Let $a>1$ be a positive integer. The sequence of natural numbers $\{a_n\}_{n\geq 1}$ is defined such that $a_1 = a$ and for all $n\geq 1$, $a_{n+1}$ is the largest prime factor of $a_n^2 - 1$. Determine the smallest possible value of $a$ such that the numbers $a_1$, $a_2$,$\ldots$, $a_7$ are all distinct.

Let $a_1,a_2,\dots$ and $b_1,b_2,\dots$ be sequences of positive real numbers such that $a_1=b_1=1$ and $b_n=b_{n-1}a_n-2$ for $n=2,3,\dots.$ Assume that the sequence $(b_j)$ is bounded. Prove that \[S=\sum_{n=1}^{\infty}\frac1{a_1\cdots a_n}\] converges, and evaluate $S.$

Each vertex of a cube is assigned an 1 or a -1, and each face is assigned the product of the numbers assigned to its vertices. Determine the possible values the sum of these 14 numbers can attain.

Consider a tetrahedron $ABCD$ in which $AD \perp BC$ and $AC \perp BD$. We denote by $E$ and $F$ the projections of point $B$ on the lines $AD$ and $AC$, respectively. If $M$ and $N$ are the midpoints of the segments $[AB]$ and $[CD]$, respectively, show that $MN \perp EF$

A lucky number is a number whose digits are only $4$ or $7.$ What is the $17$th smallest lucky number? [i]Author: Ray Li[/i] [hide="Clarifications"] [list=1][*]Lucky numbers are positive. [*]"only 4 or 7" includes combinations of 4 and 7, as well as only 4 and only 7. That is, 4 and 47 are both lucky numbers.[/list][/hide]

Square $ABCD$ has side length $68$. Let $E$ be the midpoint of segment $\overline{CD}$, and let $F$ be the point on segment $\overline{AB}$ a distance $17$ from point $A$. Point $G$ is on segment $\overline{EF}$ so that $\overline{EF}$ is perpendicular to segment $\overline{GD}$. The length of segment $\overline{BG}$ can be written as $m\sqrt{n}$ where $m$ and $n$ are positive integers, and $n$ is not divisible by the square of any prime. Find $m+n$.

Consider the functions $f$ defined on the set of integers such that \[f(x)=f(x^2+x+1)\] for all integer $x$. Find $(a)$ all even functions, $(b)$ all odd functions of this kind.

A pair of words consisting only of the letters $a$ and $b$ (with repetitions) is [i]good[/i] if it is $(a,b)$ or of one of the forms $(uv, v)$, $(u, uv)$, where $(u,v)$ is a good pair. Prove that if $(\alpha, \beta)$ is a good pair, then there exists a palindrome $\gamma$ such that $\alpha\beta = a\gamma b$.

Given a line segment $AB$ in the plane, find all possible points $C$ such that in the triangle $ABC$, the altitude from $A$ and the median from $B$ have the same length.

A positive integer number $N{}$ is divisible by 2020. All its digits are different and if any two of them are swapped, the resulting number is not divisible by 2020. How many digits can such a number $N{}$ have? [i]Sergey Tokarev[/i]

An equilateral triangle with side length 20 is subdivided using parallels to its sides into \( 20^2 = 400 \) smaller equilateral triangles of side length 1. Some segments of length 1, which are edges of these small triangles, must be colored red in such a way that no small triangle has all three of its edges colored red. Determine the maximum number of segments of length 1 that can be colored red.

Let $m$ be a positive integer. Find polynomials $P(x)$ with real coefficients such that $$(x-m)P(x+2023) = xP(x)$$ is satisfied for all real numbers $x.$

In triangle $ABC$, we have $\angle ABC=\angle ACB=44^o$. Point $M$ is in its interior such that $\angle MBC=16^o$ and $\angle MCB=30^o$. Prove that $\angle MAC=\angle MBC$. [i] (Andra Elena Mircea)[/i]

Positive real numbers $a, b, c$ satisfy $abc = 1$. Prove that $$\frac{a}{1+b}+\frac{b}{1+c}+\frac{c}{1+a} \ge \frac32$$

Let $m$ be an integer and $\mathbb{Z}_m$ the set of integer modulo $m$. An equivalence relation is defined in $\mathbb{Z}_m$ given by, $x \sim y$ if there exists a natural $t$ such that $y \equiv 2^tx \, (\bmod m)$ . Find al values of $m$ such that the number of equivalent classes is even.

Find all real solutions of the system : (a) $$\begin{cases}1-x_1^2=x_2 \\ 1-x_2^2=x_3\\ ...\\ 1-x_{98}^2=x_{99}\\ 1-x_{99}^2=x_1\end{cases}$$ (b)* $$\begin{cases} 1-x_1^2=x_2\\ 1-x_2^2=x_3\\ ...\\1-x_{98}^2=x_{n}\\ 1-x_{n}^2=x_1\end{cases}$$

Triangle $ABC$ has side lengths $AB=7, BC=8, $ and $CA=9.$ Circle $\omega_1$ passes through $B$ and is tangent to line $AC$ at $A.$ Circle $\omega_2$ passes through $C$ and is tangent to line $AB$ at $A.$ Let $K$ be the intersection of circles $\omega_1$ and $\omega_2$ not equal to $A.$ Then $AK=\tfrac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$

In triangle $ABC$, $AB=7$, $AC=9$, and $BC=8$. The angle bisector of $\angle{BAC}$ intersects side $BC$ at $D$, and the angle bisector of $\angle{ABC}$ intersects $AD$ at $E$. Compute $AE^2$. [i]2016 CCA Math Bonanza Lightning #2.2[/i]

There are 5 coins placed flat on a table according to the figure. What is the order of the coins from top to bottom? [asy] size(100); defaultpen(linewidth(.8pt)+fontsize(8pt)); draw(arc((0,1), 1.2, 25, 214)); draw(arc((.951,.309), 1.2, 0, 360)); draw(arc((.588,-.809), 1.2, 132, 370)); draw(arc((-.588,-.809), 1.2, 75, 300)); draw(arc((-.951,.309), 1.2, 96, 228)); label("$A$",(0,1),NW); label("$B$",(-1.1,.309),NW); label("$C$",(.951,.309),E); label("$D$",(-.588,-.809),W); label("$E$",(.588,-.809),S);[/asy] $ \textbf{(A)}\ (C, A, E, D, B) \qquad \textbf{(B)}\ (C, A, D, E, B) \qquad \textbf{(C)}\ (C, D, E, A, B) \\ [1ex] \textbf{(D)}\ (C, E, A, D, B) \qquad \textbf{(E)}\ (C, E, D, A, B)$

Let $n$ be the largest integer such that $5^n$ divides $12^{2015}+13^{2015}$. Compute the remainder when $\frac{12^{2015}+13^{2015}}{5^n}$ is divided by $1000$.

In a $5 \times 5$ grid of squares, how many nonintersecting pairs rectangles of rectangles are there? (Note sharing a vertex or edge still means the rectangles intersect.)

Given a circle of radius $\sqrt{13}$, let $A$ be a point at a distance $4 + \sqrt{13}$ from the center $O$ of the circle. Let $B$ be the point on the circle nearest to point $A$. A line passing through the point $A$ intersects the circle at points $K$ and $L$. The maximum possible area for $\triangle BKL$ can be written in the form $\tfrac{a-b\sqrt{c}}{d}$, where $a$, $b$, $c$, and $d$ are positive integers, $a$ and $d$ are relatively prime, and $c$ is not divisible by the square of any prime. Find $a+b+c+d$.

In a triangle $ABC$, let $AA', BB'$ and $CC'$ be the bisectors. Suppose $A'B' \cap CC' =P$ and $A'C' \cap BB'= Q$. Prove that $\angle PAC = \angle QAB$.

In the scalene acute triangle $ABC$, $O$ is the circumcenter. $AD, BE, CF$ are three altitudes. And $H$ is the orthocenter. Let $G$ be the reflection point of $O$ through $BC$. Draw the diameter $EK$ in $\odot (GHE)$, and the diameter $FL$ in $\odot (GHF)$. a) If $AK, AL$ and $DE, DF$ intersect at $U, V$ respectively, prove that $UV\parallel EF$. b) Suppose $S$ is the intersection of the two tangents of the circumscribed circle of $\triangle ABC$ at $B$ and $C$. $T$ is the intersection of $DS$ and $HG$. And $M,N$ are the projection of $H$ on $TE,TF$ respectively. Prove that $M,N,E,F$ are concyclic.

Suppose $x$ and $y$ are real numbers such that \[x^2+xy+y^2=2\qquad\text{and}\qquad x^2-y^2=\sqrt5.\] The sum of all possible distinct values of $|x|$ can be written in the form $\textstyle\sum_{i=1}^n\sqrt{a_i}$, where each of the $a_i$ is a rational number. If $\textstyle\sum_{i=1}^na_i=\frac mn$ where $m$ and $n$ are positive realtively prime integers, what is $100m+n$? [i] Proposed by David Altizio [/i]