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THE 4th ROMANIAN MASTER OF MATHEMATICS COMPETITION

DAY 1: FRIDAY, FEBRUARY 25, 2011, BUCHAREST

Language: English Problem 1. Prove that there exist two functions f , g : R → R, such that f ? g is strictly decreasing and g ? f is strictly increasing.

(P OLAND ) A NDRZEJ KOMISARSKI & M ARCIN K UCZMA

Problem 2. Determine all positive integers n for which there exists a polynomial f (x) with real coef?cients, with the following properties: (1) for each integer k, the number f (k) is an integer if and only if k is not divisible by n; (2) the degree of f is less than n.

(H UNGARY ) G ?ZA K?S

Problem 3. A triangle ABC is inscribed in a circle ω. A variable line chosen parallel to BC meets segments AB , AC at points D, E respectively, and meets ω at points K , L (where D lies between K and E ). Circle γ1 is tangent to the segments K D and B D and also tangent to ω, while circle γ2 is tangent to the segments LE and C E and also tangent to ω. Determine the locus, as varies, of the meeting point of the common inner tangents to γ1 and γ2 .

(RUSSIA ) VASILY M OKIN & F EDOR I VLEV

Each of the three problems is worth 7 points. 1 Time allowed 4 2 hours.

THE 4th ROMANIAN MASTER OF MATHEMATICS COMPETITION

DAY 2: SATURDAY, FEBRUARY 26, 2011, BUCHAREST

Language: English

s α

Problem 4. Given a positive integer n =

s i =1

p i i , we write ?(n) for the total num-

ber

i =1

αi of prime factors of n, counted with multiplicity. Let λ(n) = (?1)?(n)

(so, for example, λ(12) = λ(22 · 31 ) = (?1)2+1 = ?1). Prove the following two claims: i) There are in?nitely many positive integers n such that λ(n) = λ(n + 1) = +1; ii) There are in?nitely many positive integers n such that λ(n) = λ(n + 1) = ?1.

(R OMANIA ) D AN S CHWARZ

Problem 5. For every n ≥ 3, determine all the con?gurations of n distinct points X 1 , X 2 , . . . , X n in the plane, with the property that for any pair of distinct points X i , X j there exists a permutation σ of the integers {1, . . . , n}, such that d(X i , X k ) = d(X j , X σ(k) ) for all 1 ≤ k ≤ n. (We write d(X , Y ) to denote the distance between points X and Y .)

(U NITED K INGDOM ) L UKE B ETTS

Problem 6. The cells of a square 2011 × 2011 array are labelled with the integers 1, 2, . . . , 20112 , in such a way that every label is used exactly once. We then identify the left-hand and right-hand edges, and then the top and bottom, in the normal way to form a torus (the surface of a doughnut). Determine the largest positive integer M such that, no matter which labelling we choose, there exist two neighbouring cells with the difference of their labels at least M .?

(R OMANIA ) D AN S CHWARZ

Each of the three problems is worth 7 points. 1 Time allowed 4 2 hours.

Cells with coordinates (x, y) and (x , y ) are considered to be neighbours if x = x and y ? y ≡ ±1 (mod 2011), or if y = y and x ? x ≡ ±1 (mod 2011).

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