Let $x$, $y$ be two positive integers such that $\displaystyle 3x^2+x=4y^2+y$. Prove that $x-y$ is a perfect square.

Define the sequences $ a_n, b_n, c_n$ as follows. $ a_0 \equal{} k, b_0 \equal{} 4, c_0 \equal{} 1$. If $ a_n$ is even then $ a_{n \plus{} 1} \equal{} \frac {a_n}{2}$, $ b_{n \plus{} 1} \equal{} 2b_n$, $ c_{n \plus{} 1} \equal{} c_n$. If $ a_n$ is odd, then $ a_{n \plus{} 1} \equal{} a_n \minus{} \frac {b_n}{2} \minus{} c_n$, $ b_{n \plus{} 1} \equal{} b_n$, $ c_{n \plus{} 1} \equal{} b_n \plus{} c_n$. Find the number of positive integers $ k < 1995$ such that some $ a_n \equal{} 0$.

Show that no integer of the form $ xyxy$ in base $ 10$ can be a perfect cube. Find the smallest base $ b>1$ for which there is a perfect cube of the form $ xyxy$ in base $ b$.

Find all $f: \mathbb{N}\rightarrow \mathbb{N}$, for which $f(f(n)+m)=n+f(m+2014)$ for $\forall$ $m,n\in \mathbb{N}$.

[u]Round 5 [/u] [b]p13.[/b] Suppose $\vartriangle ABC$ is an isosceles triangle with $\overline{AB} = \overline{BC}$, and $X$ is a point in the interior of $\vartriangle ABC$. If $m \angle ABC = 94^o$, $m\angle ABX = 17^o$, and $m\angle BAX = 13^o$, then what is $m\angle BXC$ (in degrees)? [b]p14.[/b] Find the remainder when $\sum^{2019}_{n=1} 1 + 2n + 4n^2 + 8n^3$ is divided by $2019$. [b]p15.[/b] How many ways can you assign the integers $1$ through $10$ to the variables $a, b, c, d, e, f, g, h, i$, and $j$ in some order such that $a < b < c < d < e, f < g < h < i$, $a < g, b < h, c < i$, $f < b, g < c$, and $h < d$? [u]Round 6 [/u] [b]p16.[/b] Call an integer $n$ equi-powerful if $n$ and $n^2$ leave the same remainder when divided by 1320. How many integers between $1$ and $1320$ (inclusive) are equi-powerful? [b]p17.[/b] There exists a unique positive integer $j \le 10$ and unique positive integers $n_j$ , $n_{j+1}$, $...$, $n_{10}$ such that $$j \le n_j < n_{j+1} < ... < n_{10}$$ and $${n_{10} \choose 10}+ {n_9 \choose 9}+ ... + {n_j \choose j}= 2019.$$ Find $n_j + n_{j+1} + ... + n_{10}$. [b]p18.[/b] If $n$ is a randomly chosen integer between $1$ and $390$ (inclusive), what is the probability that $26n$ has more positive factors than $6n$? [u]Round 7[/u] [b]p19.[/b] Suppose $S$ is an $n$-element subset of $\{1, 2, 3, ..., 2019\}$. What is the largest possible value of $n$ such that for every $2 \le k \le n$, $k$ divides exactly $n - 1$ of the elements of $S$? [b]p20.[/b] For each positive integer $n$, let $f(n)$ be the fewest number of terms needed to write $n$ as a sum of factorials. For example, $f(28) = 3$ because $4! + 2! + 2! = 28$ and 28 cannot be written as the sum of fewer than $3$ factorials. Evaluate $f(1) + f(2) + ... + f(720)$. [b]p21.[/b] Evaluate $\sum_{n=1}^{\infty}\frac{\phi (n)}{101^n-1}$ , where $\phi (n)$ is the number of positive integers less than or equal to n that are relatively prime to $n$. PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c4h2788993p24519281]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find the last two digits of each of the numbers $3^{1974}$ and $7^{1974}$.

Does there exist an infinite set of triples of integers $x, y, z$ (not necessarily positive) such that $$x^2 + y^2 + z^2 = x^3 + y^3+z^3?$$ (NB Vassiliev)

Determine all positive integers which cannot be represented as $\frac{a}{b}+\frac{a+1}{b+1}$ with $a,b$ being positive integers.

Find all pairs of $p,q$ prime numbers that satisfy the equation $$p(p^4+p^2+10q)=q(q^2+3)$$

Given distinct prime numbers $p$ and $q$ and a natural number $n \geq 3$, find all $a \in \mathbb{Z}$ such that the polynomial $f(x) = x^n + ax^{n-1} + pq$ can be factored into 2 integral polynomials of degree at least 1.

Given an integer $n > 2$, give an example of a set of $n$ mutually different numbers $a_1,...,a_n$ for which the set of their pairwise sums $a_i + a_j$ ($i \ne j$) contains as few different numbers as possible; also give an example of a set of n different numbers $b_1,...,b_n$ for which the set of their pairwise sums $b_i+b_j$ ($i \ne j$) contains as many different numbers as possible;

A sequence of real numbers is called a [i]Fibonacci [/i] sequence if $$t_{n+2} = t_{n+1} + t_n$$ for $n= 1,2,3,. .$ . Two Fibonacci sequences are said to be [i]essentially different[/i] if the terms of one sequence cannot be obtained by multiplying the terms of the other by a constant. For example, the Fibonacci sequences $1,2,3,5,8,...$ and $1,3,4,7,11,...$ are essentially different, but the sequences $1,2,3,5,8,...$ and $2,4,6,10,16,...$ are not. (a) Prove that there exist real numbers $p$ and $q$ such that the sequences $1,p,p^2,p^3,...$ and $1,q,q^2,q^3,...$ are essentially different Fibonacci sequences. (b) Let $a_1,a_2,a_3,...$ and $b_1,b_2,b_3,...$ be essentially different Fibonacci sequences. Prove that for every Fibonacci sequence $t_1,t_2,t_3,...$, there exists exactly one number $\alpha$ and exactly one number $\beta$, such that: $$t_n = \alpha a_n + \beta b_n$$ for $n = 1,2,3,...$ (c) $t_1,t_2,t_3,...$, is the Fibonacci sequence with $t_1 = 1$ and $t_2= 2$. Express $t_n$ in terms of $n$.

Find all primes $p$ such that the expression \[\binom{p}1^2+\binom{p}2^2+\cdots+\binom{p}{p-1}^2\] is divisible by $p^3$.

Find all the functions $\varphi:\mathbb{Z}[x]\to\mathbb{Z}[x]$ such that $\varphi(x)=x,$ any integer polynomials $f, g$ satisfy $\varphi(f+g)=\varphi(f)+\varphi(g)$ and $\varphi(f)$ is a perfect power if and only if $f{}$ is a perfect power. [i]Note:[/i] A polynomial $f\in \mathbb{Z}[x]$ is a perfect power if $f = g^n$ for some $g\in \mathbb{Z}[x]$ and $n\geqslant 2.$ [i]Proposed by Pavel Ciurea[/i]

Qing initially writes the ordered pair $(1,0)$ on a blackboard. Each minute, if the pair $(a,b)$ is on the board, she erases it and replaces it with one of the pairs $(2a-b,a)$, $(2a+b+2,a)$ or $(a+2b+2,b)$. Eventually, the board reads $(2014,k)$ for some nonnegative integer $k$. How many possible values of $k$ are there? [i]Proposed by Evan Chen[/i]

Find all sets of not necessary distinct 2014 rationals such that:if we remove an arbitrary number in the set, we can divide remaining 2013 numbers into three sets such that each set has exactly 671 elements and the product of all elements in each set are the same.

Prove that among any 18 consecutive positive 3-digit numbers, there is at least one that is divisible by the sum of its digits!

Three different digits were used to create three different three-digit numbers forming an arithmetic progression. (In each number, all the digits are different.) What is the largest difference in this progression?

At what index the harmonic series has a fractional part of $ 1/12? $

Find all $ a, b, c \in \mathbb{Z} $, $ c \ge 0 $ such that $ a^n + 2^n | b^n + c $ for all positive integers $ n $ where $ 2ab $ is non-square.

If the fraction $\frac{an + b}{cn + d}$ may be simplified using $2$ (as a common divisor ), show that the number $ad - bc$ is even. ($a, b, c, d, n$ are natural numbers and the $cn + d$ different from zero).

Determine all integer triplets $(x,y,z)$ such that \[x^4+x^2=7^zy^2\mbox{.}\]

Find the largest $n$ for which there exists a sequence $(a_0, a_1, \ldots, a_n)$ of non-zero digits such that, for each $k$, $1 \le k \le n$, the $k$-digit number $\overline{a_{k-1} a_{k-2} \ldots a_0} = a_{k-1} 10^{k-1} + a_{k-2} 10^{k-2} + \cdots + a_0$ divides the $(k+1)$-digit number $\overline{a_{k} a_{k-1}a_{k-2} \ldots a_0}$. P.S.: This is basically the same problem as http://www.artofproblemsolving.com/Forum/viewtopic.php?f=57&t=548550.

Find all pairs of two positive integers $(m,n)$ satisfying $ 3^m - 7^n = 2 $.

Find all triples $(a,b,c)$ of positive integers such that the product of any two of them when divided by the third leaves the remainder $1$.