For a positive integer $K$, define a sequence, $\{a_n\}_n$, as following $a_1=K$, \[ a_{n+1} = \{ \begin{array} {cc} a_n-1 , & \mbox{ if } a_n \mbox{ is even} \\ \frac{a_n-1}2 , & \mbox{ if } a_n \mbox{ is odd} \end{array}, \] for all $n\geq 1$. Find the smallest value of $K$, which makes $a_{2005}$ the first term equal to 0.

i) Prove that if a function $f$ is continuous on the closed interval $[0, \pi]$ and $$ \int_{0}^{\pi} f(t) \cos t \; dt= \int_{0}^{\pi} f(t) \sin t \; dt=0,$$ then there exist points $0 < \alpha < \beta < \pi$ such that $f(\alpha) =f(\beta) =0.$ ii) Let $R$ be a bounded, convex, and open region in the Euclidean plane. Prove with the help of i) that the centroid of $R$ bisects at least three different chords of the boundary of $ R.$

Let $P$ be a set of $7$ different prime numbers and $C$ a set of $28$ different composite numbers each of which is a product of two (not necessarily different) numbers from $P$. The set $C$ is divided into $7$ disjoint four-element subsets such that each of the numbers in one set has a common prime divisor with at least two other numbers in that set. How many such partitions of $C$ are there ?

a) Divide $a^{128} - b^{128}$ by $(a + b)(a^2 + b^2)(a^4 + b^4)(a^8 + b^8)(a^{16} + b^{16})(a^{32} + b^{32})(a^{64} + b^{64}) $. b) Divide $a^{2^k} - b^{2^k}$ by $(a + b)(a^2 + b^2)(a^4 + b^4) ... (a^{2^{k-1}} + b^{2^{k-1}})$

Suppose we have a cubic polynomial $p(x)$ such that $p(0)=0,p(1)=1,$ and $p(x)\leq \sqrt x$ for $0\leq x \leq 1.$ Suppose $p(0.5)$ is maximized. What is the sum of $p(0.25)+p(0.75)?$ [i]Proposed by Ishin Shah[/i]

Find all functions f : $R \to R $such that for all $x, y \in R$: $$f(x + yf(x)) = f(xf(y)) - x + f(y + f(x)).$$

Raashan, Sylvia, and Ted play the following game. Each starts with $\$1$. A bell rings every $15$ seconds, at which time each of the players who currently have money simultaneously chooses one of the other two players independently and at random and gives $\$1$ to that player. What is the probability that after the bell has rung $2019$ times, each player will have $\$1$? (For example, Raashan and Ted may each decide to give $\$1$ to Sylvia, and Sylvia may decide to give her dollar to Ted, at which point Raashan will have $\$0$, Sylvia would have $\$2$, and Ted would have $\$1$, and and that is the end of the first round of play. In the second round Raashan has no money to give, but Sylvia and Ted might choose each other to give their $\$1$ to, and and the holdings will be the same as the end of the second [sic] round. $\textbf{(A) } \frac{1}{7} \qquad\textbf{(B) } \frac{1}{4} \qquad\textbf{(C) } \frac{1}{3} \qquad\textbf{(D) } \frac{1}{2} \qquad\textbf{(E) } \frac{2}{3}$

In $\triangle ABC$, $DB$ and $DC$ are tangent to the circumcircle of $\triangle ABC$. Let $B'$ be the reflection of $B$ with respect to the line $AC$ and similarly define $C'$. If line $BC$ intersects the circumcircle of $\triangle DB'C'$ at $E$ and $F$, prove that $AE = AF$.

A positive integer $m$ is called triangular if $m = 1 + 2 + ... + n$, for an integer $n$. Show that a positive integer $m$ is triangular if and only if $8m + 1$ is the square of an integer.

Point $M$ is the midpoint of base $AD$ of an isosceles trapezoid $ABCD$ with circumcircle $\omega$. The angle bisector of $ABD$ intersects $\omega$ at $K$. Line $CM$ meets $\omega$ again at $N$. From point $B$, tangents $BP, BQ$ are drawn to $(KMN)$. Prove that $BK, MN, PQ$ are concurrent. [i]A. Kuznetsov[/i]

Let $a,b$ and $c$ be positive real numbers. Prove that \[ \frac{a^2b(b-c)}{a+b}+\frac{b^2c(c-a)}{b+c}+\frac{c^2a(a-b)}{c+a} \ge 0. \]

Bored in an infinitely long class, Evan jots down a fraction whose numerator and denominator are both $70$-character strings, as follows: \[ r = \frac{loooloolloolloololllloloollollolllloollloloolooololooolololooooollllol} {lolooloolollollolloooooloooloololloolllooollololoooollllooolollloloool}. \] If $o=2013$ and $l=\frac{1}{50}$, find $\lceil roll \rceil$. [i]Proposed by Evan Chen[/i]

Find the least positive integer whose digits add to a multiple of 27 yet the number itself is not a multiple of 27. For example, 87999921 is one such number.

Dragos, the early ruler of Moldavia, and Maria the Oracle play the following game. Firstly, Maria chooses a set $S$ of prime numbers. Then Dragos gives an infinite sequence $x_1, x_2, ...$ of distinct positive integers. Then Maria picks a positive integer $M$ and a prime number $p$ from her set $S$. Finally, Dragos picks a positive integer $N$ and the game ends. Dragos wins if and only if for all integers $n \ge N$ the number $x_n$ is divisible by $p^M$; otherwise, Maria wins. Who has a winning strategy if the set S must be: $\hspace{5px}$a) finite; $\hspace{5px}$b) infinite? Proposed by [i]Boris Stanković, Bosnia and Herzegovina[/i]

A circle $\omega$ is inscribed in the acute-angled triangle $\vartriangle ABC$, which touches the side $BC$ at the point $K$. On the lines $AB$ and $AC$, the points $P$ and $Q$, respectively, are chosen so that $PK \perp AC$ and $QK \perp AB$. Denote by $M$ and $N$ the points of intersection of $KP$ and $KQ$ with the circle $\omega$. Prove that if $MN \parallel PQ$, then $\vartriangle ABC$ is isosceles. (S. Slobodyanyuk)

For positive real numbers $a$ and $b,$ let $f(a,b)$ denote the real number $x$ such that area of the (non-degenerate) triangle with side lengths $a,b,$ and $x$ is maximized. Find \[ \sum_{n=2}^{100}f\left(\sqrt{\tbinom{n}{2}},\sqrt{\tbinom{n+1}{2}}\right). \]

We define the [i]repetition[/i] number of a positive integer $n$ to be the number of distinct digits of $n$ when written in base $10$. Prove that each positive integer has a multiple which has a repetition number less than or equal to $2$.

Francesca and Giorgia play the following game. On a table there are initially coins piled up in some stacks, possibly in different numbers in each stack, but with at least one coin. In turn, each player chooses exactly one move between the following: (i) she chooses a stack that has an even non-zero number of coins $ 2k$ and breaks it into two identical stacks of coins, i.e. each stack contains $ k$ coins; (ii) she removes from the table the stacks with coins in an odd number, i.e. all such in odd number, not just those with a specific odd number. At each turn, a player necessarily moves: if one choice is not available, the she must take the other. Francesca moves first. The one who removes the last coin from the table wins. 1. If initially there is only one stack of coins on the table, and this stack contains $ 2008^{2008}$ coins, which of the players has a winning strategy? 2. For which initial configurations of stacks of coins does Francesca have a winning strategy?

Real numbers $x$, $y$, and $z$ satisfy the inequalities $$0<x<1,\qquad-1<y<0,\qquad\text{and}\qquad1<z<2.$$ Which of the following numbers is nessecarily positive? $\textbf{(A) } y+x^2 \qquad \textbf{(B) } y+xz \qquad \textbf{(C) }y+y^2 \qquad \textbf{(D) }y+2y^2 \qquad\\ \textbf{(E) } y+z$

Let $ABC$ be a non-isosceles triangle. Two isosceles triangles $AB'C$ with base $AC$ and $CA'B$ with base $BC$ are constructed outside of triangle $ABC$. Both triangles have the same base angle $\varphi$. Let $C_1$ be a point of intersection of the perpendicular from $C$ to $A'B'$ and the perpendicular bisector of the segment $AB$. Determine the value of $\angle AC_1B.$

Define the sequence $ (a_p)_{p\ge0}$ as follows: $ a_p\equal{}\displaystyle\frac{\binom p0}{2\cdot 4}\minus{}\frac{\binom p1}{3\cdot5}\plus{}\frac{\binom p2}{4\cdot6}\minus{}\ldots\plus{}(\minus{}1)^p\cdot\frac{\binom pp}{(p\plus{}2)(p\plus{}4)}$. Find $ \lim_{n\to\infty}(a_0\plus{}a_1\plus{}\ldots\plus{}a_n)$.

Let $n$ be an integer and $a, b$ real numbers such that $n > 1$ and $a > b > 0$. Prove that $$(a^n - b^n) \left ( \frac{1}{b^{n- 1}} - \frac{1}{a^{n -1}}\right) > 4n(n -1)(\sqrt{a} - \sqrt{b})^2$$

Geodude wants to assign one of the integers $1,2,3,\ldots,11$ to each lattice point $(x,y,z)$ in a 3D Cartesian coordinate system. In how many ways can Geodude do this if for every lattice parallelogram $ABCD$, the positive difference between the sum of the numbers assigned to $A$ and $C$ and the sum of the numbers assigned to $B$ and $D$ must be a multiple of $11$? (A [i]lattice point[/i] is a point with all integer coordinates. A [i]lattice parallelogram[/i] is a parallelogram with all four vertices lying on lattice points. Here, we say four not necessarily distinct points $A,B,C,D$ form a [i]parallelogram[/i] $ABCD$ if and only if the midpoint of segment $AC$ coincides with the midpoint of segment $BD$.) [hide="Clarifications"] [list] [*] The ``positive difference'' between two real numbers $x$ and $y$ is the quantity $|x-y|$. Note that this may be zero. [*] The last sentence was added to remove confusion about ``degenerate parallelograms.''[/list][/hide] [i]Victor Wang[/i]

Show that for any integter $a$, the number $\frac{a^5}5+\frac{a^3}3+\frac{7a}{15}$ is an integer.

Let $AD$ be the bisector of an acute-angled triangle $ABC$. The circle circumscribed around the triangle $ABD$ intersects the straight line perpendicular to $AD$ that passes through point $B$, at point $E$. Point $O$ is the center of the circumscribed circle of triangle $ABC$. Prove that the points $A, O, E$ lie on the same line.