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the mathscope
All the best from
Vietnamese Problem Solving Journals
May 10, 2006
please download for free at our websites:
 & 

translated by Phạm Văn Thuận, Eckard Specht

Vol I, Problems in Mathematics Journal for the Youth

The Mathscope is a free problem resource selected from mathematical
problem solving journals in Vietnam. This freely accessible collection
is our effort to introduce elementary mathematics problems to foreign
friends for either recreational or professional use. We would like to give
you a new taste of Vietnamese mathematical culture. Whatever the purpose, we welcome suggestions and comments from you all. More communications can be addressed to Phạm Văn Thuận of Hanoi University,
at pvthuan@vnu.edu.vn
It’s now not too hard to find problems and solutions on the Internet due
to the increasing number of websites devoted to mathematical problem
solving. It is our hope that this collection saves you considerable time
searching the problems you really want. We intend to give an outline of
solutions to the problems in the future. Now enjoy these “cakes” from
Vietnam first.

Pham Van Thuan

1

The Mathscope

Vol I, Problems in Mathematics

153. 1 (Nguyễn Đông Yên) Prove that if y ≥ y 3 + x2 + |x| + 1, then x2 +
y 2 ≥ 1. Find all pairs of (x, y) such that the first inequality holds while
equality in the second one attains.
153. 2 (Tạ Văn Tự) Given natural numbers m, n, and a real number a > 1,
prove the inequality
2n

a m − 1 ≥ n(a

n+1
m

−a

n−1
m

).

153. 3 (Nguyễn Minh Đức) Prove that for each 0 < < 1, there exists a
natural number n0 such that the coefficients of the polynomial
(x + y)n (x2 − (2 − )xy + y 2 )
are all positive for each natural number n ≥ n0 .
235. 1 (Đặng Hùng Thắng) Given real numbers x, y, z such that
a + b = 6,
ax + by = 10,
ax2 + by 2 = 24,
ax3 + by 3 = 62,
determine ax4 + by 4 .
235. 2 (Hà Đức Vượng) Let ABC be a triangle, let D be a fixed point on
the opposite ray of ray BC. A variable ray Dx intersects the sides AB, AC at
E, F , respectively. Let M and N be the midpoints of BF , CE, respectively.
Prove that the line M N has a fixed point.
235. 3 (Đàm Văn Nhỉ) Find the maximum value of
a
b
c
d
+
+
+
,
bcd + 1 cda + 1 dab + 1 abc + 1
where a, b, c, d ∈ [0, 1].
235. 4 (Trần Nam Dũng) Let M be any point in the plane of an equilateral triangle ABC. Denote by x, y, z the distances from P to the vertices and
p, q, r the distances from M to the sides of the triangle. Prove that
1
p2 + q 2 + r2 ...
the mathscope
All the best from
Vietnamese Problem Solving Journals
May 10, 2006
please download for free at our websites:
www.math4u.de, & www.imo.org.yu
translated by Phạm Văn Thuận, Eckard Specht
Vol I, Problems in Mathematics Journal for the Youth
The Mathscope is a free problem resource selected from mathematical
problem solving journals in Vietnam. This freely accessible collection
is our effort to introduce elementary mathematics problems to foreign
friends for either recreational or professional use. We would like to give
you a new taste of Vietnamese mathematical culture. Whatever the pur-
pose, we welcome suggestions and comments from you all. More commu-
nications can be addressed to Phạm Văn Thuận of Hanoi University,
at pvthuan@vnu.edu.vn
It’s now not too hard to find problems and solutions on the Internet due
to the increasing number of websites devoted to mathematical problem
solving. It is our hope that this collection saves you considerable time
searching the problems you really want. We intend to give an outline of
solutions to the problems in the future. Now enjoy these “cakes” from
Vietnam first.
Pham Van Thuan
1
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