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Baltic Way Mathematical Competition 1992 Solutions

  1. Let $p,q$ be two consecutive odd prime numbers. Prove that $p+q$ is a product of at least $3$ natural numbers greater than $1$ (not necessarily different).
  2. Denote by $ d(n)$ the number of all positive divisors of a natural number $ n$ (including $ 1$ and $ n$). Prove that there are infinitely many $ n$, such that $ n/d(n)$ is an integer.
  3. Find an infinite non-constant arithmetic progression of natural numbers such that each term is neither a sum of two squares, nor a sum of two cubes (of natural numbers).
  4. Is it possible to draw a hexagon with vertices in the knots of an integer lattice so that the squares of the lengths of the sides are six consecutive positive integers?
  5. It is given that $ a^2+b^2+(a+b)^2=c^2+d^2+(c+d)^2$. Prove that $$a^4+b^4+(a+b)^4=c^4+d^4+(c+d)^4.$$
  6. Prove that the product of the 99 numbers $ \frac{k^3-1}{k^3+1},k=2,3,\ldots,100$ is greater than $ 2/3$.
  7. Let $ a=\sqrt[1992]{1992}$. Which number is greater \[\underbrace{a^{a^{a^{\ldots^{a}}}}}_{1992}\quad\text{or}\quad 1992? \]
  8. Find all integers satisfying the equation $ 2^x\cdot(4-x)=2x+4$.
  9. A polynomial $ f(x)=x^3+ax^2+bx+c$ is such that $ b<0$ and $ ab=9c$. Prove that the polynomial $ f$ has three different real roots.
  10. Find all fourth degree polynomial $ p(x)$ such that the following four conditions are satisfied:
    • $ p(x)=p(-x)$ for all $ x$,
    • $ p(x)\ge0$ for all $ x$,
    • $ p(0)=1$
    • $ p(x)$ has exactly two local minimum points $ x_1$ and $ x_2$ such that $ |x_1-x_2|=2$.
  11. Let $ Q^+$ denote the set of positive rational numbers. Show that there exists one and only one function $f: Q^+\to Q^+$ satisfying the following conditions
    • If $ 0<q<1/2$ then $ f(q)=1+f(q/(1-2q))$,
    • If $ 1<q\le2$ then $ f(q)=1+f(q-1)$,
    • $ f(q)\cdot f(1/q)=1$ for all $ q\in Q^+$.
  12. Let $ N$ denote the set of natural numbers. Let $ \phi: N\rightarrow N$ be a bijective function and assume that there exists a finite limit \[ \lim_{n\rightarrow\infty}\frac{\phi(n)}{n}=L. \] What are the possible values of $ L$?
  13. Prove that for any positive $ x_1,x_2,\ldots,x_n,y_1,y_2,\ldots,y_n$ the inequality \[ \sum_{i=1}^n\frac1{x_iy_i}\ge\frac{4n^2}{\sum_{i=1}^n(x_i+y_i)^2} \] holds.
  14. There is a finite number of towns in a country. They are connected by one direction roads. It is known that, for any two towns, one of them can be reached from another one. Prove that there is a town such that all remaining towns can be reached from it.
  15. Noah has 8 species of animals to fit into 4 cages of the ark. He plans to put species in each cage. It turns out that, for each species, there are at most 3 other species with which it cannot share the accomodation. Prove that there is a way to assign the animals to their cages so that each species shares with compatible species.
  16. All faces of a convex polyhedron are parallelograms. Can the polyhedron have exactly 1992 faces?
  17. Quadrangle $ ABCD$ is inscribed in a circle with radius 1 in such a way that the diagonal $ AC$ is a diameter of the circle, while the other diagonal $ BD$ is as long as $ AB$. The diagonals intersect at $ P$. It is known that the length of $ PC$ is $ 2/5$. How long is the side $ CD$?
  18. Show that in a non-obtuse triangle the perimeter of the triangle is always greater than two times the diameter of the circumcircle.
  19. Let $ C$ be a circle in plane. Let $ C_1$ and $ C_2$ be nonintersecting circles touching $ C$ internally at points $ A$ and $ B$ respectively. Let $ t$ be a common tangent of $ C_1$ and $ C_2$ touching them at points $ D$ and $ E$ respectively, such that both $ C_1$ and $ C_2$ are on the same side of $ t$. Let $ F$ be the point of intersection of $ AD$ and $ BE$. Show that $ F$ lies on $ C$.
  20. Let $ a\le b\le c$ be the sides of a right triangle, and let $ 2p$ be its perimeter. Show that \[ p(p - c) = (p - a)(p - b) = S, \] where $ S$ is the area of the triangle.

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MOlympiad: Baltic Way Mathematical Competition 1992 Solutions
Baltic Way Mathematical Competition 1992 Solutions
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