A list of $8$ numbers is formed by beginning with two given numbers. Each new number in the list is the product of the two previous numbers. Find the first number if the last three are shown: \[ \text{\underline{\hspace{3 mm}?\hspace{3 mm}}\hspace{1 mm},\hspace{1 mm}\underline{\hspace{7 mm}}\hspace{1 mm},\hspace{1 mm}\underline{\hspace{7 mm}}\hspace{1 mm},\hspace{1 mm}\underline{\hspace{7 mm}}\hspace{1 mm},\hspace{1 mm}\underline{\hspace{7 mm}}\hspace{1 mm},\hspace{1 mm}\underline{\hspace{2 mm}16\hspace{2 mm}}\hspace{1 mm},\hspace{1 mm}\underline{\hspace{2 mm}64\hspace{2 mm}}\hspace{1 mm},\hspace{1 mm}\underline{\hspace{1 mm}1024\hspace{1 mm}}} \] $ \text{(A)}\ \frac{1}{64}\qquad\text{(B)}\ \frac{1}{4}\qquad\text{(C)}\ 1\qquad\text{(D)}\ 2\qquad\text{(E)}\ 4 $

Find all functions $ f: Z \rightarrow R$ that verify the folowing two conditions: (i) for each pair of integers $ (m,n)$ with $ m<n$ one has $ f(m)<f(n)$; (ii) for each pair of integers $ (m,n)$ there exists an integer $ k$ such that $ f(m)\minus{}f(n)\equal{}f(k)$.

Let $K$ and $L$ be two points on the arcs $AB$ and $BC$ of the circumcircle of a triangle $ABC$, respectively, such that $KL\parallel AC$. Show that the incenters of triangles $ABK$ and $CBL$ are equidistant from the midpoint of the arc $ABC$ of the circumcircle of triangle $ABC$.

Two circles in the plane do not intersect and do not lie inside each other. We choose diameters $A_1B_1$ and $A_2B_2$ of these circles such that the segments $A_1A_2$ and $B_1B_2'$ intersect. Let $A$ and $B$ be the midpoints of the segments $A_1A_2$ and $B_1B_2$, and $C$ be the intersection point of these segments. Prove that the orthocenter of the triangle $ABC$ belongs to a fixed line that does not depend on the choice of diameters.

Triangle $ABC$ is given with angles $\angle ABC = 60^o$ and $\angle BCA = 100^o$. On the sides AB and AC, the points $D$ and $E$ are chosen, respectively, in such a way that $\angle EDC = 2\angle BCD = 2\angle CAB$. Find the angle $\angle BED$.

Circles $\omega_a, \omega_b, \omega_c$ have centers $A, B, C$, respectively and are pairwise externally tangent at points $D, E, F$ (with $D\in BC, E\in CA, F\in AB$). Lines $BE$ and $CF$ meet at $T$. Given that $\omega_a$ has radius $341$, there exists a line $\ell$ tangent to all three circles, and there exists a circle of radius $49$ tangent to all three circles, compute the distance from $T$ to $\ell$. [i]Proposed by Andrew Gu.[/i]

Six distinct positive integers are randomly chosen between $1$ and $2011;$ inclusive. The probability that some pair of the six chosen integers has a difference that is a multiple of $5 $ is $n$ percent. Find $n.$

We work in the decimal system and the following operations are allowed to be done with a positive integer: a) append $4$ at the end of the number. b) append $0$ at the end of the number. c) divide the number by $2$ if it's even. Show that starting with $4$, we can reach every positive integer by a finite number of these operations

There are $b$ boys and $g$ girls, with $g \ge 2b-1$, at presence at a party. Each boy invites a girl for the first dance. Prove that this can be done in such a way that either a boy is dancing with a girl he knows or all the girls he knows are not dancing.

Simplify the fraction $\frac{123456788...87654321}{1234567899...987654321}$’ if the digit $8$ in the numerator occurs $2000$ times, and the digit $9$ in the denominator $1999$ occurs times (as a result you need to get an irreducible fraction).

Call admissible a set $A$ of integers that has the following property: If $x,y \in A$ (possibly $x=y$) then $x^2+kxy+y^2 \in A$ for every integer $k$. Determine all pairs $m,n$ of nonzero integers such that the only admissible set containing both $m$ and $n$ is the set of all integers. [i]Proposed by Warut Suksompong, Thailand[/i]

Let $ABC$ be a triangle and a circle $\omega$ through $C$ and its incenter $I$ meets $CA, CB$ at $P, Q$. The circumcircles $(CPQ)$ and $(ABC)$ meet at $L$. The angle bisector of $\angle ALB$ meets $AB$ at $K$. Show that, as $\omega$ varies, $\angle PKQ$ is constant.

Triangle $ ABC$ has $ \angle C \equal{} 60^{\circ}$ and $ BC \equal{} 4$. Point $ D$ is the midpoint of $ BC$. What is the largest possible value of $ \tan{\angle BAD}$? $ \textbf{(A)} \ \frac {\sqrt {3}}{6} \qquad \textbf{(B)} \ \frac {\sqrt {3}}{3} \qquad \textbf{(C)} \ \frac {\sqrt {3}}{2\sqrt {2}} \qquad \textbf{(D)} \ \frac {\sqrt {3}}{4\sqrt {2} \minus{} 3} \qquad \textbf{(E)}\ 1$

About the pentagon $ABCDE$ we know that $AB = BC = CD = DE$, $\angle C = \angle D =108^o$, $\angle B = 96^o$. Find the value in degrees of $\angle E$.

Define sequence $ (a_n)$ by $ \sum_{d|n} a_d \equal{} 2^n.$ Show that $ n|a_n.$

Let $ABC$ be a right triangle $(AB<AC)$ with heights $AD, BE,$ and $CF$ and orthocenter $H$. Let $M$ denote the midpoint of $BC$ and let $X$ be the second intersection of the circle with diameter $HM$ and line $AM.$ Given that lines $HX$ and $BC$ intersect at $T,$ prove that the circumcircles of $\triangle TFD$ and $\triangle AEF$ are tangent.

Find all positive integers $n$ such that $\sum_{k=1}^{n}\frac{n}{k!}$ is an integer.

Let triangle $\triangle ABC$ have side lengths $AB = 7, BC = 8,$ and $CA = 9$, and let $M$ and $D$ be the midpoint of $\overline{BC}$ and the foot of the altitude from $A$ to $\overline{BC}$, respectively. Let $E$ and $F$ lie on $\overline{AB}$ and $\overline{AC}$, respectively, such that $m\angle{AEM} = m\angle{AFM} = 90^{\circ}$. Let $P$ be the intersection of the angle bisectors of $\angle{AED}$ and $\angle{AFD}$. If $MP$ can be written as $\frac{a\sqrt{b}}{c}$ for positive integers $a, b,$ and $c$ with $b$ squarefree and $\gcd(a,c) = 1$, then find $a + b + c$. [i]Proposed by Andrew Wu[/i]

Let $ \left( a_n \right)_{n\ge 1} $ be an arithmetic progression with $ a_1=1 $ and natural ratio. [b]a)[/b] Prove that $$ a_n^{1/a_k} <1+\sqrt{\frac{2\left( a_n-1 \right)}{a_k\left( a_k -1 \right)}} , $$ for any natural numbers $ 2\le k\le n. $ [b]b)[/b] Calculate $ \lim_{n\to\infty } \frac{1}{a_n}\sum_{k=1}^n a_n^{1/a_k} . $ [i]Nicolae Bourbăcuț[/i]

Let $f$ be a real-valued differentiable function on the real line $\mathbb{R}$ such that $\lim_{x\to 0} \frac{f(x)}{x^2}$ exists, and is finite . Prove that $f'(0)=0$.

Let $ M$ and $ N$ be points on the side $ BC$ of triangle $ ABC$, with the point $ M$ lying on the segment $ BN$, such that $ BM \equal{} CN$. Let $ P$ and $ Q$ be points on the segments $ AN$ and $ AM$, respectively, such that $ \measuredangle PMC \equal{}\measuredangle MAB$ and $ \measuredangle QNB \equal{}\measuredangle NAC$. Prove that $ \measuredangle QBC \equal{}\measuredangle PCB$.

Elax creates a partially filled $4 \times 4$ grid, and is trying to write in positive integers such that any four cells that share no rows and columns always sum to a number $S$. Given that the sum of the numbers in the top row is also $S$, what is the missing cell number? [asy] size(100); add(grid(4,4)); label("$11$", (0.5,1.5)); label("$10$", (0.5,2.5)); label("?", (0.5,3.5)); label("$8$", (1.5,3.5)); label("$7$", (2.5,2.5)); label("$4$", (3.5,0.5)); label("$9$", (3.5,1.5)); [/asy] $\textbf{(A) }4\qquad\textbf{(B) }6\qquad\textbf{(C) }9\qquad\textbf{(D) }10\qquad\textbf{(E) }12$

For every positive integer $k>1$ prove that there exist a real number $x$ so that for every positive integer $n<1398$: $$\left\{x^n\right\}<\left\{x^{n-1}\right\} \Longleftrightarrow k\mid n.$$ [i]Proposed by Mohammad Amin Sharifi[/i]

Points $D$ and $N$ on the sides $AB$ and $BC$ and points $E,M$ on the side $AC$ of an equilateral triangle $ABC$, respectively, with $E$ between $A$ and $M$, satisfy $AD+AE=CN+CM=BD+BN+EM$. Determine the angle between the lines $DM$ and $EN$.

Let $n$ be a positive integer. Prove that \[-1<\sum_{k=1}^{n}\frac{k}{k^2+1}-\ln n\le\frac{1}{2}\]