Let $a, b, c$ be positive integers for which \[ac = b^2 + b + 1\] Prove that the equation \[ax^2 - (2b + 1)xy + cy^2 = 1\] has an integer solution.

There is an arithmetic progression of $7$ terms in which all the terms are different prime numbers. Determine the smallest possible value of the last term of such a progression. Clarification: In an arithmetic progression of difference $d$ each term is equal to the previous one plus $d$.

Let $p$ be a prime number. Find all pairs of coprime positive integers $(m,n)$ such that $$ \frac{p+m}{p+n}=\frac{m}{n}+\frac{1}{p^2}.$$

Prove that there are infinitely many quadruples of integers $(a,b,c,d)$ such that \begin{align*} a^2 + b^2 + 3 &= 4ab\\ c^2 + d^2 + 3 &= 4cd\\ 4c^3 - 3c &= a \end{align*} [i]Travis Hance.[/i]

Let $m$ and $n$ be natural numbers such that $m^3-n^3$ is a prime number. What is the remainder of the number $m^3-n^3$ when divided by $6$?

[b]p1.[/b] Find all triples of positive integers such that the sum of their reciprocals is equal to one. [b]p2.[/b] Prove that $a(a + 1)(a + 2)(a + 3)$ is divisible by $24$. [b]p3.[/b] There are $20$ very small red chips and some blue ones. Find out whether it is possible to put them on a large circle such that (a) for each chip positioned on the circle the antipodal position is occupied by a chip of different color; (b) there are no two neighboring blue chips. [b]p4.[/b] A $12$ liter container is filled with gasoline. How to split it in two equal parts using two empty $5$ and $8$ liter containers? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Is it possible to partition $\{1, 2, 3, \ldots, 28\}$ into two sets $A$ and $B$ such that both of the following conditions hold simultaneously: (i) the number of odd integers in $A$ is equal to the number of odd integers in $B$; (ii) the difference between the sum of squares of the integers in $A$ and the sum of squares of the integers in $B$ is $16$?

Find all possible triples of positive integers, $a, b, c$ so that $\frac{a+1}{b}$, $\frac{b+1}{c}$ and $\frac{c+1}{a}$ are also integers.

Find prime numbers $p$, $q$, $r$ and $s$, pairwise distinct, such that their sum is prime number and numbers $p^2+qr$ and $p^2+qs$ are perfect squares

Find all prime $p$ numbers such that $p^3-4p+9$ is perfect square.

Let $k$ be a prime, such as $k\neq 2, 5$, prove that between the first $k$ terms of the sequens $1, 11, 111, 1111,....,1111....1$, where the last term have $k$ ones, is divisible by $k$.

Prove that the numbers $${{2^n-1} \choose {i}}, i = 0, 1, . . ., 2^{n-1} - 1,$$ have pairwise different residues modulo $2^n$

Let $f(x)=x^2+x+1$. Determine, with proof, all positive integers $n$ such that $f(k)$ divides $f(n)$ whenever $k$ is a positive divisor of $n$.

Let $n$ be a positive integer. A sequence $a_1$, $a_2$,$ ...$ , $a_n$ is called [i]good [/i] if the following conditions hold: $\bullet$ For each $i \in \{1, 2, ..., n\}$, $1 \le a_i \le n$ $\bullet$ For all positive integers $i, j$ with $1 \le i \le j \le n$, the expression $a_i + a_{i+1} + ...+ a_j$ is not divisible by $ n + 1$. Find the number of good sequences (in terms of $n$).

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

Let $b$ be a positive integer such that $\gcd(b,6)=1$. Show that there are positive integers $x$ and $y$ such that $\frac1x+\frac1y=\frac3b$ if and only if $b$ is divisible by some prime number of form $6k-1$.

A diabolical combination lock has $n$ dials (each with $c$ possible states), where $n,c>1$. The dials are initially set to states $d_1, d_2, \ldots, d_n$, where $0\le d_i\le c-1$ for each $1\le i\le n$. Unfortunately, the actual states of the dials (the $d_i$'s) are concealed, and the initial settings of the dials are also unknown. On a given turn, one may advance each dial by an integer amount $c_i$ ($0\le c_i\le c-1$), so that every dial is now in a state $d_i '\equiv d_i+c_i \pmod{c}$ with $0\le d_i ' \le c-1$. After each turn, the lock opens if and only if all of the dials are set to the zero state; otherwise, the lock selects a random integer $k$ and cyclically shifts the $d_i$'s by $k$ (so that for every $i$, $d_i$ is replaced by $d_{i-k}$, where indices are taken modulo $n$). Show that the lock can always be opened, regardless of the choices of the initial configuration and the choices of $k$ (which may vary from turn to turn), if and only if $n$ and $c$ are powers of the same prime. [i]Bobby Shen.[/i]

Determine all positive integers $n < 200$, such that $n^2 + (n+ 1)^2$ is the square of an integer.

Solve in integers the equation $x + y = x^2 - xy + y^2$.

Let $\mathcal{S}$ be a set consisting of $n \ge 3$ positive integers, none of which is a sum of two other distinct members of $\mathcal{S}$. Prove that the elements of $\mathcal{S}$ may be ordered as $a_1, a_2, \dots, a_n$ so that $a_i$ does not divide $a_{i - 1} + a_{i + 1}$ for all $i = 2, 3, \dots, n - 1$.

For which integers $n \ge 2$ can we arrange numbers $1,2, \ldots, n$ in a row, such that for all integers $1 \le k \le n$ the sum of the first $k$ numbers in the row is divisible by $k$?

[u]Round 1[/u] [b]p1.[/b] Let $n$ be a two-digit positive integer. What is the maximum possible sum of the prime factors of $n^2 - 25$ ? [b]p2.[/b] Angela has ten numbers $a_1, a_2, a_3, ... , a_{10}$. She wants them to be a permutation of the numbers $\{1, 2, 3, ... , 10\}$ such that for each $1 \le i \le 5$, $a_i \le 2i$, and for each $6 \le i \le 10$, $a_i \le - 10$. How many ways can Angela choose $a_1$ through $a_{10}$? [b]p3.[/b] Find the number of three-by-three grids such that $\bullet$ the sum of the entries in each row, column, and diagonal passing through the center square is the same, and $\bullet$ the entries in the nine squares are the integers between $1$ and $9$ inclusive, each integer appearing in exactly one square. [u]Round 2 [/u] [b]p4.[/b] Suppose that $P(x)$ is a quadratic polynomial such that the sum and product of its two roots are equal to each other. There is a real number $a$ that $P(1)$ can never be equal to. Find $a$. [b]p5.[/b] Find the number of ordered pairs $(x, y)$ of positive integers such that $\frac{1}{x} +\frac{1}{y} =\frac{1}{k}$ and k is a factor of $60$. [b]p6.[/b] Let $ABC$ be a triangle with $AB = 5$, $AC = 4$, and $BC = 3$. With $B = B_0$ and $C = C_0$, define the infinite sequences of points $\{B_i\}$ and $\{C_i\}$ as follows: for all $i \ge 1$, let $B_i$ be the foot of the perpendicular from $C_{i-1}$ to $AB$, and let $C_i$ be the foot of the perpendicular from $B_i$ to $AC$. Find $C_0C_1(AC_0 + AC_1 + AC_2 + AC_3 + ...)$. [u]Round 3 [/u] [b]p7.[/b] If $\ell_1, \ell_2, ... ,\ell_{10}$ are distinct lines in the plane and $p_1, ... , p_{100}$ are distinct points in the plane, then what is the maximum possible number of ordered pairs $(\ell_i, p_j )$ such that $p_j$ lies on $\ell_i$? [b]p8.[/b] Before Andres goes to school each day, he has to put on a shirt, a jacket, pants, socks, and shoes. He can put these clothes on in any order obeying the following restrictions: socks come before shoes, and the shirt comes before the jacket. How many distinct orders are there for Andres to put his clothes on? [b]p9. [/b]There are ten towns, numbered $1$ through $10$, and each pair of towns is connected by a road. Define a backwards move to be taking a road from some town $a$ to another town $b$ such that $a > b$, and define a forwards move to be taking a road from some town $a$ to another town $b$ such that $a < b$. How many distinct paths can Ali take from town $1$ to town $10$ under the conditions that $\bullet$ she takes exactly one backwards move and the rest of her moves are forward moves, and $\bullet$ the only time she visits town $10$ is at the very end? One possible path is $1 \to 3 \to 8 \to 6 \to 7 \to 8 \to 10$. [u]Round 4[/u] [b]p10.[/b] How many prime numbers $p$ less than $100$ have the properties that $p^5 - 1$ is divisible by $6$ and $p^6 - 1$ is divisible by $5$? [b]p11.[/b] Call a four-digit integer $\overline{d_1d_2d_3d_4}$ [i]primed [/i] if 1) $d_1$, $d_2$, $d_3$, and $d_4$ are all prime numbers, and 2) the two-digit numbers $\overline{d_1d_2}$ and $\overline{d_3d_4}$ are both prime numbers. Find the sum of all primed integers. [b]p12.[/b] Suppose that $ABC$ is an isosceles triangle with $AB = AC$, and suppose that $D$ and $E$ lie on $\overline{AB}$ and $\overline{AC}$, respectively, with $\overline{DE} \parallel \overline{BC}$. Let $r$ be the length of the inradius of triangle $ADE$. Suppose that it is possible to construct two circles of radius $r$, each tangent to one another and internally tangent to three sides of the trapezoid $BDEC$. If $\frac{BC}{r} = a + \sqrt{b}$ forpositive integers $a$ and $b$ with $b$ squarefree, then find $a + b$. PS. You should use hide for answers. Rounds 5-7 have been posted [url=https://artofproblemsolving.com/community/c4h2800986p24675177]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ p$ be a prime number and let $ a_1,a_2,\ldots,a_{p \minus{} 2}$ be positive integers such that $ p$ doesn't $ a_k$ or $ {a_k}^k \minus{} 1$ for any $ k$. Prove that the product of some of the $ a_i$'s is congruent to $ 2$ modulo $ p$.

When a number is divided by $2$ it has quotient $x$ and remainder $1$. Whereas, when the same number is divided by $3$ it has quotient $y$ and remainder $2$. What is the remainder when $x+y$ is divided by $5$?

Let $d$ be any positive integer not equal to $2, 5$ or $13$. Show that one can find distinct $a,b$ in the set $\{2,5,13,d\}$ such that $ab-1$ is not a perfect square.