Matematické Fórum

stuart clark · 10. 05. 2012 18:50

How many triangles are there with integer sides and perimeter $2007$

How many of these are equilateral

How any are isosceles

How many are scalene, i.e., neither of the first two kinds?

vanok · 10. 06. 2012 14:47 — Editoval vanok (10. 06. 2012 16:05)

Hi ↑ stuart clark:,

Remark: we shall count the triangles of sides has, b, c that only once, about is their permutation of sides.
Thus let us put, without losing the generality, that a suitable triangle has the sides $a, b, c$ and that $a \geq b \geq c$

We know, thanks to the statement that $a+b+c=2007$ , where $a, b, c$ are positive, not zeros numbers with furthermore $b+c \geq a \geq b-c$

Let us notice first of $a\in \{669; 1003 \}$

Let us describe now, how form methodically such triangles: we form for one gave all the possible triangles, and we make this for everything has possible.

FORMATIONS OF ALL THE TRIANGLES FOR ONE LOOKED FIX $a$ :
We begin with the triangle,

$b =a; c=2007-2a$
then
$b=a-1; c=2008-2a$
$b=a-2; c=2009-2a$
until
$b=a-k; c=2007+k-2a$ (with k so big as possible).

So, we are on to have all the triangles of this type.
Let us show now 2 interesting properties.

PROPRETY 1:
By taking into account the condition $b \geq c$ , we have
$a - k \geq 2007+ k- 2a$
$3a \geq 2007+2k$
$k \leq \frac {3a-2007}2$

Let $K_a$ possible maximum of k.
$K_a = [\frac {3a-2007}2]$

In other words:

For $a$ odd $K_a = \frac {3a-2007}2$
and for $a$ even $K_a = \frac {3a-2006}2$

So for a looked we can form $T_a = K_a+ 1$ triangles

PROPRETY 2
$T_{a+2}-T_a=3$
(Thus $T_a; T_{a+2};...$ and $K_a; K_{a+2};...$ forms the arithmetic sequences)

Observation:
$T_{669}=1$
$T_{670} = 2$
$T_{671}=4$
…
$T_{1002}= 500$
$T_{1003}=502$
Conlusion
The sum of number of all suitable triangles is
$\sum_{a=669}^{1003} T_a$
But this is a sum of two aritmetical sequences
For a even
$T_{670}+ T_{672}... + T_{1002}= 2+ 5+... + 500= \frac {167 \cdot (2+502)}2= 42084$
For a odd
$T_{669}+T_{671}+...+ T_ {1003} =1+4+ ...+502= \frac{168 \cdot (1+502)}2= 42252$
Thus we have $84336$ of all suitable triangles.
Of which $1$ is equilateral
$334$ are isosceles (As shows it the methode used $1003- 669= 334$ )
And all the others $84098$ triangles are scalene.