An integer $n > 1$ is called $\emph{good}$ if there exists a permutation $a_1, a_2, a_3, \dots, a_n$ of the numbers $1, 2, 3, \dots, n$, such that: $(i)$ $a_i$ and $a_{i+1}$ have different parities for every $1 \leq i \leq n-1$; $(ii)$ the sum $a_1 + a_2 + \cdots + a_k$ is a quadratic residue modulo $n$ for every $1 \leq k \leq n$. Prove that there exist infinitely many good numbers, as well as infinitely many positive integers which are not good.

The integers $1, 2, \dots, 64$ are written in the squares of a $8 \times 8$ chess board, such that for each $1 \le i < 64$, the numbers $i$ and $i+1$ are in squares that share an edge. What is the largest possible sum that can appear along one of the diagonals?

Given a cyclic quadriteral $ABCD$. The circumcenter lies in the quadriteral $ABCD$. Diagonals $AC$ and $BD$ intersects at $S$. Points $P$ and $Q$ are the midpoints of $AD$ and $BC$. Let $p$ be a line perpendicular to $AC$ through $P$, $q$ perpendicular line to $BD$ through $Q$ and $s$ perpendicular to $CD$ through $S$. Prove that $p,q,s$ intersects at one point.

A four-pointed star is formed by placing for equilateral triangles of side length $4$ in a coordinate grid. The triangles are placed such that their bases lie along one of the coordinate axes, with the midpoint of the bases lying at the origin, and such that the vertices opposite the bases lie at four distinct points. Compute the area contained within the star.

Let $S$ be the set of all partitions of $2000$ (in a sum of positive integers). For every such partition $p$, we define $f (p)$ to be the sum of the number of summands in $p$ and the maximal summand in $p$. Compute the minimum of $f (p)$ when $p \in S .$

Given a set $S \subset N$ and a positive integer n, let $S\oplus \{n\} = \{s+n / s \in S\}$. The sequence $S_k$ of sets is defined inductively as follows: $S_1 = {1}$, $S_k=(S_{k-1} \oplus \{k\}) \cup \{2k-1\}$ for $k = 2,3,4, ...$ (a) Determine $N - \cup _{k=1}^{\infty} S_k$. (b) Find all $n$ for which $1994 \in S_n$.

Let $A_1,B_1,C_1$ be the midpoints of the sides $BC,CA,AB$ respectively of a triangle $ABC$. Points $K$ on segment $C_1A_1$ and $L$ on segment $A_1B_1$ are taken such that $$\frac{C_1K}{KA_1}=\frac{BC+AC}{AC+AB}\enspace\enspace\text{and}\enspace\enspace\frac{A_1L}{LB_1}=\frac{AC+AB}{BC+AB}.$$If $BK$ and $CL$ meet at $S$, prove that $\angle C_1A_1S=\angle B_1A_1S$.

Let $P_1, P_2,..., P_n$ be a convex $n$-gon. If all lines $P_iP_j$ are joined, what is the maximum possible number of intersections in terms of $n$ obtained from strictly inside the polygon?

Let $S$ be a subset of the set $\{1, 2, 3, \dots, 2015\}$ such that for any two elements $a, b \in S$, the difference $a - b$ does not divide the sum $a + b$. Find the maximum possible size of $S$.

Find all positive integers $n$ such that \[n=d_6^2+d_7^2-1,\] where $1 = d_1 < d_2 < \cdots < d_k = n$ are all positive divisors of the number $n.$

Find the largest divisor of $2800$ that is a perfect square.

A necklace contains $2016$ pearls, each of which has one of the colours black, green or blue. In each step we replace simultaneously each pearl with a new pearl, where the colour of the new pearl is determined as follows: If the two original neighbours were of the same colour, the new pearl has their colour. If the neighbours had two different colours, the new pearl has the third colour. (a) Is there such a necklace that can be transformed with such steps to a necklace of blue pearls if half of the pearls were black and half of the pearls were green at the start? (b) Is there such a necklace that can be transformed with such steps to a necklace of blue pearls if thousand of the pearls were black at the start and the rest green? (c) Is it possible to transform a necklace that contains exactly two adjacent black pearls and $2014$ blue pearls to a necklace that contains one green pearl and $2015$ blue pearls? Proposed byTheresia Eisenkölbl

Let $ x $ be a positive real number such that the numbers $ x^{-1} $, $ x $, and $ x^{2018} $ have the same fractional part: $$ \{x^{-1}\} = \{x\} = \{x^{2018}\}. $$ Prove that $ x = 1 $. [b]Note:[/b] If $ x $ is a real number, its fractional part is $ \{x\} = x - \lfloor x \rfloor $, where $ \lfloor x \rfloor $ denotes the greatest integer less than or equal to $ x $.

Find all non-negative integers $k,n$ which satisfy $2^{2k+1} + 9\cdot 2^k+5=n^2$.

Let $a$ and $ b$ be even numbers, such that $M = (a + b)^2-ab$ is a multiple of $5$. Consider the following statements: I) The unit digits of $a^3$ and $b^3$ are different. II) $M$ is divisible by $100$. Please indicate which of the above statements are true with certainty.

Line $\ell$ intersects sides $BC$ and $AD$ of cyclic quadrilateral $ABCD$ in its interior points $R$ and $S$, respectively, and intersects ray $DC$ beyond point $C$ at $Q$, and ray $BA$ beyond point $A$ at $P$. Circumcircles of the triangles $QCR$ and $QDS$ intersect at $N \neq Q$, while circumcircles of the triangles $PAS$ and $PBR$ intersect at $M\neq P$. Let lines $MP$ and $NQ$ meet at point $X$, lines $AB$ and $CD$ meet at point $K$ and lines $BC$ and $AD$ meet at point $L$. Prove that point $X$ lies on line $KL$.

Given an acute triangle $ABC$, let $M$ be the midpoint of $BC$ and $H$ the orthocentre. Let $\Gamma$ be the circle with diameter $HM$, and let $X,Y$ be distinct points on $\Gamma$ such that $AX,AY$ are tangent to $\Gamma$. Prove that $BXYC$ is cyclic.

Show that the infinite arithmetic progression $\{1,4,7,10 \ldots\}$ has infinitely many 3 -term sub sequences in harmonic progression such that for any two such triples $\{a_1, a_2 , a_3 \}$ and $\{b_1, b_2 ,b_3\}$ in harmonic progression , one has $$\frac{a_1} {b_1} \ne \frac {a_2}{b_2}$$.

In the set $ S_n \equal{} \{1, 2,\ldots ,n\}$ a new multiplication $ a*b$ is defined with the following properties: [b](i)[/b] $ c \equal{} a * b$ is in $ S_n$ for any $ a \in S_n, b \in S_n.$ [b](ii)[/b] If the ordinary product $ a \cdot b$ is less than or equal to $ n,$ then $ a*b \equal{} a \cdot b.$ [b](iii)[/b] The ordinary rules of multiplication hold for $ *,$ i.e.: [b](1)[/b] $ a * b \equal{} b * a$ (commutativity) [b](2)[/b] $ (a * b) * c \equal{} a * (b * c)$ (associativity) [b](3)[/b] If $ a * b \equal{} a * c$ then $ b \equal{} c$ (cancellation law). Find a suitable multiplication table for the new product for $ n \equal{} 11$ and $ n \equal{} 12.$

You are given three lists A, B, and C. List A contains the numbers of the form $10^{k}$ in base 10, with $k$ any integer greater than or equal to 1. Lists B and C contain the same numbers translated into base 2 and 5 respectively: \[\begin{array}{lll}A & B & C \\ 10 & 1010 & 20 \\ 100 & 1100100 & 400 \\ 1000 & 1111101000 & 13000 \\ \vdots & \vdots & \vdots \end{array}.\] Prove that for every integer $n > 1$, there is exactly one number in exactly one of the lists B or C that has exactly $n$ digits.

Let $ f(x)$ be a $ n \minus{}$degree polynomial all of whose coefficients are equal to $ \pm 1$, and having $ x \equal{} 1$ as its $ m$ multiple root. If $ m\ge 2^k (k\ge 2,k\in N)$, then $ n\ge 2^{k \plus{} 1} \minus{} 1.$

Sum the series \[\sum_{m=1}^\infty\sum_{n=1}^\infty\dfrac{m^2n}{3^m(n3^m+m3^n)}.\]

Find the smallest positive real number $k$ such that the following inequality holds $$\left|z_{1}+\ldots+z_{n}\right| \geqslant \frac{1}{k}\big(\left|z_{1}\right|+\ldots+\left|z_{n}\right|\big) .$$ for every positive integer $n \geqslant 2$ and every choice $z_{1}, \ldots, z_{n}$ of complex numbers with non-negative real and imaginary parts. [Hint: First find $k$ that works for $n=2$. Then show that the same $k$ works for any $n \geqslant 2$.]

Let $p$ be a prime number. Determine the maximal degree of a polynomial $T(x)$ whose coefficients belong to $\{ 0,1,\cdots,p-1 \}$, whose degree is less than $p$, and which satisfies \[T(n)=T(m) \; \pmod{p}\Longrightarrow n=m \; \pmod{p}\] for all integers $n, m$.

Let $d\leq n$ be positive integers and $A$ a real $d\times n$ matrix. Let $\sigma(A)$ be the supremum of $\inf_{v\in W,|v|=1}|Av|$ over all subspaces $W$ of $R^n$ with dimension $d$. For each $j \leq d$, let $r(j) \in \mathbb{R}^n$ be the $j$th row vector of $A$. Show that: \[\sigma(A) \leq \min_{i\leq d} d(r(i), \langle r(j), j\ne i\rangle) \leq \sqrt{n}\sigma(A)\] In which all are euclidian norms and $d(r(i), \langle r(j), j\ne i\rangle)$ denotes the distance between $r(i)$ and the span of $r(j), 1 \leq j \leq d, j\ne i$.