MathDB

3

Part of 2010 Romania Team Selection Test

Problems(6)

Area of intersecting rectangles

Source: Romania TST 1 2010, Problem 3

8/25/2012
Two rectangles of unit area overlap to form a convex octagon. Show that the area of the octagon is at least 12\dfrac {1} {2}.
Kvant Magazine
geometryrectanglegeometry proposed
Concurrent lines

Source: Romania TST 2 2010, Problem 3

8/25/2012
Let γ1\gamma_1 and γ2\gamma_2 be two circles tangent at point TT, and let ℓ1\ell_1 and ℓ2\ell_2 be two lines through TT. The lines ℓ1\ell_1 and ℓ2\ell_2 meet again γ1\gamma_1 at points AA and BB, respectively, and γ2\gamma_2 at points A1A_1 and B1B_1, respectively. Let further XX be a point in the complement of γ1∪γ2∪ℓ1∪ℓ2\gamma_1 \cup \gamma_2 \cup \ell_1 \cup \ell_2. The circles ATXATX and BTXBTX meet again γ2\gamma_2 at points A2A_2 and B2B_2, respectively. Prove that the lines TXTX, A1B2A_1B_2 and A2B1A_2B_1 are concurrent.
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geometrycircumcirclepower of a pointradical axisgeometry proposed
Relations involving the sum of divisors

Source: Romania TST 3 2010, Problem 3

8/25/2012
Given a positive integer aa, prove that σ(am)<σ(am+1)\sigma(am) < \sigma(am + 1) for infinitely many positive integers mm. (Here σ(n)\sigma(n) is the sum of all positive divisors of the positive integer number nn.)
Vlad Matei
modular arithmeticarithmetic sequencenumber theory proposednumber theory
Combinatorics of family of lines in the plane

Source: Romania TST 4 (All Geometry) 2010, Problem 3

8/25/2012
Let L\mathcal{L} be a finite collection of lines in the plane in general position (no two lines in L\mathcal{L} are parallel and no three are concurrent). Consider the open circular discs inscribed in the triangles enclosed by each triple of lines in L\mathcal{L}. Determine the number of such discs intersected by no line in L\mathcal{L}, in terms of ∣L∣|\mathcal{L}|.
B. Aronov et al.
geometry proposedgeometry
Irreducible rational polynomial

Source: Romania TST 5 2010, Problem 3

8/25/2012
Let pp be a prime number,let n1,n2,…,npn_1, n_2, \ldots, n_p be positive integer numbers, and let dd be the greatest common divisor of the numbers n1,n2,…,npn_1, n_2, \ldots, n_p. Prove that the polynomial Xn1+Xn2+⋯+Xnp−pXd−1\dfrac{X^{n_1} + X^{n_2} + \cdots + X^{n_p} - p}{X^d - 1} is irreducible in Q[X]\mathbb{Q}[X].
Beniamin Bogosel
algebrapolynomialIrreducibleirreducibility
Inequality involving vectors in the plane

Source: Romania TST 6 2010, Problem 3

8/25/2012
Let nn be a positive integer number. If SS is a finite set of vectors in the plane, let N(S)N(S) denote the number of two-element subsets {v,v′}\{\mathbf{v}, \mathbf{v'}\} of SS such that 4 (v⋅v′)+(∣v∣2−1)(∣v′∣2−1)<0.4\,(\mathbf{v} \cdot \mathbf{v'}) + (|\mathbf{v}|^2 - 1)(|\mathbf{v'}|^2 - 1) < 0. Determine the maximum of N(S)N(S) when SS runs through all nn-element sets of vectors in the plane.
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inequalitiesvectoralgebra proposedalgebra