Find the smallest constant $C > 0$ for which the following statement holds: among any five positive real numbers $a_1,a_2,a_3,a_4,a_5$ (not necessarily distinct), one can always choose distinct subscripts $i,j,k,l$ such that \[ \left| \frac{a_i}{a_j} - \frac {a_k}{a_l} \right| \le C. \]

Let $u$ and $v$ be real numbers such that \[ (u + u^2 + u^3 + \cdots + u^8) + 10u^9 = (v + v^2 + v^3 + \cdots + v^{10}) + 10v^{11} = 8. \] Determine, with proof, which of the two numbers, $u$ or $v$, is larger.

Determine the largest natural number $k$ such that there exists a natural number $n$ satisfying: \[ \sin(n + 1) < \sin(n + 2) < \sin(n + 3) < \ldots < \sin(n + k). \]

Given a string of length $2n$, we perform the following operation: [list] [*]Place all the even indexed positions together, and then all the odd indexed positions next. Indexing is done starting from $0$.[/*] [/list] For example, say our string is ``abcdef''. Performing our operation yields ``abcdef'' $\to$ ``acebdf''. Performing the operation again yields ``acebdf'' $\to$ ``aedcbf''. Doing this repeatedly, we have:\\ ``abcdef'' $\to$ ``acebdf'' $\to$ ``aedcbf'' $\to$ ``adbecf'' $\to$ ``abcdef''.\\\\ You can assume that the characters in the string will be unique. It can be shown that, by performing the above operation a finite number of times we can get back our original string.\\\\ Given $n$, you have to determine the minimum number of times the operation must be performed to get our original string of length $2n$ back.\\\\ In the example given above, $2n = 6$. The minimum steps required is $4$.

For an integer $n\geq 1$ let $a(n)$ denote the total number of carries which arise when adding $2017$ and $n\cdot 2017$. The first few values are given by $a(1)=1$, $a(2)=1$, $a(3)=0$, which can be seen from the following: \begin{align*} 001 &&001 && 000 \\ 2017 &&4034 &&6051 \\ +2017 &&+2017 &&+2017\\ =4034 &&=6051 &&=8068\\ \end{align*} Prove that $$a(1)+a(2)+...+a(10^{2017}-1)=10\cdot\frac{10^{2017}-1}{9}.$$

Let $m,n\geq 2$. One needs to cover the table $m \times n$ using only the following tiles: Tile 1 - A square $2 \times 2$. Tile 2 - A L-shaped tile with five cells, in other words, the square $3 \times 3$ [b]without[/b] the upper right square $2 \times 2$. Each tile 1 covers exactly $4$ cells and each tile 2 covers exactly $5$ cells. Rotation is allowed. Determine all pairs $(m,n)$, such that the covering is possible.

Find all functions $ f:\mathbb{Q}\longrightarrow\mathbb{Q} $ with the property that $$ f\left( p(x)\right) =p\left( f(x)\right) ,\quad\forall x\in\mathbb{Q} , $$ for all integer polynomials $ p. $

How many positive integers less than 2015 have exactly 9 positive integer divisors?

Let $x_1, x_2, \ldots, x_n$ be real positive numbers such that $x_1\cdot x_2\cdots x_n=1$. Prove the inequality $\frac1{x_1(x_1+1)}+\frac1{x_2(x_2+1)}+\cdots+\frac1{x_n(x_n+1)}\geq\frac n2$

The quadrilateral $ABCD$ has $AD = DC = CB < AB$ and $AB \parallel CD$. Points $E$ and $F$ lie on the sides $CD$ and $BC$ such that $\angle ADE = \angle AEF$. Prove that: (a) $4CF \le CB$. (b) If $4CF = CB$, then $AE$ is the angle bisector of $\angle DAF$.

The incircle of $\triangle ABC$ tangents $BC$, $CA$, $AB$ at $D$, $E$, $F$, respectively. The projections of $B$, $C$ to $AD$ are $U$, $V$, respectively; the projections of $C$, $A$ to $BE$ are $W$, $X$, respectively; and the projections of $A$, $B$ to $CF$ are $Y$, $Z$, respectively. Show that the circumcircle of the triangle formed by $UX$, $VY$, $WZ$ is tangent to the incircle of $\triangle ABC$.

Let $ABCD$ be a convex cyclic quadrilateral with $AD > BC$, A$B$ not being diameter and $C D$ belonging to the smallest arc $AB$ of the circumcircle. The rays $AD$ and $BC$ are cut at $K$, the diagonals $AC$ and $BD$ are cut at $P$ and the line $KP$ cuts the side $AB$ at point $L$. Prove that angle $\angle ALK$ is acute.

Let $\vartriangle ABC$ be an isosceles triangle with $AB = AC$. A circle passing through $B$ and $C$ intersects sides $AB$ and $AC$ at $D$ and $E$ respectively. A point $F$ on this circle is chosen so that $EF\perp BC$. If $BC = x$, $CF = y$, and $BF = z$, find the length of $DF$ in terms of $x, y, z$.

Find all positive integers $n$ such that $n^{2}+2^{n}$ is square of an integer.

Let $A$ be the set of four-digit natural numbers having exactly two distinct digits, none of which is zero. Interchanging the two digits of $n\in A$ yields a number $f (n) \in A$ (for instance, $f (3111) = 1333$). Find those $n \in A$ with $n > f (n)$ for which $gcd(n, f (n))$ is the largest possible.

Points $T,P,H$ lie on the side $BC,AC,AB$ respectively of triangle $ABC$, so that $BP$ and $AT$ are angle bisectors and $CH$ is an altitude of $ABC$. Given that the midpoint of $CH$ belongs to the segment $PT,$ find the value of $\cos A + \cos B$ I. Voronovich