MathDB

2

locus of the orthocenter of the touchpoints of the incircle

{ABC}

be a non-equilateral triangle. The incircle is tangent to the sides

{BC,CA,AB}

at

{A_1,B_1,C_1}

, respectively, and M is the orthocenter of triangle

{A_1B_1C_1}

. Prove that

{M}

lies on the line through the incenter and circumcenter of

{\vartriangle ABC}

.

kl positive integers

Let

k

and

l

be two given positive integers and

a_{ij}(1 \leq i \leq k, 1 \leq j \leq l)

be

kl

positive integers. Show that if

q \geq p > 0

, then

(\sum_{j=1}^{l}(\sum_{i=1}^{k}a_{ij}^{p})^{q/p})^{1/q}\leq (\sum_{i=1}^{k}(\sum_{j=1}^{l}a_{ij}^{q})^{p/q})^{1/p}.

2

binomial coeffcient and relation divisible

Prove or disprove: For any positive integer

k

there exists an integer

n > 1

such that the binomial coeffcient

\binom{n}{i}

is divisible by

k

for any

1 \leq i \leq n-1.

on $S = \{m^2 + dn^2 \}

For a given integer

d

, let us deﬁne

S = \{m^{2}+dn^{2}| m, n \in\mathbb{Z}\}

. Suppose that

p, q

are two elements of

S

, where

p

is prime and

p | q

. Prove that

r = q/p

also belongs to

S

.

1

2

$(A − A) \cap (B − B)$ is nonempty

Let

A

and

B

be two subsets of

S = \{1, 2, . . . , 2000\}

with

|A| \cdot |B| \geq 3999

. For a set

X

, let

X-X

denotes the set

\{s-t | s, t \in X, s \not = t\}

. Prove that

(A-A) \cap (B-B)

is nonempty.

partitions of $2000$ (in a sum of positive integers).

Let

S

be the set of all partitions of

2000

(in a sum of positive integers). For every such partition

p

, we deﬁne

f (p)

to be the sum of the number of summands in

p

and the maximal summand in

p

. Compute the minimum of

f (p)

when

p \in S .

2000 Hungary-Israel Binational

Subcontests

locus of the orthocenter of the touchpoints of the incircle

kl positive integers

binomial coeffcient and relation divisible

on $S = \{m^2 + dn^2 \}

$(A − A) \cap (B − B)$ is nonempty

partitions of $2000$ (in a sum of positive integers).