[url=https://artofproblemsolving.com/community/c675693][b]Macedonia JBMO TST 2017[/b][/url] [url=http://artofproblemsolving.com/community/c6h1663908p10569198][b]Problem 1[/b][/url]. Let $p$ be a prime number such that $3p+10$ is a sum of squares of six consecutive positive integers. Prove that $p-7$ is divisible by $36$. [url=http://artofproblemsolving.com/community/c6h1663916p10569261][b]Problem 2[/b][/url]. In the triangle $ABC$, the medians $AA_1$, $BB_1$, and $CC_1$ are concurrent at a point $T$ such that $BA_1=TA_1$. The points $C_2$ and $B_2$ are chosen on the extensions of $CC_1$ and $BB_2$, respectively, such that $$C_1C_2 = \frac{CC_1}{3} \quad \text{and} \quad B_1B_2 = \frac{BB_1}{3}.$$ Show that $TB_2AC_2$ is a rectangle. [url=http://artofproblemsolving.com/community/c6h1663918p10569305][b]Problem 3[/b][/url]. Let $x,y,z$ be positive reals such that $xyz=1$. Show that $$\frac{x^2+y^2+z}{x^2+2} + \frac{y^2+z^2+x}{y^2+2} + \frac{z^2+x^2+y}{z^2+2} \geq 3.$$ When does equality happen? [url=http://artofproblemsolving.com/community/c6h1663920p10569326][b]Problem 4[/b][/url]. In triangle $ABC$, the points $X$ and $Y$ are chosen on the arc $BC$ of the circumscribed circle of $ABC$ that doesn't contain $A$ so that $\measuredangle BAX = \measuredangle CAY$. Let $M$ be the midpoint of the segment $AX$. Show that $$BM + CM > AY.$$ [url=http://artofproblemsolving.com/community/c6h1663922p10569370][b]Problem 5[/b][/url]. Find all the positive integers $n$ so that $n$ has the same number of digits as its number of different prime factors and the sum of these different prime factors is equal to the sum of exponents of all these primes in factorization of $n$.

Let $ABC$ be an acute-angled scalene triangle. Inside the triangle, distinct points $A_1, B_1, C_1$ are chosen such that \[ \frac{AB_1}{CB_1} = \frac{AB}{CB} \quad \text{and} \quad \frac{AC_1}{BC_1} = \frac{AC}{BC}. \] Let $A_2, B_2, C_2$ be the reflections of $A_1, B_1, C_1$ across lines $BC, AC, AB$, respectively. These points satisfy the following conditions: [list] [*] The four points $A, A_2, B, C_2$ are concyclic. [*] The four points $A, A_2, B_2, C$ are concyclic. [*] The four points $B, B_2, C, C_2$ are concyclic. [*] The three points $A_2, B_2, C_2$ do not lie on the circumcircle of $\triangle ABC$. [/list] Prove that triangles $A_1B_1C_1$ and $A_2B_2C_2$ are similar.

If $x,y$ are positive real numbers satisfying $x^3-xy+1=y^3$, find the minimum possible value of $y$.

Let be a finite group $ G $ that has an element $ a\neq 1 $ for which exists a prime number $ p $ such that $ x^{1+p}=a^{-1}xa, $ for all $ x\in G. $ [b]a)[/b] Prove that the order of $ G $ is a power of $ p. $ [b]b)[/b] Show that $ H:=\{x\in G|\text{ord} (x)=p\}\le G $ and $ \text{ord}^2(H)>\text{ord}(G). $

Points $ A$ and $ B$ lie on a circle centered at $ O$, and $ \angle AOB\equal{}60^\circ$. A second circle is internally tangent to the first and tangent to both $ \overline{OA}$ and $ \overline{OB}$. What is the ratio of the area of the smaller circle to that of the larger circle? $ \textbf{(A)}\ \frac{1}{16} \qquad \textbf{(B)}\ \frac{1}{9} \qquad \textbf{(C)}\ \frac{1}{8} \qquad \textbf{(D)}\ \frac{1}{6} \qquad \textbf{(E)}\ \frac{1}{4}$

Let $n(n>1)$ be an odd. We define $x_k=(x^{(k)}_1,x^{(k)}_2,\cdots ,x^{(k)}_n)$ as follow: $x_0=(x^{(0)}_1,x^{(0)}_2,\cdots ,x^{(0)}_n)=(1,0,\cdots ,0,1)$; $ x^{(k)}_i =\begin{cases}0, \quad x^{(k-1)}_i=x^{(k-1)}_{i+1},\\ 1, \quad x^{(k-1)}_i\not= x^{(k-1)}_{i+1},\end{cases} $ $i=1,2,\cdots ,n$, where $x^{(k-1)}_{n+1}= x^{(k-1)}_1$. Let $m$ be a positive integer satisfying $x_0=x_m$. Prove that $m$ is divisible by $n$.

Is there any numbber $n$, such that the sum of its digits in the decimal notation is $1000$, and the sum of its square digits in the decimal notation is $1000000$?

Concentric circles $\Omega_1$ and $\Omega_2$ with radii $1$ and $100$, respectively, are drawn with center $O$. Points $A$ and $B$ are chosen independently at random on the circumferences of $\Omega_1$ and $\Omega_2$, respectively. Denote by $\ell$ the tangent line to $\Omega_1$ passing through $A$, and denote by $P$ the reflection of $B$ across $\ell$. Compute the expected value of $OP^2$. [i]Proposed by Lewis Chen[/i]

A small ship sails on an infinite coordinate sea. At the moment $t$ the ship is at the point with coordinates $(f(t), g(t))$, where $f$ and $g$ are two polynomials of third degree. Yesterday at $14:00$ the ship was at the same point as at $13:00$, and at $20:00$, it was at the same point as at $19:00$. Prove that the ship sails along a straight line.

Koshchey opened an account at the bank. Initially, it had 0 rubles. On the first day, Koshchey puts $k>0$ rubles in, and every next day adds one ruble more there than the day before. Each time after Koshchey deposits money into the account, the total amount in the account is divided by two by the bank. Find all such $k{}$ for which the amount on the account will always be an integer number of rubles. [i]Proposed by S. Berlov[/i]

Suppose that $I$ is the incenter of triangle $ABC$. The perpendicular to line $AI$ from point $I$ intersects sides $AC$ and $AB$ at points $B'$ and $C'$ respectively. Points $B_1$ and $C_1$ are placed on half lines $BC$ and $CB$ respectively, in such a way that $AB=BB_1$ and $AC=CC_1$. If $T$ is the second intersection point of the circumcircles of triangles $AB_1C'$ and $AC_1B'$, prove that the circumcenter of triangle $ATI$ lies on the line $BC$

Let $S$ be a subset of $\{0,1,2,\ldots,98 \}$ with exactly $m\geq 3$ (distinct) elements, such that for any $x,y\in S$ there exists $z\in S$ satisfying $x+y \equiv 2z \pmod{99}$. Determine all possible values of $m$.

For $n$ a positive integer, let $f(n)$ be the quotient obtained when the sum of all positive divisors of $n$ is divided by $n$. For example, \[f(14) = (1 + 2 + 7 + 14) \div 14 = \frac{12}{7}.\] What is $f(768) - f(384)?$ $\textbf{(A) }\frac{1}{768}\qquad\textbf{(B) }\frac{1}{192}\qquad\textbf{(C) }1\qquad\textbf{(D) }\frac{4}{3}\qquad\textbf{(E) }\frac{8}{3}$

In a football tournament there are $8$ teams, each of which plays exacly one match against every other team. If a team $A$ defeats team $B$, then $A$ is awarded $3$ points and $B$ gets $0$ points. If they end up in a tie, they receive $1$ point each. It turned out that in this tournament, whenever a match ended up in a tie, the two teams involved did not finish with the same final score. Find the maximum number of ties that could have happened in such a tournament.

Find the integers $a \ne 0, b$ and $c$ such that $x = 2 +\sqrt3$ would be a solution of the quadratic equation $ax^2 + bx + c = 0$.

Let $ABC$ be a triangle with $AB=13$, $BC=14$, $CA=15$. Let $O$ be the circumcenter of $ABC$. Find the distance between the circumcenters of triangles $AOB$ and $AOC$.

Let $ABC$ be an acute triangle. Construct a point $X$ on the different side of $C$ with respect to the line $AB$ and construct a point $Y$ on the different side of $B$ with respect to the line $AC$ such that $BX=AC$, $CY=AB$, and $AX=AY$. Let $A'$ be the reflection of $A$ across the perpendicular bisector of $BC$. Suppose that $X$ and $Y$ lie on different sides of the line $AA'$, prove that points $A$, $A'$, $X$, and $Y$ lie on a circle.

A sphere with center at $O$ is inscribed in a trihedral angle with vertex $S$. Prove that the plane passing through the three tangent points is perpendicular to $OS$.

Let $\omega_1$ and $\omega_2$ be two non-intersecting circles. Let one of its internal tangents touches $\omega_1$ and $\omega_2$ at $A_1$ and $A_2$, respectively, and let one of its external tangents touches $\omega_1$ and $\omega_2$ at $B_1$ and $B_2$, respectively. Prove that if $A_1B_2 = A_2B_1$, then $A_1B_2 \perp A_2B_1$.

Altitudes $AA_1$ and $BB_1$ are drawn in the acute-angled triangle $ABC$. Prove that the perpendicular drawn from the touchpoint of the inscribed circle with the side $BC$, on the line $AC$ passes through the center of the inscribed circle of the triangle $A_1CB_1$. (V. Protasov)

Find all functions $f: R \to R$ for which the image $f ([a, b])$ for all real $a \le b$ is (not necessarily closed!) interval of length $b - a$.

Suppose that $a$ and $ b$ are positive integers such that $$c = a +\frac{b}{a} -\frac{1}{b}$$ is an integer. Prove that $c$ is a perfect square.

Point $A$ is outside of a given circle $\omega$. Let the tangents from $A$ to $\omega$ meet $\omega$ at $S, T$ points $X, Y$ are midpoints of $AT, AS$ let the tangent from $X$ to $\omega$ meet $\omega$ at $R\neq T$. points $P, Q$ are midpoints of $XT, XR$ let $XY\cap PQ=K, SX\cap TK=L$ prove that quadrilateral $KRLQ$ is cyclic.

Find the smallest positive integer $n$, such that one can color every cell of a $n \times n$ grid in red, yellow or blue with all the following conditions satisfied: (1) the number of cells colored in each color is the same; (2) if a row contains a red cell, that row must contain a blue cell and cannot contain a yellow cell; (3) if a column contains a blue cell, it must contain a red cell but cannot contain a yellow cell.

Consider for a real number $t>1$, $I(t)=\int_{-4}^{4t-4} (x-4)\sqrt{x+4}\ dx.$ Find the minimum value of $I(t)\ (t>1).$