a) Let $(a_n)$ be the sequence defined by $a_n=\ln (2n^2+1)-\ln (n^2+n+1)\,\,\forall n\geq 1.$ Prove that the set $\{n\in\mathbb{N}|\,\{a_n\}<\dfrac{1}{2}\}$ is a finite set; b) Let $(b_n)$ be the sequence defined by $a_n=\ln (2n^2+1)+\ln (n^2+n+1)\,\,\forall n\geq 1$. Prove that the set $\{n\in\mathbb{N}|\,\{b_n\}<\dfrac{1}{2016}\}$ is an infinite set.

In convex quadrilateral $ABCD$, let diagonals $\overline{AC}$ and $\overline{BD}$ intersect at $E$. Let the circumcircles of $ADE$ and $BCE$ intersect $\overline{AB}$ again at $P \neq A$ and $Q \neq B$, respectively. Let the circumcircle of $ACP$ intersect $\overline{AD}$ again at $R \neq A$, and let the circumcircle of $BDQ$ intersect $\overline{BC}$ again at $S \neq B$. Prove that $A$, $B$, $R$, and $S$ are concyclic. [i]Tiger Zhang[/i]

Let $A,B\in \mathcal{M}_n(\mathbb{C})$ such that $AB=BA$ and $\det B\neq 0$. a) If $|\det(A+zB)|=1$ for any $z\in \mathbb{C}$ such that $|z|=1$, then $A^n=O_n$. b) Is the question from a) still true if $AB\neq BA$ ?

Let $ A$ be a subset of the set $ \{1, 2,\ldots,2006\}$, consisting of $ 1004$ elements. Prove that there exist $ 3$ distinct numbers $ a,b,c\in A$ such that $ gcd(a,b)$: a) divides $ c$ b) doesn't divide $ c$

Prove that there are infinitely many integers can't be written as $$\frac{p^a-p^b}{p^c-p^d}$$, with a,b,c,d are arbitrary integers and p is an arbitrary prime such that the fraction is an integer too.

$d(n)$ shows the number of positive integer divisors of positive integer $n$. For which positive integers $n$ one cannot find a positive integer $k$ such that $\underbrace{d(\dots d(d}_{k\ \text{times}} (n) \dots )$ is a perfect square.

Points $A$, $B$, and $O$ lie in the plane such that $\measuredangle AOB = 120^\circ$. Circle $\omega_0$ with radius $6$ is constructed tangent to both $\overrightarrow{OA}$ and $\overrightarrow{OB}$. For all $i \ge 1$, circle $\omega_i$ with radius $r_i$ is constructed such that $r_i < r_{i - 1}$ and $\omega_i$ is tangent to $\overrightarrow{OA}$, $\overrightarrow{OB}$, and $\omega_{i - 1}$. If \[ S = \sum_{i = 1}^\infty r_i, \] then $S$ can be expressed as $a\sqrt{b} + c$, where $a, b, c$ are integers and $b$ is not divisible by the square of any prime. Compute $100a + 10b + c$. [i]Proposed by Aaron Lin[/i]

In the triangle $ABC$ on the side $AC$ the points $F$ and $L$ are selected so that $AF = LC <\frac{1}{2} AC$. Find the angle $ \angle FBL $ if $A {{B} ^ {2}} + B {{C} ^ {2}} = A {{L} ^ {2}} + L {{C } ^ {2}}$ (Zhidkov Sergey)

A quadratic polynomial $f(x)$ is called sparse if its degree is exactly 2 , if it has integer coefficients, and if there exists a nonzero polynomial $g(x)$ with integer coefficients such that $f(x) g(x)$ has degree at most 3 and $f(x) g(x)$ has at most two nonzero coefficients. Find the number of sparse quadratics whose coefficients lie between 0 and 10, inclusive.

For positive rational $x$, if $x$ is written in the form $p/q$ with $p, q$ positive relatively prime integers, define $f(x)=p+q$. For example, $f(1)=2$. a) Prove that if $f(x)=f(mx/n)$ for rational $x$ and positive integers $m, n$, then $f(x)$ divides $|m-n|$. b) Let $n$ be a positive integer. If all $x$ which satisfy $f(x)=f(2^nx)$ also satisfy $f(x)=2^n-1$, find all possible values of $n$. [i]Anderson Wang.[/i]

Find all primes $p$ and $q$ such that $3p^{q-1}+1$ divides $11^p+17^p$ Proposed by Stanislav Dimitrov,Bulgaria

Let $n$ be a positive integer. Determine the smallest positive integer $k$ with the following property: it is possible to mark $k$ cells on a $2n \times 2n$ board so that there exists a unique partition of the board into $1 \times 2$ and $2 \times 1$ dominoes, none of which contain two marked cells.

Let $A,B\in\mathcal{M}_n(\mathbb{C})$ such that $A^2+B^2=2AB.$ Prove that for any complex number $x$\[\det(A-xI_n)=\det(B-xI_n).\][i]Mihai Opincariu and Vasile Pop[/i]

Prove that there are infinitely many positive integers $a, b$ and $c$ such that their greatest common divisor is $1$ (ie: $gcd(a, b, c) = 1$) and satisfy that: $$a^2=b^2+c^2+bc$$

If $d_1,d_2,...,d_k$ are all distinct positive divisors of $n$, we define $f_s(n) = d_1^s+d_2^s+..+d_k^s$. For example, we have $f_1(3) = 1 + 3 = 4, f_2(4) = 1 + 2^2 + 4^2 = 21$. Prove that for all positive integers $n$, $n^3f_1(n) - 2nf_9(n) + n^2f_3(n)$ is divisible by $8$.

Let $A$ and $B$ be complex Hermitian $2\times2$ matrices having the pairs of eigenvalues $(\alpha_1,\alpha_2)$ and $(\beta_1,\beta_2)$, respectively. Determine all possible pairs of eigenvalues $(\gamma_1,\gamma_2)$ of the matrix $C=A+B$. (We recall that a matrix $A=(a_{ij})$ is Hermitian if and only if $a_{ij}=\overline{a_{ji}}$ for all $i$ and $j$.)

Let $\triangle{ABC}$ exist such that $AB=6, BC=8, AC=10.$ Let $P$ lie on the circumcircle of $ABC,$ $\omega,$ such that $P$ lies strictly on the arc between $B$ and $C$ (i.e. $P \neq B, C$). Drop altitudes from $P$ to $BC, AC$ at points $J$ and $Q$ respectively. Let $l$ be a line through $B$ such that it intersects $AC$ at a point $K.$ Let $M$ be the midpoint of $BQ.$ Let $CM$ intersect line $l$ at a point $I.$ Let $AI$ intersect $JQ$ at a point $U.$ Now, $B, J, U, M$ are cyclic. Now, let $\angle{QJC}=\theta.$ If we set $y=\sin(\theta), x=\cos(\theta),$ they satisfy the equation $$768(xy)=(16-8x^2+6xy)(x^2y^2(8x-6y)^2+(8x-8xy^2+6y^3)^2)$$ The numerical values of $x,y$ are approximately: $$x=0.72951, y=0.68400$$ Let $BK$ intersect the circumcircle of $ABC,$ $\omega,$ at a point $L.$ Find the value of $BL.$ We will only look up to two decimal places for correctness.

Wu starts out with exactly one coin. Wu flips every coin he has [i]at once[/i] after each year. For each heads he flips, Wu receives a coin, and for every tails he flips, Wu loses a coin. He will keep repeating this process each year until he has $0$ coins, at which point he will stop. The probability that Wu will stop after exactly five years can be expressed as $\frac{a}{2^b}$, where $a, b$ are positive integers such that $a$ is odd. Find $a+b$. [i]Proposed by Bradley Guo[/i]

Prove that for any irrational number $\xi$, there are infinitely many rational numbers $\frac{m}{n}$ $\left( (m,n) \in \mathbb{Z}\times \mathbb{N}\right)$ such that \[\left\vert \xi-\frac{n}{m}\right\vert < \frac{1}{\sqrt{5}m^{2}}.\]

Let $y_{1}, y_{2}, y_{3},\ldots$ be a sequence such that $y_{1}=1$ and, for $k>0,$ is defined by the relationship: \[y_{2k}=\begin{cases}2y_{k}& \text{if}~k~ \text{is even}\\ 2y_{k}+1 & \text{if}~k~ \text{is odd}\end{cases}\]\[y_{2k+1}=\begin{cases}2y_{k}& \text{if}~k~ \text{is odd}\\ 2y_{k}+1 & \text{if}~k~ \text{is even}\end{cases}\]Show that the sequence takes on every positive integer value exactly once.

Find all integer solutions of the equation $x^2y^5 - 2^x5^y = 2015 + 4xy$. (Malik Talbi)

Tessa has a unit cube, on which each vertex is labeled by a distinct integer between 1 and 8 inclusive. She also has a deck of 8 cards, 4 of which are black and 4 of which are white. At each step she draws a card from the deck, and[list][*]if the card is black, she simultaneously replaces the number on each vertex by the sum of the three numbers on vertices that are distance 1 away from the vertex;[*]if the card is white, she simultaneously replaces the number on each vertex by the sum of the three numbers on vertices that are distance $\sqrt2$ away from the vertex.[/list]When Tessa finishes drawing all cards of the deck, what is the maximum possible value of a number that is on the cube?

Let $ABCD$ be a trapezoid with bases $AB = 50$ and $CD = 125$, and legs $AD = 45$ and $BC = 60$. Find the area of the intersection between the circle centered at $B$ with radius $BD$ and the circle centered at $D$ with radius $BD$. Express your answer as a common fraction in simplest radical form and in terms of $\pi$.

Determine all real numbers $a$ such that \[4\lfloor an\rfloor =n+\lfloor a\lfloor an\rfloor \rfloor \; \text{for all}\; n \in \mathbb{N}.\]

Let $T=\text{TNFTPP}$. Three distinct positive Fibonacci numbers, all greater than $T$, are in arithmetic progression. Let $N$ be the smallest possible value of their sum. Find the remainder when $N$ is divided by $2007$.