## [Shortlists & Solutions] Junior Balkan Mathematical Olympiad 2016

### Algebra

1. Let $a, b, c$ be positive real numbers such that $abc = 8$. Prove that $$\frac{ab + 4}{a + 2}+\frac{bc + 4}{b + 2}+\frac{ca + 4}{c + 2}\ge 6.$$
2. Let $a,b,c$ be positive real numbers. Prove that $$\frac{8}{(a+b)^2 + 4abc} + \frac{8}{(b+c)^2 + 4abc} + \frac{8}{(a+c)^2 + 4abc} + a^2 + b^2 + c ^2 \\ \ge \frac{8}{a+3} + \frac{8}{b+3} + \frac{8}{c+3}$$
3. Find all the pairs of integers $(m, n)$ such that $$\sqrt {n +\sqrt {2016}} +\sqrt {m-\sqrt {2016}} \in \mathbb {Q}.$$
4. If the non-negative reals $x,y,z$ satisfy $x^2+y^2+z^2=x+y+z$. Prove that $$\displaystyle\frac{x+1}{\sqrt{x^5+x+1}}+\frac{y+1}{\sqrt{y^5+y+1}}+\frac{z+1}{\sqrt{z^5+z+1}}\geq 3.$$ When does the equality occur?
5. Let $x,y,z$ be positive real numbers such that $$x+y+z=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}.$$ Prove that $x+y+z\geq \sqrt{\frac{xy+1}{2}}+\sqrt{\frac{yz+1}{2}}+\sqrt{\frac{zx+1}{2}} \ .$

### Combinatorics

1. Let $S_n$ be the sum of reciprocal values of non-zero digits of all positive integers up to (and including) $n$. For instance, $$S_{13} = \frac{1}{1}+ \frac{1}{2}+ \frac{1}{3}+ \frac{1}{4}+ \frac{1}{5}+ \frac{1}{6}+ \frac{1}{7}+ \frac{1}{8}+ \frac{1}{9}+ \frac{1}{1}+ \frac{1}{1}+ \frac{1}{1}+ \frac{1}{1}+ \frac{1}{2}+ \frac{1}{1}+ \frac{1}{3}.$$ Find the least positive integer $k$ making the number $k!\cdot S_{2016}$ an integer.
2. The natural numbers from $1$ to $50$ are written down on the blackboard. At least how many of them should be deleted, in order that the sum of any two of the remaining numbers is not a prime?
3. A $5 \times 5$ table is called regular if each of its cells contains one of four pairwise distinct real numbers, such that each of them occurs exactly one in every $2 \times 2$ subtable. The sum of all numbers of a regular table is called the total sum of the table. With any four numbers, one constructs all possible regular tables, computes their total sums and counts the distinct outcomes.Determine the maximum possible count.
4. A splitting of a planar polygon is a finite set of triangles whose interiors are pairwise disjoint, and whose union is the polygon in question. Given an integer $n \ge 3$, determine the largest integer $m$ such that no planar $n$-gon splits into less than $m$ triangles.

### Geometry

1. Let ${ABC}$ be an acute angled triangle, let ${O}$ be its circumcentre, and let $D$, $E$, $F$ be points on the sides $BC$, $CA$, $AB$, respectively. The circle ${(c_1)}$ of radius ${FA}$, centered at ${F}$, crosses the segment ${OA}$ at ${A'}$ and the circumcircle ${(c)}$ of the triangle ${ABC}$again at ${K}$. Similarly, the circle ${(c_2)}$ of radius $DB$, centered at $D$, crosses the segment $\left( OB \right)$ at ${B}'$ and the circle ${(c)}$ again at ${L}$. Finally, the circle ${(c_3)}$ of radius $EC$, centered at $E$, crosses the segment $\left( OC \right)$at ${C}'$ and the circle ${(c)}$ again at ${M}$. Prove that the quadrilaterals $BKF{A}'$, $CLD{B}'$ and $AME{C}'$ are all cyclic, and their circumcircles share a common point.
2. Let ${ABC}$ be a triangle with $\angle BAC={{60}^{{}^\circ }}$. Let $D$ and $E$ be the feet of the perpendiculars from ${A}$ to the external angle bisectors of $\angle ABC$ and $\angle ACB$, respectively. Let ${O}$ be the circumcenter of the triangle ${ABC}$. Prove that the circumcircles of the triangles ${ADE}$and ${BOC}$ are tangent to each other.
3. A trapezoid $ABCD$ ($AB || CF$, $AB > CD$) is circumscribed. The incircle of the triangle $ABC$ touches the lines $AB$ and $AC$ at the points $M$ and $N$, respectively. Prove that the incenter of the trapezoid $ABCD$ lies on the line $MN$.
4. Let ${ABC}$ be an acute angled triangle whose shortest side is ${BC}$. Consider a variable point ${P}$ on the side ${BC}$, and let ${D}$ and ${E}$ be points on ${AB}$ and ${AC}$, respectively, such that ${BD=BP}$ and ${CP=CE}$. Prove that, as ${P}$ traces ${BC}$, the circumcircle of the triangle ${ADE}$ passes through a fixed point.
5. Let $ABC$ be an acute angled triangle with orthocenter ${H}$ and circumcenter ${O}$. Assume the circumcenter ${X}$ of ${BHC}$lies on the circumcircle of ${ABC}$. Reflect O across ${X}$ to obtain ${O'}$, and let the lines ${XH}$ and ${O'A}$ meet at ${K}$. Let $L,M$ and $N$ be the midpoints of $\left[ XB \right]$, $\left[ XC \right]$ and $\left[ BC \right]$, respectively. Prove that the points $K$, $L$, $M$ and $K$, $L$, $M$, $N$ are cocyclic.
6. Given an acute triangle ${ABC}$, erect triangles ${ABD}$ and ${ACE}$ externally, so that $\angle ADB= \angle AEC=90^\circ$ and $\angle BAD= \angle CAE$. Let ${{A}_{1}}\in BC$, ${{B}_{1}}\in AC$ and ${{C}_{1}}\in AB$ be the feet of the altitudes of the triangle ${ABC}$, and let $K$ and $K$, $L$ be the midpoints of $[ B{{C}_{1}} ]$ and ${BC_1, CB_1}$, respectively. Prove that the circumcenters of the triangles $AKL$, ${{A}_{1}}{{B}_{1}}{{C}_{1}}$ and ${DEA_1}$ are collinear.
7. Let ${AB}$ be a chord of a circle ${(c)}$ centered at ${O}$, and let ${K}$ be a point on the segment ${AB}$ such that ${AK<BK}$. Two circles through ${K}$, internally tangent to ${(c)}$ at ${A}$ and ${B}$, respectively, meet again at ${L}$. Let ${P}$ be one of the points of intersection of the line ${KL}$and the circle ${(c)}$, and let the lines ${AB}$and ${LO}$meet at ${M}$. Prove that the line ${MP}$is tangent to the circle ${(c)}$.

### Number Theory

1. Determine the largest positive integer $n$ that divides $p^6 - 1$ for all primes $p > 7$.
2. Find the maximum number of natural numbers $x_1,x_2, ... , x_m$ satisfying the conditions
• No $x_i - x_j , 1 \le i < j \le m$ is divisible by $11$, and
• The sum $x_2x_3 ...x_m + x_1x_3 ... x_m + \cdot \cdot \cdot + x_1x_2... x_{m-1}$ is divisible by $11$.
3. Find all positive integers $n$ such that the number $A_n =\frac{ 2^{4n+2}+1}{65}$ is
a) an integer,
b) a prime.
4. Find all triplets of integers $(a,b,c)$ such that the number $$N = \frac{(a-b)(b-c)(c-a)}{2} + 2$$ is a power of $2016$. (A power of $2016$ is an integer of form $2016^n$ where $n$ is a non-negative integer.)
5. Determine all four-digit numbers $\overline{abcd}$ such that $$(a + b)(a + c)(a + d)(b + c)(b + d)(c + d) =\overline{abcd}$$