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[Solutions] Baltic Way Mathematical Competition 2003

  1. Find all functions $f:\mathbb{Q}^{+}\rightarrow \mathbb{Q}^{+}$ which for all $x \in \mathbb{Q}^{+}$ fulfil \[f\left(\frac{1}{x}\right)=f(x) \ \ \text{and} \ \ \left(1+\frac{1}{x}\right)f(x)=f(x+1). \]
  2. Prove that any real solution of $x^3+px+q=0$, where $p,q$ are real numbers, satisfies the inequality $4qx\le p^2$.
  3. Let $x$, $y$ and $z$ be positive real numbers such that $xyz = 1$. Prove that $$\left(1+x\right)\left(1+y\right)\left(1+z\right)\geq 2\left(1+\sqrt[3]{\frac{x}{z}}+\sqrt[3]{\frac{y}{x}}+\sqrt[3]{\frac{z}{y}}\right).$$
  4. Let $a,b,c$ be positive real numbers. Prove that \[\frac{2a}{a^{2}+bc}+\frac{2b}{b^{2}+ca}+\frac{2c}{c^{2}+ab}\leq\frac{a}{bc}+\frac{b}{ca}+\frac{c}{ab} \]
  5. The sequence $(a_n)$ is defined by $a_1=\sqrt{2}$, $a_2=2$, and $a_{n+1}=a_na_{n-1}^2$ for $n\ge 2$. Prove that for every $n\ge 1$ \[(1+a_1)(1+a_2)\cdots (1+a_n)<(2+\sqrt{2})a_1a_2\cdots a_n. \]
  6. Let $n\ge 2$ and $d\ge 1$ be integers with $d\mid n$, and let $x_1,x_2,\ldots x_n$ be real numbers such that $x_1+x_2+\cdots + x_n=0$. Show that there are at least $\binom{n-1}{d-1}$ choices of $d$ indices $1\le i_1<i_2<\cdots <i_d\le n $ such that $$x_{i_{1}}+x_{i_{2}}+\cdots +x_{i_{d}}\ge 0.$$
  7. A subset of $X$ of $\{1,2,3, \ldots 10000 \}$ has the following property: If $a,b$ are distinct elements of $X$, then $ab\not\in X$. What is the maximal number of elements in $X$?
  8. There are $2003$ pieces of candy on a table. Two players alternately make moves. A move consists of eating one candy or half of the candies on the table (the “lesser half” if there are an odd number of candies). At least one candy must be eaten at each move. The loser is the one who eats the last candy. Which player has a winning strategy?
  9. It is known that $n$ is a positive integer and $n \le 144$. Ten questions of the type “Is $n$ smaller than $a$?” are allowed. Answers are given with a delay: for $i = 1, \ldots , 9$, the $i$-th question is answered only after the $(i + 1)$-th question is asked. The answer to the tenth question is given immediately. Find a strategy for identifying $n$.
  10. A lattice point in the plane is a point with integral coordinates. The centroid of four points $(x_i,y_i )$, $i = 1, 2, 3, 4$, is the point $\left(\dfrac{x_1 +x_2 +x_3 +x_4}{4},\dfrac{y_1 +y_2 +y_3 +y_4 }{4}\right)$. Let $n$ be the largest natural number for which there are $n$ distinct lattice points in the plane such that the centroid of any four of them is not a lattice point. Prove that $n = 12$.
  11. Is it possible to select $1000$ points in the plane so that $6000$ pairwise distances between them are equal?
  12. Points $M$ and $N$ are taken on the sides $BC$ and $CD$ respectively of a square $ABCD$ so that $\angle MAN=45^{\circ}$. Prove that the circumcentre of $\triangle AMN$ lies on $AC$.
  13. In a rectangle $ABCD$ be a rectangle and $BC = 2AB$, let $E$ be the midpoint of $BC$ and $P$ an arbitrary inner point of $AD$. Let $F$ and $G$ be the feet of perpendiculars drawn correspondingly from $A$ to $BP$ and from $D$ to $CP$. Prove that the points $E,F,P,G$ are concyclic.
  14. Equilateral triangles $AMB$, $BNC$, $CKA$ are constructed on the exterior of a triangle $ABC$. The perpendiculars from the midpoints of $MN, NK, KM$ to the respective lines $CA, AB, BC$ are constructed. Prove that these three perpendiculars pass through a single point.
  15. The diagonals of a cyclic convex quadrilateral $ABCD$ intersect at $P$. A circle through $P$ touches the side $CD$ at its midpoint $M$ and intersects the segments $BD$ and $AC$ again at the points $Q$ and $R$ respectively. Let $S$ be the point on segment $BD$ such that $BS = DQ$. The line through $S$ parallel to $AB$ intersects $AC$ at $T$. Prove that $AT = RC$.
  16. Find all pairs of positive integers $(a,b)$ such that $a-b$ is a prime number and $ab$ is a perfect square.
  17. All the positive divisors of a positive integer $n$ are stored into an increasing array. Mary is writing a programme which decides for an arbitrarily chosen divisor $d > 1$ whether it is a prime. Let $n$ have $k$ divisors not greater than $d$. Mary claims that it suffices to check divisibility of $d$ by the first $\left\lceil\frac{k}{2}\right\rceil$ divisors of $n$: $d$ is prime if and only if none of them but $1$ divides $d$. Is Mary right?
  18. Every integer is to be coloured blue, green, red, or yellow. Can this be done in such a way that if $a, b, c, d$ are not all $0$ and have the same colour, then $3a-2b \neq 2c-3d$?
  19. Let $a$ and $b$ be positive integers. Show that if $a^3+b^3$ is the square of an integer, then $a + b$ is not a product of two different prime numbers.
  20. Suppose that the sum of all positive divisors of a natural number $n$, $n$ excluded, plus the number of these divisors is equal to $n$. Prove that $n = 2m^2$ for some integer $m$.

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MOlympiad: [Solutions] Baltic Way Mathematical Competition 2003
[Solutions] Baltic Way Mathematical Competition 2003
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