Junior Balkan Mathematical Olympiad 2001 [Shortlists & Solutions]

1. Find the positive integers $n$ that are not divisible by $3$ if the number $2^{n^2-10}+2133$ is a perfect cube.
2. Let $P_n \ (n=3,4,5,6,7)$ be the set of positive integers $n^k+n^l+n^m$, where $k,l,m$ are positive integers. Find $n$ such that a) In the set $P_n$ there are infinitely many squares. b) In the set $P_n$ there are no squares.
   
3. Find all the three-digit numbers $\overline{abc}$ such that the $6003$-digit number $\overline{abcabc\ldots abc}$ is divisible by $91$.
4. The discriminant of the equation $x^2-ax+b=0$ is the square of a rational number and $a$ and $b$ are integers. Prove that the roots of the equation are integers.
5. Let $x_k=\dfrac{k(k+1)}{2}$ for all integers $k\ge 1$. Prove that for any integer $n \ge 10$, between the numbers $A=x_1+x_2 + \ldots + x_{n-1}$ and $B=A+x_n$ there is at least one square.
6. Find all integers $x$ and $y$ such that $x^3\pm y^3 =2001p$, where $p$ is prime.
7. Prove that there are are no positive integers $x$ and $y$ such that $$x^5+y^5+1=(x+2)^5+(y-3)^5.$$
8. Prove that no three points with integer coordinates can be the vertices of an equilateral triangle.
9. Consider a convex quadrilateral $ABCD$ with $AB=CD$ and $\angle BAC=30^{\circ}$. Prove that if $\angle ADC=150^{\circ}$ then $\angle BCA= \angle ACD$.
10. A triangle $ABC$ is inscribed in the circle $\mathcal{C}(O,R)$. Let $\alpha <1$ be the ratio of the radii of the circles tangent to $\mathcal{C}$, and both of the rays $(AB$ and $(AC$. The numbers $\beta <1$ and $\gamma <1$ are defined analogously. Prove that $$\alpha + \beta + \gamma =1.$$
11. Consider a triangle $ABC$ with $AB=AC$, and $D$ the foot of the altitude from the vertex $A$. The point $E$ lies on the side $AB$ such that $\angle ACE= \angle ECB=18^{\circ}$. If $AD=3$, find the length of the segment $CE$.
12. Consider the triangle $ABC$ with $\angle A= 90^{\circ}$ and $\angle B \not= \angle C$. A circle $\mathcal{C}(O,R)$ passes through $B$ and $C$ and intersects the sides $AB$ and $AC$ at $D$ and $E$, respectively. Let $S$ be the foot of the perpendicular from $A$ to $BC$ and let $K$ be the intersection point of $AS$ with the segment $DE$. If $M$ is the midpoint of $BC$. Prove that $AKOM$ is a parallelogram.
13. At a conference there are $n$ mathematicians. Each of them knows exactly $k$ fellow mathematicians. Find the smallest value of $k$ such that there are at least three mathematicians that are acquainted each with the other two. Find the smallest value of $k$ such that there (always) exists $3$ mathematicians $X,Y,Z$ such that $X$ and $Y$ know each other, $X$ and $Z$ know each other and $Y$ and $Z$ know each other.

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