Let $ \left( a_n\right)_{n\ge 1} $ be a sequence defined by $ a_1>0 $ and $ \frac{a_{n+1}}{a}=\frac{a_n}{a}+\frac{a}{a_n} , $ with $ a>0. $ Calculate $ \lim_{n\to\infty} \frac{a_n}{\sqrt{n+a}} . $ [i]Florin Rotaru[/i]

Let $A,B,C,D \in \mathcal{M}_n(\mathbb{C}),$ $A$ and $C$ invertible. Prove that if $A^k B = C^k D$ for any positive integer $k,$ then $B=D.$

In connection with the formation of a stable government, the President invited all $240$ Members of Parliament to three separate consultations, where each member participated in exactly one consultation, and at each consultation there has been at least one member present. Discussions between pairs of members are to take place to discuss the consultations. Is it possible for these discussions to occur in such a way that there exists a non-negative integer $k$, such that for every two members who participated in different consultations, there are exactly $k$ members who participated in the remaining consultation, with whom each of the two members has a conversation, and exactly $k$ members who participated in the remaining consultation, with whom neither of the two has a conversation? If yes, then find all possible values of $k$.

A triangle of sides $\frac{3}{2}, \frac{\sqrt{5}}{2}, \sqrt{2}$ is folded along a variable line perpendicular to the side of $\frac{3}{2}.$ Find the maximum value of the coincident area.

Find all the functions $\varphi:\mathbb{Z}[x]\to\mathbb{Z}[x]$ such that $\varphi(x)=x,$ any integer polynomials $f, g$ satisfy $\varphi(f+g)=\varphi(f)+\varphi(g)$ and $\varphi(f)$ is a perfect power if and only if $f{}$ is a perfect power. [i]Note:[/i] A polynomial $f\in \mathbb{Z}[x]$ is a perfect power if $f = g^n$ for some $g\in \mathbb{Z}[x]$ and $n\geqslant 2.$ [i]Proposed by Pavel Ciurea[/i]

There are positive integers $x$ and $y$ that satisfy the system of equations \begin{align*} \log_{10} x + 2 \log_{10} (\gcd(x,y)) &= 60 \\ \log_{10} y + 2 \log_{10} (\text{lcm}(x,y)) &= 570. \end{align*} Let $m$ be the number of (not necessarily distinct) prime factors in the prime factorization of $x$, and let $n$ be the number of (not necessarily distinct) prime factors in the prime factorization of $y$. Find $3m+2n$.

Prove that \[\displaystyle \sum_{\{i,j,k\}=\{1,2,3\}} \csc ^{13} \frac{2^i \pi}{7}\csc ^{14} \frac{2^j \pi}{7}\csc ^{2005} \frac{2^k\pi}{7}\] is rational. Here, $(i,j,k)$ is summed over all possible permutations of $(1,2,3)$.

$$problem 2$$:A point $P$ is taken in the interior of a right triangle$ ABC$ with $\angle C = 90$ such hat $AP = 4, BP = 2$, and$ CP = 1$. Point $Q$ symmetric to $P$ with respect to $AC$ lies on the circumcircle of triangle $ABC$. Find the angles of triangle $ABC$.

Let $n$ be a fixed positive integer and let $b(n)$ be the minimum value of $$k+\frac{n}{k},$$ where $k$ is allowed to range through all positive integers. Prove that $\lfloor b(n) \rfloor= \lfloor \sqrt{4n+1} \rfloor.$

Points $W$, $X$, $Y,$ and $Z$ are chosen inside a regular octagon so that four congruent rhombuses are formed, as shown in the diagram below. If the side length of the octagon is $1$, compute the area of quadrilateral $WXY Z$. [img]https://cdn.artofproblemsolving.com/attachments/9/6/bb12385cbd9fd802b3f3960b5e449268be45d4.png[/img]

Let $F_m$ be the $m$'th Fibonacci number, defined by $F_1=F_2=1$ and $F_m = F_{m-1}+F_{m-2}$ for all $m \geq 3$. Let $p(x)$ be the polynomial of degree 1008 such that $p(2n+1)=F_{2n+1}$ for $n=0,1,2,\ldots,1008$. Find integers $j$ and $k$ such that $p(2019) = F_j - F_k$.

There are 10 people standing equally spaced around a circle. Each person knows exactly 3 of the other 9 people: the 2 people standing next to her or him, as well as the person directly across the circle. How many ways are there for the 10 people to split up into 5 pairs so that the members of each pair know each other? $\textbf{(A) } 11 \qquad \textbf{(B) } 12 \qquad \textbf{(C) } 13 \qquad \textbf{(D) } 14 \qquad \textbf{(E) } 15$

Let $P(x_1,\dots,x_n)$ denote a polynomial with real coefficients in the variables $x_1,\dots,x_n,$ and suppose that (a) $\left(\frac{\partial^2}{\partial x_1^2}+\cdots+\frac{\partial^2}{\partial x_n^2} \right)P(x_1,\dots,x_n)=0$ (identically) and that (b) $x_1^2+\cdots+x_n^2$ divides $P(x_1,\dots,x_n).$ Show that $P=0$ identically.

Let $ABC$ be a triangle with $AB = AC = 10$ and $BC = 12$. Define $\ell_A$ as the line through $A$ perpendicular to $\overline{AB}$. Similarly, $\ell_B$ is the line through $B$ perpendicular to $\overline{BC}$ and $\ell_C$ is the line through $C$ perpendicular to $\overline{CA}$. These three lines $\ell_A, \ell_B, \ell_C$ form a triangle with perimeter $m/n$ for relatively prime positive integers $m$ and $n$. Find $m + n$.

Given nonisosceles triangle $ ABC$. Consider three segments passing through different vertices of this triangle and bisecting its perimeter. Are the lengths of these segments certainly different?

Kailey starts with the number $0$, and she has a fair coin with sides labeled $1$ and $2$. She repeatedly flips the coin, and adds the result to her number. She stops when her number is a positive perfect square. What is the expected value of Kailey’s number when she stops? If E is your estimate and A is the correct answer, you will receive $\left\lfloor 25e^{-\frac{5|E-A|}{2} }\right\rfloor$ points.

Find all real-valued continuous functions defined on the interval $(-1, 1)$ such that $(1-x^{2}) f(\frac{2x}{1+x^{2}}) = (1+x^{2})^{2}f(x)$ for all $x$.

If $x,y,z$ are nonnegative numbers with $x+y+z=3$, prove that $$\sqrt x+\sqrt y+\sqrt z\ge xy+yz+xz.$$

Let $S = \left\{ 1, \dots, 100 \right\}$, and for every positive integer $n$ define \[ T_n = \left\{ (a_1, \dots, a_n) \in S^n \mid a_1 + \dots + a_n \equiv 0 \pmod{100} \right\}. \] Determine which $n$ have the following property: if we color any $75$ elements of $S$ red, then at least half of the $n$-tuples in $T_n$ have an even number of coordinates with red elements. [i]Ray Li[/i]

How many ordered pairs of real numbers $(x, y)$ are there such that $x^2+y^2 = 200$ and \[\sqrt{(x-5)^2+(y-5)^2}+\sqrt{(x+5)^2+(y+5)^2}\] is an integer?

The angle formed by the hands of a clock at $ 2: 15$ is: $ \textbf{(A)}\ 30^\circ \qquad \textbf{(B)}\ 27\frac{1}{2}^\circ\qquad \textbf{(C)}\ 157\frac{1}{2}^\circ\qquad \textbf{(D)}\ 172\frac{1}{2}^\circ\qquad \textbf{(E)}\ \text{none of these}$

Let $ABCD$ be a cyclic quadrilatetral inscribed in a circle $k$. Let $M$ and $N$ be the midpoints of the arcs $AB$ and $CD$ which do not contain $C$ and $A$ respectively. If $MN$ meets side $AB$ at $P$, then show that $\frac{AP}{BP}=\frac{AC+AD}{BC+BD}$

Let $A_1A_2A_3A_4A_5$ be a convex 5-gon in which the coordinates of all of it's vertices are rational. For each $1\leq i \leq 5$ define $B_i$ the intersection of lines $A_{i+1}A_{i+2}$ and $A_{i+3}A_{i+4}$. ($A_i=A_{i+5}$) Prove that at most 3 lines from the lines $A_iB_i$ ($1\leq i \leq 5$) are concurrent. Time allowed for this problem was 75 minutes.

Take a sequence $(a_n)_{n=1}^\infty$ such that $a_1=3$ $a_n=a_1a_2a_3...a_{n-1}-1$ [b]a)[/b] Prove that there exists infitely many primes that divides at least 1 term of the sequence. [b]b)[/b] Prove that there exists infitely many primes that doesn't divide any term of the sequence.

Prove: for every positive integer there exists a positive integer having n digits, all of them being $1$'s and $2$'s only, such that this number is divisible by $2^{n}$. Is this still true in base $4$ or $6$¿