For arbitrary non-constant polynomials $f_1(x),\ldots,f_{2018}(x)\in\mathbb Z[x]$, is it always possible to find a polynomial $g(x)\in\mathbb Z[x]$ such that $$f_1(g(x)),\ldots,f_{2018}(g(x))$$are all reducible.

Is it possible to paint the plane in $4$ colors so that inside any circle are the dots of all four colors?

Let $C$ be a real number, and let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a three times differentiable function such that $$ \lim_{x \to \infty} f(x)=C, \;\; \; \lim_{x \to \infty} f'''(x)=0.$$ Prove that $$ \lim_{x \to \infty} f'(x) =0 \;\; \text{and} \;\; \lim_{x \to \infty} f''(x)=0.$$

Given a function $f\colon \mathbb{R}\rightarrow \mathbb{R} $ with $f(x)^2\le f(y)$ for all $x,y\in\mathbb{R} $, $x>y$, prove that $f(x)\in [0,1] $ for all $x\in \mathbb{R}$.

Let $m$ and $n$ be positive integers. In Philand, the Kingdom of Olymphics, with $m$ cities, and the Kingdom of Mathematicians for Fun, with $n$ cities, fight a battle in rounds. Some cities in the country are connected by roads, so that it is possible to travel through all the cities via the roads. In each round of the battle, if all cities neighboring, that is, connected directly by a road, a city in one of the kingdoms are from the other kingdom, that city is conquered in the next round and switches to the other kingdom. Knowing that between the first and second round, at least one city is not conquered, show that at some point the battle must end, i.e., no city can be captured by another kingdom.

[u]Round 1[/u] [b]p1.[/b] In a chemical lab there are three vials: one that can hold $1$ oz of fluid, another that can hold $2$ oz, and a third that can hold $3$ oz. The first is filled with grape juice, the second with sulfuric acid, and the third with water. There are also $3$ empty vials in the cupboard, also of sizes $1$ oz, $2$ oz, and $3$ oz. In order to save the world with grape-flavored acid, James Bond must make three full bottles, one of each size, filled with a mixture of all three liquids so that each bottle has the same ratio of juice to acid to water. How can he do this, considering he was silly enough not to bring any equipment? [b]p2.[/b] Twelve people, some are knights and some are knaves, are sitting around a table. Knaves always lie and knights always tell the truth. At some point they start up a conversation. The first person says, “There are no knights around this table.” The second says, “There is at most one knight at this table.” The third – “There are at most two knights at the table.” And so on until the $12$th says, “There are at most eleven knights at the table.” How many knights are at the table? Justify your answer. [b]p3.[/b] Aquaman has a barrel divided up into six sections, and he has placed a red herring in each. Aquaman can command any fish of his choice to either ‘jump counterclockwise to the next sector’ or ‘jump clockwise to the next sector.’ Using a sequence of exactly $30$ of these commands, can he relocate all the red herrings to one sector? If yes, show how. If no, explain why not. [img]https://cdn.artofproblemsolving.com/attachments/0/f/956f64e346bae82dee5cbd1326b0d1789100f3.png[/img] [b]p4.[/b] Is it possible to place $13$ integers around a circle so that the sum of any $3$ adjacent numbers is exactly $13$? [b]p5.[/b] Two girls are playing a game. The first player writes the letters $A$ or $B$ in a row, left to right, adding one letter on her turn. The second player switches any two letters after each move by the first player (the letters do not have to be adjacent), or does nothing, which also counts as a move. The game is over when each player has made $2011$ moves. Can the second player plan her moves so that the resulting letters form a palindrome? (A palindrome is a sequence that reads the same forward and backwards, e.g. $AABABAA$.) [u]Round 2[/u] [b]p6.[/b] Eight students participated in a math competition. There were eight problems to solve. Each problem was solved by exactly five people. Show that there are two students who solved all eight problems between them. [b]p7.[/b] There are $3n$ checkers of three different colors: $n$ red, $n$ green and $n$ blue. They were used to randomly fill a board with $3$ rows and $n$ columns so that each square of the board has one checker on it. Prove that it is possible to reshuffle the checkers within each row so that in each column there are checkers of all three colors. Moving checkers to a different row is not allowed. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

How many rectangles can be formed by the vertices of a cube? (Note: square is also a special rectangle). A. $6$ B. $8$ C. $12$ D. $18$ E. $16$

Ryan Alweiss storms into the Fine Hall common room with a gigantic eraser and erases all integers $n$ in the interval $[2, 728]$ such that $3^t$ doesn’t divide $n!$, where $t = \left\lceil \frac{n-3}{2} \right\rceil$. Find the sum of the leftover integers in that interval modulo $1000$.

Farmer James invents a new currency, such that for every positive integer $n\le 6$, there exists an $n$-coin worth $n!$ cents. Furthermore, he has exactly $n$ copies of each $n$-coin. An integer $k$ is said to be [i]nice[/i] if Farmer James can make $k$ cents using at least one copy of each type of coin. How many positive integers less than 2018 are nice?

At Easter Day, $4$ children and their mothers participated in a game in which they had to find hidden chocolate eggs. Augustine found $4$ eggs, Bruno found $6$, Carlos found $9$ and Daniel found $12$. Mrs. Gómez found the same number of eggs as her son, Mrs. Junco found twice as many eggs as her son, Mrs. Messi found three times as many eggs as her son, and Mrs. Núñez found five times as many eggs as her son. At the end of the day, they put all the eggs in boxes, with $18$ eggs in each box, and only one egg was left over. Determine who the mother of each child is.

A sled loaded with children starts from rest and slides down a snowy $25^\circ$ (with respect to the horizontal) incline traveling $85$ meters in $17$ seconds. Ignore air resistance. What is the coefficient of kinetic friction between the sled and the slope? $ \textbf {(A) } 0.36 \qquad \textbf {(B) } 0.40 \qquad \textbf {(C) } 0.43 \qquad \textbf {(D) } 1.00 \qquad \textbf {(E) } 2.01 $

The leader of an IMO team chooses positive integers $n$ and $k$ with $n > k$, and announces them to the deputy leader and a contestant. The leader then secretly tells the deputy leader an $n$-digit binary string, and the deputy leader writes down all $n$-digit binary strings which differ from the leader’s in exactly $k$ positions. (For example, if $n = 3$ and $k = 1$, and if the leader chooses $101$, the deputy leader would write down $001, 111$ and $100$.) The contestant is allowed to look at the strings written by the deputy leader and guess the leader’s string. What is the minimum number of guesses (in terms of $n$ and $k$) needed to guarantee the correct answer?

Prove that if a sphere can be inscribed in a convex polyhedron and each face of this polyhedron can be painted in one of two colors such that any two faces sharing a common edge are of different colors, then the sum of the areas of the faces of one color is equal to the sum of the areas of the faces of the other color.

How many positive integer solutions does the equation have $$\left\lfloor\frac{x}{10}\right\rfloor= \left\lfloor\frac{x}{11}\right\rfloor + 1?$$ ($\lfloor x \rfloor$ denotes the integer part of $x$, for example $\lfloor 2\rfloor = 2$, $\lfloor \pi\rfloor = 3$, $\lfloor \sqrt2 \rfloor =1$)

Find all pairs of positive integers $(n,t)$ such that $6^n+1=n^2t$, and $(n,29 \times 197)=1$

For a finite set $A$ of positive integers and its subset $B$, call $B$ a [i]half subset[/i] of $A$ when it satisfies the equation $\sum_{a\in A}a=2\sum_{b\in B}b$. For example, if $A=\{1,2,3\}$, then $\{1,2\}$ and $\{3\}$ are half subset of $A$. Determine all positive integers $n$ such that there exists a finite set $A$ which has exactly $n$ half subsets.

Given a non-negative integer \( n \), determine the values of \( c \) for which the sequence of numbers \[ a_n = 4^n c + \frac{4^n - (-1)^n}{5} \] contains at least one perfect square.

Find all functions $f,g : \mathbb{R} \times \mathbb{R} \mapsto \mathbb{R}^{+}$ such that \[ ( \sum_{j=1}^{n}a_{j}b_{j})^2 \leq (\sum_{j=1}^{n} f({a_{j},b_{j}))(\sum_{j=1}^{n} g({a_{j},b_{j})) \leq (\sum_{j=1}^{n} (a_j)^2 )(\sum_{j=1}^{n} (b_j)^2 ) }}\] for any two sets $a_j$ and $b_j$ of real numbers.

Determine all strictly increasing functions $f : R \to R$ such that for all $x \ne y$ to hold $$\frac{2\left[f(y)-f\left(\frac{x+y}{2}\right) \right]}{f(x)-f(y)}=\frac{f(x)-f(y)}{2\left[f\left(\frac{x+y}{2}\right)-f(x) \right]}$$

Hey, This problem is from the VTRMC 2006. 3. Recall that the Fibonacci numbers $ F(n)$ are defined by $ F(0) \equal{} 0$, $ F(1) \equal{} 1$ and $ F(n) \equal{} F(n \minus{} 1) \plus{} F(n \minus{} 2)$ for $ n \geq 2$. Determine the last digit of $ F(2006)$ (e.g. the last digit of 2006 is 6). As, I and a friend were working on this we noticed an interesting relationship when writing the Fibonacci numbers in "mod" notation. Consider the following, 01 = 1 mod 10 01 = 1 mod 10 02 = 2 mod 10 03 = 3 mod 10 05 = 5 mod 10 08 = 6 mod 10 13 = 3 mod 10 21 = 1 mod 10 34 = 4 mod 10 55 = 5 mod 10 89 = 9 mod 10 Now, consider that between the first appearance and second apperance of $ 5 mod 10$, there is a difference of five terms. Following from this we see that the third appearance of $ 5 mod 10$ occurs at a difference 10 terms from the second appearance. Following this pattern we can create the following relationships. $ F(55) \equal{} F(05) \plus{} 5({2}^{2})$ This is pretty much as far as we got, any ideas?

Given is an integer $n \geq 1$ and an $n \times n$ board, whose all cells are initially white. Peter the painter walks around the board and recolors the visited cells according to the following rules. Each walk of Peter starts at the bottom-left corner of the board and continues as follows: - if he is standing on a white cell, he paints it black and moves one cell up (or walks off the board if he is in the top row); - if he is standing on a black cell, he paints it white and moves one cell to the right (or walks off the board if he is in the rightmost column). Peter’s walk ends once he walks off the board. Determine the minimum positive integer $s$ with the following property: after exactly $s$ walks all the cells of the board will become white again.

For which integers $n>1$ does there exist a rectangle that can be subdivided into $n$ pairwise noncongruent rectangles similar to the original rectangle?

The five tires of a car (four road tires and a full-sized spare) were rotated so that each tire was used the same number of miles during the first $30,000$ miles the car traveled. For how many miles was each tire used? $\text{(A) }6000\qquad\text{(B) }7500\qquad\text{(C) }24,000\qquad\text{(D) }30,000\qquad\text{(E) }37,500$

Let $a$, $b$, $c$ be the lengths of the sides of a triangle. Prove that \[ \sqrt{a+b-c} + \sqrt{b+c-a} + \sqrt{c+a-b} \leq \sqrt{a} + \sqrt{b} + \sqrt{c} \] and determine when equality occurs.

Some natural numbers are marked. It is known that on every a segment of the number line of length $1999$ has a marked number. Prove that there is a pair of marked numbers, one of which is divisible by the other.