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Đề thi HSG Hà Nội (HOMO 2012)

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Hanoi Open Mathematical Olympiad

•••



HEXAGON®
inspiring minds always
Hanoi Mathematical Olympiad 2012
Senior Section

1. Let x =

√ √
√
6+2 √
5+ 6−2 5
.
20

√

√

2. Arrange the numbers p = 2

311

Find the value of (1 + x5 − x7 )2012
2,

1+ √1

q = 3, t = 2

2

in increasing order.

3. Let ABCD be a trapezoid with AD parallel to BC and BC = 3 cm, DA = 6 cm. Find the
length of the line segment EF parallel to the two bases and passing through the intersection of
the two diagonals AC, BD, E is on CD, F on AB.
√
√
3
3
4. What is the largest integer less than or equal to 4x3 −3x, where x = 21 ( 2 + 3+ 2 − 3).
5. Let f (x) be a function such that f (x) + 2f
of f (2012).

x+2010
x−1

= 4020 − x for all x = 1. Find the value

6. For every n = 2, 3, . . . , let
An =

1−

1
1+2

× 1−

1
1+2+3

Determine all positive integers n such that

1
An

× ··· × 1 −

1
1 + 2 + ··· + n

.

is an integer.

7. Prove that a = 1 . . . 1 5 . . . 5 6 is a perfect square.
2012

2011

8. Determine the greatest number m such that the system
x2 + y 2 = 1, |x3 − y 3 | + |x − y| = m3
has a solution.

9. Let P be the intersection of the three internal angle bisectors of a triangle ABC. The line
passing through P and perpendicular to CP intersects AC and BC at M, N respectively. If
AP = 3 cm, BP = 4 cm, find the value of AM/BN .
10. Suppose that the equation x3 + px2 + qx + 1√= 0, with p, q being some rational numbers, has
three real rooots x1 , x2 , x3 , where x3 = 2 + 5. Find the values of p, q.
11. Suppose that the equation x3 + px2 + qx + r = 0 has three real roots x1 , x2 , x3 where p, q, r
are integers. :et Sn = xn1 + xn2 + xn3 , for n = 1, 2, . . . ,. Prove that S2012 is an integer.
Copyright c 2011 HEXAGON

1

13. A cube with sides of length 3 cm is painted red and then cut into 3 × 3 × 3 = 27 cubes with
sides of length 1 cm. If a denotes the number of small cubes of side-length 1 cm that are not
painted at all, b the number of cubes painted on one side, c the number of cubes painted on two
sides, and d the number of cubes painted on three sides, determine the value of a − b − c + d.
14. Sovle the equation in the set of integers 16x + 1 = (x2 − y 2 )2 .
15. Determine the smallest value of the expression s = xy−yz−zx, where x, y, z are real numbers
satisfying the condition x2 + 2y 2 + 5z 2 = 22.

Hanoi Open Mathematical Olympiad

•••



12. Let M be a point on the side BC of an isosceles triangle ABC with BC = BA. Let O be the
circumcenter of ...
Hanoi Open Mathematical Olympiad www.hexagon.edu.vn
H
E
XAGO
inspiring minds always
Hanoi Mathematical Olympiad 2012
Senior Section
1. Let x =
6+2
5+
62
5
20
. Find the value of (1 + x
5
x
7
)
2012
311
2. Arrange the numbers p = 2
2
, q = 3, t = 2
1+
1
2
in increasing order.
3. Let ABCD be a trapezoid with AD parallel to BC and BC = 3 cm, DA = 6 cm. Find the
length of the line segment EF parallel to the two bases and passing through the intersection of
the two diagonals AC, BD, E is on CD, F on AB.
4. What is the largest integer less than or equal to 4x
3
3x, where x =
1
2
(
3
2 +
3+
3
2
3).
5. Let f(x) be a function such that f(x) + 2f
x+2010
x1
= 4020 x for all x = 1. Find the value
of f(2012).
6. For every n = 2, 3, . . . , let
A
n
=
1
1
1 + 2
×
1
1
1 + 2 + 3
× ··· ×
1
1
1 + 2 + ··· + n
.
Determine all positive integers n such that
1
A
n
is an integer.
7. Prove that a =
1 . . . 1

2012
5 . . . 5

2011
6 is a perfect square.
8. Determine the greatest number m such that the system
x
2
+ y
2
= 1, |x
3
y
3
| + |x y| = m
3
has a solution.
9. Let P be the intersection of the three internal angle bisectors of a triangle ABC. The line
passing through P and perpendicular to CP intersects AC and BC at M, N respectively. If
AP = 3 cm, BP = 4 cm, find the value of AM/BN .
10. Suppose that the equation x
3
+ px
2
+ qx + 1 = 0, with p, q being some rational numbers, has
three real rooots x
1
, x
2
, x
3
, where x
3
= 2 +
5. Find the values of p, q.
11. Suppose that the equation x
3
+ px
2
+ qx + r = 0 has three real roots x
1
, x
2
, x
3
where p, q, r
are integers. :et S
n
= x
n
1
+ x
n
2
+ x
n
3
, for n = 1, 2, . . . ,. Prove that S
2012
is an integer.
Copyright
c
2011 H
E
XAGON
1
Đề thi HSG Hà Nội (HOMO 2012) - Trang 2
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