A hollow cylinder with a very thin wall (like a toilet paper tube) and a block are placed at rest at the top of a plane with inclination $\theta$ above the horizontal. The cylinder rolls down the plane without slipping and the block slides down the plane; it is found that both objects reach the bottom of the plane simultaneously. What is the coefficient of kinetic friction between the block and the plane? (A) $0$ (B) $\frac{1}{3}\tan \theta$ (C) $\frac{1}{2}\tan \theta$ (D) $\frac{2}{3}\tan \theta$ (E) $\tan \theta$

Consider a sequence of circles $K_1,K_2,K_3,K_4, \ldots$ of radii $r_1, r_2, r_3, r_4, \ldots$ , respectively, situated inside a triangle $ABC$. The circle $K_1$ is tangent to $AB$ and $AC$; $K_2$ is tangent to $K_1$, $BA$, and $BC$; $K_3$ is tangent to $K_2$, $CA$, and $CB$; $K_4$ is tangent to $K_3$, $AB$, and $AC$; etc. (a) Prove the relation \[r_1 \cot \frac 12 A+ 2 \sqrt{r_1r_2} + r_2 \cot \frac 12 B = r \left(\cot \frac 12 A + \cot \frac 12 B \right) \] where $r$ is the radius of the incircle of the triangle $ABC$. Deduce the existence of a $t_1$ such that \[r_1=r \cot \frac 12 B \cot \frac 12 C \sin^2 t_1\] (b) Prove that the sequence of circles $K_1,K_2, \ldots $ is periodic.

Oleksiy wrote several distinct positive integers on the board and calculated all their pairwise sums. It turned out that all digits from $0$ to $9$ appear among the last digits of these sums. What could be the smallest number of integers that Oleksiy wrote? [i]Proposed by Oleksiy Masalitin[/i]

Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that $ \forall x\notin\{-1,1\}$ holds: \[\displaystyle{f\Big(\frac{x-3}{x+1}\Big)+f\Big(\frac{3+x}{1-x}\Big)=x}\]

Two sets $A=\{x\in\mathbb{R}|x^2-4x+3<0\},B=\{x\in\mathbb{R}|2^{1-x}+a\leq0,x^2-2(a+7)x+5\leq0\}$. If $A\subseteq B$, then the range value of real number $a$ is________.

Let $ABC$ be an acute triangle, $H$ its orthocentre, $D$ a point on the side $[BC]$, and $P$ a point such that $ADPH$ is a parallelogram. Show that $\angle BPC > \angle BAC$.

Let $E$ be a finite set of points such that $E$ is not contained in a plane and no three points of $E$ are collinear. Show that at least one of the following alternatives holds: (i) $E$ contains five points that are vertices of a convex pyramid having no other points in common with $E;$ (ii) some plane contains exactly three points from $E.$

Given trapezoid $ ABCD$ with parallel sides $ AB$ and $ CD$, assume that there exist points $ E$ on line $ BC$ outside segment $ BC$, and $ F$ inside segment $ AD$ such that $ \angle DAE \equal{} \angle CBF$. Denote by $ I$ the point of intersection of $ CD$ and $ EF$, and by $ J$ the point of intersection of $ AB$ and $ EF$. Let $ K$ be the midpoint of segment $ EF$, assume it does not lie on line $ AB$. Prove that $ I$ belongs to the circumcircle of $ ABK$ if and only if $ K$ belongs to the circumcircle of $ CDJ$. [i]Proposed by Charles Leytem, Luxembourg[/i]

Let $ABCD$ be a convex quadrilateral with perpendicular diagonals. If $AB = 20, BC = 70$ and $CD = 90$, then what is the value of $DA$?

A set of balls contains $ n$ balls which are labeled with numbers $ 1,2,3,\ldots,n.$ We are given $ k > 1$ such sets. We want to colour the balls with two colours, black and white in such a way, that (a) the balls labeled with the same number are of the same colour, (b) any subset of $ k\plus{}1$ balls with (not necessarily different) labels $ a_{1},a_{2},\ldots,a_{k\plus{}1}$ satisfying the condition $ a_{1}\plus{}a_{2}\plus{}\ldots\plus{}a_{k}\equal{} a_{k\plus{}1}$, contains at least one ball of each colour. Find, depending on $ k$ the greatest possible number $ n$ which admits such a colouring.

Let $\Gamma_1$ and $\Gamma_2$ be two non-overlapping circles. $A,C$ are on $\Gamma_1$ and $B,D$ are on $\Gamma_2$ such that $AB$ is an external common tangent to the two circles, and $CD$ is an internal common tangent to the two circles. $AC$ and $BD$ meet at $E$. $F$ is a point on $\Gamma_1$, the tangent line to $\Gamma_1$ at $F$ meets the perpendicular bisector of $EF$ at $M$. $MG$ is a line tangent to $\Gamma_2$ at $G$. Prove that $MF=MG$.

Two triangles intersect to form seven finite disjoint regions, six of which are triangles with area 1. The last region is a hexagon with area \(A\). Compute the minimum possible value of \(A\). [i]Proposed by Karthik Vedula[/i]

Sinclair starts with the number $1$. Every minute, he either squares his number or adds $1$ to his number, both with equal probability. What is the expected number of minutes until his number is divisible by $3$? [i]Proposed by Nathan Xiong[/i]

Suppose $ f(x)$ is defined for all real numbers $ x$; $ f(x)>0$ for all $ x$, and $ f(a)f(b)\equal{}f(a\plus{}b)$ for all $ a$ and $ b$. Which of the following statements is true? $ \text{I. } f(0)\equal{}1$ $ \text{II. } f(\minus{}a)\equal{}1/f(a) \text{ for all } a$ $ \text{III. } f(a) \equal{} \sqrt[3]{f(3a)} \text{ for all } a$ $ \text{IV. } f(b)>f(a) \text{ if } b>a$ $ \textbf{(A)}\ \text{III and IV only} \qquad$ $ \textbf{(B)}\ \text{I, III, and IV only} \qquad$ $ \textbf{(C)}\ \text{I, II, and IV only} \qquad$ $ \textbf{(D)}\ \text{I, II, and III only} \qquad$ $ \textbf{(E)}\ \text{All are true.}$

Part of an "$n$-pointed regular star" is shown. It is a simple closed polygon in which all $2n$ edges are congruent, angles $A_{1}$, $A_{2}$, $\ldots$, $A_{n}$ are congruent and angles $B_{1}$, $B_{2}$, $\ldots$, $B_{n}$ are congruent. If the acute angle at $A_{1}$ is $10^{\circ}$ less than the acute angle at $B_{1}$, then $n = $ [asy] size(200); defaultpen(linewidth(0.7)+fontsize(10)); pair A=dir(90-2*36), B=dir(90-36), C=dir(90), D=dir(90+36), E=dir(90+2*36); pair F=2*dir(90-1.5*36), G=2*dir(90-0.5*36), H=2*dir(90+0.5*36), I=2*dir(90+1.5*36); draw(A--F--B--G--C--H--D--I--E); label("$B_2$", B, -0.3*dir(B)); label("$B_1$", C, -0.3*dir(C)); label("$B_n$", D, -0.3*dir(D)); label("$A_3$", F, dir(F)); label("$A_2$", G, dir(G)); label("$A_1$", H, dir(H)); label("$A_n$", I, dir(I)); [/asy] $ \textbf{(A)}\ 12\qquad\textbf{(B)}\ 18\qquad\textbf{(C)}\ 24\qquad\textbf{(D)}\ 36\qquad\textbf{(E)}\ 60 $

Suppose that $ a_1$, $ a_2$, $ \ldots$, $ a_n$ are integers such that $ n\mid a_1 \plus{} a_2 \plus{} \ldots \plus{} a_n$. Prove that there exist two permutations $ \left(b_1,b_2,\ldots,b_n\right)$ and $ \left(c_1,c_2,\ldots,c_n\right)$ of $ \left(1,2,\ldots,n\right)$ such that for each integer $ i$ with $ 1\leq i\leq n$, we have \[ n\mid a_i \minus{} b_i \minus{} c_i \] [i]Proposed by Ricky Liu & Zuming Feng, USA[/i]