Các bài toán vô địch Bankan

Prove that ABCD is a square
and O its center. (Yugoslavia)
2. Let A = {A1, A2, . . . , Ak} be a collection of subsets of an n-element set
S. If for any two elements x, y ∈ S there is a subset Ai ∈ A containing
exactly one of the two elements x, y, prove that 2k ≥ n. (Yugoslavia)
3. Circles C1 and C2 touch each other externally at D, and touch a circle Γ
internally at B and C, respectively. Let A be an intersection point of Γ
and the common tangent to C1 and C2 at D. Lines AB and AC meet C1
and C2 again at K and L, respectively, and the line BC meets C1 again at
M and C2 again at N . Show that the lines AD,KM,LN are concurrent.
(Greece)
4. Determine all functions f : R→ R that satisfy
f(xf(x) + f(y)) = f(x)2 + y for all x, y. (Bulgaria)
15-th Balkan Mathematical Olympiad
Nicosia, Cyprus – May 5, 1998
1. Consider the finite sequence
[
k2
1998
]
, k = 1, 2, . . . , 1997. How many distinct
terms are there in this sequence? (Greece)
2. Let n ≥ 2 be an integer, and let 0 < a1 < a2 < · · · < a2n+1 be real
numbers. Prove the inequality
n
√
a1 − n√a2 + n√a3 − · · ·+√a2n+1 < n
√
a1 − a2 + a3 − · · ·+ a2n+1.
(Romania)
3. Let S denote the set of points inside or on the border of a triangle ABC,
without a fixed point T inside the triangle. Show that S can be partitioned
into disjoint closed segemnts. (Yugoslavia)
4. Prove that the equation y2 = x5 − 4 has no integer solutions. (Bulgaria)
16-th Balkan Mathematical Olympiad
Ohrid, Macedonia – May 8, 1999
1. Let D be the midpoint of the shorter arc BC of the circumcircle of an
acute-angled triangle ABC. The points symmetric to D with respect to
BC and the circumcenter are denoted by E and F , respectively. Let K
be the midpoint of EA.
(a) Prove that the circle passing through the midpoints of the sides of
4ABC also passes through K.
(b) The line through K and the midpoint of BC is perpendicular to AF .
2. Let p > 2 be a prime number with 3 | p− 2. Consider the set
S = {y2 − x3 − 1 | x, y ∈ Z, 0 ≤ x, y ≤ p− 1}.
Prove that at most p− 1 elements of S are divisible by p.
3. Let M,N,P be the orthogonal projections of the centroid G of an acute-
angled triangle ABC onto AB,BC,CA, respectively. Prove that
4
27
<
SMNP
SABC
≤ 1
4
.
4. Let 0 ≤ x0 ≤ x1 ≤ x2 ≤ · · · be a sequence of nonnegative integers such
that for every k ≥ 0 the number of terms of the sequence which do not
exceed k is finite, say yk. Prove that for all positive integers m,n,
n∑
i=0
xi +
m∑
j=0
yj ≥ (n+ 1)(m+ 1).
17-th Balkan Mathematical Olympiad
Chis¸inaˇu, Moldova – May 5, 2000
1. [BMO 1997#4] Determine all functions f : R→ R that satisfy
f(xf(x) + f(y)) = f(x)2 + y for all x, y. (Albania)
2. Let ABC be a scalene triangle and E be a point on the median AD. Point
F is the orthogonal projection of E onto BC. Let M be a point on the
segment EF , and N,P be the orthogonal projections of M onto AC and
AB respectively. Prove that the bisectors of the angles PMN and PEN
are parallel.
3. Find the maximal number of rectangles 1× 10√2 that can be cut off from
a rectangle 50×90 by using cuts parallel to the edges of the big rectangle.
(Yugoslavia)
4. A positive integer is a power if it is of the form ts for some integers t, s ≥ 2.
Prove that for any natural number n there exists a set A of positive integers
with the following properties:
(i) A has n elements;
(ii) Every element of A is a power;
(iii) For any 2 ≤ k ≤ n and any r1, . . . , rk ∈ A, r1 + · · ·+ rk
k
is a power.
18-th Balkan Mathematical Olympiad
Belgrade, Yugoslavia – May 5, 2001
1. Let n be a positive integer. Prove that if a, b are integers greater than 1
such that ab = 2n − 1, then the number ab − (a − b) − 1 is of the form
k · 22m, where k is odd and m a positive integer.
2. Prove that a convex pentagon that satisfies the following two conditions
must be regular:
(i) All its interior angles are equal;
(ii) The lengths of all its sides are rational numbers.
3. Let a, b, c be positive real numbers such that a+ b+ c ≥ abc. Prove that
a2 + b2 + c2 ≥ abc
√
3.
4. A cube of edge 3 is divided into 27 unit cube cells. One of these cells is
empty, while in the other cells there are unit cubes which are arbitrarily
denoted by 1, 2, . . . , 26. An legal move consists of moving a unit cube into
a neighboring empty cell (two cells are neighboring if they share a face).
Does there exist a finite sequence of legal moves after which any two cubes
denoted by k and 27− k (k = 1, 2, . . . , 13) will exchange their positions?
19-th Balkan Mathematical Olympiad
Antalya, Turkey – April 27, 2002
1. Points A1, A2, . . . , An (n ≥ 4), no three of which are collinear, are given
on the plane. Some pairs of distinct points among them are connected by
segments such that every point is connected to at least three other points.
Prove that there exist an integer k > 1 and distinct points X1, X2, . . . , X2k
from the set {A1, . . . , An} such that Xi is connected to Xi+1 for i =
1, 2, . . . , 2k, where X2k+1 ≡ X1.
2. The sequence (an) is defined by a1 = 20, a2 = 30 and an+2 = 3an+1 − an
for every n ≥ 1. Find all positive integers n for which 1 + 5anan+1 is a
perfect square.
3. Two circles with different radii intersect at A and B. Their common
tangents MN and ST touch the first circle at M and S and the second
circle at N and T . Show that the orthocenters of triangles AMN , AST ,
BMN , and BST are the vertices of a rectangle.
4. Determine all functions f : N→ N such that for all positive integers n
2n+ 2001 ≤ f(f(n)) + f(n) ≤ 2n+ 2002.
20-th Balkan Mathematical Olympiad
Tirana, Albania – May 4, 2003
1. Does there exist a set B of 4004 distinct natural numbers, such that for
any subset A of B containing 2003 elements, the sum of the elements of
A is not divisible by 2003? (FYR Macedonia)
2. Let ABC be a triangle with AB 6= AC. The tangent at A to the cir-
cumcircle of the triangle ABC meets the line BC at D. Let E and F
be the points on the perpendicular bisectors of the segments AB and AC
respectively, such that BE and CF are both perpendicular to BC. Prove
that the points D,E, and F are collinear. (Romania)
3. Find all functions f : Q→ R which satisfy the following conditions:
(i) f(x+ y)− yf(x)− xf(y) = f(x)f(y)− x− y + xy for all x, y ∈ Q;
(ii) f(x) = 2f(x+ 1) + 2 +X for all x ∈ Q;
(iii) f(1) + 1 > 0. (Cyprus)
4. Let m and n be coprime odd positive integers. A rectangle ABCD with
AB = m and AD = n is divided into mn unit squares. Let A1, A2, . . . , Ak
be the consecutive points of intersection of the diagonal AC with the sides
of the unit squares (where A1 = A and Ak = C). Prove that
k−1∑
j=1
(−1)j+1AjAj+1 =
√
m2 + n2
mn
. (Bulgaria)
21-st Balkan Mathematical Olympiad
Pleven, Bulgaria – May 7, 2004
1. A sequence of real numbers a0, a2, a2, . . . satisfies the condition
am+n + am−n −m+ n− 1 = a2m + a2n2
for all m,n ∈ N with m ≥ n. If a1 = 3, determine a2004. (Cyprus)
2. Find all solutions in the set of prime numbers of the equation
xy − yx = xy2 − 19.
(Albania)
3. Let O be an interior point of an acute-angled triangle ABC. The circles
centered at the midpoints of the sides of the triangle ABC and passing
through point O, meet in points K,L,M different from O. Prove that O
is the incenter of the triangle KLM if and only if O is the circumcenter
of the triangle ABC. (Romania)
4. A plane is divided into regions by a finite number of lines, no three of
which are concurrent. We call two regions neighboring if their common
boundary is either a segment, a ray, or a line. One should write an integer
in each of the regions so as to fulfil the following two conditions:
(a) The product of the numbers from two neighboring regions is less than
their sum;
(b) The sum of all the numbers in the halfplane determined by any of
the lines is equal to zero.
Prove that this can be done if and only if not all the lines are parallel.
(Serbia and Montenegro)
22-nd Balkan Mathematical Olympiad
Ias¸i, Romania – May 6, 2005
1. The incircle of an acute-angled triangle ABC touches AB at D and AC
at E. Let the bisectors of the angles ∠ACB and ∠ABC intersect the line
DE at X and Y respectively, and let Z be the midpoint of BC. Prove
that the triangle XY Z is equilateral if and only if ∠A = 60◦. (Bulgaria)
2. Find all primes p such that p2 − p+ 1 is a perfect cube. (Albania)
3. If a, b, c are positive real numbers, prove the inequality
a2
b
+
b2
c
+
c2
a
≥ a+ b+ c+ 4(a− b)
2
a+ b+ c
.
When does equality occur? (Serbia and Montenegro)
4. Let n ≥ 2 be an integer, and let S be a subset of {1, 2, . . . , n} such that S
neither contains two coprime elements, nor does it contain two elements,
one of which divides the other. What is the maximum possible number of
elements of S? (Romania)
23-rd Balkan Mathematical Olympiad
Agros, Cyprus – April 29, 2006
1. If a, b, c are positive numbers, prove the inequality
1
a(1 + b)
+
1
b(1 + c)
+
1
c(1 + a)
≥ 3
1 + abc
.
2. A line m intersects the sides AB, AC and the extension of BC beyond
C of the triangle ABC at points D,F,E, respectively. The lines through
points A,B,C which are parallel to m meet the circumcircle of triangle
ABC again at points A1, B1, C1, respectively. Show that the lines A1E,
B1F , C1D are concurrent.
3. Determine all triples (m,n, p) of positive rational numbers such that the
numbers
m+
1
np
, n+
1
pm
, p+
1
mn
are integers.
4. Given a positive integer m, consider the sequence (an) of positive integers
defined by the initial term a0 = a and the recurrent relation
an+1 =
{ an
2
if an is even,
an +m if an is odd.
Find all values of a for which this sequence is periodic (i.e. there exists
d > 0 such that an+d = an for all n).
24-th Balkan Mathematical Olympiad
Rhodes, Greece – April 28, 2007
1. In a convex quadrilateral ABCD with AB = BC = CD, the diagonals
AC and BD are of different length and intersect at