In a plane a set of $n\geq 3$ points is given. Each pair of points is connected by a segment. Let $d$ be the length of the longest of these segments. We define a diameter of the set to be any connecting segment of length $d$. Prove that the number of diameters of the given set is at most $n$.

Find all odd prime $p$ such that $1+k(p-1)$ is prime for all integer $k$ where $1 \le k \le \dfrac{p-1}{2}$.

For each positive integer $k$, let $S(k)$ the sum of digits of $k$ in decimal system. Show that there is an integer $k$, with no $9$ in it's decimal representation, such that: $$S(2^{24^{2017}}k)=S(k)$$

A $4\times 4$ window is made out of $16$ square windowpanes. How many ways are there to stain each of the windowpanes, red, pink, or magenta, such that each windowpane is the same color as exactly two of its neighbors? Two different windowpanes are neighbors if they share a side.

The base of a new rectangle equals the sum of the diagonal and the greater side of a given rectangle, while the altitude of the new rectangle equals the difference of the diagonal and the greater side of the given rectangle. The area of the new rectangle is: $ \textbf{(A)}$ greater than the area of the given rectangle $ \textbf{(B)}$ equal to the area of the given rectangle $ \textbf{(C)}$ equal to the area of a square with its side equal to the smaller side of the given rectangle $ \textbf{(D)}$ equal to the area of a square with its side equal to the greater side of the given rectangle $ \textbf{(E)}$ equal to the area of a rectangle whose dimensions are the diagonal and the shorter side of the given rectangle

(a) Find all positive integers $ n$ for which $ 2^n\minus{}1$ is divisible by $ 7$. (b) Prove that there is no positive integer $ n$ for which $ 2^n\plus{}1$ is divisible by $ 7$.

Let P be a finite, partially ordered set with one largest element, which is the only upper bound of the set of minimal elements. Prove that any monotonic function $f : P^n\to P$ can be written in the form $g( x_1 , x_2 , ..., x_n , c_1 , ..., c_m )$, where $c_i\in P$ and g is a monotonic, idempotent function. (g is idempotent iff $g(x , x , ..., x) = x\,\forall x\in P$)

Let $f:\mathbb{R}\to\mathbb{R}$ be a continuous function such that for all $x\in\mathbb{R}$ and for all $t\geqslant 0$, $$f(x)=f(e^tx)$$ Show that $f$ is a constant function.

For two positive integers $a$ and $b$, Ivica and Marica play the following game: Given two piles of $a$ and $b$ cookies, on each turn a player takes $2n$ cookies from one of the piles, of which he eats $n$ and puts $n$ of them on the other pile. Number $n$ is arbitrary in every move. Players take turns alternatively, with Ivica going first. The player who cannot make a move, loses. Assuming both players play perfectly, determine all pairs of numbers $(a, b)$ for which Marica has a winning strategy. Proposed by Petar Orlić

The sequence of real numbers $a_1,a_2,...,a_{2015}$ is such that the 2015 equations: $a_1^3=a_1^2;a_1^3+a_2^3=(a_1+a_2 )^2;...;a_1^3+a_2^3+...+a_{2015}^3=(a_1+a_2+...+a_{2015} )^2$ are true. Prove that $a_1,a_2,…,a_{2015}$ are integers.

In triangle $ABC$ $(\angle A\neq 90^\circ)$, let $O$, $H$ be the circumcenter and the foot of the altitude from $A$ respectively. Suppose $M$, $N$ are the midpoints of $BC$, $AH$ respectively. Let $D$ be the intersection of $AO$ and $BC$ and let $H'$ be the reflection of $H$ about $M$. Suppose that the circumcircle of $OH'D$ intersects the circumcircle of $BOC$ at $E$. Prove that $NO$ and $AE$ are concurrent on the circumcircle of $BOC$. [i]Proposed by Mehran Talaei[/i]

Let a triangle $ABC$ be given with $AB <AC$. Let the inscribed center of the triangle be $I$. The perpendicular bisector of side $BC$ intersects the angle bisector of $BAC$ at point $S$ and the angle bisector of $CBA$ at point $T$. Prove that the points $C, I, S$ and $T$ lie on a circle. (Karl Czakler)

Let $ABC$ be a triangle with incentre $I$ and circumcircle $\Omega$. Let $D, E, F$ be the midpoints of the arcs $\stackrel{\frown}{BC}, \stackrel{\frown}{CA}, \stackrel{\frown}{AB}$ of $\Omega$ not containing $A, B, C$ respectively. Let $D'$ be the point of $\Omega$ diametrically opposite to $D$. Show that $I, D'$ and the midpoint $M$ of $EF$ lie on a line.

Let $d$ be one of the common tangent lines of externally tangent circles $k_1$ and $k_2$. $d$ touches $k_1$ at $A$. Let $[AB]$ be a diameter of $k_1$. The tangent from $B$ to $k_2$ touches $k_2$ at $C$. If $|AB|=8$ and the diameter of $k_2$ is $7$, then what is $|BC|$? $ \textbf{(A)}\ 7 \qquad\textbf{(B)}\ 6\sqrt 2 \qquad\textbf{(C)}\ 10 \qquad\textbf{(D)}\ 8 \qquad\textbf{(E)}\ 5\sqrt 3 $

If the coefficient of the third term in the binomial expansion of $(1 - 3x)^{1/4}$ is $-a/b$, where $ a$ and $b$ are relatively prime integers, find $a+b$.

Find all solutions of the following system of $n$ equations in $n$ variables: \[\begin{array}{c}\ x_1|x_1| - (x_1 - a)|x_1 - a| = x_2|x_2|,x_2|x_2| - (x_2 - a)|x_2 - a| = x_3|x_3|,\ \vdots \ x_n|x_n| - (x_n - a)|x_n - a| = x_1|x_1|\end{array}\] where $a$ is a given number.

The numbers $1,2,...,n$ ($n \ge 5$) are written on the circle in the clockwise order. Per move it is allowed to exchange any couple of consecutive numbers $a, b$ to the couple $\frac{a+b}{2}, \frac{a+b}{2}$. Is it possible to make all numbers equal using these operations?

Johan is swimming laps in the pool. At $\text{12:17}$ he realized that he had just finished one-third of his workout. By $\text{12:22}$ he had completed eight more laps, and he realized that he had just finished five-elevenths of his workout. After $\text{12:22}$ how many more laps must Johan swim to complete his workout?

A wheel is rolled without slipping through $15$ laps on a circular race course with radius $7$. The wheel is perfectly circular and has radius $5$. After the three laps, how many revolutions around its axis has the wheel been turned through?

There is a safe that can be opened by entering a secret code consisting of $n$ digits, each of them is $0$ or $1$. Initially, $n$ zeros were entered, and the safe is closed (so, all zeros is not the secret code). In one attempt, you can enter an arbitrary sequence of $n$ digits, each of them is $0$ or $1$. If the entered sequence matches the secret code, the safe will open. If the entered sequence matches the secret code in more positions than the previously entered sequence, you will hear a click. In any other cases the safe will remain locked and there will be no click. Find the smallest number of attempts that is sufficient to open the safe in all cases.

How many two-digit positive integers $ N$ have the property that the sum of $ N$ and the number obtained by reversing the order of the digits of $ N$ is a perfect square? $ \textbf{(A)}\ 4\qquad \textbf{(B)}\ 5\qquad \textbf{(C)}\ 6\qquad \textbf{(D)}\ 7\qquad \textbf{(E)}\ 8$

Let $a_1,a_2,a_3,a_4$ be positive real numbers satisfying \[\sum_{i<j}a_ia_j=1.\]Prove that \[\sum_{\text{sym}}\frac{a_1a_2}{1+a_3a_4}\geq\frac{6}{7}.\][i]* * *[/i]

Find the number of triples $(a, b, c)$ of positive integers such that (a) $a b$ is a prime; (b) $b c$ is a product of two primes; (c) $a b c$ is not divisible by square of any prime and (d) $a b c \leq 30$.

Prove that there do not exist pairwise distinct complex numbers $a, b, c,$ and $d$ such that $$a^3-bcd=b^3-acd=c^3-abd=d^3-abc.$$

On a $49\times 69$ rectangle formed by a grid of lattice squares, all $50\cdot 70$ lattice points are colored blue. Two persons play the following game: In each step, a player colors two blue points red, and draws a segment between these two points. (Different segments can intersect in their interior.) Segments are drawn this way until all formerly blue points are colored red. At this moment, the first player directs all segments drawn - i. e., he takes every segment AB, and replaces it either by the vector $\overrightarrow{AB}$, or by the vector $\overrightarrow{BA}$. If the first player succeeds to direct all the segments drawn in such a way that the sum of the resulting vectors is $\overrightarrow{0}$, then he wins; else, the second player wins. Which player has a winning strategy?