Vietnam Team Selection Tests

y in each of the following cases:
a) n0 = 120;
b) n0 =
32002 − 1
2
;
c) n0 =
32002 + 1
2
.
3 Let m be a given positive integer which has a prime divisor greater than
√
2m+ 1. Find the
minimal positive integer n such that there exists a finite set S of distinct positive integers
satisfying the following two conditions:
I. m ≤ x ≤ n for all x ∈ S;
II. the product of all elements in S is the square of an integer.
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Team Selection Tests
2002
Day 2
1 Let n ≥ 2 be an integer and consider an array composed of n rows and 2n columns. Half of
the elements in the array are colored in red. Prove that for each integer k, 1 < k ≤
⌊n
2
⌋
+ 1,
there exist k rows such that the array of size k × 2n formed with these k rows has at least
k!(n− 2k + 2)
(n− k + 1)(n− k + 2) · · · (n− 1)
columns which contain only red cells.
2 Find all polynomials P (x) with integer coefficients such that the polynomial
Q(x) = (x2 + 6x+ 10) · P 2(x)− 1
is the square of a polynomial with integer coefficients.
3 Prove that there exists an integer n, n ≥ 2002, and n distinct positive integers a1, a2, . . . , an
such that the number N = a21a
2
2 · · · a2n − 4(a21 + a22 + · · ·+ a2n) is a perfect square.
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Team Selection Tests
2003
Day 1
1 Let be four positive integers m,n, p, q, with p < m given and q < n. Take four points
A(0; 0), B(p; 0), C(m; q) and D(m;n) in the coordinate plane. Consider the paths f from A
to D and the paths g from B to C such that when going along f or g, one goes only in
the positive directions of coordinates and one can only change directions (from the positive
direction of one axe coordinate into the the positive direction of the other axe coordinate) at
the points with integral coordinates. Let S be the number of couples (f, g) such that f and
g have no common points. Prove that
S =
(
n
m+ n
)
·
(
q
m+ q − p
)
−
(
q
m+ q
)
·
(
n
m+ n− p
)
.
2 Given a triangle ABC. Let O be the circumcenter of this triangle ABC. Let H, K, L be the
feet of the altitudes of triangle ABC from the vertices A, B, C, respectively. Denote by A0,
B0, C0 the midpoints of these altitudes AH, BK, CL, respectively. The incircle of triangle
ABC has center I and touches the sides BC, CA, AB at the points D, E, F , respectively.
Prove that the four lines A0D, B0E, C0F and OI are concurrent. (When the point O concides
with I, we consider the line OI as an arbitrary line passing through O.)
3 Let f(0, 0) = 52003, f(0, n) = 0 for every integer n 6= 0 and f(m,n) = f(m − 1, n) − 2 ·[
f(m− 1, n)
2
]
+
[
f(m− 1, n− 1)
2
]
+
[
f(m− 1, n+ 1)
2
]
for every natural number m > 0 and
for every integer n.
Prove that there exists natural number M such that f(M,n) = 1 for all integers n such that
|n| ≤ (5
2003 − 1)
2
and f(M,n) = 0 for all integers n such that |n| > 5
2003 − 1
2
. (Here [x]
denotes the integral part of real number x).
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Team Selection Tests
2003
Day 2
1 On the sides of triangle ABC take the pointsM1, N1, P1 such that each lineMM1, NN1, PP1
divides the perimeter of ABC in two equal parts (M,N,P are respectively the midpoints of
the sides BC,CA,AB).
I. Prove that the lines MM1, NN1, PP1 are concurrent at a point K. II. Prove that among
the ratios
KA
BC
,
KB
CA
,
KC
AB
there exist at least a ratio which is not less than
1√
3
.
2 Let A be the set of all permutations a = (a1, a2, . . . , a2003) of the 2003 first positive integers
such that each permutation satisfies the condition: there is no proper subset S of the set
{1, 2, . . . , 2003} such that {ak|k ∈ S} = S.
For each a = (a1, a2, . . . , a2003) ∈ A, let d(a) =
2003∑
k=1
(ak − k)2 .
I. Find the least value of d(a). Denote this least value by d0. II. Find all permutations a ∈ A
such that d(a) = d0.
3 Let n be a positive integer. Prove that the number 2n + 1 has no prime divisor of the form
8 · k − 1, where k is a positive integer.
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Team Selection Tests
2004
Day 1
1 Let us consider a set S = {a1 < a2 < . . . < a2004}, satisfying the following properties:
f(ai) < 2003 and f(ai) = f(aj) ∀i, j from {1, 2, . . . , 2004}, where f(ai) denotes number of
elements which are relatively prime with ai. Find the least positive integer k for which in
every k-subset of S, having the above mentioned properties there are two distinct elements
with greatest common divisor greater than 1.
2 Find all real values of α, for which there exists one and only one function f : R 7→ R and
satisfying the equation
f(x2 + y + f(y)) = (f(x))2 + α · y
for all x, y ∈ R.
3 In the plane, there are two circles Γ1,Γ2 intersecting each other at two points A and B.
Tangents of Γ1 at A and B meet each other at K. Let us consider an arbitrary point M
(which is different of A and B) on Γ1. The line MA meets Γ2 again at P . The line MK
meets Γ1 again at C. The line CA meets Γ2 again at Q. Show that the midpoint of PQ lies
on the line MC and the line PQ passes through a fixed point when M moves on Γ1.
{[Moderator edit: This problem was also discussed on 
.]
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Team Selection Tests
2004
Day 2
1 Let {xn}, with n = 1, 2, 3, . . ., be a sequence defined by x1 = 603, x2 = 102 and xn+2 =
xn+1 + xn + 2
√
xn+1 · xn − 2 ∀n ≥ 1. Show that:
(1) The number xn is a positive integer for every n ≥ 1.
(2) There are infinitely many positive integers n for which the decimal representation of xn
ends with 2003.
(3) There exists no positive integer n for which the decimal representation of xn ends with
2004.
2 Let us consider a convex hexagon ABCDEF. Let A1, B1, C1, D1, E1, F1 be midpoints of the
sides AB,BC,CD,DE,EF, FA respectively. Denote by p and p1, respectively, the perimeter
of the hexagon ABCDEF and hexagon A1B1C1D1E1F1. Suppose that all inner angles of
hexagon A1B1C1D1E1F1 are equal. Prove that
p ≥ 2 ·
√
3
3
· p1.
When does equality hold ?
3 Let S be the set of positive integers in which the greatest and smallest elements are relatively
prime. For natural n, let Sn denote the set of natural numbers which can be represented as
sum of at most n elements (not necessarily different) from S. Let a be greatest element from
S. Prove that there are positive integer k and integers b such that |Sn| = a · n + b for all
n > k.
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Team Selection Tests
2005
Day 1
1 Let (I), (O) be the incircle, and, respectiely, circumcircle of ABC. (I) touches BC,CA,AB
in D,E, F respectively. We are also given three circles ωa, ωb, ωc, tangent to (I), (O) in D,K
(for ωa), E,M (for ωb), and F,N (for ωc).
a) Show that DK,EM,FN are concurrent in a point P ;
b) Show that the orthocenter of DEF lies on OP .
2 Given n chairs around a circle which are marked with numbers from 1 to n .There are k,
k ≤ 4 · n students sitting on those chairs .Two students are called neighbours if there is no
student sitting between them. Between two neighbours students ,there are at less 3 chairs.
Find the number of choices of k chairs so that k students can sit on those and the condition
is satisfied.
3 Find all functions f : Z 7→ Z satisfying the condition: f(x3+y3+z3) = f(x)3+f(y)3+f(z)3.
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Team Selection Tests
2005
Day 2
1 Let be given positive reals a, b, c. Prove that:
a3
(a+ b)3
+
b3
(b+ c)3
+
c3
(c+ a)3
≥ 3
8
.
2 Let p ∈ P, p > 3. Calcute:
a)S =
p−1
2∑
k=1
[
2k2
p
]
− 2 ·
[
k2
p
]
if p ≡ 1 mod 4
b) T =
p−1
2∑
k=1
[
k2
p
]
if p ≡ 1 mod 8
3 n is called diamond 2005 if n = ...ab999...99999cd..., e.g. 2005×9. Let {an} : an < C ·n, {an}
is increasing. Prove that {an} contain infinite diamond 2005.
Compare with [url=]this problem.[/url]
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Team Selection Tests
2006
Day 1
1 Given an acute angles triangle ABC, and H is its orthocentre. The external bisector of the
angle ∠BHC meets the sides AB and AC at the points D and E respectively. The internal
bisector of the angle ∠BAC meets the circumcircle of the triangle ADE again at the point
K. Prove that HK is through the midpoint of the side BC.
2 Find all pair of integer numbers (n, k) such that n is not negative and k is greater than 1,
and satisfying that the number:
A = 172006n + 4.172n + 7.195n
can be represented as the product of k consecutive positive integers.
3 In the space, given 2006 distinct points such that no 4 of them are coplanar. One draws each
pair of points by a segment. A natural number m is called ”good” if one can put on each
of these segments a number not greater than m sothat every triangle whose three points are
in the 2006 points given has the following property: Two of this triangle’s sides are put two
equal numbers, and the other a greater number. Find the minimum value of the ”good”
number m.
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Team Selection Tests
2006
Day 2
1 Prove that for all real numbers x, y, z ∈ [1, 2] the following inequality always holds:
(x+ y + z)(
1
x
+
1
y