Prove among any $14$ consecutive positive integers there exist $6$ which are pairwise relatively prime.

For $n \ge 1$, let $a_n$ be the largest odd divisor of $n$, and let $b_n = a_1+a_2+...+a_n$. Prove that $b_n \ge \frac{ n^2 + 2}{3}$, and determine for which $n$ equality holds. For example, $a_1 = 1, a_2 = 1, a_3 = 3, a_4 = 1, a_5 = 5, a_6 = 3$, thus $b_6 = 1 + 1 + 3 + 1 + 5 + 3 = 14 \ge \frac{ 6^2 + 2}{3}= 12\frac23$ .

Let $p$ be a prime number. Let $\mathbb F_p$ denote the integers modulo $p$, and let $\mathbb F_p[x]$ be the set of polynomials with coefficients in $\mathbb F_p$. Define $\Psi : \mathbb F_p[x] \to \mathbb F_p[x]$ by \[ \Psi\left( \sum_{i=0}^n a_i x^i \right) = \sum_{i=0}^n a_i x^{p^i}. \] Prove that for nonzero polynomials $F,G \in \mathbb F_p[x]$, \[ \Psi(\gcd(F,G)) = \gcd(\Psi(F), \Psi(G)). \] Here, a polynomial $Q$ divides $P$ if there exists $R \in \mathbb F_p[x]$ such that $P(x) - Q(x) R(x)$ is the polynomial with all coefficients $0$ (with all addition and multiplication in the coefficients taken modulo $p$), and the gcd of two polynomials is the highest degree polynomial with leading coefficient $1$ which divides both of them. A non-zero polynomial is a polynomial with not all coefficients $0$. As an example of multiplication, $(x+1)(x+2)(x+3) = x^3+x^2+x+1$ in $\mathbb F_5[x]$. [i]Proposed by Mark Sellke[/i]

Prove that the equation $x+x^2=y+y^2+y^3$ do not have any solutions in positive integers.

Let $\{F_n\}_{n\ge1}$ be the Fibonacci sequence defined by $F_1=F_2=1$ and for all $n\ge3$, $F_n=F_{n-1}+F_{n-2}$. Prove that among the first $10000000000000002$ terms of the sequence there is one term that ends up with $8$ zeroes. [i]Proposed by José Luis Díaz-Barrero[/i]

Each integer is colored with exactly one of the following colors: red, blue, or orange, and all three colors are used in the coloring. The coloring also satisfies the following properties: 1. The sum of a red number and an orange number results in a blue-colored number, 2. The sum of an orange and blue number results in an orange-colored number; 3. The sum of a blue number and a red number results in a red-colored number. (a) Prove that $0$ and $1$ must have distinct colors. (b) Determine all possible colorings of the integers which also satisfy the properties stated above.

How many zeroes appear at the end of $209$ factorial?

Let $S$ be the set of integers which are both a multiple of $70$ and a factor of $630{,}000$. A random element $c$ of $S$ is selected. If the probability that there exists an integer $d$ with $\gcd (c,d) = 70$ and $\operatorname{lcm} (c,d) = 630{,}000$ is $\frac mn$ for some relatively prime integers $m$ and $n$, compute $100m+n$. [i]Proposed by Eugene Chen[/i]

Suppose that $m,n$ are integers greater than $1$ such that $m+n-1$ divides $m^2+n^2-1$. Prove that $m+n-1$ cannot be a prime number.

Given an integer ${n>1}$, denote by $P_{n}$ the product of all positive integers $x$ less than $n$ and such that $n$ divides ${x^2-1}$. For each ${n>1}$, find the remainder of $P_{n}$ on division by $n$. [i]Proposed by John Murray, Ireland[/i]

Let $a_{10} = 10$, and for each integer $n >10$ let $a_n = 100a_{n - 1} + n$. Find the least $n > 10$ such that $a_n$ is a multiple of $99$.

Consider all positive integers written in a row: $123456789101112131415...$ Find the $206788$-th digit from the left.

Define $a_1 = 1979$ and $a_{n+1} = 9^{a_n}$ for $n = 1,2,3,...$. Determine the last two digits of $a_{1979}$.

Natalia and Marcela count $1$ by $1$ starting together at $1$, but Marcela's speed is triple that of Natalia (when Natalia says her second number, Marcela says the fourth number). When the difference of the numbers that they say in unison is any of the multiples of $ 29$, between $500$ and $600$, Natalia continues counting normally and Marcela begins to count downwards in such a way that, at one point, the two say in unison the same number. What is said number?

$ f(x)$ and $ g(x)$ are two polynomials with nonzero degrees and integer coefficients, such that $ g(x)$ is a divisor of $ f(x)$ and the polynomial $ f(x)\plus{}2009$ has $ 50$ integer roots. Prove that the degree of $ g(x)$ is at least $ 5$.

Each different letter in the following addition represents a different decimal digit. The sum is a six-digit integer whose digits are all equal. $$\begin{tabular}{ccccccc} & P & U & R & P & L & E\\ + & & C & O & M & E & T \\ \hline \\ \end{tabular}$$ Find the greatest possible value that the five-digit number $COMET$ could represent.

How many zeros are there in the last digits of the following number $P = 11\times12\times ...\times 88\times 89$ ? (A): $16$, (B): $17$, (C): $18$, (D): $19$, (E) None of the above.

Prove that For all positive integers $m$ and $n$ , one has $| n \sqrt{2005} - m | > \frac{1}{90n}$

Does there exist such infinite set of positive integers $S$ that satisfies the condition below? *for all $a,b$ in $S$, there exists an odd integer $k$ that $a$ divides $b^k+1$.

Let $p{}$ be a prime number. Prove that the equation has $x^2-x+3-ps=0$ with $x,s\in\mathbb{Z}$ has solutions if and only if the equation $y^2-y+25-pt=0$ with $y,t\in\mathbb{Z}$ has solutions.

The sequence $(x_n)$ is defined as follows: $$x_0=2,\, x_1=1,\, x_{n+2}=x_{n+1}+x_n$$ for every non-negative integer $n$. a. For each $n\geq 1$, prove that $x_n$ is a prime number only if $n$ is a prime number or $n$ has no odd prime divisors b. Find all non-negative pairs of integers $(m,n)$ such that $x_m|x_n$.

$ \forall n > 0, n \in \mathbb{Z},$ there exists uniquely determined integers $ a_n, b_n, c_n \in \mathbb{Z}$ such \[ \left(1 \plus{} 4 \cdot \sqrt[3]{2} \minus{} 4 \cdot \sqrt[3]{4} \right)^n \equal{} a_n \plus{} b_n \cdot \sqrt[3]{2} \plus{} c_n \cdot \sqrt[3]{4}.\] Prove that $ c_n \equal{} 0$ implies $ n \equal{} 0.$

Determine all non negative integers $x$ and $y$ such that $6^x$ + $2^y$ + 2 is a perfect square.

[u]Round 1[/u] [b]p1.[/b] Every angle of a regular polygon has degree measure $179.99$ degrees. How many sides does it have? [b]p2.[/b] What is $\frac{1}{20} + \frac{1}{1}+ \frac{1}{5}$ ? [b]p3.[/b] If the area bounded by the lines $y = 0$, $x = 0$, and $x = 3$ and the curve $y = f(x)$ is $10$ units, what is the area bounded by $y = 0$, $x = 0$, $x = 6$, and $y = f(x/2)$? [u]Round 2[/u] [b]p4.[/b] How many ways can $42$ be expressed as the sum of $2$ or more consecutive positive integers? [b]p5.[/b] How many integers less than or equal to $2015$ can be expressed as the sum of $2$ (not necessarily distinct) powers of two? [b]p6.[/b] $p,q$, and $q^2 - p^2$ are all prime. What is $pq$? [u]Round 3[/u] [b]p7.[/b] Let $p(x) = x^2 + ax + a$ be a polynomial with integer roots, where $a$ is an integer. What are all the possible values of $a$? [b]p8.[/b] In a given right triangle, the perimeter is $30$ and the sum of the squares of the sides is $338$. Find the lengths of the three sides. [b]p9.[/b] Each of the $6$ main diagonals of a regular hexagon is drawn, resulting in $6$ triangles. Each of those triangles is then split into $4$ equilateral triangles by connecting the midpoints of the $3$ sides. How many triangles are in the resulting figure? [u]Round 4[/u] [b]p10.[/b] Let $f = 5x+3y$, where $x$ and $y$ are positive real numbers such that $xy$ is $100$. Find the minimum possible value of $f$. [b]p11.[/b] An integer is called "Awesome" if its base $8$ expression contains the digit string $17$ at any point (i.e. if it ever has a $1$ followed immediately by a $7$). How many integers from $1$ to $500$ (base $10$) inclusive are Awesome? [b]p12.[/b] A certain pool table is a rectangle measuring $15 \times 24$ feet, with $4$ holes, one at each vertex. When playing pool, Joe decides that a ball has to hit at least $2$ sides before getting into a hole or else the shot does not count. What is the minimum distance a ball can travel after being hit on this table if it was hit at a vertex (assume it only stops after going into a hole) such that the shot counts? PS. You should use hide for answers. Rounds 5-8 have been posted [url=https://artofproblemsolving.com/community/c3h3157013p28696685]here [/url] and 9-12 [url=https://artofproblemsolving.com/community/c3h3158564p28715928]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Positive integer $n$ when divided with number $3$ gives remainder $a$, when divided with $5$ has remainder $b$ and when divided with $7$ gives remainder $c$. Find remainder when dividing number $n$ with $105$ if $4a+3b+2c=30$