A sequence of real numbers is called a [i]Fibonacci [/i] sequence if $$t_{n+2} = t_{n+1} + t_n$$ for $n= 1,2,3,. .$ . Two Fibonacci sequences are said to be [i]essentially different[/i] if the terms of one sequence cannot be obtained by multiplying the terms of the other by a constant. For example, the Fibonacci sequences $1,2,3,5,8,...$ and $1,3,4,7,11,...$ are essentially different, but the sequences $1,2,3,5,8,...$ and $2,4,6,10,16,...$ are not. (a) Prove that there exist real numbers $p$ and $q$ such that the sequences $1,p,p^2,p^3,...$ and $1,q,q^2,q^3,...$ are essentially different Fibonacci sequences. (b) Let $a_1,a_2,a_3,...$ and $b_1,b_2,b_3,...$ be essentially different Fibonacci sequences. Prove that for every Fibonacci sequence $t_1,t_2,t_3,...$, there exists exactly one number $\alpha$ and exactly one number $\beta$, such that: $$t_n = \alpha a_n + \beta b_n$$ for $n = 1,2,3,...$ (c) $t_1,t_2,t_3,...$, is the Fibonacci sequence with $t_1 = 1$ and $t_2= 2$. Express $t_n$ in terms of $n$.

There are $100$ people standing in a line from left to right. Half of them are randomly chosen to face right (with all ${100 \choose 50}$ possible choices being equally likely), and the others face left. Then, while there is a pair of people who are facing each other and have no one between them, the leftmost such pair leaves the line. Compute the expected number of people remaining once this process terminates.

Let $a$ and $b$ be positive real numbers that satisfy \begin{align*} \sqrt{a-ab}+\sqrt{b-ab}=\frac{\sqrt{6}+\sqrt{2}}{4} \,\,\, \text{and}\,\,\, \sqrt{a-a^2}+\sqrt{b-b^2}=\left(\frac{\sqrt{6}+\sqrt{2}}{4}\right)^2. \end{align*} Find the ordered pair $(a, b)$ such that $a>b$ and $a+b$ is maximal.

Under what conditions is $\sqrt{a^2+b^2}=a+b$ true, where $a$ and $b$ are real numbers? $\textbf{(A) }$ It is never true. $\textbf{(B) }$ It is true if and only if $ab=0$. $\textbf{(C) }$ It is true if and only if $a+b\ge 0$. $\textbf{(D) }$ It is true if and only if $ab=0$ and $a+b\ge 0$. $\textbf{(E) }$ It is always true.

A kingdom consists of $12$ cities located on a one-way circular road. A magician comes on the $13$th of every month to cast spells. He starts at the city which was the 5th down the road from the one that he started at during the last month (for example, if the cities are numbered $1–12$ clockwise, and the direction of travel is clockwise, and he started at city #$9$ last month, he will start at city #$2$ this month). At each city that he visits, the magician casts a spell if the city is not already under the spell, and then moves on to the next city. If he arrives at a city which is already under the spell, then he removes the spell from this city, and leaves the kingdom until the next month. Last Thanksgiving the capital city was free of the spell. Prove that it will be free of the spell this Thanksgiving as well.

Let $M$ be the midpoint of side $BC$ of triangle $ABC$ ($AB>AC$), and let $AL$ be the bisector of the angle $A$. The line passing through $M$ perpendicular to $AL$ intersects the side $AB$ at the point $D$. Prove that $AD+MC$ is equal to half the perimeter of triangle $ABC$.

Let $P$ and $Q$ be on segment $BC$ of an acute triangle $ABC$ such that $\angle PAB=\angle BCA$ and $\angle CAQ=\angle ABC$. Let $M$ and $N$ be the points on $AP$ and $AQ$, respectively, such that $P$ is the midpoint of $AM$ and $Q$ is the midpoint of $AN$. Prove that the intersection of $BM$ and $CN$ is on the circumference of triangle $ABC$. [i]Proposed by Giorgi Arabidze, Georgia.[/i]

In a qualification group with $15$ volleyball teams, each team plays with all the other teams exactly once. Since there is no tie in volleyball, there is a winner in every match. After all matches played, a team would be qualified if its total number of losses is not exceeding $N$. If there are at least $7$ teams qualified, find the possible least value of $N$.

A student wrote down the following sequence of numbers : the first number is 1, the second number is 2, and after that, each number is obtained by adding together all the previous numbers. Determine the 12th number in the sequence.

Is the number \[\sqrt[3]{\sqrt 5 + 2} + \sqrt[3]{\sqrt 5 - 2}\] rational or irrational?

A three-digit $\overline{abc}$ number is called [i]Ecuadorian [/i] if it meets the following conditions: $\bullet$ $\overline{abc}$ does not end in $0$. $\bullet$ $\overline{abc}$ is a multiple of $36$. $\bullet$ $\overline{abc} - \overline{cba}$ is positive and a multiple of $36$. Determine all the Ecuadorian numbers.

Let $Z=\{ z_1,\ldots, z_{n-1}\}$, $n\ge 2$, be a set of different complex numbers such that $Z$ contains the conjugate of any its element. a) Show that there exists a constant $C$, depending on $Z$, such that for any $\varepsilon\in (0,1)$ there exists an algebraic integer $x_0$ of degree $n$, whose algebraic conjugates $x_1, x_2, \ldots, x_{n-1}$ satisfy $|x_1-z_1|\le \varepsilon, \ldots, |x_{n-1}-z_{n-1}|\le \varepsilon$ and $|x_0|\le \frac{C}{\varepsilon}$. b) Show that there exists a set $Z=\{ z_1, \ldots, z_{n-1}\}$ and a positive number $c_n$ such that for any algebraic integer $x_0$ of degree $n$, whose algebraic conjugates satisfy $|x_1-z_1|\le \varepsilon,\ldots, |x_{n-1}-z_{n-1}|\le \varepsilon$, it also holds that $|x_0|>\frac{c_n}{\varepsilon}$. (translated by L. Erdős)

The point $M$ is the center of a regular pentagon $ABCDE$. The point $P$ is an inner point of the line segment $[DM]$. The circumscribed circle of triangle $\vartriangle ABP$ intersects the side $[AE]$ at point $Q$ (different from $A$). The perpendicular from $P$ on $CD$ intersects the side $[AE] $ at point $S$. Prove that $PS$ is the bisector of $\angle APQ$.

Let $P$ be a regular $99$-gon. Assign integers between $1$ and $99$ to the vertices of $P$ such that each integer appears exactly once. (If two assignments coincide under rotation, treat them as the same. ) An [i]operation[/i] is a swap of the integers assigned to a pair of adjacent vertices of $P$. Find the smallest integer $n$ such that one can achieve every other assignment from a given one with no more than $n$ operations. [i]Proposed by Zhenhua Qu[/i]

Given is a set of $2n$ cards numbered $1,2, \cdots, n$, each number appears twice. The cards are put on a table with the face down. A set of cards is called good if no card appears twice. Baron Munchausen claims that he can specify $80$ sets of $n$ cards, of which at least one is sure to be good. What is the maximal $n$ for which the Baron's words could be true?

In terms of the fixed non-negative integers $\alpha$ and $\beta$ determine the least upper bound of the ratio (or show that it is unbounded) \[ \frac{S(n)}{S(2^{\alpha}5^{\beta}n)} \] as $n$ varies through the positive integers, where $S(\cdot)$ denotes sum of digits in decimal representation.

Consider the addition problem: \begin{tabular}{ccccc} &C&A&S&H\\ +&&&M&E\\ \hline O&S&I&D&E \end{tabular} where each letter represents a base-ten digit, and $C,M,O \ne 0.$ (Distinct letters are allowed to represent the same digit.) How many ways are there to assign values to the letters so that the addition problem is true?

On the planet Automory, there are infinitely many inhabitants. Every Automorian loves exactly one Automorian and honours exactly one Automorian. Additionally, the following can be noticed: $\bullet$ each Automorian is loved by some Automorian; $\bullet$ if Automorian $A$ loves Automorian $B$, then also all Automorians honouring $A$ love $B$, $\bullet$if Automorian $A$ honours Automorian $B$, then also all Automorians loving $A$ honour $B$. Is it correct to claim that every Automorian honours and loves the same Automorian?

An $8 \times 11$ rectangle of unit squares somehow becomes disassembled into $21$ contiguous parts . Prove that at least two of these parts, except for rotations and reflections have the same shape.

A linear form in $k$ variables is an expression of the form $P(x_1,...,x_k)=a_1x_1+...+a_kx_k$ with real constants $a_1,...,a_k$. Prove that there exist a positive integer $n$ and linear forms $P_1,...,P_n$ in $2017$ variables such that the equation $$x_1\cdot x_2\cdot ... \cdot x_{2017}=P_1(x_1,...,x_{2017})^{2017}+...+P_n(x_1,...,x_{2017})^{2017}$$ holds for all real numbers $x_1,...,x_{2017}$.

Let $ABC$ be an acute scalene triangle with incenter $I$ and orthocenter $H$. Let $M$ be the midpoint of $AB$. On the line $AH$ we consider points $D$ and $E$, such that the line $MD$ is parallel to $CI$ and $ME$ is perpendicular to $CI$. Prove that $AE=DH$.

For the real number $a$ it holds that there is exactly one square whose vertices are all on the graph with the equation $y = x^3 + ax$. Find the side length of this square.

Seven classmates are comparing their end-of-year grades in $ 12$ subjects. They observe that for any two of them, there is some subject out of the $ 12$ where the two students got different grades. It is possible to choose n subjects out of the $ 12$ such that if the seven students only compare their grades in these $n$ subjects, it will still be true that for any two, there is some subject out of the n where they got different grades. What is the smallest value of $n$ for which such a selection is surely possible? Note: In Hungarian high schools, students receive an integer grade from $ 1$ to $5$ in each subject at the end of the year.

Evaluate the following definite integral. \[ 2^{2009}\frac {\int_0^1 x^{1004}(1 \minus{} x)^{1004}\ dx}{\int_0^1 x^{1004}(1 \minus{} x^{2010})^{1004}\ dx}\]

Show that the circumradius $R$ of a triangle $ABC$ equals the arithmetic mean of the oriented distances from its incenter $I$ and three excenters $I_a,I_b, I_c$ to any tangent $\tau$ to its circumcircle. In other words, if $\delta(P)$ denotes the distance from a point $P$ to $\tau$, then with appropriate choices of signs, we have $$\delta(I) \pm \delta_(I_a) \pm \delta_(I_b) \pm \delta_(I_c) = 4R$$ Luis González