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Baltic Way Mathematical Competition 1999 Solutions

  1. Determine all real numbers $a,b,c,d$ that satisfy the following equations \[\begin{cases} abc + ab + bc + ca + a + b + c = 1\\ bcd + bc + cd + db + b + c + d = 9\\ cda + cd + da + ac + c + d + a = 9\\ dab + da + ab + bd + d + a + b = 9\end{cases}\]
  2. Determine all positive integers $n$ with the property that the third root of $n$ is obtained by removing its last three decimal digits.
  3. Determine all positive integers $n\ge 3$ such that the inequality \[a_1a_2+a_2a_3+\ldots a_{n-1}a_n\le 0\] holds for all real numbers $a_1,a_2,\ldots , a_n$ which satisfy $a_1+a_2+\ldots +a_n=0$.
  4. For all positive real numbers $x$ and $y$ let \[f(x,y)=\min\left( x,\frac{y}{x^2+y^2}\right) \] Show that there exist $x_0$ and $y_0$ such that $f(x, y)\le f(x_0, y_0)$ for all positive $x$ and $y$, and find $f(x_0,y_0)$.
  5. The point $(a,b)$ lies on the circle $x^2+y^2=1$. The tangent to the circle at this point meets the parabola $y=x^2+1$ at exactly one point. Find all such points $(a,b)$.
  6. What is the least number of moves it takes a knight to get from one corner of an $n\times n$ chessboard, where $n\ge 4$, to the diagonally opposite corner?
  7. Two squares on an $8\times 8$ chessboard are called adjacent if they have a common edge or common corner. Is it possible for a king to begin in some square and visit all squares exactly once in such a way that all moves except the first are made into squares adjacent to an even number of squares already visited?
  8. We are given $1999$ coins. No two coins have the same weight. A machine is provided which allows us with one operation to determine, for any three coins, which one has the middle weight. Prove that the coin that is the $1000$th by weight can be determined using no more than $1000000$ operations and that this is the only coin whose position by weight can be determined using this machine.
  9. A cube with edge length $3$ is divided into $27$ unit cubes. The numbers $1, 2,\ldots ,27$ are distributed arbitrarily over the unit cubes, with one number in each cube. We form the $27$ possible row sums (there are nine such sums of three integers for each of the three directions parallel with the edges of the cube). At most how many of the $27$ row sums can be odd?
  10. May the points of a disc of radius $1$ (including its circumference) be partitioned into three subsets in such a way that no subset contains two points separated by a distance $1$?
  11. Prove that for any four points in the plane, no three of which are collinear, there exists a circle such that three of the four points are on the circumference and the fourth point is either on the circumference or inside the circle.
  12. In a triangle $ABC$ it is given that $2AB=AC+BC$. Prove that the incentre of $\triangle ABC$, the circumcentre of $\triangle ABC$, and the midpoints of $AC$ and $BC$ are concyclic.
  13. The bisectors of the angles $A$ and $B$ of the triangle $ABC$ meet the sides $BC$ and $CA$ at the points $D$ and $E$, respectively. Assuming that $AE+BD=AB$, determine the angle $C$.
  14. Let $ABC$ be an isosceles triangle with $AB=AC$. Points $D$ and $E$ lie on the sides $AB$ and $AC$, respectively. The line passing through $B$ and parallel to $AC$ meets the line $DE$ at $F$. The line passing through $C$ and parallel to $AB$ meets the line $DE$ at $G$. Prove that \[\frac{[DBCG]}{[FBCE]}=\frac{AD}{DE} \]
  15. Let $ABC$ be a triangle with $\angle C=60^\circ$ and $AC<BC$. The point $D$ lies on the side $BC$ and satisfies $BD=AC$. The side $AC$ is extended to the point $E$ where $AC=CE$. Prove that $AB=DE$.
  16. Find the smallest positive integer $k$ which is representable in the form $k=19^n-5^m$ for some positive integers $m$ and $n$.
  17. Does there exist a finite sequence of integers $c_1,c_2,\ldots ,c_n$ such that all the numbers $a+c_1,a+c_2,\ldots ,a+c_n$ are primes for more than one but not infinitely many different integers $a$?
  18. Let $m$ be a positive integer such that $m=2\pmod{4}$. Show that there exists at most one factorization $m=ab$ where $a$ and $b$ are positive integers satisfying \[0<a-b<\sqrt{5+4\sqrt{4m+1}}\]
  19. Prove that there exist infinitely many even positive integers $k$ such that for every prime $p$ the number $p^2+k$ is composite.
  20. Let $a,b,c$ and $d$ be prime numbers such that $a>3b>6c>12d$ and $a^2-b^2+c^2-d^2=1749$. Determine all possible values of $$a^2+b^2+c^2+d^2.$$

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MOlympiad: Baltic Way Mathematical Competition 1999 Solutions
Baltic Way Mathematical Competition 1999 Solutions
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