Juniorska balkanska matematička olimpijada

Junior Balkan MO 2012 - Problem 4

Find all positive integers $x,y,z$ and $t$ such that $2^x3^y+5^z=7^t$

Junior Balkan MO 2010

Find all integers $n,n\ge 1$ ,such that $n\cdot 2^{n+1}+1$ is a perfect square

Junior Balkan MO 1997 - Problem 5

Let $n_1, n_2, \ldots, n_{1998}$ be positive integers such that $n_1^2 + n_2^2 + \cdots + n_{1997}^2 = n_{1998}^2.$ Show that at least two of the numbers are even.

Junior Balkan MO 2013 - Problem 1

Find all ordered pairs $(a,b)$ of positive integers for which the numbers $\dfrac{a^3b-1}{a+1}$ and $\dfrac{b^3a+1}{b-1}$ are both positive integers.

Junior Balkan MO 2013 - Problem 3

Let $ABC$ be an acute-angled triangle with $AB<AC$ and let $O$ be the centre of its circumcircle $\omega$ . Let $D$ be a point on the line segment $BC$ such that $\angle BAD = \angle CAO$ . Let $E$ be the second point of intersection of $\omega$ and the line $AD$ . If $M, N$ and $P$ are the midpoints of the line segments $BE, OD$ and $AC$ , respectively, show that the points $M, N$ and $P$ are collinear.

Junior Balkan MO 2013 - Problem 2

Show that
$\left(a+2b+\dfrac{2}{a+1}\right)\left(b+2a+\dfrac{2}{b+1}\right)\geq 16$
for all positive real numbers $a$ and $b$ such that $ab\geq 1$ .

Junior Balkan MO 2013 - Problem 4

Let $n$ be a positive integer. Two players, Alice and Bob, are playing the following game:
- Alice chooses n real numbers; not necessarily distinct.
- Alice writes all pairwise sums on a sheet of paper and gives it to Bob. (There are $\frac{n(n-1)}{2}$ such sums; not necessarily distinct.)
- Bob wins if he finds correctly the initial n numbers chosen by Alice with only one guess.
Can Bob be sure to win for the following cases?

a. $n=5$
b. $n=6$
c. $n=8$

Justify your answer(s).

[For example, when n=4, Alice may choose the numbers 1, 5, 7, 9, which have the same pairwise sums as the numbers 2, 4, 6, 10, and hence Bob cannot be sure to win.]

Junior Balkan MO 2012 - Problem 3

On a board there are $n$ nails, each two connected by a rope. Each rope is colored in one of $n$ given distinct colors. For each three distinct colors, there exist three nails connected with ropes of these three colors.
a) Can $n$ be $6$ ?
b) Can $n$ be $7$ ?

Junior Balkan MO 2012 - Problem 2

Let the circles $k_1$ and $k_2$ intersect at two points $A$ and $B$ , and let t be a common tangent of $k_1$ and $k_2$ that touches $k_1$ and $k_2$ at $M$ and $N$ respectively. If $t\perp AM$ and $MN=2AM$ , evaluate the angle $NMB$ .

Junior Balkan MO 2012 - Problem 1

Let $a,b,c$ be positive real numbers such that $a+b+c=1$ . Prove that $\frac{a}{b} + \frac{a}{c} + \frac{c}{b} + \frac{c}{a} + \frac{b}{c} + \frac{b}{a} + 6 \geq 2\sqrt{2} \cdot \left( \sqrt{\frac{1-a}{a}}+\sqrt{\frac{1-b}{b}}+\sqrt{\frac{1-c}{c}} \right)$ When does equality hold?

Junior Balkan MO 1997 - Problem 1

Show that given any $9$ points inside a square of side length $1$ we can always find $3$ that form a triangle with area less than $\frac{1}{8}$ .

Junior Balkan MO 1997 - Problem 2

Let $\frac{x^2+y^2}{x^2-y^2} + \frac{x^2-y^2}{x^2+y^2} = k$ . Compute the following expression in terms of $k$ : $E(x,y) = \frac{x^8 + y^8}{x^8-y^8} - \frac{ x^8-y^8}{x^8+y^8} \text.$

Junior Balkan MO 1997 - Problem 3

Let $ABC$ be a triangle and let $I$ be the incenter. Let $N$ , $M$ be the midpoints of the sides $AB$ and $CA$ respectively. The lines $BI$ and $CI$ meet $MN$ at $K$ and $L$ respectively. Prove that $|AI|+|BI|+|CI|>|BC|+|KL|$ .

Junior Balkan MO 1997 - Problem 4

Determine the triangle with sides $a,b,c$ and circumradius $R$ for which $R(b+c) = a\sqrt{bc}$ .

Junior Balkan MO 1998 - Problem 1

Prove that the number $\underbrace{111\ldots 11}_{1997}\underbrace{22\ldots 22}_{1998}5$ (which has 1997 of $1$ -s and 1998 of $2$ -s) is a perfect square.

Junior Balkan MO 1998 - Problem 2

Let $ABCDE$ be a convex pentagon such that $AB=AE=CD=1$ , $\angle ABC=\angle DEA=90^\circ$ and $|BC|+|DE|=1$ . Compute the area of the pentagon.

Junior Balkan MO 1998 - Problem 3

Find all pairs of positive integers $(x,y)$ such that $x^y = y^{x - y}.$

Junior Balkan MO 1998 - Problem 4

Does there exist $16$ three digit numbers, using only three different digits in all, so that the all numbers give different residues when divided by $16$ ?

Junior Balkan MO 1999 - Problem 1

Let $a,b,c,x,y$ be five real numbers such that $a^3 + ax + y = 0$ , $b^3 + bx + y = 0$ and $c^3 + cx + y = 0$ . If $a,b,c$ are all distinct numbers, prove that their sum is zero.

Junior Balkan MO 1999 - Problem 2

For each nonnegative integer $n$ we define $A_n = 2^{3n}+3^{6n+2}+5^{6n+2}$ . Find the greatest common divisor of the numbers $A_0,A_1,\ldots, A_{1999}$ .

Junior Balkan MO 1999 - Problem 3

Let $S$ be a square with the side length 20 and let $M$ be the set of points formed with the vertices of $S$ and another 1999 points lying inside $S$ . Prove that there exists a triangle with vertices in $M$ and with area at most equal with $\frac{1}{10}$ .

Junior Balkan MO 2000 - Problem 1

Let $x$ and $y$ be positive reals such that $x^3 + y^3 + (x + y)^3 + 30xy = 2000 \text.$ Show that $x + y = 10$ .

Junior Balkan MO 2000 - Problem 2

Find all positive integers $n\geq 1$ such that $n^2+3^n$ is the square of an integer.

Junior Balkan MO 2000 - Problem 3

A half-circle of diameter $\overline{EF}$ is placed on the side $\overline{BC}$ of a triangle $ABC$ and it is tangent to the sides $\overline{AB}$ and $\overline{AC}$ in the points $Q$ and $P$ respectively. Prove that the intersection point $K$ between the lines $EP$ and $FQ$ lies on the altitude from $A$ of the triangle $ABC$ .

Junior Balkan MO 2000 - Problem 4

At a tennis tournament there were $2n$ boys and $n$ girls participating. Every player played every other player. The boys won $\frac 75$ times as many matches as the girls. It is known that there were no draws. Find $n$ .

Junior Balkan MO 2001 - Problem 1

Solve the equation $a^3 + b^3 + c^3 = 2001$ in positive integers.

Junior Balkan MO 2001 - Problem 2

Let $ABC$ be a triangle with $\angle C = 90^\circ$ and $|CA| \ne |CB|$ . Let $\overline{CH}$ be an altitude and $\overline{CL}$ be an interior angle bisector. Show that for $X \ne C$ on the line $\overline{CL}$ , we have $\angle XAC \ne \angle XBC$ . Also show that for $Y \ne C$ on the line $\overline{CH}$ we have $\angle YAC \ne \angle YBC$ .

Junior Balkan MO 2001 - Problem 3

Let $ABC$ be an equilateral triangle and $D,E$ on the sides $\overline{AB}$ and $\overline{AC}$ respectively. If $\overline{DF},\overline{EF}$ (with $F \in \overline{AE}, G \in \overline{AD}$ ) are the interior angle bisectors of the angles of the triangle $ADE$ , prove that the sum of the areas of the triangles $DEF$ and $DEG$ is at most equal with the area of the triangle $ABC$ . When does the equality hold?

Junior Balkan MO 2001 - Problem 4

Let $N$ be a convex polygon with 1415 vertices and perimeter $2001$ . Prove that we can find 3 vertices of $N$ which form a triangle of area smaller than $1$ .

Junior Balkan MO 2002 - Problem 1

The triangle $ABC$ has $|CA| = |CB|$ . $P$ is a point on the circumcircle between $A$ and $B$ (and on the opposite side of the line $\overline{AB}$ to $C$ ). $D$ is the foot of the perpendicular from $C$ to $PB$ . Show that $|PA| + |PB| = 2 \cdot |PD|$ .