[Shortlist] Junior Balkan Mathematical Olympiad 2007

Algebra

  1. Let $a$ be positive real number such that $a^{3}=6(a+1)$. Prove that the equation $$x^{2}+ax+a^{2}-6=0$$ has no real solution.
  2. Prove that for all positive reals $a,b,c$ we have $$\frac{a^2-bc}{2a^2+bc}+\frac{b^2-ca}{2b^2+ca}+\frac{c^2-ab}{2c^2+ab}\leq 0$$
  3. Let $A$ be a set of positive integers containing the number $1$ and at least one more element. Given that for any two different elements $m, n$ of A the number $$\dfrac{m+1 }{(m+1,n+1) }$$ is also an element of $A$. Prove that $A$ coincides with the set of positive integers.
  4. Let $a$ and $ b$ be positive integers bigger than $2$. Prove that there exists a positive integer $k$ and a sequence $n_1, n_2, ..., n_k$ consisting of positive integers, such that $n_1 = a$, $n_k = b$ and $$(n_i + n_{i+1}) \mid n_in_{i+1},\,\forall i = 1,2,..., k - 1.$$
  5. The real numbers $x,y,z, m, n$ are positive, such that $m + n \ge 2$. Prove that $$x\sqrt{yz(x + my)(x + nz)} + y\sqrt{xz(y + mx)(y + nz)} + \\ + z\sqrt{xy(z + mx)(x + ny) } \le \frac{3(m + n)}{8} (x + y)(y + z)(z + x)$$

Combinatorics

  1. We call a tiling of an $m \times n$ rectangle with corners "regular" if there is no sub-rectangle which is tiled with corners. Prove that if for some $m$ and $n$ there exists a "regular" tiling of the $m \times n$ rectangular then there exists a "regular" tiling also for the $2m \times 2n $ rectangle.
  2. Given are $50$ points in the plane, no three of them belonging to a same line. Each of these points is colored using one of four given colors. Prove that there is a color and at least $130$ scalene triangles with vertices of that color.
  3. The nonnegative integer $n$ and $ (2n + 1) \times (2n + 1)$ chessboard with squares colored alternatively black and white are given. For every natural number $m$ with $1 < m < 2n+1$, an $m \times m$ square of the given chessboard that has more than half of its area colored in black, is called a $B$-square. If the given chessboard is a $B$-square, nd in terms of $n$ the total number of $B$-squares of this chessboard.

Geometry

  1. Let $M$ be interior point of the triangle $ABC$ with $\angle BAC=70^\circ$, $\angle ABC=80^\circ$, $\angle ACM=10$ and $\angle CBM=20^\circ$. Prove that $AB=MC$.
  2. Let $ABCD$ be a convex quadrilateral with $\angle{DAC}= \angle{BDC}= 36^\circ$, $\angle{CBD}= 18^\circ$ and $\angle{BAC}= 72^\circ$. The diagonals and intersect at point $P$. Determine the measure of $\angle{APD}$.
  3. Let the inscribed circle of the triangle $\vartriangle ABC$ touch side $BC$ at $M$, side $CA$ at $N$ and side $AB$ at $P$. Let $D$ be a point from $\left[ NP \right]$ such that $$\frac{DP}{DN}=\frac{BD}{CD}.$$ Show that $DM \perp PN$ .
  4. Let $S$ be a point inside $\angle pOq$, and let $k$ be a circle which contains $S$ and touches the legs $Op$ and $Oq$ in points $P$ and $Q$ respectively. Straight line $s$ parallel to $Op$ from $S$ intersects $Oq$ in a point $R$. Let $T$ be the point of intersection of the ray $PS$ and circumscribed circle of $\vartriangle SQR$ and $T \ne S$. Prove that $OT || SQ$ and $OT$ is a tangent of the circumscribed circle of $\vartriangle SQR$.

Number Theory

  1. Find all the pairs positive integers $(x, y)$ such that $$\frac{1}{x}+\frac{1}{y}+\frac{1}{[x, y]}+\frac{1}{(x, y)}=\frac{1}{2}$$ where $(x, y)$ is the greatest common divisor of $x, y$ and $[x, y]$ is the least common multiple of $x, y$.
  2. Prove that the equation $$x^{2006} - 4y^{2006} -2006 = 4y^{2007} + 2007y$$ has no solution in the set of the positive integers.
  3. Let $n > 1$ be a positive integer and $p$ a prime number such that $n \mid (p - 1) $and $p \mid (n^6 - 1)$. Prove that at least one of the numbers $p- n$ and $p + n$ is a perfect square.
  4. Let $a, b$ be two co-prime positive integers. A number is called good if it can be written in the form $ax + by$ for non-negative integers $x, y$. Define the function $f : \mathbb Z \to \mathbb Z $as $f(n) = n - n_a - n_b$, where $s_t$ represents the remainder of $s$ upon division by $t$. Show that an integer $n$ is good if and only if the infinite sequence $n, f(n), f(f(n)), ...$ contains only non-negative integers.
  5. Prove that if $ p$ is a prime number, then $ 7p+3^{p}-4$ is not a perfect square.

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