MathDB

1

Minimum value of sum over sequence elements

n \geq 2

determine the minimum value that the sum \sum^n_{i\equal{}0} a_i can take for nonnegative numbers

a_0, a_1, \ldots, a_n

satisfying the condition a_0 \equal{} 1, a_i \leq a_{i\plus{}1} \plus{} a_{i\plus{}2} for i \equal{} 0, \ldots, n \minus{} 2.

18

1

Circumcircle of triangle PQR passes through midpoint of BC

The altitudes through the vertices

A,B,C

of an acute-angled triangle

ABC

meet the opposite sides at

D,E, F,

respectively. The line through

D

parallel to

EF

meets the lines

AC

and

AB

at

Q

and

R,

respectively. The line

EF

meets

BC

at

P.

Prove that the circumcircle of the triangle

PQR

passes through the midpoint of

BC.

16

1

Collinear iff the points P, Q, and O are collinear

In an acute-angled triangle

ABC,

let

AD,BE

be altitudes and

AP,BQ

internal bisectors. Denote by

I

and

O

the incenter and the circumcentre of the triangle, respectively. Prove that the points

D, E,

and

I

are collinear if and only if the points

P, Q,

and

O

are collinear.

13

1

Girls from town A can dance with r boys from their own town

In town

A,

there are

n

girls and

n

boys, and each girl knows each boy. In town

B,

there are

n

girls

g_1, g_2, \ldots, g_n

and 2n \minus{} 1 boys b_1, b_2, \ldots, b_{2n\minus{}1}. The girl

g_i,

i \equal{} 1, 2, \ldots, n, knows the boys b_1, b_2, \ldots, b_{2i\minus{}1}, and no others. For all r \equal{} 1, 2, \ldots, n, denote by

A(r),B(r)

the number of different ways in which

r

girls from town

A,

respectively town

B,

can dance with

r

boys from their own town, forming

r

pairs, each girl with a boy she knows. Prove that A(r) \equal{} B(r) for each r \equal{} 1, 2, \ldots, n.

11

1

(1 + x)^n*P(x) is polynomial with nonnegative coefficients

Let

P(x)

be a polynomial with real coefficients such that

P(x) > 0

for all

x \geq 0.

Prove that there exists a positive integer n such that (1 \plus{} x)^n \cdot P(x) is a polynomial with nonnegative coefficients.

9

1

Prove that the circumcentres of the triangles are collinear

Let

A_1A_2A_3

be a non-isosceles triangle with incenter

I.

Let

C_i,

i \equal{} 1, 2, 3, be the smaller circle through

I

tangent to A_iA_{i\plus{}1} and A_iA_{i\plus{}2} (the addition of indices being mod 3). Let B_i, i \equal{} 1, 2, 3, be the second point of intersection of C_{i\plus{}1} and C_{i\plus{}2}. Prove that the circumcentres of the triangles

A_1 B_1I,A_2B_2I,A_3B_3I

are collinear.

6

1

Prove that there exist infinitely many triples (x, y, z)

(a) Let

n

be a positive integer. Prove that there exist distinct positive integers

x, y, z

such that x^{n\minus{}1} \plus{} y^n \equal{} z^{n\plus{}1}. (b) Let

a, b, c

be positive integers such that

a

and

b

are relatively prime and

c

is relatively prime either to

a

or to

b.

Prove that there exist infinitely many triples

(x, y, z)

of distinct positive integers

x, y, z

such that x^a \plus{} y^b \equal{} z^c.

5

1

Regular tetrahedron and M,N distinct points in the planes

Let

ABCD

be a regular tetrahedron and

M,N

distinct points in the planes

ABC

and

ADC

respectively. Show that the segments

MN,BN,MD

are the sides of a triangle.

3

1

Sets of 4 and 5 vectors that have 8 and 10 maximal subsets

For each finite set

U

of nonzero vectors in the plane we define

l(U)

to be the length of the vector that is the sum of all vectors in

U.

Given a finite set

V

of nonzero vectors in the plane, a subset

B

of

V

is said to be maximal if

l(B)

is greater than or equal to

l(A)

for each nonempty subset

A

of

V.

(a) Construct sets of 4 and 5 vectors that have 8 and 10 maximal subsets respectively. (b) Show that, for any set

V

consisting of

n \geq 1

vectors the number of maximal subsets is less than or equal to

2n.

2

1

kth term from the left in R_n is equal to 1

Let

R_1,R_2, \ldots

be the family of finite sequences of positive integers defined by the following rules: R_1 \equal{} (1), and if R_{n - 1} \equal{} (x_1, \ldots, x_s), then
R_n \equal{} (1, 2, \ldots, x_1, 1, 2, \ldots, x_2, \ldots, 1, 2, \ldots, x_s, n).
For example, R_2 \equal{} (1, 2), R_3 \equal{} (1, 1, 2, 3), R_4 \equal{} (1, 1, 1, 2, 1, 2, 3, 4). Prove that if

n > 1,

then the

k

th term from the left in

R_n

is equal to 1 if and only if the

k

th term from the right in

R_n

is different from 1.

10

1

Polynomial

Find all positive integers

k

for which the following statement is true: If

F(x)

is a polynomial with integer coefficients satisfying the condition

0 \leq F(c) \leq k

for each c\in \{0,1,\ldots,k \plus{} 1\}, then F(0) \equal{} F(1) \equal{} \ldots \equal{} F(k \plus{} 1).

12

1

Prove that d >= p-1

Let

p

be a prime number and

f

an integer polynomial of degree

d

such that

f(0) = 0,f(1) = 1

and

f(n)

is congruent to

0

or

1

modulo

p

for every integer

n

. Prove that

d\geq p - 1

.

23

1

Trig condition for cyclic ABCD

Let

ABCD

be a convex quadrilateral. The diagonals

AC

and

BD

intersect at

K

. Show that

ABCD

is cyclic if and only if AK \sin A \plus{} CK \sin C \equal{} BK \sin B \plus{} DK \sin D.

25

1

three lines through midpoints of arcs

Let

X,Y,Z

be the midpoints of the small arcs

BC,CA,AB

respectively (arcs of the circumcircle of

ABC

).

M

is an arbitrary point on

BC

, and the parallels through

M

to the internal bisectors of

\angle B,\angle C

cut the external bisectors of

\angle C,\angle B

in

N,P

respectively. Show that

XM,YN,ZP

concur.

19

1

Square roots

Let a_1\geq \cdots \geq a_n \geq a_{n \plus{} 1} \equal{} 0 be real numbers. Show that \sqrt {\sum_{k \equal{} 1}^n a_k} \leq \sum_{k \equal{} 1}^n \sqrt k (\sqrt {a_k} \minus{} \sqrt {a_{k \plus{} 1}}). Proposed by Romania

14

1

Same divisor

Let

b, m, n

be positive integers such that

b > 1

and

m \neq n.

Prove that if b^m \minus{} 1 and b^n \minus{} 1 have the same prime divisors, then b \plus{} 1 is a power of 2.

20

1

ISL Shortlist 1997 Hard geometry

A quick solution: Let R be the foot of the perpend. from X to BC. Let's assume Q and R are in the interior of the segms AC and BC (respectively) and P in the ext of AD. P, R, Q are colinear (Simson's thm). PQ tangent to circle XRD iff XRQ=XDR iff Pi-XCA=XDR iff XBA=XDR=XDC=ADB iff XBC+ABC=ADB=DAC+ACB iff XAC+ABC=DAC+ACD iff ABC=ACD=ACB iff AB=AC. It's the same for all the other cases.

22

1

Functional equation, IMO 1997 Shortlist

Does there exist functions

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

such that f(g(x)) \equal{} x^2 and g(f(x)) \equal{} x^k for all real numbers

x

a) if k \equal{} 3? b) if k \equal{} 4?

15

1

ISL Shortlist 1997 - Arithmetic progression

An infinite arithmetic progression whose terms are positive integers contains the square of an integer and the cube of an integer. Show that it contains the sixth power of an integer.

7

1

Problem from IMO Jury [hexagon inequality]

The lengths of the sides of a convex hexagon

ABCDEF

satisfy AB \equal{} BC, CD \equal{} DE, EF \equal{} FA. Prove that: \frac {BC}{BE} \plus{} \frac {DE}{DA} \plus{} \frac {FA}{FC} \geq \frac {3}{2}.

1997 IMO Shortlist

Subcontests

Minimum value of sum over sequence elements

Circumcircle of triangle PQR passes through midpoint of BC

Collinear iff the points P, Q, and O are collinear

Girls from town A can dance with r boys from their own town

(1 + x)^n*P(x) is polynomial with nonnegative coefficients

Prove that the circumcentres of the triangles are collinear

Prove that there exist infinitely many triples (x, y, z)

Regular tetrahedron and M,N distinct points in the planes

Sets of 4 and 5 vectors that have 8 and 10 maximal subsets

kth term from the left in R_n is equal to 1

Polynomial

Prove that d >= p-1

Trig condition for cyclic ABCD

three lines through midpoints of arcs

Square roots

Same divisor

ISL Shortlist 1997 Hard geometry

Functional equation, IMO 1997 Shortlist

ISL Shortlist 1997 - Arithmetic progression

Problem from IMO Jury [hexagon inequality]

1997 IMO Shortlist

Subcontests

Minimum value of sum over sequence elements

Circumcircle of triangle PQR passes through midpoint of BC

Collinear iff the points P, Q, and O are collinear

Girls from town A can dance with r boys from their own town

(1 + x)^n*P(x) is polynomial with nonnegative coefficients

Prove that the circumcentres of the triangles are collinear

Prove that there exist infinitely many triples (x, y, z)

Regular tetrahedron and M,N distinct points in the planes

Sets of 4 and 5 vectors that have 8 and 10 maximal subsets

kth term from the left in R_n is equal to 1

Polynomial

Prove that d &gt;= p-1

Trig condition for cyclic ABCD

three lines through midpoints of arcs

Square roots

Same divisor

ISL Shortlist 1997 Hard geometry

Functional equation, IMO 1997 Shortlist

ISL Shortlist 1997 - Arithmetic progression

Problem from IMO Jury [hexagon inequality]

Prove that d >= p-1