Arithmetic progression $a_1, a_2, . . . , $ consisting of natural numbers is such that for any $n$ the product $a_n \cdot a_{n+31}$ is divisible by $2005$. Is it possible to say that all terms of the progression are divisible by $2005$?

Let \[ f(n)= \sum_{p|n , \;p^{\alpha} \leq n < p^{\alpha+1} \ } p^{\alpha} .\] Prove that \[ \limsup_{n \rightarrow \infty}f(n) \frac{ \log \log n}{n \log n}=1 .\] [i]P. Erdos[/i]

Let $k > 1$ be a fixed odd number, and for non-negative integers $n$ let $$f_n=\sum_{\substack{0\leq i\leq n\\ k\mid n-2i}}\binom{n}{i}.$$ Prove that $f_n$ satisfy the following recursion: $$f_{n}^2=\sum_{i=0}^{n} \binom{n}{i}f_{i}f_{n-i}.$$

Given that $n$ is a natural number such that the leftmost digits in the decimal representations of $2^n$ and $3^n$ are the same, find all possible values of the leftmost digit.

Let $a_0$ be an arbitrary positive integer. Consider the infinite sequence $(a_n)_{n\geq 1}$, defined inductively as follows: given $a_0, a_1, ..., a_{n-1}$ define the term $a_n$ as the smallest positive integer such that $a_0+a_1+...+a_n$ is divisible by $n$. Prove that there exist a positive integer a positive integer $M$ such that $a_{n+1}=a_n$ for all $n\geq M$.

Let $n > 1$ be a given integer. Prove that infinitely many terms of the sequence $(a_k )_{k\ge 1}$, defined by \[a_k=\left\lfloor\frac{n^k}{k}\right\rfloor,\] are odd. (For a real number $x$, $\lfloor x\rfloor$ denotes the largest integer not exceeding $x$.) [i]Proposed by Hong Kong[/i]

Let $p > 2$ be a prime number. Prove that there is a permutation $k_1, k_2, ..., k_{p-1}$ of numbers $1,2,...,p-1$ such that the number $1^{k_1}+2^{k_2}+3^{k_3}+...+(p-1)^{k_{p-1}}$ is divisible by $p$. Note: The numbers $k_1, k_2, ..., k_{p-1}$ are a permutation of the numbers $1,2,...,p-1$ if each of of numbers $1,2,...,p-1$ appears exactly once among the numbers $k_1, k_2, ..., k_{p-1}$.

The sequence $(x_n)$ is defined by formulas $$ x_1=c,\; x_{n+1} = cx_n + \sqrt{(c^2-1)(x_n^2-1)} \quad\text{ for }\quad n=1,2,\ldots$$ Prove that if $ c $ is a natural number, then all numbers $ x_n $ are natural.

Determine all triplets $(x, y, z)$ of positive integers with $x \le y \le z$ that satisfy $\left(1+\frac1x \right)\left(1+\frac1y \right)\left(1+\frac1z \right) = 3$

Find the smallest positive prime that divides $n^2 + 5n + 23$ for some integer $n$.

Prove that it is impossible to arrange the numbers $1,2,\ldots,1997$ around a circle in such a way that, if $x$ and $y$ are any two neighboring numbers, then $499\le|x-y|\le997$.

Define a sequence by $s_0 = 1$ and for $d \geq 1$, $s_d = s_{d-1} + X_d$, where $X_d$ is chosen uniformly at random from the set $\{1, 2, \dots, d\}$. What is the probability that the sequence $s_0, s_1, s_2, \dots$ contains infinitely many primes?

Find all pairs of integers $(x,y)$ satisfying the following equality $8x^2y^2 + x^2 + y^2 = 10xy$

Prove that there exists a positive integer $k$ such that $k\cdot2^n+1$ is composite for every integer $n$.

Define an $n$-digit pair cycle to be a number with $n^2 + 1$ digits between $1$ and $n$ with every possible pair of consecutive digits. For instance, $11221$ is a 2-digit pair cycle since it contains the consecutive digits $11$, $12$, $22$, and $21$. How many $3$-digit pair cycles exist?

Natural number $N$ is given. Let $p_N$ - biggest prime, that $ \leq N$. On every move we replace $N$ by $N-p_N$. We repeat this until we get $0$ or $1$. If we get $1$ then $N$ is called as good, else is bad. For example, $95$ is good because we get $95 \to 6 \to 1$. Prove that among numbers from $1$ to $1000000$ there are between one quarter and half good numbers

For positive integers $n,d$, define $n \% d$ to be the unique value of the positive integer $r < d$ such that $n = qd + r$, for some positive integer $q$. What is the smallest value of $n$ not divisible by $5,7,11,13$ for which $n^2 \% 5 < n^2 \% 7 < n^2 \% 11 < n^2 \% 13$?

$32$ knights live in a kingdom. Some of them are servants of others. A servant may have only one master and any master is more wealthy than any of his servants. A knight having not less than $4$ servants is called a baron. What is the maximum number of barons? (The kingdom is ruled by the law: “My servant’s servant is not my servant”. (A. Tolpygo, Kiev)

Kelvin and $15$ other frogs are in a meeting, for a total of $16$ frogs. During the meeting, each pair of distinct frogs becomes friends with probability $\frac{1}{2}$. Kelvin thinks the situation after the meeting is [I]cool[/I] if for each of the $16$ frogs, the number of friends they made during the meeting is a multiple of $4$. Say that the probability of the situation being cool can be expressed in the form $\frac{a}{b}$, where $a$ and $b$ are relatively prime. Find $a$.

Find all positive integers $n$ such that there exist $n$ consecutive positive integers whose sum is a perfect square.

Given positive integers $m$ and $n$ so there is a chessboard with $mn$ $1 \times 1$ grids. Colour the grids into red and blue (Grids that have a common side are not the same colour and the grid in the left corner at the bottom is red). Now the diagnol that goes from the left corner at the bottom to the top right corner is coloured into red and blue segments (Every segment has the same colour with the grid that contains it). Find the sum of the length of all the red segments.

The smallest positive integer which can be expressed as sum of positive perfect cubes (possibly with repetition and/or with a single element sum) in at least two different ways in $$(A) 8$$ $$(B) 1729$$ $$(C) 2023$$ $$(D) 2024$$

Consider the sequence of integers $0, 1, 2, 4, 6, 9, 12,...$ obtained by starting with zero, adding $1$, then adding $1$ again, then adding $2$, and adding $2$ again, then adding $3$, and adding $3$ again, and so on. If we call the subsequent terms of this sequence $a_0, a_1, a_2, ...$, then we have $a_0 = 0$, and $a_{2n-1} = a_{2n-2} + n$ , $a_{2n} = a_{2n-1} + n$ for all integers $n \ge 1$. Find all integers $k \ge 0$ for which $a_k$ is the square of an integer.

[b]p1.[/b] What is the maximum number of points of intersection between a square and a triangle, assuming that no side of the triangle is parallel to any side of the square? [b]p2.[/b] Two angles of an isosceles triangle measure $80^o$ and $x^o$. What is the sum of all the possible values of $x$? [b]p3.[/b] Let $p$ and $q$ be prime numbers such that $p + q$ and p + $7q$ are both perfect squares. Find the value of $pq$. [b]p4.[/b] Anna, Betty, Carly, and Danielle are four pit bulls, each of which is either wearing or not wearing lipstick. The following three facts are true: (1) Anna is wearing lipstick if Betty is wearing lipstick. (2) Betty is wearing lipstick only if Carly is also wearing lipstick. (3) Carly is wearing lipstick if and only if Danielle is wearing lipstick The following five statements are each assigned a certain number of points: (a) Danielle is wearing lipstick if and only if Carly is wearing lipstick. (This statement is assigned $1$ point.) (b) If Anna is wearing lipstick, then Betty is wearing lipstick. (This statement is assigned $6$ points.) (c) If Betty is wearing lipstick, then both Anna and Danielle must be wearing lipstick. (This statement is assigned $10$ points.) (d) If Danielle is wearing lipstick, then Anna is wearing lipstick. (This statement is assigned $12$ points.) (e) If Betty is wearing lipstick, then Danielle is wearing lipstick. (This statement is assigned $14$ points.) What is the sum of the points assigned to the statements that must be true? (For example, if only statements (a) and (d) are true, then the answer would be $1 + 12 = 13$.) [b]p5.[/b] Let $f(x)$ and $g(x)$ be functions such that $f(x) = 4x + 3$ and $g(x) = \frac{x + 1}{4}$. Evaluate $g(f(g(f(42))))$. [b]p6.[/b] Let $A,B,C$, and $D$ be consecutive vertices of a regular polygon. If $\angle ACD = 120^o$, how many sides does the polygon have? [b]p7.[/b] Fred and George have a fair $8$-sided die with the numbers $0, 1, 2, 9, 2, 0, 1, 1$ written on the sides. If Fred and George each roll the die once, what is the probability that Fred rolls a larger number than George? [b]p8.[/b] Find the smallest positive integer $t$ such that $(23t)^3 - (20t)^3 - (3t)^3$ is a perfect square. [b]p9.[/b] In triangle $ABC$, $AC = 8$ and $AC < AB$. Point $D$ lies on side BC with $\angle BAD = \angle CAD$. Let $M$ be the midpoint of $BC$. The line passing through $M$ parallel to $AD$ intersects lines $AB$ and $AC$ at $F$ and $E$, respectively. If $EF =\sqrt2$ and $AF = 1$, what is the length of segment $BC$? (See the following diagram.) [img]https://cdn.artofproblemsolving.com/attachments/2/3/4b5dd0ae28b09f5289fb0e6c72c7cbf421d025.png[/img] [b]p10.[/b] There are $2011$ evenly spaced points marked on a circular table. Three segments are randomly drawn between pairs of these points such that no two segments share an endpoint on the circle. What is the probability that each of these segments intersects the other two? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find all integer solutions of the equation $14^x - 3^y = 2015$. (Malik Talbi)