a Find all distinct permutations of the word TOANHOCTUOITRE.. Prove that the line MN passes through a fixed point.. Important: Answer all 12 questions.. Enter your answers on the answer
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f7/ 405 OAB C is a trirectangular tetrah d
O 0 , '" t h ' e ron
at vertex n IS e altitude from 0 of
tetrahedron Let R be the circumradius of
tr!'angle ABC Prove that OJ{ ~ RJ2
''''-does equality occur?
2 "Ilen
f 8 / 4 05 a) Find all distinct permutations of
the word TOANHOCTUOITRE
b) How many permutations are there that has
three consecutive T - TIT?
c) How many permutations are there without
adjacent Ts?
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19 / 405 Let k be a positive integer, a is
arbitrary real number Find the limit
sequence (a.) where
does not exceed x TlO/40S Find all functions/: N' -+N' which satisfy the following conditions
i) f is strictly increasing;
ii) f(f(n» = 4n + 9 for all n e N';
iii) f(f(n)-n)=2n+9 for all ne N'
Tll/40S Does there exist a positive integer
n~2sothat
f(x) = I + 4x+ 4x' + + 4x",
is a perfect square polynomial?
Tl2l40S Let ABC be a triangle inscribed the circle (0) and A ' is a fixed point on (0) P
moves on BC, K belongs to AC so that PK is
always parallel to a fixed line d The
circumcircle of triangle APK cuts the circle
(0) at a second point E AE cuts BC at M A 'P
cuts tbe circle (0) at a second point N Prove that the line MN passes through a fixed point
Translated by LE MINH HA
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Important:
Answer all 12 questions
Enter your answers on the answer sheet provi
For the multiple choice questions, enter only
letters (A D C D or E) corresponding to
correct answers in the answer sheet
No calculators are allowed
'un "" F ,?ruan _ O 2011 U8M <
'Iultiple Choice Questions
Qt An integer is called "octal" ifit is divisible
by 8 or If at least one of its digits is 8 How
many mtegers between I and 100 are octal?
Q5 Let a b c be positive integers sucb
a+2b+3c=100 Find the greatest value
M=abc
Q6 Find all pairs (x, y) of real
satislYing the system
{
X+y=2
x' - y' =Sx-3y
100 such that 4a'+3a+5 isdivilibleby61 Q8 Find the minimum value of
280 ~~_~ O SuppoIe tbIt BD.CE
that ON.l BC.1'roYe AB.AC- 2HB.IIC
MIC.-A - 90', AS - c IIId AC - lAc P • fC