50-th Belarusian Mathematical Olympiad 2000: Final Round

50-th Belarusian Mathematical Olympiad 2000
Final Round
Category D
First Day
1. Find all pairs of integers (x, y) satisfying 3xy − x − 2y = 8.

2. Points M and K are marked on the sides BC and CD of a square ABCD, respec-
tively. The segments MD and BK intersect at P. Prove that AP ⊥ MK if and only
if MC = KD.
3. The roots of the quadratic equation ax2 − 4bx + 4c = 0 (where a > 0) all lie in
the segment [2, 3]. Prove that:
(a) a ≤ b ≤ c < a + b;
a b c
(b) + > .
a+c b+a c+b
4. Cross-shaped tiles are to be placed on a 8 × 8 square grid without overlap-

ping. Find the largest possible number of tiles that can be placed.
Second Day
5. On a mathematical olympiad, problem 5 for category D was worth 4 points. It

turned out after the olympiad that the number of students who scored 3 points
on this problem was equal to the number of those who scored 2 points on the
problem. Each student scored at least 1 point on this problem. Given that the
total number of points gained on this problem was 30 greater than the number of
students, find the number of students who scored at least 3 points.
6. Consider the function f (x) = {x} + {1/x}.
(a) Prove that f (x) < 1.5 for x > 0 and f (x) < 2 for x < 0.
1
(b) Prove that for all n ∈ N there exists x0 such that f (x0 ) > 2 − .
n
7. On the side AB of a triangle ABC with BC < AC < AB, points B1 and C2 are
marked so that AC2 = AC and BB1 = BC. Points B2 on side AC and C1 on the
extension of CB are marked so that CB2 = CB and CC1 = CA. Prove that the
lines C1C2 and B1 B2 are parallel.
8. Seven points are given on a plane, no three of which lie on a line. Any two of
these points are joined by a segment. Is it possible to color these segments by
several colors in such a way that, for each color, there are exactly three segments
of that color and these three segments form a triangle?
The IMO Compendium Group,

D. Djukić, V. Janković, I. Matić, N. Petrović
www.imomath.com
Category C
First Day
1. Find all pairs of integers (x, y) satisfying the equality
y(x2 + 36) + x(y2 − 36) + y2(y − 12) = 0.
2. In a triangle ABC with a right angle at C, the altitude CD intersects the angle
bisector AE at F. Lines ED and BF meet at G. Prove that the area of the
quadrilateral CEGF is equal to the area of the triangle BDF.
3. A set S consists of k sequences of 0, 1, 2 of length n. For any two sequences
ai + bi + 1
(ai ), (bi ) ∈ S we can construct a new sequence (ci ) such that ci =
2
and include it in S. Assume that after performing finitely many such operations
we obtain all the 3n sequences of 0, 1, 2 of length n. Find the smallest possible
value of k.
4. Cross-shaped tiles are to be placed on a 9 × 9 square grid without

overlapping. Find the largest possible number of tiles that can be placed.
Second Day
5. Find the number of pairs of positive integers (p, q) such that the roots of the
equation x2 − px − q = 0 do not exceed 10.
6. The equilateral triangles ABF and CAG are constructed in the exterior of a right-
angled triangle ABC with ∠C = 90◦ . Let M be the midpoint of BC. Given that
MF = 11 and MG = 7, find the length of BC.
7. Tom and Jerry play the following game. They alternately put pawns onto empty
cells of a 20 × 20 square board. Tom plays first. A player wins if after his move
some four pawns are at the vertices of a rectangle with sides parallel to the sides
of the board. Determine who of them has a winning strategy.
8. Suppose that real numbers a, b, c, d satisfy

a b b c c d d a a c b d ac bd
+ + + + + + + = + + + + + + 2.
b a c b d c a d c a d b bd ac
Prove that at least two of the numbers are equal.
Category B
First Day

www.imomath.com
1. Find the locus of points M on the Cartesian plane such that the tangents from M
to the parabola y = x2 are perpendicular.
2. Find all pairs of positive integers (m, n) such that
(m − n)2 (n2 − m) = 4m2 n.
3. The diagonals of a convex quadrilateral ABCD intersect at M. The bisector of

∠ACD intersects the extension of BA over A at K. Prove that if MA · MC + MA ·
CD = MB · MD, then ∠BKC = ∠CDB.
4. An equilateral triangle of side n is divided into n2 equilateral triangles of side
1. Each of the vertices of the small triangles is labelled with number 1, except
for one that is labelled with -1. Per move one can choose a line containing a
side of a small triangle and change the signs of the numbers at points on this
line. Determine all initial positions (the value of n and the position of the -1) for
which one can achieve that all the labels equal 1 using the described operations.
Second Day
5. Tom and Jerry play the following game. They alternately put pawns onto empty
cells of a 25 × 25 square board. Tom plays first. A player wins if after his move
some four pawns are at the vertices of a rectangle with sides parallel to the sides
of the board. Determine who of them has a winning strategy.
6. A rectangle ABCD and a point X are given on plane.
(a) Prove that among the segments XA, XB, XC, XD, some three are sides of a
triangle.
(b) Does (a) necessarily hold if ABCD is a parallelogram?
a
7. Find all positive integers a and b such that aa = bb .
8. A set R of nonzero vectors on a plane is called concordant if it satisfies the
following conditions:
(i) For any vectors a, b ∈ R (may be equal) the vector Sb (a) symmetric to a
with respect to the line perpendicular to b belongs to R;
(ii) For any a, b ∈ R there exists an integer k such that a − Sb(a) = kb.
(a) Prove that for any two non-parallel and non-perpendicular vectors a, b ∈ R,
either a − b or a + b is in R.
(b) Does there exist an infinite concordant set? Find the largest possible cardi-
nality of a finite concordant set.
Category A
First Day

www.imomath.com
1. Pit and Bill play the following game. First Pit writes down a number a, then Bill
writes a number b, then Pit writes a number c. Can Pit always play so that the
three equations
x3 + ax2 + bx + c = 0, x3 + bx2 + cx + a = 0, x3 + cx2 + ax + b = 0
have: (a) a common real root; (b) a common negative root?

2. Find the number of pairs (n, q), where n is a positive integer
andq a non-integer
n!
rational number with 0 < q < 2000, that satisfy {q2 } = .
2000
3. Let N ≥ 5 be given. Consider all sequences (e1 , e2 , . . . , eN ) with each ei equal to
1 or −1. Per move one can choose any five consecutive terms and change their
signs. Two sequences are said to be similar if one of them can be transformed
into the other in finitely many moves. Find the maximum number of pairwise
non-similar sequences of length N.
4. The lateral sides and diagonals of a trapezoid intersect a line l, determining three
equal segments on it. Must l be parallel to the bases of the trapezoid?
Second Day
5. Nine points are given on a plane, no three of which lie on a line. Any two of
these points are joined by a segment. Is it possible to color these segments by
several colors in such a way that, for each color, there are exactly three segments
of that color and these three segments form a triangle?
6. A vertex of a tetrahedron is called perfect if the three edges at this vertex are
sides of a certain triangle. How many perfect vertices can a tetrahedron have?
7. (a) Find all positive integers n for which the equation (aa )n = bb has a solution
in positive integers a, b greater than 1.
(b) Find all positive integers a, b satisfying (aa )5 = bb .
8. To any triangle with side lengths a, b, c and the corresponding angles α , β , γ
(measured in radians), the 6-tuple (a, b, c, α , β , γ ) is assigned. Find the minimum
possible number n of distinct terms in 6-tuple assigned to a scalene triangle.

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50-th Belarusian Mathematical Olympiad 2000

1. Find all pairs of integers (x, y) satisfying 3xy − x − 2y = 8.

4. Cross-shaped tiles are to be placed on a 8 × 8 square grid without overlap-

5. On a mathematical olympiad, problem 5 for category D was worth 4 points. It

The IMO Compendium Group,

1. Find all pairs of integers (x, y) satisfying the equality

y(x2 + 36) + x(y2 − 36) + y2(y − 12) = 0.

4. Cross-shaped tiles are to be placed on a 9 × 9 square grid without

8. Suppose that real numbers a, b, c, d satisfy

The IMO Compendium Group,

3. The diagonals of a convex quadrilateral ABCD intersect at M. The bisector of

The IMO Compendium Group,

x3 + ax2 + bx + c = 0, x3 + bx2 + cx + a = 0, x3 + cx2 + ax + b = 0

have: (a) a common real root; (b) a common negative root?

The IMO Compendium Group,

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