MathDB

1

a_{2n}-ba_n is square, if a_n is positive integer with n digits all =1, b digit

n

and

b

with

0 < b < 10

such that if

a_n

is the positive integer with

n

digits, all of them

1

, then

a_{2n} - b a_n

is a square.

4

1

12 lines containing the sides of a cube meet at plane p in 12 points

C

is a cube side

1

. The

12

lines containing the sides of the cube meet at plane

p

in

12

points. What can you say about the

12

points?

2

1

(a) 9/n, 25/ (n+1), (b)12/n, 165/(n+1) , (c) 9/n, 25/(n+1), 4/(n+2)

i) How many integers

n

are there such that

n

is divisible by

9

and

n+1

is divisible by

25

? ii) How many integers

n

are there such that

n

is divisible by

21

and

n+1

is divisible by

165

? iii) How many integers

n

are there such that

n

is divisible by

9, n + 1

is divisible by

25

, and

n + 2

is divisible by

4

?

3

1

point on line connecting PQ, projections of H on AB,AC whereas AH altitude

Let

ABC

be a triangle with

A = 90^o, AH

the altitude,

P,Q

the feet of the perpendiculars from

H

to

AB,AC

respectively. Let

M

be a variable point on the line

PQ

. The line through

M

perpendicular to

MH

meets the lines

AB,AC

at

R, S

respectively. i) Prove that circumcircle of

ARS

always passes the fixed point

H

. ii) Let

M_1

be another position of

M

with corresponding points

R_1, S_1

. Prove that the ratio

RR_1/SS_1

is constant. iii) The point

K

is symmetric to

H

with respect to

M

. The line through

K

perpendicular to the line

PQ

meets the line

RS

at

D

. Prove that

\angle BHR = \angle DHR, \angle DHS = \angle CHS

.

1974 Vietnam National Olympiad

Subcontests

a_{2n}-ba_n is square, if a_n is positive integer with n digits all =1, b digit

12 lines containing the sides of a cube meet at plane p in 12 points

(a) 9/n, 25/ (n+1), (b)12/n, 165/(n+1) , (c) 9/n, 25/(n+1), 4/(n+2)

point on line connecting PQ, projections of H on AB,AC whereas AH altitude