equally spaced along a one-way railroad track, labeled in oder

1,2, \ldots, N, N \geq 2.

As a safety rule, a train is not allowed to pass a signal if any other train is in motion on the length of track between it and the following signal. However, there is no limit to the number of trains that can be parked motionless at a signal, one behind the other. (Assume the trains have zero length.) A series of

K

freight trains must be driven from Signal 1 to Signal

N.

Each train travels at a distinct but constant spped at all times when it is not blocked by the safety rule. Show that, regardless of the order in which the trains are arranged, the same time will elapse between the first train's departure from Signal 1 and the last train's arrival at Signal

N.

28

1

Prove that terms of UK sequence are odd

The sequence

\{a_n\}

of integers is defined by a_1 \equal{} 2, a_2 \equal{} 7 and \minus{} \frac {1}{2} < a_{n \plus{} 1} \minus{} \frac {a^2_n}{a_{n \minus{} 1}} \leq \frac {}{}, n \geq 2. Prove that

a_n

is odd for all

n > 1.

25

1

Block of digits does not commence with 0

A positive integer is called a double number if its decimal representation consists of a block of digits, not commencing with 0, followed immediately by an identical block. So, for instance, 360360 is a double number, but 36036 is not. Show that there are infinitely many double numbers which are perfect squares.

24

1

Swedish shortlisted inequality

Let

\{a_k\}^{\infty}_1

be a sequence of non-negative real numbers such that: a_k \minus{} 2 a_{k \plus{} 1} \plus{} a_{k \plus{} 2} \geq 0 and \sum^k_{j \equal{} 1} a_j \leq 1 for all k \equal{} 1,2, \ldots. Prove that: 0 \leq a_{k} \minus{} a_{k \plus{} 1} < \frac {2}{k^2} for all k \equal{} 1,2, \ldots.

23

1

Geometric Singapore Equality

Let

Q

be the centre of the inscribed circle of a triangle

ABC.

Prove that for any point

P,

a(PA)^2 \plus{} b(PB)^2 \plus{} c(PC)^2 \equal{} a(QA)^2 \plus{} b(QB)^2 \plus{} c(QC)^2 \plus{} (a \plus{} b \plus{} c)(QP)^2, where a \equal{} BC, b \equal{} CA and c \equal{} AB.

22

1

Product of two consecutive integers greater than 2

Let

p

be the product of two consecutive integers greater than 2. Show that there are no integers

x_1, x_2, \ldots, x_p

satisfying the equation \sum^p_{i \equal{} 1} x^2_i \minus{} \frac {4}{4 \cdot p \plus{} 1} \left( \sum^p_{i \equal{} 1} x_i \right)^2 \equal{} 1 OR Show that there are only two values of

p

for which there are integers

x_1, x_2, \ldots, x_p

satisfying \sum^p_{i \equal{} 1} x^2_i \minus{} \frac {4}{4 \cdot p \plus{} 1} \left( \sum^p_{i \equal{} 1} x_i \right)^2 \equal{} 1

21

1

Who remembers IMO 2001?

Forty-nine students solve a set of 3 problems. The score for each problem is a whole number of points from 0 to 7. Prove that there exist two students

A

and

B

such that, for each problem,

A

will score at least as many points as

B.

20

1

Two non-intersecting subsets

Find the least natural number

n

such that, if the set

\{1,2, \ldots, n\}

is arbitrarily divided into two non-intersecting subsets, then one of the subsets contains 3 distinct numbers such that the product of two of them equals the third.

19

1

Function defined on the set of all positive integers

Let

f(n)

be a function defined on the set of all positive integers and having its values in the same set. Suppose that f(f(n) \plus{} f(m)) \equal{} m \plus{} n for all positive integers

n,m.

Find the possible value for

f(1988).

17

1

IMO ShortList 1988 regular pentagon

In the convex pentagon

ABCDE,

the sides

BC, CD, DE

are equal. Moreover each diagonal of the pentagon is parallel to a side (

AC

is parallel to

DE

,

BD

is parallel to

AE

etc.). Prove that

ABCDE

is a regular pentagon.

15

1

Constructed through the vertices A,B and C

Let

ABC

be an acute-angled triangle. The lines

L_{A}

,

L_{B}

and

L_{C}

are constructed through the vertices

A

,

B

and

C

respectively according the following prescription: Let

H

be the foot of the altitude drawn from the vertex

A

to the side

BC

; let

S_{A}

be the circle with diameter

AH

; let

S_{A}

meet the sides

AB

and

AC

at

M

and

N

respectively, where

M

and

N

are distinct from

A

; then let

L_{A}

be the line through

A

perpendicular to

MN

. The lines

L_{B}

and

L_{C}

are constructed similarly. Prove that the lines

L_{A}

,

L_{B}

and

L_{C}

are concurrent.

14

1

Sums obtained by summing the elements of the rows/columns

For what values of

n

does there exist an

n \times n

array of entries -1, 0 or 1 such that the

2 \cdot n

sums obtained by summing the elements of the rows and the columns are all different?

12

1

IMO ShortList 1988, Geometric Inequality

In a triangle

ABC,

choose any points

K \in BC, L \in AC, M \in AB, N \in LM, R \in MK

and

F \in KL.

If

E_1, E_2, E_3, E_4, E_5, E_6

and

E

denote the areas of the triangles

AMR, CKR, BKF, ALF, BNM, CLN

and

ABC

respectively, show that

E \geq 8 \cdot \sqrt [6]{E_1 E_2 E_3 E_4 E_5 E_6}.

11

1

Lock of a safe consists of 3 wheels

The lock of a safe consists of 3 wheels, each of which may be set in 8 different ways positions. Due to a defect in the safe mechanism the door will open if any two of the three wheels are in the correct position. What is the smallest number of combinations which must be tried if one is to guarantee being able to open the safe (assuming the "right combination" is not known)?

10

1

Simultaneously separating and covering

Let N \equal{} \{1,2 \ldots, n\}, n \geq 2. A collection F \equal{} \{A_1, \ldots, A_t\} of subsets

A_i \subseteq N,

i \equal{} 1, \ldots, t, is said to be separating, if for every pair

\{x,y\} \subseteq N,

there is a set

A_i \in F

so that

A_i \cap \{x,y\}

contains just one element.

F

is said to be covering, if every element of

N

is contained in at least one set

A_i \in F.

What is the smallest value

f(n)

of

t,

so there is a set F \equal{} \{A_1, \ldots, A_t\} which is simultaneously separating and covering?

8

1

Vectors with zero sum in the plane

Let

u_1, u_2, \ldots, u_m

be

m

vectors in the plane, each of length

\leq 1,

with zero sum. Show that one can arrange

u_1, u_2, \ldots, u_m

as a sequence

v_1, v_2, \ldots, v_m

such that each partial sum v_1, v_1 \plus{} v_2, v_1 \plus{} v_2 \plus{} v_3, \ldots, v_1, v_2, \ldots, v_m has length less than or equal to

\sqrt {5}.

7

1

[a^1788] and [a^1988] divisible by 17

Let

a

be the greatest positive root of the equation x^3 \minus{} 3 \cdot x^2 \plus{} 1 \equal{} 0. Show that

\left[a^{1788} \right]

and

\left[a^{1988} \right]

are both divisible by 17. Here

[x]

denotes the integer part of

x.

6

1

Tedrahedron ABCD

In a given tedrahedron

ABCD

let

K

and

L

be the centres of edges

AB

and

CD

respectively. Prove that every plane that contains the line

KL

divides the tedrahedron into two parts of equal volume.

4

1

n x n numbered chessboard

An

n \times n, n \geq 2

chessboard is numbered by the numbers

1, 2, \ldots, n^2

(and every number occurs). Prove that there exist two neighbouring (with common edge) squares such that their numbers differ by at least

n.

3

1

Interior bisectors of the angles A,B and C

The triangle

ABC

is inscribed in a circle. The interior bisectors of the angles

A,B

and

C

meet the circle again at

A', B'

and

C'

respectively. Prove that the area of triangle

A'B'C'

is greater than or equal to the area of triangle

ABC.

2

1

(x^2 + x + 1)^n

Let

n

be a positive integer. Find the number of odd coefficients of the polynomial u_n(x) \equal{} (x^2 \plus{} x \plus{} 1)^n.

27

1

Geometric inequality 2

Let

ABC

be an acute-angled triangle. Let

L

be any line in the plane of the triangle

ABC

. Denote by

u

,

v

,

w

the lengths of the perpendiculars to

L

from

A

,

B

,

C

respectively. Prove the inequality u^2\cdot\tan A \plus{} v^2\cdot\tan B \plus{} w^2\cdot\tan C\geq 2\cdot S, where

S

is the area of the triangle

ABC

. Determine the lines

L

for which equality holds.

1988 IMO Shortlist

Subcontests

2^k divides a_n = 2*a_(n-1) + a_(n-2)

After a break they sit again around the circular table

USS cot triangle equality [equal-inradius cevian length]

American one-way railroad track

Prove that terms of UK sequence are odd

Block of digits does not commence with 0

Swedish shortlisted inequality

Geometric Singapore Equality

Product of two consecutive integers greater than 2

Who remembers IMO 2001?

Two non-intersecting subsets

Function defined on the set of all positive integers

IMO ShortList 1988 regular pentagon

Constructed through the vertices A,B and C

Sums obtained by summing the elements of the rows/columns

IMO ShortList 1988, Geometric Inequality

Lock of a safe consists of 3 wheels

Simultaneously separating and covering

Vectors with zero sum in the plane

[a^1788] and [a^1988] divisible by 17

Tedrahedron ABCD

n x n numbered chessboard

Interior bisectors of the angles A,B and C

(x^2 + x + 1)^n

Geometric inequality 2