The first term of a sequence is $2014$. Each succeeding term is the sum of the cubes of the digits of the previous term. What is the $2014$ th term of the sequence?

Find all positive integers $n$ for which the following statement holds: For any two polynomials $P(x)$ and $Q(x)$ of degree $n$ there exist monomials $ax^k$ and $bx^{ell}, 0 \le k,\ ell \le n$, such that the graphs of $P(x) + ax^k$ and $Q(x) + bx^{ell}$ have no common points.

Prove that $\frac{5^{125}-1}{5^{25}-1}$ is a composite number.

Let $S$ be a set of positive integers such that $\lfloor \sqrt{x}\rfloor =\lfloor \sqrt{y}\rfloor $ for all $x, y \in S$. Show that the products $xy$, where $x, y \in S$, are pairwise distinct.

[b]p1.[/b] Let $S$ be the sum of the $99$ terms: $$(\sqrt1 + \sqrt2)^{-1},(\sqrt2 + \sqrt3)^{-1}, (\sqrt3 + \sqrt4)^{-1},..., (\sqrt{99} + \sqrt{100})^{-1}.$$ Prove that $S$ is an integer. [b]p2.[/b] Determine all pairs of positive integers $x$ and $y$ for which $N=x^4+4y^4$ is a prime. (Your work should indicate why no other solutions are possible.) [b]p3.[/b] Let $w,x,y,z$ be arbitrary positive real numbers. Prove each inequality: (a) $xy \le \left(\frac{x+y}{2}\right)^2$ (b) $wxyz \le \left(\frac{w+x+y+z}{4}\right)^4$ (c) $xyz \le \left(\frac{x+y+z}{3}\right)^3$ [b]p4.[/b] Twelve points $P_1$,$P_2$, $...$,$P_{12}$ are equally spaaed on a circle, as shown. Prove: that the chords $\overline{P_1P_9}$, $\overline{P_4P_{12}}$ and $\overline{P_2P_{11}}$ have a point in common. [img]https://cdn.artofproblemsolving.com/attachments/d/4/2eb343fd1f9238ebcc6137f7c84a5f621eb277.png[/img] [b]p5.[/b] Two very busy men, $A$ and $B$, who wish to confer, agree to appear at a designated place on a certain day, but no earlier than noon and no later than $12:15$ p.m. If necessary, $A$ will wait $6$ minutes for $B$ to arrive, while $B$ will wait $9$ minutes for $A$ to arrive but neither can stay past $12:15$ p.m. Express as a percent their chance of meeting. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Determine the minimum value of $a^{2} + b^{2}$ when $(a,b)$ traverses all the pairs of real numbers for which the equation \[ x^{4} + ax^{3} + bx^{2} + ax + 1 = 0 \] has at least one real root.

[b]p1[/b]. Is it possible to find $n$ positive numbers such that their sum is equal to $1$ and the sum of their squares is less than $\frac{1}{10}$? [b]p2.[/b] In the country of Sepulia, there are several towns with airports. Each town has a certain number of scheduled, round-trip connecting flights with other towns. Prove that there are two towns that have connecting flights with the same number of towns. [b]p3.[/b] A $4 \times 4$ magic square is a $4 \times 4$ table filled with numbers $1, 2, 3,..., 16$ - with each number appearing exactly once - in such a way that the sum of the numbers in each row, in each column, and in each diagonal is the same. Is it possible to complete $\begin{bmatrix} 2 & 3 & * & * \\ 4 & * & * & *\\ * & * & * & *\\ * & * & * & * \end{bmatrix}$ to a magic square? (That is, can you replace the stars with remaining numbers $1, 5, 6,..., 16$, to obtain a magic square?) [b]p4.[/b] Is it possible to label the edges of a cube with the numbers $1, 2, 3, ... , 12$ in such a way that the sum of the numbers labelling the three edges coming into a vertex is the same for all vertices? [b]p5.[/b] (Bonus) Several ants are crawling along a circle with equal constant velocities (not necessarily in the same direction). If two ants collide, both immediately reverse direction and crawl with the same velocity. Prove that, no matter how many ants and what their initial positions are, they will, at some time, all simultaneously return to the initial positions. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Prove that if the equation $$x^4 + ax + b = 0$$ has two equal roots, then $$\left( \frac{a}{4} \right)^4 =\left( \frac{b}{3} \right)^3.$$

Solve system of equations $$\begin{cases} x+\dfrac{x+y}{x^2+y^2}=1 \\ x+\dfrac{x-y}{x^2+y^2}=2 \end{cases}$$

Let $n$ be a fixed natural number. Find all $n$ tuples of natural pairwise distinct and coprime numbers like $a_1,a_2,\ldots,a_n$ such that for $1\leq i\leq n$ we have \[ a_1+a_2+\ldots+a_n|a_1^i+a_2^i+\ldots+a_n^i \]

Prove that for all integers $n > 1$, $$\frac{1}{n}+\frac{1}{n + 1}+ ...+\frac{1}{n^2- 1}+\frac{1}{n^2} > 1$$

Find the smallest value that the expression can take $$|a-1|+|b-2|+c-3|+|3a+2b+c|$$ ($a$, $b$ and $c$ are arbitrary numbers).

$a, b, c, d$ are integers with $ad \ne bc$. Show that $1/((ax+b)(cx+d))$ can be written in the form $ r/(ax+b) + s/(cx+d)$. Find the sum $1/1\cdot 4 + 1/4\cdot 7 + 1/7\cdot 10 + ... + 1/2998 \cdot 3001$.

Let $a$, $b$, $c$ be positive real numbers with $abc \leq a+b+c$. Show that \[ a^2 + b^2 + c^2 \geq \sqrt 3 abc. \] [i]Cristinel Mortici, Romania[/i]

Find all nonempty sets $S$ of integers such that $3m-2n \in S$ for all (not necessarily distinct) $m,n \in S$.

Equation $(1-\text{i})x^2+(\lambda+\text{i})x+(1+\text{i}\lambda)=0(\lambda\in\mathbb{R})$ has two imaginary roots, then the range value of $\lambda$ is________.

Both roots of the quadratic equation $ x^2 \minus{} 63x \plus{} k \equal{} 0$ are prime numbers. The number of possible values of $ k$ is $ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 3 \qquad \textbf{(E)}\ \textbf{more than four}$

Consider the parametric function $ f_k:\mathbb{R}\longrightarrow\mathbb{R}, f(x)=x+k\lfloor x \rfloor . $ [b]a)[/b] For which integer values of $ k $ the above function is injective? [b]b)[/b] For which integer values of $ k $ the above function is surjective? [b]c)[/b] Given two natural numbers $ n,m, $ create two bijective functions: $$ \phi : f_m (\mathbb{R} )\cap [0,\infty )\longrightarrow f_n(\mathbb{R})\cap [0,\infty ) $$ $$ \psi : \left(\mathbb{R}\setminus f_m (\mathbb{R})\right)\cap [0,\infty )\longrightarrow\left(\mathbb{R}\setminus f_n (\mathbb{R})\right)\cap [0,\infty ) $$ [i]Cristinel Mortici[/i]

Find all one-to-one mappings $f:\mathbb{N}\to\mathbb{N}$ such that for all positive integers $n$ the following relation holds: \[ f(f(n)) \leq \frac {n+f(n)} 2 . \]

Let $P(x)$ be a polynomial with the coefficients being $0$ or $1$ and degree $2023$. If $P(0)=1$, then prove that every real root of this polynomial is less than $\frac{1-\sqrt{5}}{2}$.

Goodnik writes 10 numbers on the board, then Nogoodnik writes 10 more numbers, all 20 of the numbers being positive and distinct. Can Goodnik choose his 10 numbers so that no matter what Nogoodnik writes, he can form 10 quadratic trinomials of the form $x^2 +px+q$, whose coeficients $p$ and $q$ run through all of the numbers written, such that the real roots of these trinomials comprise exactly 11 values? [i]I. Rubanov[/i]

Determine the maximum possible value of \[\frac{\left(x^2+5x+12\right)\left(x^2+5x-12\right)\left(x^2-5x+12\right)\left(-x^2+5x+12\right)}{x^4}\] over all non-zero real numbers $x$. [i]2019 CCA Math Bonanza Lightning Round #3.4[/i]

Let $Q(x) = x^2 + 2x + 3$, and suppose that $P(x)$ is a polynomial such that $$P(Q(x)) = x^6 + 6x^5 + 18x^4 + 32x^3 + 35x^2 + 22x + 8.$$ Compute $P(2)$.

The sequences $(a_{n})$, $(b_{n})$ are defined by $a_{1} = \alpha$, $b_{1} = \beta$, $a_{n+1} = \alpha a_{n} - \beta b_{n}$, $b_{n+1} = \beta a_{n} + \alpha b_{n}$ for all $n > 0.$ How many pairs $(\alpha, \beta)$ of real numbers are there such that $a_{1997} = b_{1}$ and $b_{1997} = a_{1}$?

Given a polynomial $f(x) = x^4+ax^3+bx^2+cx$. It is known that each of the equations $f(x) = 1$ and $f(x) = 2$ has four real roots (not necessarily distinct). Prove that if the roots of the first equation satisfy the equality $x_1 + x_2 = x_3 + x_4$, then the same equation holds for the roots of the second equation