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Đề thi học sinh giỏi Toán toàn quốc năm
1998
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Dịch sang tiếng
Anh
| A1. Define
the sequence x1, x2,
x3, ... by x1 = a < 1,
xn+1 = 1 +
ln(xn2/(1 +
ln(xn) ). Show that the sequence
converges and find the limit. |
| A2. Let O be
the circumcenter of the tetrahedron ABCD. Let A',
B', C', D' be points on the circumsphere such that
AA', BB', CC' and DD' are diameters. Let A" be the
centroid of the triangle BCD. Define B", C", D"
similarly. Show that the lines A'A", B'B", C'C",
D'D" are concurrent. Suppose they meet at X. Show
that the line through X and the midpoint of AB is
perpendicular to CD. |
| A3. The
sequence a0, a1,
a2, ... is defined by a0=
20, a1 = 100, an+2 =
4an+1 + 5an + 20. Find the
smallest m such that am -
a0, am+1 - a1,
am+2 - a2, ... are all
divisible by 1998. |
| B1. Does
there exist an infinite real sequence
x1, x2, x3, ...
such that | xn | ≤ 0.666, and |
xm - xn | ≥ 1/(n2
+ n + m2 + m) for all distinct m, n?
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| B2. What is
the minimum value of √( (x+1)2 +
(y-1)2) + √( (x-1)2 +
(y+1)2) + √( (x+2)2 +
(y+2)2)? |
| B3. Find all
positive integers n for which there is a
polynomial p(x) with real coefficients such that
p(x1998 - x-1998) =
(xn - x-n) for all x.
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