Find all triples $(a,b,c)$ of positive integers such that (i) $a \leq b \leq c$; (ii) $\text{gcd}(a,b,c)=1$; and (iii) $a^3+b^3+c^3$ is divisible by each of the numbers $a^2b, b^2c, c^2a$.

Compute the number of positive integers $n < 2012$ that share exactly two positive factors with 2012. [i]Proposed by Aaron Lin[/i]

Using each of the digits $1,2,3,\ldots ,8,9$ exactly once,we form nine,not necassarily distinct,nine-digit numbers.Their sum ends in $n$ zeroes,where $n$ is a non-negative integer.Determine the maximum possible value of $n$.

Let $P$ be a polynomial with integer coefficients. Suppose that for $n=1,2,3,\ldots ,1998$ the number $P(n)$ is a three-digit positive integer. Prove that the polynomial $P$ has no integer roots.

Let $p=4k+3$ be a prime number. Find the number of different residues mod p of $(x^{2}+y^{2})^{2}$ where $(x,p)=(y,p)=1.$

Show that for each non-negative integer $n$ there are unique non-negative integers $x$ and $y$ such that we have \[n=\frac{(x+y)^2+3x+y}{2}.\]

Positive integers $a,b,c$ satisfy that $a+b=b(a-c)$ and c+1 is a square of a prime. Prove that $a+b$ or $ab$ is a square.

Let $p$ be a prime number such that $p\equiv 1\pmod{4}$. Determine $\sum_{k=1}^{\frac{p-1}{2}}\left \lbrace \frac{k^2}{p} \right \rbrace$, where $\{x\}=x-[x]$.

Let $p$ be the probability that, in the process of repeatedly flipping a fair coin, one will encounter a run of 5 heads before one encounters a run of 2 tails. Given that $p$ can be written in the form $m/n$ where $m$ and $n$ are relatively prime positive integers, find $m+n$.

Determine the digits $0\leqslant c\leqslant 9$ such that for any positive integer $k{}$ there exists a positive integer $n$ such that the last $k{}$ digits of $n^9$ are equal to $c{}.$

Let $\epsilon$  be a positive real number. A positive integer will be called $\epsilon$-squarish if it is the product of two integers $a$ and $b$ such that $1 < a < b < (1 +\epsilon )a$. Prove that there are infinitely many occurrences of six consecutive $\epsilon$ -squarish integers.

Show that it is possible to color the points of $\mathbb Q\times\mathbb Q$ in two colors in such a way that any two points having distance $1$ have distinct colors.

Find all functions $f: \mathbb{Z}^+\rightarrow \mathbb{Z}^+$ such that for all positive integers $m,n$ with $m\ge n$, $$f(m\varphi(n^3)) = f(m)\cdot \varphi(n^3).$$ Here $\varphi(n)$ denotes the number of positive integers coprime to $n$ and not exceeding $n$.

Show that there do not exist polynomials $p(x)$ and $q(x)$ each having integer coefficients and of degree greater than or equal to 1 such that \[ p(x)q(x) = x^5 +2x +1 . \]

For a nonnegative integer $n$, define $a_n$ to be the positive integer with decimal representation \[1\underbrace{0\ldots0}_{n}2\underbrace{0\ldots0}_{n}2\underbrace{0\ldots0}_{n}1\mbox{.}\] Prove that $\frac{a_n}{3}$ is always the sum of two positive perfect cubes but never the sum of two perfect squares. [i](4th Middle European Mathematical Olympiad, Team Competition, Problem 7)[/i]

How many positive integers $n$ are there such that there are exactly $20$ positive odd integers that are less than $n$ and relatively prime with $n$? $ \textbf{(A)}\ 5 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 2 \qquad\textbf{(E)}\ \text{None of above} $

Fibonacci sequences is defined as $f_1=1$,$f_2=2$, $f_{n+1}=f_{n}+f_{n-1}$ for $n \ge 2$. a) Prove that every positive integer can be represented as sum of several distinct Fibonacci number. b) A positive integer is called [i]Fib-unique[/i] if the way to represent it as sum of several distinct Fibonacci number is unique. Example: $13$ is not Fib-unique because $13 = 13 = 8 + 5 = 8 + 3 + 2$. Find all Fib-unique.

Show that there exists an integer divisible by $1996$ such that the sum of the its decimal digits is $1996$.

For a given positive integer $ k$ denote the square of the sum of its digits by $ f_1(k)$ and let $ f_{n\plus{}1}(k) \equal{} f_1(f_n(k)).$ Determine the value of $ f_{1991}(2^{1990}).$

$a_1=-5$, $a_2=-6$ and for all $n \geq 2$ the ${(a_n)^\infty}_{n=1}$ sequence defined as, \[a_{n+1}=a_n+(a_1+1)(2a_2+1)(3a_3+1)\cdots((n-1)a_{n-1}+1)((n^2+n)a_n+2n+1)).\] If a prime $p$ divides $na_n+1$ for a natural number n, prove that there is a integer $m$ such that $m^2\equiv5(modp)$

For any integer $n$ greater than $1$, the number of prime numbers greater than $n!+1$ and less than $n!+n$ is: $\textbf{(A) }0\qquad \textbf{(B) }1\qquad \textbf{(C) }\dfrac n2\text{ for }n\text{ even,}\,\dfrac{n+1}2\text{ for }n\text{ odd}$ $\textbf{(D) }n-1\qquad \textbf{(E) }n$

Determine whether or not there exist numbers $x_1,x_2,\ldots ,x_{2009}$ from the set $\{-1,1\}$, such that: \[x_1x_2+x_2x_3+x_3x_4+\ldots+x_{2008}x_{2009}+x_{2009}x_1=999\]

[u]Round 6 [/u] [b]p16.[/b] Let $a$ be a solution to $x^3 -x +1 = 0$. Find $a^6 -a^2 +2a$. [b]p17.[/b] For a positive integer $n$, $\phi (n)$ is the number of positive integers less than $n$ that are relatively prime to $n$. Compute the sum of all $n$ for which $\phi (n) = 24$. [b]p18.[/b] Let $x$ be a positive integer such that $x^2 \equiv 57$ (mod $59$). Find the least possible value of $x$. [u]Round 7[/u] [b]p19.[/b] In the diagram below, find the number of ways to color each vertex red, green, yellow or blue such that no two vertices of a triangle have the same color. [img]https://cdn.artofproblemsolving.com/attachments/1/e/01418af242c7e2c095a53dd23e997b8d1f3686.png[/img] [b]p20.[/b] In a set with $n$ elements, the sum of the number of ways to choose $3$ or $4$ elements is a multiple of the sumof the number of ways to choose $1$ or $2$ elements. Find the number of possible values of $n$ between $4$ and $120$ inclusive. [b]p21.[/b] In unit square $ABCD$, let $\Gamma$ be the locus of points $P$ in the interior of $ABCD$ such that $2AP < BP$. The area of $\Gamma$ can be written as $\frac{a\pi +b\sqrt{c}}{d}$ for integers $a,b,c,d$ with $c$ squarefree and $gcd(a,b,d) = 1$. Find $1000000a +10000b +100c +d$. [u]Round 8 [/u] [b]p22.[/b] Ephram, GammaZero, and Orz walk into a bar. Each write some permutation of the letters “LMT” once, then concatenate their permutations one after the other (i.e. LTMTLMTLM would be a possible string, but not LLLMMMTTT). Suppose that the probability that the string “LMT” appears in that order among the new $9$-character string can be written as $\frac{A}{B}$ for relatively prime positive integers $A$ and $B$. Find $1000A+B$. [b]p23.[/b] In $\vartriangle ABC$ with side lengths $AB = 27$, $BC = 35$, and $C A = 32$, let $D$ be the point at which the incircle is tangent to $BC$. The value of $\frac{\sin \angle C AD }{\sin\angle B AD}$ can be expressed as $\frac{A}{B}$ for relatively prime positive integers $A$ and $B$. Find $1000A+B$. [b]p24.[/b] Let $A$ be the greatest possible area of a square contained in a regular hexagon with side length $1$. Let B be the least possible area of a square that contains a regular hexagon with side length $1$. The value of $B-A$ can be expressed as $a\sqrt{b}-c$ for positive integers $a$, $b$, and $c$ with $b$ squarefree. Find $10000a +100b +c$. [u]Round 9[/u] [b]p25.[/b] Estimate how many days before today this problem was written. If your estimation is $E$ and the actual answer is $A$, you will receive $\max \left( \left \lfloor 10 - \left| \frac{E-A}{2} \right| \right \rfloor , 0 \right)$ points. [b]p26.[/b] Circle $\omega_1$ is inscribed in unit square $ABCD$. For every integer $1 < n \le 10,000$, $\omega_n$ is defined as the largest circle which can be drawn inside $ABCD$ that does not overlap the interior of any of $\omega_1$,$\omega_2$, $...$,$\omega_{n-1}$ (If there are multiple such $\omega_n$ that can be drawn, one is chosen at random). Let r be the radius of ω10,000. Estimate $\frac{1}{r}$ . If your estimation is $E$ and the actual answer is $A$, you will receive $\max \left( \left \lfloor 10 - \left| \frac{E-A}{200} \right| \right \rfloor , 0 \right)$ points. [b]p27.[/b] Answer with a positive integer less than or equal to $20$. We will compare your response with the response of every other team that answered this problem. When two equal responses are compared, neither team wins. When two unequal responses $A > B$ are compared, $A$ wins if $B | A$, and $B$ wins otherwise. If your team wins n times, you will receive $\left \lfloor \frac{n}{2} \right \rfloor$ points. PS. You should use hide for answers. Rounds 1-5 have been posted [url=https://artofproblemsolving.com/community/c3h3167135p28823324]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Palmer correctly computes the product of the first $1,001$ prime numbers. Which of the following is NOT a factor of Palmer's product? $\text{(A) }2,002\qquad\text{(B) }3,003\qquad\text{(C) }5,005\qquad\text{(D) }6,006\qquad\text{(E) }7,007$

Prove that for any natural number $n$, the number $\dbinom{2n}{n}$ divides the least common multiple of the numbers $1, 2,\cdots, 2n -1, 2n$.