Nguyễn Quang Hưng
(sonnet)
New Member
Các bạn trẻ khối Tóan-Tin và các lớp khác thử sức đề thi toán QT năm nay ở Hy Lạp. Chúc các tài tử Ams học nhẹ nhàng chơi là chính giải được ít nhất 3 bài trong vòng 8h - không cần qua các lò luyện gà chọi như của Tổng Hợp vs Sư Phạm - có huy chương đem về treo tường làm kỷ niệm 
=;
Paper 1 (12 July 2004)
1. Let ABC be an acute-angled triangle with AB ≠ AC. The circle with diameter BC intersects the sides AB and AC at M and N respectively. Denote by O the midpoint of the side BC. The bisectors of the angles ∠BAC and ∠MON intersect at R. Prove that the circumcircles of the triangles BMR and CNR have a common point lying on the side BC.
2. Find all polynomials f with real coefficients such that for all reals a, b, c such that ab+bc+ca = 0 we have the following relation:
f(a-b) + f(b-c) + f(c-a) = 2f(a+b+c).
3. Define a “hook” to be a figure made up of six unit squares as shown below in the picture, or any of the figures obtained by applying rotations and reflections to this figure.
|------|------|------|
|------|------|------|
|------|>- |------|
|------|
Determine all m × n rectangles that can be covered without gaps and without overlaps with hooks such that
Paper 2 (13 July 2004)
4. Let n ≥ 3 be an integer. Let t_1, t_2, ..., t_n be positive real numbers such that
n^2 + 1 > (t_1 + t_2 + ... + t_n) (1/t_1 + 1/t_2 + ... + 1/t_n).
Show that t_i, t_j, t_k are side lengths of a triangle for all i, j, k with 1 ≤ i < j < k ≤ n.
5. In a convex quadrilateral ABCD the diagonal BD bisects neither the angle ABC nor the angle CDA. A point P lies inside ABCD and satisfies
∠PBC = ∠DBA and ∠PDC = ∠BDA.
Prove that ABCD is a cyclic quadrilateral if and only if AP = CP.
6. We call a positive integer alternating if every two consecutive digits in its decimal representation are of different parity.
Find all positive integers n such that n has a multiple which is alternating.
=; Paper 1 (12 July 2004)
1. Let ABC be an acute-angled triangle with AB ≠ AC. The circle with diameter BC intersects the sides AB and AC at M and N respectively. Denote by O the midpoint of the side BC. The bisectors of the angles ∠BAC and ∠MON intersect at R. Prove that the circumcircles of the triangles BMR and CNR have a common point lying on the side BC.
2. Find all polynomials f with real coefficients such that for all reals a, b, c such that ab+bc+ca = 0 we have the following relation:
f(a-b) + f(b-c) + f(c-a) = 2f(a+b+c).
3. Define a “hook” to be a figure made up of six unit squares as shown below in the picture, or any of the figures obtained by applying rotations and reflections to this figure.
|------|------|------|
|------|------|------|
|------|
>- |------||------|
Determine all m × n rectangles that can be covered without gaps and without overlaps with hooks such that
the rectangle is covered without gaps and without overlaps
no part of a hook covers area outside the rectangle.
Paper 2 (13 July 2004)
4. Let n ≥ 3 be an integer. Let t_1, t_2, ..., t_n be positive real numbers such that
n^2 + 1 > (t_1 + t_2 + ... + t_n) (1/t_1 + 1/t_2 + ... + 1/t_n).
Show that t_i, t_j, t_k are side lengths of a triangle for all i, j, k with 1 ≤ i < j < k ≤ n.
5. In a convex quadrilateral ABCD the diagonal BD bisects neither the angle ABC nor the angle CDA. A point P lies inside ABCD and satisfies
∠PBC = ∠DBA and ∠PDC = ∠BDA.
Prove that ABCD is a cyclic quadrilateral if and only if AP = CP.
6. We call a positive integer alternating if every two consecutive digits in its decimal representation are of different parity.
Find all positive integers n such that n has a multiple which is alternating.
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