There are $2017$ jars in a row on a table, initially empty. Each day, a nice man picks ten consecutive jars and deposits one coin in each of the ten jars. Later, Kelvin the Frog comes back to see that $N$ of the jars all contain the same positive integer number of coins (i.e. there is an integer $d>0$ such that $N$ of the jars have exactly $d$ coins). What is the maximum possible value of $N$?

Let $\Gamma$ be a circle with centre $I$, and $A B C D$ a convex quadrilateral such that each of the segments $A B, B C, C D$ and $D A$ is tangent to $\Gamma$. Let $\Omega$ be the circumcircle of the triangle $A I C$. The extension of $B A$ beyond $A$ meets $\Omega$ at $X$, and the extension of $B C$ beyond $C$ meets $\Omega$ at $Z$. The extensions of $A D$ and $C D$ beyond $D$ meet $\Omega$ at $Y$ and $T$, respectively. Prove that \[A D+D T+T X+X A=C D+D Y+Y Z+Z C.\] [i]Proposed by Dominik Burek, Poland and Tomasz Ciesla, Poland[/i]

Let $ABC$ a triangle with $AB=BC$ and incircle $\omega$. Let $M$ the mindpoint of $BC$; $P, Q$ points in the sides $AB, AC$ such that $PQ\parallel BC$, $PQ$ is tangent to $\omega$ and $\angle CQM=\angle PQM$. Find the perimeter of triangle $ABC$ knowing that $AQ=1$.

A collection of regular polygons with sides of equal length is said to "fit" if, when arranged around a common vertex, they exactly complete the surrounding area of the point on the plane. For example, a square fits with two octagons. Determine all possible collections of regular polygons that fit.

For a given number $\alpha{}$ let $f_\alpha$ be a function defined as \[f_\alpha(x)=\left\lfloor\alpha x+\frac{1}{2}\right\rfloor.\]Let $\alpha>1$ and $\beta=1/\alpha$. Prove that for any natural $n{}$ the relation $f_\beta(f_\alpha(n))=n$ holds. [i]Proposed by I. Dorofeev[/i]

In an old script found in ruins of Perspolis is written: [code] This script has been finished in a year whose 13th power is 258145266804692077858261512663 You should know that if you are skilled in Arithmetics you will know the year this script is finished easily.[/code] Find the year the script is finished. Give a reason for your answer.

A set $A$ of integers is called [i]sum-full[/i] if $A \subseteq A + A$, i.e. each element $a \in A$ is the sum of some pair of (not necessarily different) elements $b,c \in A$. A set $A$ of integers is said to be [i]zero-sum-free[/i] if $0$ is the only integer that cannot be expressed as the sum of the elements of a finite nonempty subset of $A$. Does there exist a sum-full zero-sum-free set of integers? [i]Romania (Dan Schwarz)[/i]

If $a$ is a real root of $x^5 - x^3 + x - 2 = 0$, show that $[a^6] =3$

Suppose there are $4n$ line segments of unit length inside a circle of radius $n$. Furthermore, a straight line $L$ is given. Prove that there exists a straight line $L'$ that is either parallel or perpendicular to $L$ and that $L'$ cuts at least two of the given line segments.

Let $x_1,x_2,\ldots, x_{2007}$ be real numbers such that $-1\leq x_i\leq 1$ for $1\leq i\leq 2007$, and \[\sum_{i=1}^{2007}x_i^3 = 0.\] Find the maximum possible value of $\Big\lfloor\sum_{i=1}^{2007}x_i\Big\rfloor$.

Lines $l$ and $m$ are perpendicular. Line $l$ partitions a convex polygon into two parts of equal area, and partitions the projection of the polygon onto $m$ into two line segments of length $a$ and $b$ respectively. Determine the maximum value of $\left\lfloor \frac{1000a}{b} \right\rfloor$. (The floor notation $\lfloor x \rfloor$ denotes largest integer not exceeding $x$)

A net for hexagonal pyramid is constructed by placing a triangle with side lengths $x$, $x$, and $y$ on each side of a regular hexagon with side length $y$. What is the maximum volume of the pyramid formed by the net if $x+y=20$?

For some 2011 natural numbers, all the $\frac{2010\cdot 2011}{2}$ possible sums were written out on a board. Could it have happened that exactly one third of the written numbers were divisible by three and also exactly one third of them give a remainder of one when divided by three?

How many distinct triangles are there with prime side lengths and perimeter $100$? [i]Proposed by Muztaba Syed[/i] [hide=Solution][i]Solution.[/i] $\boxed{0}$ As the perimeter is even, $1$ of the sides must be $2$. Thus, the other $2$ sides are congruent by Triangle Inequality. Thus, for the perimeter to be $100$, both of the other sides must be $49$, but as $49$ is obviously composite, the answer is thus $\boxed{0}$.[/hide]

We are given twelve coins, one of which is a fake with a different mass from the other eleven. Determine that coin with three weighings and whether it is heavier or lighter than the others.

Without using a calculator, prove that $$2^{1995} > 5^{856}$$

A polygonal prism is made from a flexible material such that the two bases are regular $2^n$-gons $(n>1)$ of the same size. The prism is bent to join the two bases together without twisting, giving a figure with $2^n$ faces. The prism is then repeatedly twisted so that each edge of one base becomes aligned with each edge of the base exactly once. For an arbitrary $n$, what is the sum of the number of faces over all of these configurations (including the non-twisted case)?

Let $[ABC]$ be an equilateral triangle and $P$ be a point on $AC$ such that $\overline{PC}= 7$. The straight line that passes through $P$ and is perpendicular to $AC$, intersects $CB$ at point $M$ and intersects $AB$ at point $Q$. The midpoint $N$ of $[MQ]$ is such that $\overline{BN} = 14$. Determine the side of the triangle $[ABC]$.

Let $I$ be the incenter of triangle $ABC$ and let $D$ the foot of altitude from $I$ to $BC$. Suppose the reflection point $D’$ of $D$ with respect to $I$ satisfying $\overline{AD’} = \overline{ID’}$. Let $\Gamma$ be the circle centered at $D’$ that passing through $A$ and $I$, and let $X$, $Y\neq A$ be the intersection of $\Gamma$ and $AB$, $AC$, respectively. Suppose $Z$ is a point on $\Gamma$ so that $AZ$ is perpendicular to $BC$. Prove that $AD$, $D’Z$, $XY$ concurrent at a point.

In an equilateral triangle ${ABC}$ the segments ${ED}$ and ${GF}$ are drawn to obtain two equilateral triangles ${ADE}$ and ${GFC}$ with sides 1 and 100 (points $E$ and $G$ are on the side ${AC}$ ). The segments ${EF}$ and ${DG}$ meet at point $O$ so that the angle ${EOG}$ is equal to ${120}^{ \circ }$ . What is the length of the side of the triangle ${ABC}$ ? Mikhail Evdokimov

A nonnegative integer $n$ is said to be $\textit{squarish}$ is it satisfies the following conditions: $\textbf{(i)}$ it contains no digits of $9$; $\textbf{(ii)}$ no matter how we increase exactly one of its digits with $1$, the outcome is a square. Find all squarish nonnegative integers. $\textit{(Vlad Robu)}$

There are $m$ real numbers $x_i \geq 0$ ($i=1,2,...,m$), $n \geq 2$, $\sum_{i=1}^{m} x_i=S$. Prove that\\ \[ \sum_{i=1}^{m} \sqrt[n]{\frac{x_i}{S-x_i}} \geq 2, \] The equation holds if and only if there are exactly two of $x_i$ are equal(not equal to $0$), and the rest are equal to $0$.

Find all sequences of positive integers $\{a_n\}_{n=1}^{\infty}$, for which $a_4=4$ and \[\frac{1}{a_1a_2a_3}+\frac{1}{a_2a_3a_4}+\cdots+\frac{1}{a_na_{n+1}a_{n+2}}=\frac{(n+3)a_n}{4a_{n+1}a_{n+2}}\] for all natural $n \geq 2$. [i]Peter Boyvalenkov[/i]

Let $ABC$ be triangle with circumcircle $(O)$ of fixed $BC$, $AB \ne AC$ and $BC$ not a diameter. Let $I$ be the incenter of the triangle $ABC$ and $D = AI \cap BC, E = BI \cap CA, F = CI \cap AB$. The circle passing through $D$ and tangent to $OA$ cuts for second time $(O)$ at $G$ ($G \ne A$). $GE, GF$ cut $(O)$ also at $M, N$ respectively. a) Let $H = BM \cap CN$. Prove that $AH$ goes through a fixed point. b) Suppose $BE, CF$ cut $(O)$ also at $L, K$ respectively and $AH \cap KL = P$. On $EF$ take $Q$ for $QP = QI$. Let $J$ be a point of the circimcircle of triangle $IBC$ so that $IJ \perp IQ$. Prove that the midpoint of $IJ$ belongs to a fixed circle.

Suppose $\triangle ABC$ has angles $\angle BAC = 84^\circ, \angle ABC=60^\circ,$ and $\angle ACB = 36^\circ.$ Let $D, E,$ and $F$ be the midpoints of sides $\overline{BC}, \overline{AC},$ and $\overline{AB},$ respectively. The circumcircle of $\triangle DEF$ intersects $\overline{BD}, \overline{AE},$ and $\overline{AF}$ at points $G, H,$ and $J,$ respectively. The points $G, D, E, H, J,$ and $F$ divide the circumcircle of $\triangle DEF$ into six minor arcs, as shown. Find $\overarc{DE}+2\cdot \overarc{HJ} + 3\cdot \overarc{FG},$ where the arcs are measured in degrees. [asy] import olympiad; size(6cm); defaultpen(fontsize(10pt)); pair B = (0, 0), A = (Cos(60), Sin(60)), C = (Cos(60)+Sin(60)/Tan(36), 0), D = midpoint(B--C), E = midpoint(A--C), F = midpoint(A--B); guide circ = circumcircle(D, E, F); pair G = intersectionpoint(B--D, circ), J = intersectionpoints(A--F, circ)[0], H = intersectionpoints(A--E, circ)[0]; draw(B--A--C--cycle); draw(D--E--F--cycle); draw(circ); dot(A); dot(B); dot(C); dot(D); dot(E); dot(F); dot(G); dot(H); dot(J); label("$A$", A, (0, .8)); label("$B$", B, (-.8, -.8)); label("$C$", C, (.8, -.8)); label("$D$", D, (0, -.8)); label("$E$", E, (.8, .2)); label("$F$", F, (-.8, .2)); label("$G$", G, (0, .8)); label("$H$", H, (-.2, -1)); label("$J$", J, (.2, -.8)); [/asy]