Točno
17. kolovoza 2012. 22:35 (11 godine, 11 mjeseci)
A tetrahedron is called a MEMO-tetrahedron if all six sidelengths are different positive integers where one of them is
and one of them is
. Let
be the sum of the sidelengths of the tetrahedron
.
(a) Find all positive integers
so that there exists a MEMO-Tetrahedron
with
.
(b) How many pairwise non-congruent MEMO-tetrahedrons
satisfying
exist? Two tetrahedrons are said to be non-congruent if one cannot be obtained from the other by a composition of reflections in planes, translations and rotations. (It is not neccessary to prove that the tetrahedrons are not degenerate, i.e. that they have a positive volume).
![2](/media/m/e/e/e/eeef773d19a3b3f7bdf4c64f501e0291.png)
![3](/media/m/b/8/2/b82f544df38f2ea97fa029fc3f9644e0.png)
![l(T)](/media/m/f/d/0/fd0afa964801afa9e13b14962101f607.png)
![T](/media/m/0/1/6/016d42c58f7f5f06bdf8af6b85141914.png)
(a) Find all positive integers
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
![T](/media/m/0/1/6/016d42c58f7f5f06bdf8af6b85141914.png)
![l(T)=n](/media/m/3/5/8/358aa5a62c575781e127d502ca2deef4.png)
(b) How many pairwise non-congruent MEMO-tetrahedrons
![T](/media/m/0/1/6/016d42c58f7f5f06bdf8af6b85141914.png)
![l(T)=2007](/media/m/6/1/5/615f6705dc2854fc198bf1a38ece7435.png)
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Promatrati cemo dva slucaja, kada su stranice duljina
I
u istome trokutu, i kada nisu.
U prvom slucaju imamo trokut sa stranicama
i
pa iz nejednakosti trokuta dobivamo da je treca stranica (nazovimo je
)
![a<2+3 \newline a<5 \newline a+2>3 \newline a>1](/media/m/8/d/2/8d226d9c08a8740f8f84b2f7e7bf4c47.png)
dakle![a=4](/media/m/1/1/8/11841b46a6d22b7277e6bdfcd0a3e4c7.png)
Neka je najmanja od preostale tri stranice tetraedra
. Postoje tri mogucnosti iz kojega ce vrha
izlaziti.
1. Ako postoji vrh sa stranicama
.
U trokutu sa stranicama
i
je treca stranica
.
U trokutu sa stranicama
i
treca stranica zato mora biti
.
Dakle takvi tetraedri mogu imati
jedino oblika ![3x+12](/media/m/f/4/2/f42c380f9ff220b2a0bcc3a3a8721fef.png)
2.Ako postoji vrh sa stranicama![x,2,4](/media/m/f/0/0/f00b9775ba71ff2d95880709da7a5191.png)
U trokutu sa stranicama
i
zadnja je stranica
.
U trokutu sa stranicama
i
posaljednja stranica sada moze biti
ili
.
Dakle takvi tetraedri mogu imati
oblika
ili
.
3. Ako postoji vrh![x,3,4](/media/m/e/b/4/eb42644fc00e91b5f0de65b14a06f040.png)
U trokutu sa stranicama
i
zadnja je stranica ili
ili
.
Neka je posljednja stranica u tetraedru
. U trokutu imamo
i
, pa znamo da
ili
.
Dakle takvi tetraedri mogu imati
oblika
ili oblika
na dva nekongruentna nacina.
Drugi slucaj:stranice nisu u istom trokutu.
Neka je od preostalih stranica najmanja
. U trokutu sa stranicama
i
, posljednja stranica mora biti
zbog nejednakosti trokuta. U trokutu sa stranicama
i
, posljednja stranica mora biti
, jer nemoze biti
zbog uvjeta da su stranice razlicite. U trokutu sa stranicama
i
zadnja je stranica
. Svi tetraedri u kojima
i
nisu u istom trokutu moraju biti ovoga oblika.
takvog tetraedra je oblika
, a broj
moguce je prikayati kao
. Najmanji
kojeg ovakav tetraedar moze poprimiti poprima se za
i jedank je
.
U prvom slucaju najmanji mogci
postize se za
i jednak je
. Svi brojevi veci ili jednaki
mogu se postici jer su zapisivi u obliku
ili
ili
.
U prvom slucaju imamo 4 nekongruentna nacina na koje mozemo napraviti tetraedar
takav da
, a moguce je i na jedan nacin iz drugog slucaja pa ukupno postoji
takvih nekongruentnih tetraedara.
![2](/media/m/e/e/e/eeef773d19a3b3f7bdf4c64f501e0291.png)
![3](/media/m/b/8/2/b82f544df38f2ea97fa029fc3f9644e0.png)
U prvom slucaju imamo trokut sa stranicama
![2](/media/m/e/e/e/eeef773d19a3b3f7bdf4c64f501e0291.png)
![3](/media/m/b/8/2/b82f544df38f2ea97fa029fc3f9644e0.png)
![a](/media/m/6/d/2/6d2832265560bb67cf117009608524f6.png)
![a<2+3 \newline a<5 \newline a+2>3 \newline a>1](/media/m/8/d/2/8d226d9c08a8740f8f84b2f7e7bf4c47.png)
dakle
![a=4](/media/m/1/1/8/11841b46a6d22b7277e6bdfcd0a3e4c7.png)
Neka je najmanja od preostale tri stranice tetraedra
![x](/media/m/f/1/8/f185adeed9bd346bc960bca0147d7aae.png)
![x](/media/m/f/1/8/f185adeed9bd346bc960bca0147d7aae.png)
1. Ako postoji vrh sa stranicama
![x,2,3](/media/m/8/d/8/8d8610cb7b1c825657f59cfd7119236b.png)
U trokutu sa stranicama
![x](/media/m/f/1/8/f185adeed9bd346bc960bca0147d7aae.png)
![2](/media/m/e/e/e/eeef773d19a3b3f7bdf4c64f501e0291.png)
![x+1](/media/m/d/e/0/de098a179aba20a11154296253a54cc2.png)
U trokutu sa stranicama
![x](/media/m/f/1/8/f185adeed9bd346bc960bca0147d7aae.png)
![3](/media/m/b/8/2/b82f544df38f2ea97fa029fc3f9644e0.png)
![x+2](/media/m/2/f/2/2f284faf7502063c7c185e1af9062d80.png)
Dakle takvi tetraedri mogu imati
![l](/media/m/e/e/9/ee975101080f37986f56baaf4c3cdcd2.png)
![3x+12](/media/m/f/4/2/f42c380f9ff220b2a0bcc3a3a8721fef.png)
2.Ako postoji vrh sa stranicama
![x,2,4](/media/m/f/0/0/f00b9775ba71ff2d95880709da7a5191.png)
U trokutu sa stranicama
![x](/media/m/f/1/8/f185adeed9bd346bc960bca0147d7aae.png)
![2](/media/m/e/e/e/eeef773d19a3b3f7bdf4c64f501e0291.png)
![x+1](/media/m/d/e/0/de098a179aba20a11154296253a54cc2.png)
U trokutu sa stranicama
![x](/media/m/f/1/8/f185adeed9bd346bc960bca0147d7aae.png)
![3](/media/m/b/8/2/b82f544df38f2ea97fa029fc3f9644e0.png)
![x+2](/media/m/2/f/2/2f284faf7502063c7c185e1af9062d80.png)
![x+3](/media/m/7/a/6/7a6ff6057dcf35b745646724680a6f0c.png)
Dakle takvi tetraedri mogu imati
![l](/media/m/e/e/9/ee975101080f37986f56baaf4c3cdcd2.png)
![3x+12](/media/m/f/4/2/f42c380f9ff220b2a0bcc3a3a8721fef.png)
![3x+13](/media/m/d/4/2/d42d3ae2dd5967b272457badd535bfaf.png)
3. Ako postoji vrh
![x,3,4](/media/m/e/b/4/eb42644fc00e91b5f0de65b14a06f040.png)
U trokutu sa stranicama
![x](/media/m/f/1/8/f185adeed9bd346bc960bca0147d7aae.png)
![3](/media/m/b/8/2/b82f544df38f2ea97fa029fc3f9644e0.png)
![x+1](/media/m/d/e/0/de098a179aba20a11154296253a54cc2.png)
![x+2](/media/m/2/f/2/2f284faf7502063c7c185e1af9062d80.png)
Neka je posljednja stranica u tetraedru
![k](/media/m/f/1/3/f135be660b73381aa6bec048f0f79afc.png)
![2](/media/m/e/e/e/eeef773d19a3b3f7bdf4c64f501e0291.png)
![k](/media/m/f/1/3/f135be660b73381aa6bec048f0f79afc.png)
![x+1 \Rightarrow k=x+2 \newline x+2 \Rightarrow k=x+3](/media/m/6/0/1/6012f4b66b47e2c8973f0ebeb7d8df48.png)
![k=x+1](/media/m/b/4/b/b4b03ea0971f3f4d72537d17da0d50e9.png)
Dakle takvi tetraedri mogu imati
![l](/media/m/e/e/9/ee975101080f37986f56baaf4c3cdcd2.png)
![3x+14](/media/m/f/f/b/ffbb79ecac7a2be45da6fcab8f962e18.png)
![3x+12](/media/m/f/4/2/f42c380f9ff220b2a0bcc3a3a8721fef.png)
Drugi slucaj:stranice nisu u istom trokutu.
Neka je od preostalih stranica najmanja
![x](/media/m/f/1/8/f185adeed9bd346bc960bca0147d7aae.png)
![x](/media/m/f/1/8/f185adeed9bd346bc960bca0147d7aae.png)
![2](/media/m/e/e/e/eeef773d19a3b3f7bdf4c64f501e0291.png)
![x+1](/media/m/d/e/0/de098a179aba20a11154296253a54cc2.png)
![x](/media/m/f/1/8/f185adeed9bd346bc960bca0147d7aae.png)
![3](/media/m/b/8/2/b82f544df38f2ea97fa029fc3f9644e0.png)
![x+2](/media/m/2/f/2/2f284faf7502063c7c185e1af9062d80.png)
![x+1](/media/m/d/e/0/de098a179aba20a11154296253a54cc2.png)
![x+1](/media/m/d/e/0/de098a179aba20a11154296253a54cc2.png)
![3](/media/m/b/8/2/b82f544df38f2ea97fa029fc3f9644e0.png)
![x+3](/media/m/7/a/6/7a6ff6057dcf35b745646724680a6f0c.png)
![2](/media/m/e/e/e/eeef773d19a3b3f7bdf4c64f501e0291.png)
![3](/media/m/b/8/2/b82f544df38f2ea97fa029fc3f9644e0.png)
![l](/media/m/e/e/9/ee975101080f37986f56baaf4c3cdcd2.png)
![4x + 11](/media/m/8/8/1/881da8943f8ec94cc780a8d463b85ac3.png)
![2007](/media/m/3/f/4/3f405241bc274df2b17a4f30ef472364.png)
![4x+11](/media/m/7/8/c/78c03c38faca1cd963c8643cdde98c2d.png)
![l](/media/m/e/e/9/ee975101080f37986f56baaf4c3cdcd2.png)
![x=4](/media/m/0/f/a/0fa2c13fe4e8bbf6560bb4e20facebaa.png)
![27](/media/m/6/b/c/6bc0c17b780852c2576fe3d7dffa8f61.png)
![(a)](/media/m/a/7/f/a7fedf50ce0b917a00dd07d5233906f1.png)
![l](/media/m/e/e/9/ee975101080f37986f56baaf4c3cdcd2.png)
![x=5](/media/m/d/8/0/d80130bcdb8e8b881432220ee53e3f47.png)
![27](/media/m/6/b/c/6bc0c17b780852c2576fe3d7dffa8f61.png)
![27](/media/m/6/b/c/6bc0c17b780852c2576fe3d7dffa8f61.png)
![3x+12](/media/m/f/4/2/f42c380f9ff220b2a0bcc3a3a8721fef.png)
![3x+13](/media/m/d/4/2/d42d3ae2dd5967b272457badd535bfaf.png)
![3x+14](/media/m/f/f/b/ffbb79ecac7a2be45da6fcab8f962e18.png)
![(b)](/media/m/9/2/7/92773ef234467079b4efc86655fdc459.png)
![T](/media/m/0/1/6/016d42c58f7f5f06bdf8af6b85141914.png)
![l(T)=2007](/media/m/6/1/5/615f6705dc2854fc198bf1a38ece7435.png)
![5](/media/m/e/a/3/ea36c795dac330f34d395d8364d379b6.png)