Tam gi6c SS,A, vu6ng tai S, c6 dulng cao S,A
n€n SA.SA, = SSi . Tuong t.u
SB.SBI =SSf ; SC.SC, =SSf ; SD.SDI=SSi. X6t cilc tam giitc S.4B vd SB.A, ta co
sA.sA, =sB.s4 (=ssl) =#=ff "u g6c o dinh
S chung n6n hai tam gi6c ndy d6ng dang. Suy ra BA sA sA'sB 4!4 6o do BA=B.A..SB's4.
Br4 =sB, =sBsB, = ss; r sl
sum of any 3 consecutive numbers on any of these
strings is always positive. Prove that
'(3\^ '"'f (#l 'Yn23'
\L)
trlrohlern T1Il.{75. Find all functions /:]k-+1k such *", r(*). f@Ot)=f(t<*t), rorevery x>y>o.
Fr$hlenr "{121475. Given a triangle ABC. Tl'rc incircle (I) of ABC is tangent to BC, CA, and, AB at D, E, and F
respectively. Let B, Gesp. C,) be the intersection between the lines which go through AB and DE (resp.
AC and Dfl. Let H and K respectively be the
orthocenter of ABC ard ABp1. Prove that the line which goes through IIK contains the point L
Translated by NGUYEN PHU HOANG LAN
(College of Science-Vietnam National (Jniversity, Hanoi)
t6n tpi di6m Mthu6c dudng trdn ngoai ti€p th gi6c
sao cho u (eCno) = -1 " trong d6 M (ACBD) = -r
ld ki hi6u cho chtm MA,MC,MB,MD ld chirm tli6u hda; hodc ta c6 thil ttinh nghia "Ttir gidc nQi ti€p ABCD duqc gpi ld diiu hda n€u AB.CD = AD.BC "
hay " T* gidc diiu hda ld tu gidc nQi ti€p c6 tich cdc cfip canh aAi ai4n bdng nhau". Tri d6 chimg minh m6t
sO tinh ch6t co bdn cua tfi giitc di6u hda xem nhu bo dd
l$i gini to6n. Ching h4n c6c tinh ch6t hay duoc sri dung sau:
"Trong ta giac n6i tt€p ABCD ta c6 AB.CD =
AD.BC khi vd chi khi cdc dadng phdn gidc ctia cdc
g6c BAD vd BCD cilng di qua mQt diAm t€n BD hodc AB.CD=AD.BC khi vd chi khi cdc dudng phdn gidc ctia cdc gdc fra vd fii di qua m6t
dtem lren AL".
"Trong tilr giac n6t ndp ABCD, n€u dudng cheo AC
kh6ng di qua tdm dadng trdn ngoai ti€p tr:r gidc thi AB.CD=AD.BC khi vd chi khi cdc ti€p tuy\n vcri
dadng trdn ngogi ti€p ta gidc tqi A vd C cdt nhau tr€n BD; n€u dtdng chdo BD kh6ng di qua tdm
dadng trdn ngoai fidp nir gidc thi AB.CD = \O.AC
khi vd chi khi cdc ti€p tuy€n vcri dadng trdn ngoqi ti€p ta gidc tqi B vd D cdt nhou tr€n AC'. Cicbqn
c6 th6 dtng c6c ki6n thric quen thudc cua s5ch giSo
khoa m6n To6n d6 chimg minh c6c tinh ch6t tren holc c6 th6 xem 6 tdi li6u "M6t s6 chuydn dA hinh hqc phing UOi auOn-g hgc sinh gi6i trung hoc ph6
th6ng" ci;atdc gi6D6 Thanh Son.
e-611'rk-'e1z*5ry#$$ ary
Tucnrg W BC =r,C,.ff, DA=D.4 P#