The Fibonacci numbers are defined by $ F_0 \equal{} 1, F_1 \equal{} 1, F_{n\plus{}2} \equal{} F_{n\plus{}1} \plus{} F_n$. The positive integers $ A, B$ are such that $ A^{19}$ divides $ B^{93}$ and $ B^{19}$ divides $ A^{93}$. Show that if $ h < k$ are consecutive Fibonacci numbers then $ (AB)^h$ divides $ (A^4 \plus{} B^8)^k$

Let $k\ge 1$ be a positive integer. We consider $4k$ chips, $2k$ of which are red and $2k$ of which are blue. A sequence of those $4k$ chips can be transformed into another sequence by a so-called move, consisting of interchanging a number (possibly one) of consecutive red chips with an equal number of consecutive blue chips. For example, we can move from $r\underline{bb}br\underline{rr}b$ to $r\underline{rr}br\underline{bb}b$ where $r$ denotes a red chip and $b$ denotes a blue chip. Determine the smallest number $n$ (as a function of $k$) such that starting from any initial sequence of the $4k$ chips, we need at most $n$ moves to reach the state in which the first $2k$ chips are red.

It is known that $m$ and $n$ are positive integers, $m > n^{n-1}$, and all the numbers $m+1$, $m+2$, \dots, $m+n$ are composite. Prove that there exist such different primes $p_1$, $p_2$, \dots, $p_n$ that $p_k$ divides $m+k$ for $k = 1$, 2, \dots, $n$. [i]Proposed by C. A. Grimm [/i]

The set $\{1, 2, \cdots, 49\}$ is divided into three subsets. Prove that at least one of these subsets contains three different numbers $a, b, c$ such that $a + b = c$.

Prove that for all natural numbers $n$, \[ \sum_{k=1}^{n^2} \left\{ \sqrt{k} \right\} \le \frac{n^2-1}{2}. \] Here, $\{x\}$ denotes the fractional part of $x$.

It is known that some \(d\) distinct divisors of a positive integer number \(n\) form an arithmetic progression. Prove that the number \(n\) has at least \(2d - 2\) divisors. [i]Proposed by Anton Trygub[/i]

$a,b,c$ are positive integer. Solve the equation: $ 2^{a!}+2^{b!}=c^3 $

Let $n$ be a positive integer. Consider a triangular array of nonnegative integers as follows: \[ \begin{array}{rccccccccc} \text{Row } 1: &&&&& a_{0,1} &&&& \smallskip\\ \text{Row } 2: &&&& a_{0,2} && a_{1,2} &&& \smallskip\\ &&& \vdots && \vdots && \vdots && \smallskip\\ \text{Row } n-1: && a_{0,n-1} && a_{1,n-1} && \cdots && a_{n-2,n-1} & \smallskip\\ \text{Row } n: & a_{0,n} && a_{1,n} && a_{2,n} && \cdots && a_{n-1,n} \end{array} \] Call such a triangular array [i]stable[/i] if for every $0 \le i < j < k \le n$ we have \[ a_{i,j} + a_{j,k} \le a_{i,k} \le a_{i,j} + a_{j,k} + 1. \] For $s_1, \ldots s_n$ any nondecreasing sequence of nonnegative integers, prove that there exists a unique stable triangular array such that the sum of all of the entries in row $k$ is equal to $s_k$.

The product $(8)(888\dots8)$, where the second factor has $k$ digits, is an integer whose digits have a sum of $1000$. What is $k$? ${ \textbf{(A)}\ 901\qquad\textbf{(B)}\ 911\qquad\textbf{(C)}\ 919\qquad\textbf{(D)}}\ 991\qquad\textbf{(E)}\ 999 $

Let $\ell_1, \ell_2$ be two non-parallel lines and $d_1, d_2$ be positive reals. The set of points $X$, such that $dist(X, \ell_i)$ is a multiple of $d_i$ is called a $\textit{grid}$. Let $A$ be finite set of points, not all collinear. A triangle with vertices in $A$ is called $\textit{empty}$ if no points from $A$ lie inside or on the sides of the triangle. Given that all empty triangles have the same area, show that $A$ is the intersection of a grid $L$ and a convex polygon $F$.

Peter Pan and Crocodile are each getting hired for a job. Peter wants to get paid 6.4 dollars daily, but Crocodile demands to be paid 10 cents on day 1, 20 cents on day 2, 40 cents on day 3, 80 cents on day 4, and so on. After how many whole days will Crocodile's total earnings exceed that of Peter's?

Among the $n$ inhabitants of an island, where $n$ is even, every two are either friends or enemies. Some day, the chief of the island orders that each inhabitant (including himself) makes and wears a necklace consisting of marbles, in such a way that two necklaces have a marble of the same type if and only if their owners are friends. (a) Show that the chief’s order can be achieved by using $n^2/4$ different types of stones. (b) Prove that this is not necessarily true with less than $n^2/4$ types.

Let $ C$ be a simple arc with monotone curvature such that $ C$ is congruent to its evolute. Show that under appropriate differentiability conditions, $ C$ is a part of a cycloid or a logarithmic spiral with polar equation $ r\equal{}ae^{\vartheta}$. [i]J. Szenthe[/i]

There are $20$ balls in a bag. $a$ of them are red, $b$ of them are white, and $c$ of them are black. It is known that $ \bullet$ if we double the white balls, the probability of drawing one red ball is $\dfrac 1{25}$ less than the probability of drawing one red ball at the beginning, and $ \bullet$ if we remove all red balls, the probability of drawing one white ball is $\dfrac 1{16}$ more than the probability of drawing one white ball at the beginning. Find $a,b,c$.

Let $ABC$ a triangle and $P$ a point such that the triangles $PAB, PBC, PCA$ have the same area and the same perimeter. Prove that if: a) $P$ is in the interior of the triangle $ABC$ then $ABC$ is equilateral. b) $P$ is in the exterior of the triangle $ABC$ then $ABC$ is right angled triangle.

Source: Lusophon MO 2016 Prove that any positive power of $2$ can be written as: $$5xy-x^2-2y^2$$ where $x$ and $y$ are odd numbers.

Find the number of functions $f : \{1, 2, . . . , n\} \to \{1995, 1996\}$ such that $f(1) + f(2) + ... + f(1996)$ is odd.

Let $\triangle ABC$ be an equilateral triangle with side length $55$. Points $D$, $E$, and $F$ lie on sides $\overline{BC}$, $\overline{CA}$, and $\overline{AB}$, respectively, with $BD=7$, $CE=30$, and $AF=40$. A unique point $P$ inside $\triangle ABC$ has the property that \[\measuredangle AEP=\measuredangle BFP=\measuredangle CDP.\] Find $\tan^{2}\left(\measuredangle AEP\right)$.

Let $0\leq x_1\leq x_2\leq \cdots \leq x_n\leq 1 $ $(n\geq 2).$ Prove that $$\sqrt[n]{x_1x_2 \cdots x_n}+ \sqrt[n]{(1-x_1)(1-x_2)\cdots (1-x_n)}\leq \sqrt[n]{1-(x_1- x_n)^2}.$$

In each cell of an $n\times n$ board is a lightbulb. Initially, all of the lights are off. Each move consists of changing the state of all of the lights in a row or of all of the lights in a column (off lights are turned on and on lights are turned off). Show that if after a certain number of moves, at least one light is on, then at this moment at least $n$ lights are on.

Let $AM$ and $AN$ be the lines tangent to a circle $\Gamma$ drawn from a point $A$ $(M$ and $N$ belong to the circle). A line through $A$ cuts $\Gamma$ at $B$ and $C$ with $B$ between $A$ and $C$, and $\frac{AB}{BC} =\frac23$. If $P$ is the intersection point of $AB$ and $MN$, calculate $\frac{AP}{CP}$.

Let $ABC$ be a triangle. Let $T$ be the point of tangency of the circumcircle of triangle $ABC$ and the $A$-mixtilinear incircle. The incircle of triangle $ABC$ has center $I$ and touches sides $BC,CA$ and $AB$ at points $D,E$ and $F,$ respectively. Let $N$ be the midpoint of line segment $DF.$ Prove that the circumcircle of triangle $BTN,$ line $TI$ and the perpendicular from $D$ to $EF$ are concurrent. [i]Proposed by Diaconescu Tashi, Romania[/i]

Let $M$ be the midpoint of the side $AB$ of a triangle $ABC$. A circle through point $C$ that has a point of tangency to the line $AB$ at point $A$ and a circle through point $C$ that has a point of tangency to the line $AB$ at point $B$ intersect the second time at point $N$. Prove that $|CM|^2 + |CN|^2 - |MN|^2 = |CA|^2 + |CB|^2 - |AB|^2$.

Let $ABC$ be an acute triangle.The lines $l_1$ and $l_2$ are perpendicular to $AB$ at the points $A$ and $B$, respectively.The perpendicular lines from the midpoint $M$ of $AB$ to the lines $AC$ and $BC$ intersect $l_1$ and $l_2$ at the points $E$ and $F$, respectively.If $D$ is the intersection point of the lines $EF$ and $MC$, prove that \[\angle ADB = \angle EMF.\]

A set of three points is randomly chosen from the grid shown. Each three point set has the same probability of being chosen. What is the probability that the points lie on the same straight line? [asy]unitsize(.5cm); defaultpen(linewidth(.8pt)); dotfactor=3; pair[] dotted={(0,0),(0,1),(0,2),(1,0),(1,1),(1,2),(2,0),(2,1),(2,2)}; dot(dotted);[/asy]$ \textbf{(A)}\ \frac {1}{21}\qquad \textbf{(B)}\ \frac {1}{14}\qquad \textbf{(C)}\ \frac {2}{21}\qquad \textbf{(D)}\ \frac {1}{7}\qquad \textbf{(E)}\ \frac {2}{7}$