Let $a_1<a_2<a_3<\dots$ be positive integers such that $a_{k+1}$ divides $2(a_1+a_2+\dots+a_k)$ for every $k\geqslant 1$. Suppose that for infinitely many primes $p$, there exists $k$ such that $p$ divides $a_k$. Prove that for every positive integer $n$, there exists $k$ such that $n$ divides $a_k$.

Let $n \geq 3$ be a fixed integer. The number $1$ is written $n$ times on a blackboard. Below the blackboard, there are two buckets that are initially empty. A move consists of erasing two of the numbers $a$ and $b$, replacing them with the numbers $1$ and $a+b$, then adding one stone to the first bucket and $\gcd(a, b)$ stones to the second bucket. After some finite number of moves, there are $s$ stones in the first bucket and $t$ stones in the second bucket, where $s$ and $t$ are positive integers. Find all possible values of the ratio $\frac{t}{s}$.

Find all prime numbers $p$ for which number $p^2 + 11$ has exactly six different divisors (counting $1$ and itself).

How many positive integers $n$ are there such that $n!(2n+1)$ and $221$ are relatively prime? $ \textbf{(A)}\ 10 \qquad\textbf{(B)}\ 11 \qquad\textbf{(C)}\ 12 \qquad\textbf{(D)}\ 13 \qquad\textbf{(E)}\ \text{None of the above} $

Numbers $1, 2,\cdots, 16$ are written in a $4\times 4$ square matrix so that the sum of the numbers in every row, every column, and every diagonal is the same and furthermore that the numbers $1$ and $16$ lie in opposite corners. Prove that the sum of any two numbers symmetric with respect to the center of the square equals $17$.

Find the greatest common divisor of all numbers of the form $(2^{a^2}\cdot 19^{b^2} \cdot 53^{c^2} + 8)^{16} - 1$ where $a,b,c$ are integers.

Let be a sequence $ \left( b_n \right)_{n\ge 1} $ of integers, having the following properties: $ \text{(i)} $ the sequence $ \left( \frac{b_n}{n} \right)_{n\ge 1} $ is convergent. $ \text{(ii)} m-n|b_m-b_n, $ for any natural numbers $ m>n. $ Prove that there exists an index from which the sequence $ \left( b_n \right)_{n\ge 1} $ is an arithmetic one. [i]Cristinel Mortici[/i]

A three-digit number is written $xyz$ in the base $7$ system and $zyx$ in the base $9$ system . What is the number?

Show that for any integer $a\ge 5$ there exist integers $b$ and $c$, $c\ge b\ge a$, such that $a,b,c$ are the lengths of the sides of a right-angled triangle.

Find the least positive integer $ N$ such that the sum of its digits is 100 and the sum of the digits of $ 2N$ is 110.

Find the largest integer $N$ satisfying the following two conditions: [b](i)[/b] $\left[ \frac N3 \right]$ consists of three equal digits; [b](ii)[/b] $\left[ \frac N3 \right] = 1 + 2 + 3 +\cdots + n$ for some positive integer $n.$

Prove that there exists a positive integer $n$ such that in the decimal representation of each of the numbers $\sqrt{n}$, $\sqrt[3]{n},..., \sqrt[10]{n}$ digits $2015$ stand immediately after the decimal point. [i]A.Golovanov [/i]

Let $u_1=1,u_2=1$ and for all $k \geq 1$'s $$u_{k+2}=u_{k+1}+u_{k}$$ Prove that for all $m \geq 1$'s $5$ divides $u_{5m}$

As evidence that the correct answer does not mean the correctness of the proof, the teacher cited next example. Let's take the fraction $\frac{19}{95}$. After crossing out $9$ in the numerator and denominator (“reduction” by $9$), we get $\frac{1}{5}$ which is the correct answer. In the same way, a fraction $\frac{1999}{9995}$ can be “reduced” by three nines (cross out $999$ in the numerator and denominator). Is it possible that as a result of such a “reduction” we also get the correct answer, equal to $\frac13$ ? (We consider fractions of the form $\frac{1a}{a3}$. Here, with the letter $a$ we denote several numbers that follow in the same order in the numerator after $1$, and in the denominator before $3$. “Reduce” by $a$.)

[u]Round 1[/u] [b]p1.[/b] A $\$100$ TV has its price increased by $10\%$. The new price is then decreased by $10\%$. What is the current price of the TV? [b]p2.[/b] If $9w + 8x + 7y = 42$ and $w + 2x + 3y = 8$, then what is the value of $100w + 101x + 102y$? [b]p3.[/b] Find the number of positive factors of $37^3 \cdot 41^3$. [u]Round 2[/u] [b]p4.[/b] Three hoses work together to fill up a pool, and each hose expels water at a constant rate. If it takes the first, second, and third hoses 4, 6, and 12 hours, respectively, to fill up the pool alone, then how long will it take to fill up the pool if all three hoses work together? [b]p5.[/b] A semicircle has radius $1$. A smaller semicircle is inscribed in the larger one such that the two bases are parallel and the arc of the smaller is tangent to the base of the larger. An even smaller semicircle is inscribed in the same manner inside the smaller of the two semicircles, and this procedure continues indefinitely. What is the sum of all of the areas of the semicircles? [b]p6.[/b] Given that $P(x)$ is a quadratic polynomial with $P(1) = 0$, $P(2) = 0$, and $P(0) = 2012$, find $P(-1)$. [u]Round 3[/u] [b]p7.[/b] Darwin has a paper circle. He labels one point on the circumference as $A$. He folds $A$ to every point on the circumference on the circle and undoes it. When he folds $A$ to any point $P$, he makes a blue mark on the point where $\overline{AP}$ and the made crease intersect. If the area of Darwin paper circle is 80, then what is the area of the region surrounded by blue? [b]p8.[/b] Α rectangular wheel of dimensions $6$ feet by $8$ feet rolls for $28$ feet without sliding. What is the total distance traveled by any corner on the rectangle during this roll? [b]p9[/b]. How many times in a $24$-hour period do the minute hand and hour hand of a $12$-hour clock form a right angle? [u]Round 4[/u] The answers in this section all depend on each other. Find smallest possible solution set. [b]p10.[/b] Let B be the answer to problem $11$. Right triangle $ACD$ has a right angle at $C$. Squares $ACEF$ and $ADGH$ are drawn such that points $D$ and $E$ do not coincide and points $E$ and $H$ do not coincide. The midpoints of the sides of $ADGH$ are connected to form a smaller square with area $B.$ If the area of $ACEF$ is also $B$, then find the length $CD$ rounded up to the nearest integer. [b]p11.[/b] Let $C$ be the answer to problem $12$. Find the sum of the digits of $C$. [b]p12.[/b] Let $A$ be the answer to problem $10$. Given that $a_0 = 1$, $a_1 = 2$, and that $a_n = 3a_{n-1 }-a_{n-2}$ for $n \ge 2$, find $a_A$. PS. You should use hide for answers.Rounds 5-8 are [url=https://artofproblemsolving.com/community/c3h3134466p28406321]here [/url] and 9-12 [url=https://artofproblemsolving.com/community/c3h3134489p28406583]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $S$ be a finite set of positive integers which has the following property:if $x$ is a member of $S$,then so are all positive divisors of $x$. A non-empty subset $T$ of $S$ is [i]good[/i] if whenever $x,y\in T$ and $x<y$, the ratio $y/x$ is a power of a prime number. A non-empty subset $T$ of $S$ is [i]bad[/i] if whenever $x,y\in T$ and $x<y$, the ratio $y/x$ is not a power of a prime number. A set of an element is considered both [i]good[/i] and [i]bad[/i]. Let $k$ be the largest possible size of a [i]good[/i] subset of $S$. Prove that $k$ is also the smallest number of pairwise-disjoint [i]bad[/i] subsets whose union is $S$.

Pablo says: “I add $2$ to my birthday and multiply the result by $2$. I add to the number obtained $4$ and multiply the result by $5$. To the new number obtained I add the number of the month of my birthday (for example, if it's June, I add $6$) and I get $342$. " What is Pablo's birthday date? Give all the possibilities

Let $T$ be a finite set of squarefree integers. (a) Show that there exists an integer polynomial $P(x)$ such that the set of squarefree numbers in the range of $P(n)$ across all $n \in \mathbb{Z}$ is exactly $T$. (b) Suppose that $T$ is allowed to be infinite. Is it still true that for all choices of $T$, such an integer polynomial $P(x)$ exists? [i]Allen Wang[/i]

Let $p$ be an odd prime number and $M$ a set derived from $\frac{p^2 + 1}{2}$ square numbers. Investigate whether $p$ elements can be selected from this set whose arithmetic mean is an integer. (Walther Janous)

Let $n$ be a natural number. A sequence is $k-$complete if it contains all residues modulo $n^k$. Let $Q(x)$ be a polynomial with integer coefficients. For $k\ge 2$, define $Q^k(x)=Q(Q^{k-1}(x))$, where $Q^1(x)=Q(x)$. Show that if $$0,Q(0),Q^2(0),Q^3(0),\ldots $$is $2018-$complete, then it is $k-$complete for all positive integers $k$. [i]Proposed by Ma Zhao Yu[/i]

Find all positive integer pairs $(m, n)$ such that $m- n$ is a positive prime number and $mn$ is a perfect square. Justify your answer.

Which positive integers are missing in the sequence $ \left\{a_n\right\}$, with $ a_n \equal{} n \plus{} \left[\sqrt n\right] \plus{}\left[\sqrt [3]n\right]$ for all $ n \ge 1$? ($ \left[x\right]$ denotes the largest integer less than or equal to $ x$, i.e. $ g$ with $ g \le x < g \plus{} 1$.)

Let $(a_n)_{n\ge 0}$ be the sequence of rational numbers with $a_0 = 2016$ and $a_{n+1} = a_n + \frac{2}{a_n}$ for all $n \ge 0$. Show that the sequence does not contain a square of a rational number. Proposed by Theresia Eisenkölbl

The complete list of the three-digit palindrome numbers is written in ascending order: $$101, 111, 121, 131,... , 979, 989, 999.$$ Then eight consecutive palindrome numbers are eliminated and the numbers that remain in the list are added, obtaining $46.150$. Determine the eight erased palindrome numbers .

Given is a natural number $a$ with $54$ digits, each digit equal to $0$ or $1$. Prove the remainder of $a$ when divide by $ 33\cdot 34\cdots 39 $ is larger than $100000$. [hide](It's mean: $a \equiv r \pmod{33\cdot 34\cdots 39 }$ with $ 0<r<33\cdot 34\cdots 39 $ then prove that $r>100000$ )[/hide] M. Antipov