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[Solutions] Romanian Team Selection Tests For Junior Balkan Mathematical Olympiad 2008

  1. Let $ p$ be a prime number, $ p\not = 3$, and integers $ a,b$ such that $p\mid a+b$ and $ p^2\mid a^3 + b^3$. Prove that $ p^2\mid a + b$ or $ p^3\mid a^3 + b^3$.
  2. Prove that for every $ n \in \mathbb{N}^*$ exists a multiple of $ n$, having sum of digits equal to $ n$.
  3. Let $ ABC$ be an acute-angled triangle. We consider the equilateral triangle $ A'UV$, where $ A' \in (BC)$, $ U\in (AC)$ and $ V\in(AB)$ such that $ UV \parallel BC$. We define the points $ B'$, $C'$ in the same way. Prove that $ AA'$, $ BB'$ and $ CC'$ are concurrent.
  4. Let $ ABC$ be a triangle, and $ D$ the midpoint of the side $ BC$. On the sides $ AB$ and $ AC$ we consider the points $ M$ and $ N$, respectively, both different from the midpoints of the sides, such that \[ AM^2+AN^2 =BM^2 + CN^2 \textrm{ and } \angle MDN = \angle BAC.\] Prove that $ \angle BAC = 90^\circ$.
  5. Let $ n$ be an integer, $ n\geq 2$, and the integers $ a_1,a_2,\ldots,a_n$, such that $ 0 < a_k\leq k$, for all $ k = 1,2,\ldots,n$. Knowing that the number $ a_1 + a_2 + \cdots + a_n$ is even, prove that there exists a choosing of the signs $ +$, respectively $ -$, such that \[ a_1 \pm a_2 \pm \cdots \pm a_n= 0. \]
  6. Consider the acute-angled triangle $ ABC$, altitude $ AD$ and point $ E$ - intersection of $ BC$ with diameter from $ A$ of circumcircle. Let $ M$, $N$ be symmetric points of $ D$ with respect to the lines $ AC$ and $ AB$ respectively. Prove that $ \angle{EMC} = \angle{BNE}$.
  7. In a sequence of natural numbers $ a_1,a_2,...,a_n$ every number $ a_k$ represents sum of the multiples of the $ k$ from sequence. Find all possible values for $ n$.
  8. Let $ n$ be a positive integer and let $ a_1,a_2,\ldots,a_n$ be positive real numbers such that: \[ \sum^n_{i = 1} a_i = \sum^n_{i = 1} \frac {1}{a_i^2}. \] Prove that for every $ i = 1,2,\ldots,n$ we can find $ i$ numbers with sum at least $ i$.
  9. Let $ a,b$ be real nonzero numbers, such that number $ \lfloor an + b \rfloor$ is an even integer for every $ n \in \mathbb{N}$. Prove that $ a$ is an even integer.
  10. From numbers $ 1,2,3,...,37$ we randomly choose 10 numbers. Prove that among these exist four distinct numbers, such that sum of two of them equals to the sum of other two.
  11. Let $ a,b,c$ be positive reals with $ ab + bc + ca = 3$. Prove that: \[ \frac {1}{1 + a^2(b + c)} + \frac {1}{1 + b^2(a + c)} + \frac {1}{1 + c^2(b + a)}\le \frac {1}{abc}. \]
  12. Solve in prime numbers $ 2p^q - q^p = 7$.
  13. Let $ d$ be a line and points $ M,N$ on the $ d$. Circles $ \alpha,\beta,\gamma,\delta$ with centers $ A,B,C,D$ are tangent to $ d$, circles $ \alpha,\beta$ are externally tangent at $ M$, and circles $ \gamma,\delta$ are externally tangent at $ N$. Points $ A,C$ are situated in the same half-plane, determined by $ d$. Prove that if exists an circle, which is tangent to the circles $ \alpha,\beta,\gamma,\delta$ and contains them in its interior, then lines $ AC,BD,MN$ are concurrent or parallel.
  14. Let $ ABCD$ be a convex quadrilateral with opposite side not parallel. The line through $ A$ parallel to $ BD$ intersect line $ CD$ in $ F$, but parallel through $ D$ to $ AC$ intersect line $ AB$ at $ E$. Denote by $ M,N,P,Q$ midpoints of the segments $ AC,BD,AF,DE$. Prove that lines $ MN,PQ$ and $ AD$ are concurrent.
  15. Let $ m,n$ be two natural nonzero numbers and sets $ A = \{ 1,2,...,n\}$, $B = \{1,2,...,m\}$. We say that subset $ S$ of Cartesian product $ A \times B$ has property $ (j)$ if $ (a - x)(b - y)\le 0$ for each pairs $ (a,b), (x,y) \in S$. Prove that every set $ S$ with propery $ (j)$ has at most $ m + n - 1$ elements. The statement was edited, in order to reflect the actual problem asked. The sign of the inequality was inadvertently reversed into $ (a - x)(b - y)\ge 0$, and that accounts for the following two posts.
  16. Find all pairs $ (m,n)$ of integer numbers $ m,n > 1$ with property that $ mn - 1\mid n^3 - 1$.
  17. Determine the maximum possible real value of the number $ k$, such that \[ (a + b + c)\left (\frac {1}{a + b} + \frac {1}{c + b} + \frac {1}{a + c} - k \right )\ge k\] for all real numbers $ a,b,c\ge 0$ with $ a + b + c = ab + bc + ca$.

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MOlympiad: [Solutions] Romanian Team Selection Tests For Junior Balkan Mathematical Olympiad 2008
[Solutions] Romanian Team Selection Tests For Junior Balkan Mathematical Olympiad 2008
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