MathDB

1

inequalities with distances of G to sides of triangle

G

of a triangle

ABC

to its sides

a,b,c

are denoted

g_a,g_b,g_c

respectively. Let

r

be the inradius of the triangle. Prove that: a)

g_a,g_b,g_c \ge \frac{2}{3}r

b)

g_a+g_b+g_c \ge 3r

6

1

smallest no of parallel lines that divide a plane in N>1999 regions

A plane is divided into

N

regions by three families of parallel lines. No three lines pass through the same point. What is the smallest number of lines needed so that

N > 1999

?

4

1

smallest number at least 3 cards have equal sums of digits (100-999)

A box contains

900

cards, labeled from

100

to

999

. Cards are removed one at a time without replacement. What is the smallest number of cards that must be removed to guarantee that the labels of at least three removed cards have equal sums of digits?

3

1

removing tokens on a triangular board

A one player game is played on the triangular board shown on the picture. A token is placed on each circle. Each token is white on one side and black on the other. Initially, the token at one vertex of the triangle has the black side up, while the others have the white sides up. A move consists of removing a token with the black side up and turning over the adjacent tokens (two tokens are adjacent if they are joined by a segment). Is it possible to remove all the tokens by a sequence of moves? https://cdn.artofproblemsolving.com/attachments/d/2/aabf82a0ddd6907482f27e6e0f1e1b56cd931d.png

2

1

a_1^2+a_2^2+...+a_n^2 a perfect square

Prove that there exists a sequence of positive integers

a_1,a_2,a_3, ...

such that

a_1^2+a_2^2+...+a_n^2

is a perfect square for all positive integers

n

.

1

area of a triangle with two vertices in parabola

The lines

t

and

t'

, tangent to the parabola

y = x^2

at points

A

and

B

respectively, intersect at point

C

. The median of triangle

ABC

from

C

has length

m

. Find the area of

\triangle ABC

in terms of

m

.

1999 Spain Mathematical Olympiad

Subcontests

inequalities with distances of G to sides of triangle

smallest no of parallel lines that divide a plane in N>1999 regions

smallest number at least 3 cards have equal sums of digits (100-999)

removing tokens on a triangular board

a_1^2+a_2^2+...+a_n^2 a perfect square

area of a triangle with two vertices in parabola

1999 Spain Mathematical Olympiad

Subcontests

inequalities with distances of G to sides of triangle

smallest no of parallel lines that divide a plane in N&gt;1999 regions

smallest number at least 3 cards have equal sums of digits (100-999)

removing tokens on a triangular board

a_1^2+a_2^2+...+a_n^2 a perfect square

area of a triangle with two vertices in parabola

smallest no of parallel lines that divide a plane in N>1999 regions