MathDB

1

functional with f (mn) = f (m) f (n) / f (d)

f: N \rightarrow N

such that:

\bullet

f (m) = 1 \iff m = 1

;

\bullet

If

d = \gcd (m, n)

, then

f (mn) = \frac {f (m) f (n)} {f (d)}

; and

\bullet

\forall m \in N

, we have

f ^ {2012} (m) = m

.
Clarification:

f^n (a) = f (f^{n-1} (a))

4

1

PO = HQ wanted, <BAC = 60^o, circumcenter, orthocenter

Given a

\vartriangle ABC

with

AB> AC

and

\angle BAC = 60^o

. Denote the circumcenter and orthocenter as

O

and

H

respectively. We also have that

OH

intersects

AB

in

P

and

AC

in

Q

. Prove that

PO = HQ

.

5

1

min no of pair s(i, j) \in S_n x S_n , 1 <| x_i-x_j | <2

Let

n \in N

. Let's define

S_n = \{1, ..., n \}

. Let

x_1 <x_2 <\cdots <x_n

be any real. Determine the largest possible number of pairs

(i, j) \in S_n \times S_n

with

i \not = j

, for which it is true that

1 <| x_i-x_j | <2

and justify why said value cannot be higher.

3

1

<BPC=? equilateral ABC, PA ^2 = PB ^2 + PC ^2

In the interior of an equilateral triangle

ABC

a point

P

is chosen such that

PA ^2 = PB ^2 + PC ^2

. Find the measure of

\angle BPC

.

1

numbers around a circle

A circle is divided into

n

equal parts. Marceline sets out to assign whole numbers from

1

to

n

to each of these pieces so that the distance between two consecutive numbers is always the same. The numbers

887

,

217

and

1556

occupy consecutive positions. How many parts was the circumference divided into?

OIFMAT II 2012

Subcontests

functional with f (mn) = f (m) f (n) / f (d)

PO = HQ wanted, <BAC = 60^o, circumcenter, orthocenter

min no of pair s(i, j) \in S_n x S_n , 1 <| x_i-x_j | <2

<BPC=? equilateral ABC, PA ^2 = PB ^2 + PC ^2

numbers around a circle

OIFMAT II 2012

Subcontests

functional with f (mn) = f (m) f (n) / f (d)

PO = HQ wanted, &lt;BAC = 60^o, circumcenter, orthocenter

min no of pair s(i, j) \in S_n x S_n , 1 &lt;| x_i-x_j | &lt;2

&lt;BPC=? equilateral ABC, PA ^2 = PB ^2 + PC ^2

numbers around a circle

PO = HQ wanted, <BAC = 60^o, circumcenter, orthocenter

min no of pair s(i, j) \in S_n x S_n , 1 <| x_i-x_j | <2

<BPC=? equilateral ABC, PA ^2 = PB ^2 + PC ^2