Romanian Olympiad 2004 Junior BMO Team Selection Tests 1st Test

Romanian Olympiad 2004
Junior BMO Team Selection Tests
1st Test - April 7, 2004
1. Find all positive reals a, b, c which fulfill the following relation
4(ab + bc + ca) − 1 ≥ a 2 + b 2 + c 2 ≥ 3(a 3 + b 3 + c 3 ).
Laurenţiu Panaitopol
2. For each positive integer n ≤ 49 we define the numbers a n = 3n + √ n 2 − 1 and
b n = 2( √ n 2 + n + √ n 2 − n). Prove that there exist two integers A, B such that
√
a1 − b 1 + √ a 2 − b 2 + · · · + √ a 49 − b 49 = A + B √ 2.
Titu Andreescu
3. Let V be a point in the exterior of a circle of center O, and let T 1 , T 2 be the points
where the tangents from V touch the circle. Let T be an arbitrary point on the small
arc T 1 T 2 . The tangent in T at the circle intersects the line V T 1 in A, and the lines
T T 1 and V T 2 intersect in B. We denote by M the intersection of the lines T T 1 and
AT 2 .
Prove that the lines OM and AB are perpendicular.
Mircea Fianu
4. In two vertices M, N of a cube the number 1 is written, and in all the other six vertices
the number 0 is written. A movement is defined by choosing a vertex and adding one
unit to the numbers written in the three adiacent vertices 1 .
Prove that there exists a finite set of movements such that in each of the 8 vertices
the same number is written if and only if MN is not a diagonal of a face of the cube.
Dinu Şerbănescu
Work time: 4 hours.
L A TEX(c) 2004 Valentin Vornicu - MathLinks.ro
1 a vertex is adiacent with another vertex if the segment determined by them is a side of the cube

2nd Test - May 1, 2004
1. Let ABC be a triangle, having no right angles, and let D be a point on the side BC.
Let E and F be the feet of the perpendicular lines drawn from D to AB and AC
respectively. Let P be the point of intersection between the lines BF and CE. Prove
that AP is the an altitude of the triangle ABC if and only if AD is an interior angle
bisector of the triangle ABC.
Generalization of JBkMO 2002
2. Let ABC be a triangle with side lengths a, b, c, such that a is the longest side. Prove
that ∠BAC = 90 ◦ if and only if
( √ a + b + √ a − b)( √ a + c + √ a − c) = (a + b + c) √ 2.
Virgil Nicula
3. Let A be a 8×8 array with entries from the set {−1, 1} such that any 2×2 sub-square
of the array has the absolute value of the sum of its element equal with 2. Prove that
the array must have at least two identical lines.
Marius Ghergu
4. Find all positive integers n for which there exist distinct positive integers a 1 , a 2 , . . . , a n
such that
1
+ 2 + · · · + n = a 1 + a 2 + · · · + a n
.
a 1 a 2 a n n
Work time: 4 hours.
L A TEX(c) 2004 Valentin Vornicu - MathLinks.ro

3rd Test - May 2, 2004
1. At a chess tourney, each player played with all the other players two matches, one
time with the white pieces, and one time with the black pieces. One point was given
for a victory, and 0,5 points were given for a tied game. In the end of the tourney all
the players had the same number of points.
a) Prove that there exist two players with the same number of tied games;
b) Prove that there exist two players which have the same number of lost games when
playing with the white pieces.
2. Let ABC be a triangle with AB = AC and let M be a mobile point of the line BC,
such that B lies between M and C. Prove that the sum of the inradius of the triangle
AMB and the sum of the exradius of the excircle tangent to AC of the triangle AMC
is constant.
3. Let p, q, r be three prime numbers and n a positive integer such that
p n + q n = r 2 .
Prove that n = 1.
Laurenţiu Panaitopol
4. One considers the positive integers a < b ≤ c < d such that ad = bc and √ d − √ a ≤ 1.
Prove that a is a perfect square.
Work time: 4 hours.
L A TEX(c) 2004 Valentin Vornicu - MathLinks.ro

4th Test - May 22, 2004
1. Let ABC be a triangle inscribed in the circle C and let M be a point on the arc BC
which does not contain A. The tangents from M to the inscribed circle in the triangle
ABC intersect C in the points N and P .
Prove that if ∠BAC = ∠NMP , then the triangles ABC and MNP are congruent.
2. The real numbers a 1 , a 2 , . . . , a 100 satisfy the relationship
a 2 1 + a 2 2 + · · · + a 2 100 + (a 1 + a 2 + · · · + a 100 ) 2 = 101.
Valentin Vornicu
Prove that |a k | ≤ 10, for all k = 1, 2, . . . , 100.
Dinu Şerbănescu
3. A finite set of positive integers is called isolated if the sum of the numbers in any
given proper subset is co-prime with the sum of the elements of the set.
a) Prove that the set A = {4, 9, 16, 25, 36, 49} is isolated;
b) Determine the composite numbers n for which there exist the positive integers a, b
such that the set
A = {(a + b) 2 , (a + 2b) 2 , . . . , (a + nb) 2 }
is isolated.
Gabriel Dospinescu
4. Consider a regular polygon with 1000 sides, whose vertices are colored with red, yellow,
blue. A move consists of choosing two neighboring vertices, having different colors,
and recoloring them with the third color.
Prove that there exists a finite number of moves such that the resulting polygon is
monocolor.
Marius Ghergu
Work time: 4 hours.
L A TEX(c) 2004 Valentin Vornicu - MathLinks.ro

5th Test - May 23, 2004
1. We consider the following triangular array
which is defined by the conditions
0 1 1 2 3 5 8 . . .
0 1 1 2 3 5 . . .
2 3 5 8 13 . . .
4 7 11 18 . . .
12 19 31 . . .
i) on the first two lines, each element, starting with the third one, is the sum of the
preceding two elements;
ii) on the other lines each element is the sum of the two numbers found on the same
column above it.
a) Prove that all the lines satisfy the first condition i);
b) Let a, b, c, d be the first elements of 4 consecutive lines in the array. Find d as a
function of a, b, c.
Dinu Şerbănescu
2. Let M, N, P be the midpoints of the sides BC, CA, AB respectively, of a triangle
ABC. Let G be the centroid of the triangle ABC. Prove that if 4BN 2 = 3AB 2 , and
the quadrilateral BMGP is cyclic, then ABC is an equilateral triangle.
3. Let A be a set of positive integers such that
a) if a ∈ A, the all the positive divisors of a are also in A;
b) if a, b ∈ A, with 1 < a < b, then 1 + ab ∈ A.
Prove that if A has at least 3 elements, then A is the set of all positive integers.
4. Given is a convex polygon with n ≥ 5 sides. Prove that there exist at most
triangles of area 1 with the vertices among the vertices of the polygon.
Valentin Vornicu
n(2n − 5)
3
Andrei Neguţ
Work time: 4 hours.
L A TEX(c) 2004 Valentin Vornicu - MathLinks.ro