Toán học, HSG Quốc gia bảng A, 2002-2003
ĐỀ THI HỌC SINH GIỎI QUỐC GIA
Môn
thi:
TOÁN
(Bảng
A)
Năm
học:
2002-2003
A1. Let R be the reals and f: R ® R a function such that f( cot x ) = cos 2x + sin 2x for all 0 < x < p. Define g(x) = f(x) f(1-x) for -1 <= x <= 1. Find the maximum and minimum values of f on the closed interval [-1, 1].
A2. The circles C1 and C2 touch externally at M and the radius of C2 is larger than that of C1. A is any point on C2 which does not lie on the line joining the centers of the circles. B and C are points on C1 such that AB and AC are tangent to C1. The lines BM, CM intersect C2 again at E, F respectively. D is the intersection of the tangent at A and the line EF. Show that the locus of D as A varies is a straight line.
A3. Let Sn be the number of permutations (a1, a2, ... , an) of (1, 2, ... , n) such that 1 <= |ak - k | <= 2 for all k. Show that ¾ Sn-1 < Sn < 2 Sn-1 for n > 6.
B1.
Find
the
largest
positive
integer
n
such
that
the
following
equations
have
integer
solutions
in
x,
y1,
y2,
...
,
yn:
(x
+
1)2
+
y12
=
(x
+
2)2
+
y22
=
...
=
(x
+
n)2
+
yn2.
B2. Define p(x) = 4x3 - 2x2 - 15x + 9, q(x) = 12x3 + 6x2 - 7x + 1. Show that each polynomial has just three distinct real roots. Let A be the largest root of p(x) and B the largest root of q(x). Show that A2 + 3 B2 = 4.
B3. Let R+ be the set of positive reals and let F be the set of all functions f : R+ ® R+ such that f(3x) >= f( f(2x) ) + x for all x. Find the largest A such that f(x) >= A x for all f in F and all x in R+.
Download
bản
tiếng
Việt:
http://library.hn-ams.org/wiki/skins/common/images/icons/fileicon-pdf.png
