Romanian IMO Team Selection Tests 2002
First Test – March 21, 2002.
Time: 4 hours
1. Find all pairs A, B of sets satisfying the following conditions:
(i) A ∪ B = Z;
(ii) if x ∈ A then x − 1 ∈ B;
(iii) if x, y ∈ B then x + y ∈ A.
2. The sequence (an ) is defined by
a0 = a1 = 1 and an+1 = 14an − an−1 for all n ≥ 1.
Prove that 2an − 1 is a perfect square for any n ≥ 0.
3. In an acute triangle ABC, let M, N be the midpoints of AB and AC respectively,
P be the projection of N on BC and A1 be the midpoint of MP. Points B1 and
C1 are constructed similarly. Prove that if AA1 , BB1 and CC1 are concurrent then
△ABC is isosceles.
4. For any n ∈ N let f (n) be the number of choices of signs +/− in the expression
E = ±1 ± 2 ± · · ·± n which yield the value E = 0. Prove that:
(a) if n ≡ 1, 2 (mod 4) then f (n) = 0;
(b) if n ≡ 0, 3 (mod 4) then
√ n−2
2 ≤ f (n) < 2n − 2[n/2]+1.
Second Test – April 13, 2002.
Time: 4 hours
1. Let M and N be points in the interior of a square ABCD such that the line MN
contains no vertex of the square. Denote by s(M, N) the smallest area of a tri-
angle with vertices in the set {A, B,C, D, M, N}. Find the smallest real number k
such that for any such points M, N it holds that s(M, N) ≤ k.
2. Assume that P and Q are polynomials with coefficients in the set {1, 2002} such
that P divides Q, prove that then deg P + 1 divides deg Q + 1.
3. Given positive real numbers a, b, define xn (n ∈ N) as the sum of digits of [an + b].
Prove that there exists a positive integer which occurs in the sequence infinitely
often.
The IMO Compendium Group,
D. Djukić, V. Janković, I. Matić, N. Petrović
www.imomath.com
4. At an international conference there are four official languages. Any two partic-
ipants can talk to each other in at least one of the official languages. Prove that
there is a language which is spoken by at least 60 percents of the participants.
Third Test – April 14, 2002.
Time: 4 hours
1. A pentagon ABCD inscribed in a circle with center O has angles ∠B = ∠C =
120◦ , ∠D = 130◦ , ∠E = 100◦ . Prove that the intersection point of BD and CE
lies on AO.
2. Let a1 , a2 , . . . , an be positive real numbers (n ≥ 3) such that a21 + · · · + a2n = 1.
Prove the inequality
a1 a2 an 4 √ √
+ + ···+ 2 ≥ (a1 a1 + · · · + an an )2 .
a22 + 1 a23 + 1 a1 + 1 5
3. For an even positive integer n, let S denote the set of natural numbers a, 1 < a < n
for which aa−1 − 1 is divisible by n. If S = {n − 1}, prove that n = 2p for some
prime number p.
4. Suppose f : Z → {1, 2, . . . , n} is a function such that f (x) 6= f (y) whenever |x− y|
is 2, 3 or 5. Prove that n ≥ 3.
Fourth Test – June 1, 2002.
Time: 4 hours
1. Given p0 , p1 ∈ N, define pn+2 (n ≥ 0) inductively to be the smallest prime divisor
of pn + pn+1 . Prove that the real number whose decimal representation is given
by x = 0.p0 p1 p2 . . . is rational.
2. Consider a unit square A1 A2 A3 A4 . Determine the smallest real number a > 0
with the following property: For any positive reals r1 , r2 , r3 , r4 with sum a there
exist points Xi in the plane satisfying Xi Ai ≤ ri (1 ≤ i ≤ 4) such that one of the
triangles with vertices in X1 , X2 , X3 , X4 is equilateral.
3. In a parliament there are several parties, and each member of the parliament has
a constant absolute rating. Within a party, each member has a relative rating
which is equal to the ratio of his/her rating to the sum of all the ratings in the
party. A member of the parliament may change the party only if that would
increase his/her relative rating. Prove that after finitely many changes of parties
no more changes will be possible.
The IMO Compendium Group,
D. Djukić, V. Janković, I. Matić, N. Petrović
www.imomath.com
Fifth Test – June 2, 2002.
Time: 4 hours
1. Let m and n be positive integers, not of the same parity, such that m < n < 5m.
Show that the set {1, 2, . . . , 4mn} can be partitioned into pairs of numbers so that
the sum in each pair is a square.
2. Let a triangle ABC with AB < AC 6= BC be inscribed in a circle C . The tangent
at A to C intersects BC at D. The circle tangent to segments BD, AD and circle
C meets BC at M. Prove that ∠DAM = ∠MAB if and only if AC = CM.
3. We are given np cards. In each of n colors exactly p cards, numbered 1, 2, . . . , p,
are colored. There are n players playing the following game. Each of them
initially receives p cards. The game is glayed in p rounds after the following
rules:
(i) In each round he first player puts down a card; every other player there-
after puts down a card of the same color if he/she has any, and any card
otherwise.
(ii) In each round, the player who put down the card of the initial color which
is numbered with the biggest number wins the round.
(iii) The player who wins a round starts the next round.
(iv) The first round is started by a random player and after each round the cards
player will be taken out of the game.
Assume that all cards numbered 1 won the rounds in which they were put down.
Prove that p ≥ 2n.
The IMO Compendium Group,
D. Djukić, V. Janković, I. Matić, N. Petrović
www.imomath.com
