- Ltri gicri c'tru tndi biri vid trAn mot td'giá riAng vt
Tinh; Gia l.ai: Nguy€n Anh Phtic, 10T2,
TIIPÍ chuy€n HLng Vuong; BGn Tret Cao
'l'hdnh Chuong, Vd At{inh T'rf, Vo Duy Thúc,
I Íl','flIP.f chuyOn BOn Trẹ
Moi cdc ban ti6p tuc guibhi cho dgt [I, III nhd!
TIITT
PROIILEMS.. . t.'[iop trang l7)
where x, ), are non-negative real numbers
suchthat-{+y= l.
T71398.I-et ABC be an acute trianglé Prove
that
cosA cosR cosC .- -
J--- +---RCCAAI] RCCAAI]
COS ---2- 2 COS - COS'-_ 2 2 COS.. COS 2 -.COS .2
T8/398. Let ABC be a trianglẹ A straight line cr:t the lines BC, CA and AB at Á, R'and C' respectivelỵ LeL A", B" and C" be the points reflection of Á, B'and Cl' with centers at A, B and C respectivelỵ Prove that the area of the
triangle A"B"C" is 6 times the area of
triangle ABC.
'l'( )wARDS Mr[IIIEMATICAL OLYMPIAI)
T9l398. The positive integers are colored
with either black or white such that the sum
of two numbers with different color is painted black, and there are infinitely many numbers
with white color. I-et q (q > I ) be the smallest positive integer with black color. Prove that q
is primẹ
T10/398. Find all functions /: N. -+ N. such
that
f(f'z(m)+zf'(n))=m'+2n2, for all ru,ne N..
T11/398. The sequence (xn) @ > 1) of real
numbers is defined inductively as follows:
xt=a (ae IR.) and ,{,,n1 =2x1,-5xl+4x,, for
all n > 1. Find all possible values of c such
that the sequence (-r, ) has finite limit.Determine the limit of (x,) with respect to Determine the limit of (x,) with respect to
each such value of ạ
Tl2l398.Let ABCD be a tetrahedron. Find all
points P inside the tetrahedron such that
xdơydưzdg+tdr:c' where x, i, z, t, c ate
given positive constants and do,du,dc,do
are respectively the distances from P to the
four faces IICD, CDA, DAB, ABC of the
tetrahedron.
Translated by LE MINFI HA
PHAN THANH QỊJANG (7'P. HO Ch( Minh)
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