MathDB

2

Part of 2002 District Olympiad

Problems(6)

Examination

Source:

6/27/2012
A group of 6767 students pass their examination consisting of 66 questions, labeled with the numbers 11 to 66. A correct answer to question nn is quoted nn points and for an incorrect answer to the same question a student loses nn point.
a) Find the least possible positive difference between any 22 final scores b) Show that at least 44 participants have the same final score c) Show that at least 22 students gave identical answer to all six questions.
invariantpigeonhole principle
x is rational if x^2+x, x^3+2x are rationals 2002 Romania District VIII p2

Source:

8/15/2024
a) Let xx be a real number such that x2+xx^2+x and x3+2xx^3+2x are rational numbers. Show that xx is a rational number.
b) Show that there exist irrational numbers xx such that x2+xx^2+xand x3−2xx^3-2x are rational.
algebrarationalirrational number
parallelograms and proportions resulted from inscriptible quadrilateral

Source: Romanian Distrct Olympiad 2002, Grade IX, Problem 2

10/7/2018
Let ABCD ABCD be an inscriptible quadrilateral and M M be a point on its circumcircle, distinct from its vertices. Let H1,H2,H3,H4 H_1,H_2,H_3,H_4 be the orthocenters of MAB,MBC,MCD, MAB,MBC, MCD, respectively, MDA, MDA, and E,F, E,F, the midpoints of the segments AB, AB, respectivley, CD. CD. Prove that:
a) H1H2H3H4 H_1H_2H_3H_4 is a parallelogram. b) H1H3=2⋅EF. H_1H_3=2\cdot EF.
geometrycircumcircleparallelogram
Symmetric system of equations: x(x-y)(x-z)=3

Source: Romanian District Olympiad 2002, Grade X, Problem 2

10/7/2018
Solve in C3 \mathbb{C}^3 the following chain of equalities: x(x−y)(x−z)=y(y−x)(y−z)=z(z−x)(z−y)=3. x(x-y)(x-z)=y(y-x)(y-z)=z(z-x)(z-y)=3.
algebrasystem of equations
Romania District Olympiad 2002 - Grade XI

Source:

3/18/2011
In the xOyxOy system, consider the points An(n,n3)A_n(n,n^3) with n∈N∗n\in \mathbb{N}^* and the point B(0,1)B(0,1). Prove that
a) for any positive integers k>j>i≥1k>j>i\ge 1, the points Ai,Aj,AkA_i,A_j,A_k cannot be collinear. b) for any positive integers ik>ik−1>…>i1≥1i_k>i_{k-1}>\ldots>i_1\ge 1, we have μ(Ai1OB^)+μ(Ai2OB^)+⋯+μ(AikOB^)<π2\mu(\widehat{A_{i_1}OB})+\mu(\widehat{A_{i_2}OB})+\cdots+\mu(\widehat{A_{i_k}OB})<\frac{\pi}{2} ***
trigonometrygeometry proposedgeometry
Z_2...Z_2,Z_2...Z_2 homomorphic;

Source: Romanian District Olympiad, Grade XII, Problem 2

10/7/2018
a) Show that, for any distinct natural numbers m,n, m,n, the rings Z2×⋯⏟m times×Z2,Z2×⋯⏟n times×Z2 \mathbb{Z}_2\times \underbrace{\cdots}_{m\text{ times}} \times\mathbb{Z}_2,\mathbb{Z}_2\times \underbrace{\cdots}_{n\text{ times}} \times\mathbb{Z}_2 are homomorphic, but not isomorphic.
b) Show that there are infinitely many pairwise nonhomomorphic rings of same order.
superior algebramorphismsproduct ringsRing Theory