[Shortlist] Junior Balkan Mathematical Olympiad 2000

1. Prove that there are at least $666$ positive composite numbers with $2006$ digits, having a digit equal to $7$ and all the rest equal to $1$.
2. Find all the positive perfect cubes that are not divisible by $10$ such that the number obtained by erasing the last three digits is also a perfect cube.
3. Find the greatest positive integer $x$ such that $23^{6+x}$ divides $2000!$
4. Find all the integers written as $\overline{abcd}$ in decimal representation and $\overline{dcba}$ in base $7$.
5. Find all pairs of integers $(m,n)$ such that the numbers $$\begin{align*}A&=n^2+2mn+3m^2+2,\\ B&=2n^2+3mn+m^2+2,\\ C&=3n^2+mn+2m^2+1\end{align*}$$ have a common divisor greater than $1$.
6. Find all four-digit numbers such that when decomposed into prime factors, each number has the sum of its prime factors equal to the sum of the exponents.
7. Find all the pairs of positive integers $(m,n)$ such that the numbers $$\begin{align*}A&=n^2+2mn+3m^2+3n,\\ B&=2n^2+3mn+m^2,\\ C&=3n^2+mn+2m^2\end{align*}$$ are consecutive in some order.
8. Find all positive integers $a$, $b$ for which $a^4+4b^4$ is a prime number.
9. Find all the triples $(x,y,z)$ of positive integers such that $$xy+yz+zx-xyz=2.$$
10. Prove that there are no integers $x,y,z$ such that \[x^4+y^4+z^4-2x^2y^2-2y^2z^2-2z^2x^2=2000 \]
11. Prove that for any integer $n$ one can find integers $a$ and $b$ such that \[n=\left[ a\sqrt{2}\right]+\left[ b\sqrt{3}\right] \]
12. Consider a sequence of positive integers $x_n$ such that $$x_{2n+1}=4x_n+2n+2,\, x_{3n+2}=3x_{n+1}+6x_n,\,\forall n\ge 0.$$ Prove that $$x_{3n-1}=x_{n+2}-2x_{n+1}+10x_n,\,\forall n\ge 0.$$
13. Prove that \[ \sqrt{(1^k+2^k)(1^k+2^k+3^k)\ldots (1^k+2^k+\ldots +n^k)}\] \[ \ge 1^k+2^k+\ldots +n^k-\frac{2^{k-1}+2\cdot 3^{k-1}+\ldots + (n-1)\cdot n^{k-1}}{n}\] for all integers $n,k \ge 2$.
14. Let $m$ and $n$ be positive integers with $m\le 2000$ and $k=3-\frac{m}{n}$. Find the smallest positive value of $k$.
15. Let $x,y,a,b$ be positive real numbers such that $x\not= y$, $x\not= 2y$, $y\not= 2x$, $a\not=3b$ and $\dfrac{2x-y}{2y-x}=\frac{a+3b}{a-3b}$. Prove that $$\frac{x^2+y^2}{x^2-y^2}\ge 1.$$
16. Find all the triples $(x,y,z)$ of real numbers such that \[2x\sqrt{y-1}+2y\sqrt{z-1}+2z\sqrt{x-1} \ge xy+yz+zx \]
17. A triangle $ABC$ is given. Find all the pairs of points $X,Y$ so that $X$ is on the sides of the triangle, $Y$ is inside the triangle, and four non-intersecting segments $XY$, $AX$, $AY$, $BX$, $BY$, $CX$, $CY$ divide the triangle $ABC$ into four triangles with equal areas.
18. A triangle $ABC$ is given. Find all the segments $XY$ that lie inside the triangle such that $XY$ and five of the segments $XA$, $XB$, $XC$, $YA$, $YB$, $YC$ divide the triangle $ABC$ into $5$ regions with equal areas. Furthermore, prove that all the segments $XY$ have a common point.
19. Let $ABC$ be a triangle. Find all the triangles $XYZ$ with vertices inside triangle $ABC$ such that $XY$, $YZ$, $ZX$ and six non-intersecting segments from the following $AX$, $AY$, $AZ$, $BX$, $BY$, $BZ$, $CX$, $CY$, $CZ$ divide the triangle $ABC$ into seven regions with equal areas.
20. Let $ABC$ be a triangle and let $a,b,c$ be the lengths of the sides $BC, CA, AB$ respectively. Consider a triangle $DEF$ with the side lengths $EF=\sqrt{au}$, $FD=\sqrt{bu}$, $DE=\sqrt{cu}$. Prove that $$\angle A >\angle B >\angle C \implies \angle A >\angle D >\angle E >\angle F >\angle C.$$
21. All the angles of the hexagon $ABCDEF$ are equal. Prove that \[AB-DE=EF-BC=CD-FA \]
22. Consider a quadrilateral with $$\angle DAB=60^{\circ},\quad \angle ABC=90^{\circ},\quad \angle BCD=120^{\circ}.$$ The diagonals $AC$ and $BD$ intersect at $M$. If $MB=1$ and $MD=2$, find the area of the quadrilateral $ABCD$.
23. The point $P$ is inside of an equilateral triangle with side length $10$ so that the distance from $P$ to two of the sides are $1$ and $3$. Find the distance from $P$ to the third side.