Let $f$ be a quadratic polynomial with real coefficients, and let $g_1, g_2, g_3, \ldots$ be a geometric progression of real numbers. Define $a_n=f(n)+g_n.$ Given that $a_1, a_2, a_3, a_4,$ and $a_5$ are equal to $1, 2, 3, 14,$ and $16,$ respectively, compute $\tfrac{g_2}{g_1}.$

A sequence of numbers is defined recursively by $a_1 = 1$, $a_2 = \frac{3}{7}$, and $$a_n=\frac{a_{n-2} \cdot a_{n-1}}{2a_{n-2} - a_{n-1}}$$for all $n \geq 3$ Then $a_{2019}$ can be written as $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive inegers. What is $p+q ?$ $\textbf{(A) } 2020 \qquad\textbf{(B) } 4039 \qquad\textbf{(C) } 6057 \qquad\textbf{(D) } 6061 \qquad\textbf{(E) } 8078$

Some checkers placed on an $n \times n$ checkerboard satisfy the following conditions: (a) every square that does not contain a checker shares a side with one that does; (b) given any pair of squares that contain checkers, there is a sequence of squares containing checkers, starting and ending with the given squares, such that every two consecutive squares of the sequence share a side. Prove that at least $(n^{2}-2)/3$ checkers have been placed on the board.

Let be two distinct continuous functions $ f,g:[0,1]\longrightarrow (0,\infty ) $ corelated by the equality $ \int_0^1 f(x)dx =\int_0^1 g(x)dx , $ and define the sequence $ \left( x_n \right)_{n\ge 0} $ as $$ x_n=\int_0^1 \frac{\left( f(x) \right)^{n+1}}{\left( g(x) \right)^n} dx . $$ [b]a)[/b] Show that $ \infty =\lim_{n\to\infty} x_n. $ [b]b)[/b] Demonstrate that the sequence $ \left( x_n \right)_{n\ge 0} $ is monotone.

Let $n$ be a positive integer. We define $f(n)$ as the number of finite sequences $(a_1, a_2, \ldots , a_k)$ of positive integers such that $a_1 < a_2 < a_3 < \cdots < a_k$ and $$a_1+a_2^2+a_3^3+\cdots + a_k^k \leq n.$$ Determine the positive constants $\alpha$ and $C$ such that $$\lim\limits_{n\rightarrow \infty} \frac{f(n)}{n^\alpha}=C.$$

P6. It is given the integer $Y$ with $Y = 2018 + 20118 + 201018 + 2010018 + \cdots + 201 \underbrace{00 \ldots 0}_{\textrm{100 digits}} 18.$ Determine the sum of all the digits of such $Y$. (It is implied that $Y$ is written with a decimal representation.) P7. Three groups of lines divides a plane into $D$ regions. Every pair of lines in the same group are parallel. Let $x, y$ and $z$ respectively be the number of lines in groups 1, 2, and 3. If no lines in group 3 go through the intersection of any two lines (in groups 1 and 2, of course), then the least number of lines required in order to have more than 2018 regions is .... P8. It is known a frustum $ABCD.EFGH$ where $ABCD$ and $EFGH$ are squares with both planes being parallel. The length of the sides of $ABCD$ and $EFGH$ respectively are $6a$ and $3a$, and the height of the frustum is $3t$. Points $M$ and $N$ respectively are intersections of the diagonals of $ABCD$ and $EFGH$ and the line $MN$ is perpendicular to the plane $EFGH$. Construct the pyramids $M.EFGH$ and $N.ABCD$ and calculate the volume of the 3D figure which is the intersection of pyramids $N.ABCD$ and $M.EFGH$. P9. Look at the arrangement of natural numbers in the following table. The position of the numbers is determined by their row and column numbers, and its diagonal (which, the sequence of numbers is read from the bottom left to the top right). As an example, the number $19$ is on the 3rd row, 4th column, and on the 6th diagonal. Meanwhile the position of the number $26$ is on the 3rd row, 5th column, and 7th diagonal. (Image should be placed here, look at attachment.) a) Determine the position of the number $2018$ based on its row, column, and diagonal. b) Determine the average of the sequence of numbers whose position is on the "main diagonal" (quotation marks not there in the first place), which is the sequence of numbers read from the top left to the bottom right: 1, 5, 13, 25, ..., which the last term is the largest number that is less than or equal to $2018$. P10. It is known that $A$ is the set of 3-digit integers not containing the digit $0$. Define a [i]gadang[/i] number to be the element of $A$ whose digits are all distinct and the digits contained in such number are not prime, and (a [i]gadang[/i] number leaves a remainder of 5 when divided by 7. If we pick an element of $A$ at random, what is the probability that the number we picked is a [i]gadang[/i] number?

Five identical empty buckets of $2$-liter capacity stand at the vertices of a regular pentagon. Cinderella and her wicked Stepmother go through a sequence of rounds: At the beginning of every round, the Stepmother takes one liter of water from the nearby river and distributes it arbitrarily over the five buckets. Then Cinderella chooses a pair of neighbouring buckets, empties them to the river and puts them back. Then the next round begins. The Stepmother goal's is to make one of these buckets overflow. Cinderella's goal is to prevent this. Can the wicked Stepmother enforce a bucket overflow? [i]Proposed by Gerhard Woeginger, Netherlands[/i]

For what largest $n$ are there $n$ seven-digit numbers that are successive members of one geometric progression?

$D_n$ is a set of domino pieces. For each pair of non-negative integers $(a, b)$ with $a \le b \le n$, there is one domino, denoted $[a, b]$ or $[b, a]$ in $D_n$. A [i]ring [/i] is a sequence of dominoes $[a_1, b_1], [a_2, b_2], ... , [a_k, b_k]$ such that $b_1 = a_2, b_2 = a_3, ... , b_{k-1} = a_k$ and $b_k = a_1$. Show that if $n$ is even there is a ring which uses all the pieces. Show that for n odd, at least $(n+1)/2$ pieces are not used in any ring. For $n$ odd, how many different sets of $(n+1)/2$ are there, such that the pieces not in the set can form a ring?

Tweaking a convex $n$-gon means the following: choose two adjacent sides $AB$ and $BC$ and replaces them with the line segment $AM$, $MN$, $NC$, where $M \in AB$ and $N \in BC$ are arbitrary points inside these segments. In other words, you cut off a corner and get an $(n+1)$-corner. Starting from a regular hexagon $P_6$ with area $1$, by continuous Tweaks a sequence $P_6,P_7,P_8, ...$ convex polygons. Show that Area of $​​P_n$ for all $n\ge 6$ greater than $\frac1 2$ is, regardless of how tweaks takes place.

Show that we cannot form more than $4096$ binary sequences of length $24$ so that any two differ in at least $8$ positions.

A sequence $ (S_n), n \geq 1$ of sets of natural numbers with $ S_1 = \{1\}, S_2 = \{2\}$ and \[{ S_{n + 1} = \{k \in }\mathbb{N}|k - 1 \in S_n \text{ XOR } k \in S_{n - 1}\}. \] Determine $ S_{1024}.$

Fibonacci numbers are defined as follows: $F_0 = F_1 = 1, F_{n+2} = F_{n+1}+F_n, n \geq 0$. Let $a_n$ be the number of words that consist of $n$ letters $0$ or $1$ and contain no two letters $1$ at distance two from each other. Express $a_n$ in terms of Fibonacci numbers.

Let $f:\mathbb{R}\rightarrow\mathbb{R}, f(x,y)=x^2-2y.$ Define the sequences $(a_n)_{n\geq1}$ and $(b_n)_{n\geq1}$ such that $a_{n+1}=f(a_n,b_n), b_{n+1}=f(b_n,a_n).$ If $4a_1-2b_1=7 :$ a) find the smallest $k\in\mathbb{N}$ for which the number $p=2^k\cdot(2^{512}a_9-b_9)$ is an integer. b) prove that $2^{2^{10}}+2^{2^9}+1$ divides $p.$

A positive real number sequence $a_1, a_2, a_3,\dots $ and a positive integer \(s\) is given. Let $f_n(0) = \frac{a_n+\dots+a_1}{n}$ and for each $0<k<n$ \[f_n(k)=\frac{a_n+\dots+a_{k+1}}{n-k}-\frac{a_k+\dots+a_1}{k}\] Then for every integer $n\geq s,$ the condition \[a_{n+1}=\max_{0\leq k<n}(f_n(k))\] is satisfied. Prove that this sequence must be eventually constant.