MathDB

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Part of 2010 District Olympiad

Problems(6)

xy - x - y + 1., ab = 0 if |a + b| > |1 + ab| - 2010 Romania District VII p1

Source:

9/1/2024
a) Factorize xy−x−y+1xy - x - y + 1.
b) Prove that if integers aa and bb satisfy ∣a+b∣>∣1+ab∣ |a + b| > |1 + ab|, then ab=0ab = 0.
number theoryalgebra
cannot each vertex of a cube$ 8$ distinct numbers from 1-12

Source: 2010 Romania District VIII p1

9/1/2024
a) Prove that one cannot assign to each vertex of a cube 8 8 distinct numbers from the set {0,1,2,3,...,11,12}\{0, 1, 2, 3, . . . , 11, 12\} such that, for every edge, the sum of the two numbers assigned to its vertices is even.
b) Prove that one can assign to each vertex of a cube 88 distinct numbers from the set {0,1,2,3,...,11,12}\{0, 1, 2, 3, . . . , 11, 12\} such that, for every edge, the sum of the two numbers assigned to its vertices is divisible by 33.
combinatoricsnumber theory
Romanian District Olympiad 2010

Source: Grade IX

3/13/2010
A right that passes through the incircle I I of the triangle ΔABC \Delta ABC intersects the side AB AB and CA CA in P P, respective Q Q. We denote BC\equal{}a\ , \ AC\equal{}b\ ,\ AB\equal{}c and \frac{PB}{PA}\equal{}p\ ,\ \frac{QC}{QA}\equal{}q. i) Prove that: a(1\plus{}p)\cdot \overrightarrow{IP}\equal{}(a\minus{}pb)\overrightarrow{IB}\minus{}pc\overrightarrow{IC} ii) Show that a\equal{}bp\plus{}cq. iii) If a^2\equal{}4bcpq, then the rights AI , BQ AI\ ,\ BQ and CP CP are concurrents.
geometryvectorgeometry proposed
Romania District Olympiad 2010

Source: Grade X

3/13/2010
Prove the following equalities of sets: \text{i)} \{x\in \mathbb{R}\ |\ \log_2 \lfloor x \rfloor \equal{} \lfloor \log_2 x\rfloor \} \equal{} \bigcup_{m\in \mathbb{N}} \left[2^m,2^m \plus{} 1\right) \text{ii)} \{x\in \mathbb{R}\ |\ 2^{\lfloor x\rfloor} \equal{} \left\lfloor 2^x\right\rfloor \} \equal{} \bigcup_{m\in \mathbb{N}} \left[m, \log_2 (2^m \plus{} 1) \right)
logarithmsfloor functionalgebra proposedalgebra
Romanian District Olympiad

Source: Grade XI

3/17/2010
Prove that any continuos function f:R→R f: \mathbb{R}\rightarrow \mathbb{R} with f(x)\equal{}\left\{ \begin{aligned} a_1x\plus{}b_1\ ,\ \text{for } x\le 1 \\ a_2x\plus{}b_2\ ,\ \text{for } x>1 \end{aligned} \right. where a1,a2,b1,b2∈R a_1,a_2,b_1,b_2\in \mathbb{R}, can be written as: f(x)\equal{}m_1x\plus{}n_1\plus{}\epsilon|m_2x\plus{}n_2|\ ,\ \text{for } x\in \mathbb{R} where m1,m2,n1,n2∈R m_1,m_2,n_1,n_2\in \mathbb{R} and \epsilon\in \{\minus{}1,\plus{}1\}.
functionalgebra unsolvedalgebra
Romanian District Olympiad

Source: Grade XII

3/17/2010
Let S S be the sum of the inversible elements of a finite ring. Prove that S^2\equal{}S or S^2\equal{}0.
superior algebrasuperior algebra unsolved