Two infinite arithmetic progressions are called considerable different if the do not only differ by the absence of finitely many members at the beginning of one of the sequences. How many pairwise considerable different non-constant arithmetic progressions of positive integers that contain an infinite non-constant geometric progression $ (b_n)_{n\ge0}$ with $ b_2\equal{}40 \cdot 2009$ are there?

Let $m, n$ be positive integers. There are $n$ piles of gold coins, so that pile $i$ has $a_i > 0$ coins in it $(i = 1, ..., n)$. Consider the following game: Step 1. Nadech picks sets $B_1, B_2, ... , B_n$, where each $B_i$ is a nonempty subset of $\{1, 2, . . . , m\}$, and gives them to Yaya. Step 2. Yaya picks a set $S$ which is also a nonempty subset of $\{1, 2, . . . , m\}$. Step 3. For each $i = 1, 2, . . . , n$, Nadech wins the coins in pile $i$ if $B_i \cap S$ has an even number of elements, and Yaya wins the coins in pile $i$ if $B_i \cap S$ has an odd number of elements. Show that, no matter how Nadech picks the sets $B_1, B_2, . . . , B_n$, Yaya can always pick $S$ so that she ends up with more gold coins than Nadech

Let $P$ be a convex polygon, and let $n \ge 3$ be a positive integer. On each side of $P$, erect a regular $n$-gon that shares that side of $P$, and is outside $P$. If none of the interiors of these regular n-gons overlap, we call P $n$-[i]good[/i]. (a) Find the largest value of $n$ such that every convex polygon is $n$-[i]good[/i]. (b) Find the smallest value of $n$ such that no convex polygon is $n$-[i]good[/i].

Find all pairs of positive integers $m, n$ such that $9^{|m-n|}+3^{|m-n|}+1$ is divisible by $m$ and $n$ simultaneously.

Let $n$ be a positive integer. Show that if p is prime dividing $5^{4n}-5^{3n}+5^{2n}-5^{n}+1$, then $p\equiv 1 \;(\bmod\; 4)$.

An $8$ by $2\sqrt{2}$ rectangle has the same center as a circle of radius $2$. The area of the region common to both the rectangle and the circle is $ \textbf{(A)}\ 2\pi \qquad\textbf{(B)}\ 2\pi+2 \qquad\textbf{(C)}\ 4\pi-4 \qquad\textbf{(D)}\ 2\pi+4 \qquad\textbf{(E)}\ 4\pi-2 $

The circles $k_1$ and $k_2$ intersect at points $A$ and $B$, and $k_1$ passes through the center $O$ of the circle $k_2$. The line $p$ intersects $k_1$ at the points $K ,O$ and $k_2$ at the points $L ,M$ so that $L$ lies between $K$ and $O$. The point $P$ is the projection of $L$ on the line $AB$. Prove that $KP$ is parallel to the median of triangle $ABM$ drawn from the vertex $M$.

Three fair six-sided dice are rolled. The expected value of the median of the numbers rolled can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime integers. Find $m+n$. [i]Proposed by [b]AOPS12142015[/b][/i]

Given strictly increasing sequence $ a_1<a_2<\dots$ of positive integers such that each its term $ a_k$ is divisible either by 1005 or 1006, but neither term is divisible by $ 97$. Find the least possible value of maximal difference of consecutive terms $ a_{i\plus{}1}\minus{}a_i$.

[color=darkred]Let $f\colon [0,\infty)\to\mathbb{R}$ be a continuous function such that $\int_0^nf(x)f(n-x)\ \text{d}x=\int_0^nf^2(x)\ \text{d}x$ , for any natural number $n\ge 1$ . Prove that $f$ is a periodic function.[/color]

Two players play the following game starting with one pile of at least two stones. A player in turn chooses one of the piles and divides it into two or three nonempty piles. The player who cannot make a legal move loses the game. Which player has a winning strategy?

An integer is between $0$ and $999999$ (inclusive) is chosen, and the digits of its decimal representation are summed. What is the probability that the sum will be $19$?

Find all positive integers $k \geq 2$ for which there exists some positive integer $n$ such that the last $k$ digits of the decimal representation of $10^{10^n} - 9^{9^n}$ are the same. [i]Proposed by Andrew Wen[/i]

Let $a_1<a_2 \dots <a_n$ be positive integers, with $n\geq 2$. An invisible frog lies on the real line, at a positive integer point. Initially, the hunter chooses a number $k$, and then, once every minute, he can check if the frog currently lies in one of $k$ points of his choosing, after which the frog goes from its point $x$ to one of the points $x+a_1, x+a_2 \dots x+a_n$. Based on the values of $a_1, a_2 \dots a_n$, what is the smallest value of $k$ such that the hunter can guarantee to find the frog within a finite number of minutes, no matter where it initially started? [i]Proposed by David Anghel[/i]

In quadrilateral $ ABCD$, $ AB \equal{} 5$, $ BC \equal{} 17$, $ CD \equal{} 5$, $ DA \equal{} 9$, and $ BD$ is an integer. What is $ BD$? [asy]unitsize(4mm); defaultpen(linewidth(.8pt)+fontsize(8pt)); dotfactor=4; pair C=(0,0), B=(17,0); pair D=intersectionpoints(Circle(C,5),Circle(B,13))[0]; pair A=intersectionpoints(Circle(D,9),Circle(B,5))[0]; pair[] dotted={A,B,C,D}; draw(D--A--B--C--D--B); dot(dotted); label("$D$",D,NW); label("$C$",C,W); label("$B$",B,E); label("$A$",A,NE);[/asy]$ \textbf{(A)}\ 11 \qquad \textbf{(B)}\ 12 \qquad \textbf{(C)}\ 13 \qquad \textbf{(D)}\ 14 \qquad \textbf{(E)}\ 15$

Let \( n \) be a natural number. The numbers \( 1, 2, \ldots, n \) are written in a row in some order. For each pair of adjacent numbers, their greatest common divisor (GCD) is calculated and written on a sheet. What is the maximum possible number of distinct values among the \( n - 1 \) GCDs obtained? \\

Let $M$ be the centroid of $\Delta ABC$ Prove the inequality $\sin \angle CAM + \sin\angle CBM \le \frac{2}{\sqrt 3}$  (a) if the circumscribed circle of $\Delta AMC$ is tangent to the line $AB$ (b) for any $\Delta ABC$

A plastic ball with a radius of 45 mm has a circular hole made in it. The hole is made to fit a ball with a radius of 35 mm, in such a way that the distance between their centers is 60 mm. Calculate the radius of the hole.

How many positive integers $n$ less than $1000$ have the property that the number of positive integers less than $n$ which are coprime to $n$ is exactly $\tfrac n3$?

Consider the triangle $ABC{}$ and let $I_A{}$ be its $A{}$-excenter. Let $M,N$ and $P{}$ be the projections of $I_A{}$ onto the lines $AC,BC{}$ and $AB{}$ respectively. Prove that if $\overrightarrow{I_AM}+\overrightarrow{I_AP}=\overrightarrow{I_AN}$ then $ABC{}$ is an equilateral triangle.

Find all integers $n\geq 2$ for which there exist the real numbers $a_k, 1\leq k \leq n$, which are satisfying the following conditions: \[\sum_{k=1}^n a_k=0, \sum_{k=1}^n a_k^2=1 \text{ and } \sqrt{n}\cdot \Bigr(\sum_{k=1}^n a_k^3\Bigr)=2(b\sqrt{n}-1), \text{ where } b=\max_{1\leq k\leq n} \{a_k\}.\]

Put $ M\equal{}\begin{pmatrix}p&q&r\\ r&p&q\\q&r&p\end{pmatrix}$ where $ p,q,r>0$ and $ p\plus{}q\plus{}r\equal{}1$. Prove that $ \lim_{n\rightarrow \infty}M^n\equal{}\begin{bmatrix}\frac13&\frac13&\frac13\\ \frac13&\frac13&\frac13\\\frac13&\frac13&\frac13\end{bmatrix}$

Let $a$ and $b$ be integer solutions to $17a+6b=13$. What is the smallest possible positive value for $a-b$?

Denote by $ \mathcal{H}$ a set of sequences $ S\equal{}\{s_n\}_{n\equal{}1}^{\infty}$ of real numbers having the following properties: (i) If $ S\equal{}\{s_n\}_{n\equal{}1}^{\infty}\in \mathcal{H}$, then $ S'\equal{}\{s_n\}_{n\equal{}2}^{\infty}\in \mathcal{H}$; (ii) If $ S\equal{}\{s_n\}_{n\equal{}1}^{\infty}\in \mathcal{H}$ and $ T\equal{}\{t_n\}_{n\equal{}1}^{\infty}$, then $ S\plus{}T\equal{}\{s_n\plus{}t_n\}_{n\equal{}1}^{\infty}\in \mathcal{H}$ and $ ST\equal{}\{s_nt_n\}_{n\equal{}1}^{\infty}\in \mathcal{H}$; (iii) $ \{\minus{}1,\minus{}1,\dots,\minus{}1,\dots\}\in \mathcal{H}$. A real valued function $ f(S)$ defined on $ \mathcal{H}$ is called a quasi-limit of $ S$ if it has the following properties: If $ S\equal{}{c,c,\dots,c,\dots}$, then $ f(S)\equal{}c$; If $ s_i\geq 0$, then $ f(S)\geq 0$; $ f(S\plus{}T)\equal{}f(S)\plus{}f(T)$; $ f(ST)\equal{}f(S)f(T)$, $ f(S')\equal{}f(S)$ Prove that for every $ S$, the quasi-limit $ f(S)$ is an accumulation point of $ S$.

Let $n$ be a positive integer and let $a_1, a_2, \cdots a_k$ be all numbers less than $n$ and coprime to $n$ in increasing order. Find the set of values the function $f(n)=gcd(a_1^3-1, a_2^3-1, \cdots, a_k^3-1)$.