The Fibonacci sequence $F_n$ is defined by $F_0=0,F_1=1$ and the recurrence relation $F_n=F_{n-1}+F_{n-2}$ for all integers $n\geq2$. Let $p\geq3$ be a prime number. [list=a] [*] Prove that $F_{p-1}+F_{p+1}-1$ is divisible by $p$. [*] Prove that $F_{p^{k+1}-1}+F_{p^{k+1}+1}-\left(F_{p^k-1}+F_{p^k+1}\right)$ is divisible by $p^{k+1}$ for any positive integer $k$. [/list]

Three mutually externally tangent circles are internally tangent to a circle with radius $1$. If two of the inner circles have radius $\frac{1}{3}$, the largest possible radius of the third inner circle can be expressed in the form $\frac{a+b\sqrt{c}}{d}$ where $c$ is squarefree and $\gcd(a,b,d)=1$. Find $a+b+c+d$.

Let $A_1, A_2, ... , A_k$ be the subsets of $\left\{1,2,3,...,n\right\}$ such that for all $1\leq i,j\leq k$:$A_i\cap A_j \neq \varnothing$. Prove that there are $n$ distinct positive integers $x_1,x_2,...,x_n$ such that for each $1\leq j\leq k$: $$lcm_{i \in A_j}\left\{x_i\right\}>lcm_{i \notin A_j}\left\{x_i\right\}$$ [i]Proposed by Morteza Saghafian, Mahyar Sefidgaran[/i]

The positive real $x, y, z$ are such that $x^2+y^2+z^2 = 3$. Prove that$$\frac{x^2+yz}{x^2+yz +1}+\frac{y^2+zx}{y^2+zx+1}+\frac{z^2+xy}{z^2+xy+1}\leq 2$$

Prove that for every composite number $ n>4$, numbers $ kn$ divides $ (n\minus{}1)!$ for every integer $ k$ such that $ 1\le k\le \lfloor \sqrt{n\minus{}1} \rfloor$.

There are 7 different numbers on the board, their sum is $10$. For each number on the board, Petya wrote the product of this number and the sum of the remaining 6 numbers in his notebook. It turns out that the notebook only has 4 distinct numbers in it. Determine one of the numbers that is written on the board.

Let $a_1,a_2,\ldots,a_n$ ($n\geq 2$) be nonnegative real numbers whose sum is $\frac{n}{2}$. For every $i=1,\ldots,n$ define $$b_i=a_i+a_ia_{i+1}+a_ia_{i+1}a_{i+2}+\cdots+ a_ia_{i+1}\cdots a_{i+n-2}+2a_ia_{i+1}\cdots a_{i+n-1}$$ where $a_{j+n}=a_j$ for every $j$. Prove that $b_i\geq 1$ holds for at least one index $i$.

There are $200$ numbers on a blackboard: $ 1! , 2! , 3! , 4! , ... ... , 199! , 200!$. Julia erases one of the numbers. When Julia multiplies the remaining $199$ numbers, the product is a perfect square. Which number was erased?

Is there positive integer $n$, such that $n+2$ divides $S=1^{2019}+2^{2019}+...+n^{2019}$

Define the sequence of real numbers $\{x_n\}_{n \geq 1}$, where $x_1$ is any real number and \[x_n = 1 - x_1x_2\ldots x_{n-1} \text{ for all } n > 1.\] Show that $x_{2011} > \frac{2011}{2012}$.

Given a triangle $XYZ$ with $\angle Y = 90^{\circ}, XY=1,$ and $XZ=2,$ mark a point $Q$ on $YZ$ such that $\frac{ZQ}{ZY}=\frac{1}{3}.$ A laser beam is shot from $Q$ perpendicular to $YZ,$ and it reflects off the sides of $XYZ$ indefinitely. How many bounces does it take for the laser beam to get back to $Q$ for the first time (not including the release from $Q$ and the return to $Q$)?

Let $n$ and $k$ be positive integers such that $k\leq 2^n$. Banana and Corona are playing the following variant of the guessing game. First, Banana secretly picks an integer $x$ such that $1\leq x\leq n$. Corona will attempt to determine $x$ by asking some questions, which are described as follows. In each turn, Corona chooses $k$ distinct subsets of $\{1, 2, \ldots, n\}$ and, for each chosen set $S$, asks the question "Is $x$ in the set $S$?''. Banana picks one of these $k$ questions and tells both the question and its answer to Corona, who can then start another turn. Find all pairs $(n,k)$ such that, regardless of Banana's actions, Corona could determine $x$ in finitely many turns with absolute certainty. [i]Pitchayut Saengrungkongka, Thailand[/i]

$AB$ is the diameter of a circle centered at $O$. $C$ is a point on the circle such that angle $BOC$ is $60^\circ$. If the diameter of the circle is $5$ inches, the length of chord $AC$, expressed in inches, is: $\text{(A)} \ 3 \qquad \text{(B)} \ \frac{5\sqrt{2}}{2} \qquad \text{(C)} \frac{5\sqrt3}{2} \ \qquad \text{(D)} \ 3\sqrt3 \qquad \text{(E)} \ \text{none of these}$

Eva draws an equilateral triangle and its altitudes. In a first step she draws the center triangle of the equilateral triangle, in a second step the center triangle of this center triangle and so on. After each step Eva counts all triangles whose sides lie completely on drawn lines. What is the minimum number of center triangles she must have drawn so that the figure contains more than 2022 such triangles?

[color=darkred]A circle $ \Omega_1$ is tangent outwardly to the circle $ \Omega_2$ of bigger radius. Line $ t_1$ is tangent at points $ A$ and $ D$ to the circles $ \Omega_1$ and $ \Omega_2$ respectively. Line $ t_2$, parallel to $ t_1$, is tangent to the circle $ \Omega_1$ and cuts $ \Omega_2$ at points $ E$ and $ F$. Point $ C$ belongs to the circle $ \Omega_2$ such that $ D$ and $ C$ are separated by the line $ EF$. Denote $ B$ the intersection of $ EF$ and $ CD$. Prove that circumcircle of $ ABC$ is tangent to the line $ AD$.[/color]

Consider triangle $ABC$ with $AB<AC<BC$, inscribed in triangle $\Gamma_1$ and the circles $\Gamma_2 (B,AC)$ and $\Gamma_2 (C,AB)$. A common point of circle $\Gamma_2$ and $\Gamma_3$ is point $E$, a common point of circle $\Gamma_1$ and $\Gamma_3$ is point $F$ and a common point of circle $\Gamma_1$ and $\Gamma_2$ is point $G$, where the points $E,F,G$ lie on the same semiplane defined by line $BC$, that point $A$ doesn't lie in. Prove that circumcenter of triangle $EFG$ lies on circle $\Gamma_1$. Note: By notation $\Gamma (K,R)$, we mean random circle $\Gamma$ has center $K$ and radius $R$.

Let $P$ be a point in the plane of $\triangle ABC$, and $\gamma$ a line passing through $P$. Let $A', B', C'$ be the points where the reflections of lines $PA, PB, PC$ with respect to $\gamma$ intersect lines $BC, AC, AB$ respectively. Prove that $A', B', C'$ are collinear.

Bob sends a secret message to Alice using her RSA public key $n = 400000001.$ Eve wants to listen in on their conversation. But to do this, she needs Alice's private key, which is the factorization of $n.$ Eve knows that $n = pq,$ a product of two prime factors. Find $p$ and $q.$

A convex hexagon is circumscribed about a circle of radius $1$. Consider the three segments joining the midpoints of its opposite sides. Find the greatest real number $r$ such that the length of at least one segment is at least $r.$

Let $0<a<\frac {\pi}2.$ Find $\lim_{a\rightarrow +0} \frac{1}{a^3}\int_0^a \ln\ (1+\tan a\tan x)\ dx.$

The audience chooses two of twenty-nine cards, numbered from $1$ to $29$ respectively. The assistant of a magician chooses two of the remaining twenty-seven cards, and asks a member of the audience to take them to the magician, who is in another room. The two cards are presented to the magician in an arbitrary order. By an arrangement with the assistant beforehand, the magician is able to deduce which two cards the audience has chosen only from the two cards he receives. Explain how this may be done.

There are any number of stamps with the values ​​$134$, $135$, $...$, $142$ and $143$ cents available. Find the largest integer value (in cents) that cannot be represented by these stamps. (G. Woeginger, TU Eindhoven, The Netherlands) [hide=original wording]Es stehen beliebig viele Briefmarken mit den Werten 134, 135. . .., 142 und 143 Cent zur Verfügung. Man bestimme den größten ganzzahligen Wert (in Cent), der nicht durch diese Briefmarken dargestellt werden kann.[/hide]

Let $G$ be a group and $H$ a proper subgroup. If there exist three group homomorphisms $f,g,h:G\rightarrow G$ such that $f(xy)=g(x)h(y)$ for all $x,y\in G\setminus H$, prove that: [list=a] [*] $g=h$. [*] If $G$ is noncommutative and $H=Z(G)$, then $f=g=h$.

$n$ cars are racing. At first they have a particular order. At each moment a car may overtake another car. No two overtaking actions occur at the same time, and except moments a car is passing another, the cars always have an order. A set of overtaking actions is called "small" if any car overtakes at most once. A set of overtaking actions is called "complete" if any car overtakes exactly once. If $F$ is the set of all possible orders of the cars after a small set of overtaking actions and $G$ is the set of all possible orders of the cars after a complete set of overtaking actions, prove that \[\mid F\mid=2\mid G\mid\] (20 points) [i]Proposed by Morteza Saghafian[/i]

Prove that there exists a positive integer $n$ such that in the decimal representation of each of the numbers $\sqrt{n}$, $\sqrt[3]{n},..., \sqrt[10]{n}$ digits $2015$ stand immediately after the decimal point. [i]A.Golovanov [/i]