The sequence $a_{n}$ defined as follows: $a_{1}=4, a_{2}=17$ and for any $k\geq1$ true equalities $a_{2k+1}=a_{2}+a_{4}+...+a_{2k}+(k+1)(2^{2k+3}-1)$ $a_{2k+2}=(2^{2k+2}+1)a_{1}+(2^{2k+3}+1)a_{3}+...+(2^{3k+1}+1)a_{2k-1}+k$ Find the smallest $m$ such that $(a_{1}+...a_{m})^{2012^{2012}}-1$ divided $2^{2012^{2012}}$

Let $a$ and $b$ be positive integers such that $(a^3 - a^2 + 1)(b^3 - b^2 + 2) = 2020$. Find $10a + b$.

For positive integers $ n$, $ f_n \equal{} \lfloor2^n\sqrt {2008}\rfloor \plus{} \lfloor2^n\sqrt {2009}\rfloor$. Prove there are infinitely many odd numbers and infinitely many even numbers in the sequence $ f_1,f_2,\ldots$.

Do there exist positive integers $a,b,c$ and $d$ such that $$\begin{cases} \dfrac{a}{b} + \dfrac{c}{d} = 1\\ \\ \dfrac{a}{d} + \dfrac{c}{b} = 2008\end{cases}$$ ?

Let $k,n\ge 1$ be relatively prime integers. All positive integers not greater than $k+n$ are written in some order on the blackboard. We can swap two numbers that differ by $k$ or $n$ as many times as we want. Prove that it is possible to obtain the order $1,2,\dots,k+n-1, k+n$.

Determine all natural numbers $n$ and $k > 1$ such that $k$ divides each of the numbers$\binom{n}{1}$,$\binom{n}{2}$,..........,$\binom{n}{n-1}$

Given $m\in\mathbb{Z}^+$. Find all natural numbers $n$ that does not exceed $10^m$ satisfying the following conditions: i) $3|n.$ ii) The digits of $n$ in decimal representation are in the set $\{2,0,1,5\}$.

[b]p1.[/b] Let $a_0, a_1,...,a_n$ be such that $a_n \ne 0$ and $$(1 + x + x^3)^{341}(1 + 2x + x^2 + 2x^3 + 2x^4 + x^6)^{342} =\sum^n_{i=0}a_ix^i,$$ Find the number of odd numbers in the sequence a0; a1; : : : an. [b]p2.[/b] Let $F_0 = 1$, $F_1 = 1$ and F$_k = F_{k-1} + F_{k-2}$. Let $P(x) =\sum^{99}_{k=0} x^{F_k}$ . The remainder when $P(x)$ is divided by $x^3 - 1$ can be expressed as $ax^2 + bx + c$. Find $2a + b$. [b]p3.[/b] Let $a_n$ be the number of permutations of the numbers $S = \{1, 2,...,n\}$ such that for all $k$ with $1 \le k \le n$, the sum of $k$ and the number in the $k$th position of the permutation is a power of $2$. Compute $a_{2^0} + a_{2^1} +... + a_{2^{20}}$ . [b]p4.[/b] Three identical balls are painted white and black, so that half of each sphere is a white hemisphere, and the other half is a black one. The three balls are placed on a plane surface, each with a random orientation, so that each ball has a point of contact with the other two. What is the probability that at at least one point of contact between two of the balls, both balls are the same color? [b]p5.[/b] Compute the greatest positive integer $n$ such that there exists an odd integer $a$, for which $\frac{a^{2^n}-1}{4^{4^4}}$ is not an integer. [b]p6.[/b] You are blind and cannot feel the difference between a coin that is heads up or tails up. There are $100$ coins in front of you and are told that exactly $10$ of them are heads up. On the back of this paper, explain how you can split the otherwise indistinguishable coins into two groups so that both groups have the same number of heads. [b]p7.[/b] On the back of this page, write the best math pun you can think of. You’ll get a point if we chuckle. [b]p8.[/b] Pick an integer between $1$ and $10$. If you pick $k$, and $n$ total teams pick $k$, then you’ll receive $\frac{k}{10n}$ points. [b]p9.[/b] There are four prisoners in a dungeon. Tomorrow, they will be separated into a group of three in one room, and the other in a room by himself. Each will be given a hat to wear that is either black or white – two will be given white and two black. None of them will be able to communicate with each other and none will see his or her own hat color. The group of three is lined up, so that the one in the back can see the other two, the second can see the first, but the first cannot see the others. If anyone is certain of their hat color, then they immediately shout that they know it to the rest of the group. If they can secretly prove it to the guard, they are saved. They only say something if they’re sure. Which person is sure to survive? [b]p10.[/b] Down the road, there are $10$ prisoners in a dungeon. Tomorrow they will be lined up in a single room and each given a black or white hat – this time they don’t know how many of each. The person in the back can see everyone’s hat besides his own, and similarly everyone else can only see the hats of the people in front of them. The person in the back will shout out a guess for his hat color and will be saved if and only if he is right. Then the person in front of him will have to guess, and this will continue until everyone has the opportunity to be saved. Each person can only say his or her guess of “white” or “black” when their turn comes, and no other signals may be made. If they have the night before receiving the hats to try to devise some sort of code, how many people at a minimum can be saved with the most optimal code? Describe the code on the back of this paper for full points. [b]p11.[/b] A few of the problems on this mixer contest were taken from last year’s event. One of them had fewer than $5$ correct answers, and most of the answers given were the same incorrect answer. Half a point will be given if you can guess the number of the problem on this test that corresponds to last year’s question, and another $.5$ points will be given if you can guess the very common incorrect answer. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Alice has an integer $N > 1$ on the blackboard. Each minute, she deletes the current number $x$ on the blackboard and writes $2x+1$ if $x$ is not the cube of an integer, or the cube root of $x$ otherwise. Prove that at some point of time, she writes a number larger than $10^{100}$. [i]Proposed by Anant Mudgal and Rohan Goyal[/i]

How many pairs $(a, b)$ for integers $a, b \ge 2$ which exist the sequence $x_1, x_2, . . . , x_{1000}$ which satisfy conditions as below? 1.Terms $x_1, x_2, . . . , x_{1000}$ are sorting of $1, 2, . . . , 1000$. 2.For each integers $1 \le i < 1000$, the sequence forms $x_{i+1} = x_i + a$ or $x_{i+1} = x_i - b$.

Let $ p_1,p_2,\dots,p_r$ be distinct primes, and let $ n_1,n_2,\dots,n_r$ be arbitrary natural numbers. Prove that the number of pairs of integers $ (x, y)$ such that \[ x^3 \plus{} y^3 \equal{} p_1^{n_1}p_2^{n_2}\cdots p_r^{n_r}\] does not exceed $ 2^{r\plus{}1}$.

Let $p$ be a prime number. Find all polynomials $P$ with integer coefficients with the following properties: $(a)$ $P(x)>x$ for all positive integers $x$. $(b)$ The sequence defined by $p_0:=p$, $p_{n+1}:=P(p_n)$ for all positive integers $n$, satisfies the property that for all positive integers $m$ there exists some $l\geq 0$ such that $m\mid p_l$.

[b]p1. [/b]The longest diagonal of a regular hexagon is 12 inches long. What is the area of the hexagon, in square inches? [b]p2.[/b] When Al and Bob play a game, either Al wins, Bob wins, or they tie. The probability that Al does not win is $\frac23$ , and the probability that Bob does not win is $\frac34$ . What is the probability that they tie? [b]p3.[/b] Find the sum of the $a + b$ values over all pairs of integers $(a, b)$ such that $1 \le a < b \le 10$. That is, compute the sum $$(1 + 2) + (1 + 3) + (1 + 4) + ...+ (2 + 3) + (2 + 4) + ...+ (9 + 10).$$ [b]p4.[/b] A $3 \times 11$ cm rectangular box has one vertex at the origin, and the other vertices are above the $x$-axis. Its sides lie on the lines $y = x$ and $y = -x$. What is the $y$-coordinate of the highest point on the box, in centimeters? [b]p5.[/b] Six blocks are stacked on top of each other to create a pyramid, as shown below. Carl removes blocks one at a time from the pyramid, until all the blocks have been removed. He never removes a block until all the blocks that rest on top of it have been removed. In how many different orders can Carl remove the blocks? [img]https://cdn.artofproblemsolving.com/attachments/b/e/9694d92eeb70b4066b1717fedfbfc601631ced.png[/img] [b]p6.[/b] Suppose that a right triangle has sides of lengths $\sqrt{a + b\sqrt{3}}$,$\sqrt{3 + 2\sqrt{3}}$, and $\sqrt{4 + 5\sqrt{3}}$, where $a, b$ are positive integers. Find all possible ordered pairs $(a, b)$. [b]p7.[/b] Farmer Chong Gu glues together $4$ equilateral triangles of side length $ 1$ such that their edges coincide. He then drives in a stake at each vertex of the original triangles and puts a rubber band around all the stakes. Find the minimum possible length of the rubber band. [b]p8.[/b] Compute the number of ordered pairs $(a, b)$ of positive integers less than or equal to $100$, such that a $b -1$ is a multiple of $4$. [b]p9.[/b] In triangle $ABC$, $\angle C = 90^o$. Point $P$ lies on segment $BC$ and is not $B$ or $C$. Point $I$ lies on segment $AP$. If $\angle BIP = \angle PBI = \angle CAB = m^o$ for some positive integer $m$, find the sum of all possible values of $m$. [b]p10.[/b] Bob has $2$ identical red coins and $2$ identical blue coins, as well as $4$ distinguishable buckets. He places some, but not necessarily all, of the coins into the buckets such that no bucket contains two coins of the same color, and at least one bucket is not empty. In how many ways can he do this? [b]p11.[/b] Albert takes a $4 \times 4$ checkerboard and paints all the squares white. Afterward, he wants to paint some of the square black, such that each square shares an edge with an odd number of black squares. Help him out by drawing one possible configuration in which this holds. (Note: the answer sheet contains a $4 \times 4$ grid.) [b]p12.[/b] Let $S$ be the set of points $(x, y)$ with $0 \le x \le 5$, $0 \le y \le 5$ where $x$ and $y$ are integers. Let $T$ be the set of all points in the plane that are the midpoints of two distinct points in $S$. Let $U$ be the set of all points in the plane that are the midpoints of two distinct points in $T$. How many distinct points are in $U$? (Note: The points in $T$ and $U$ do not necessarily have integer coordinates.) [b]p13.[/b] In how many ways can one express $6036$ as the sum of at least two (not necessarily positive) consecutive integers? [b]p14.[/b] Let $a, b, c, d, e, f$ be integers (not necessarily distinct) between $-100$ and $100$, inclusive, such that $a + b + c + d + e + f = 100$. Let $M$ and $m$ be the maximum and minimum possible values, respectively, of $$abc + bcd + cde + def + ef a + f ab + ace + bdf.$$ Find $\frac{M}{m}$. [b]p15.[/b] In quadrilateral $ABCD$, diagonal $AC$ bisects diagonal $BD$. Given that $AB = 20$, $BC = 15$, $CD = 13$, $AC = 25$, find $DA$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

$\textbf{N4:} $ Let $a,b$ be two positive integers such that for all positive integer $n>2020^{2020}$, there exists a positive integer $m$ coprime to $n$ with \begin{align*} \text{ $a^n+b^n \mid a^m+b^m$} \end{align*} Show that $a=b$ [i]Proposed by ltf0501[/i]

For every positive integer $n$, let $\sigma(n)$ denote the sum of all positive divisors of $n$ ($1$ and $n$, inclusive). Show that a positive integer $n$, which has at most two distinct prime factors, satisfies the condition $\sigma(n)=2n-2$ if and only if $n=2^k(2^{k+1}+1)$, where $k$ is a non-negative integer and $2^{k+1}+1$ is prime.

There are given $1000$ integers $a_1,... , a_{1000}$. Their squares $a^2_1, . . . , a^2_{1000}$ are written in a circle. It so happened that the sum of any $41$ consecutive numbers on this circle is a multiple of $41^2$. Is it necessarily true that every integer $a_1,... , a_{1000}$ is a multiple of $41$? (Boris Frenkin)

How many integers $100 \le x \le 999$ have the property that, among the six digits in $\lfloor 280 + \frac{x}{100} \rfloor$ and $x$, exactly two are identical?

Let $k \in Z_{>0}$ be the smallest positive integer with the property that $k\frac{gcd(x,y)gcd(y,z)}{lcm (x,y^2,z)}$ is a positive integer for all values $1 \le x \le y \le z \le 121$. If k' is the number of divisors of $k$, find the number of divisors of $k'$.

Consider a nonconstant arithmetic progression $a_1, a_2,\cdots, a_n,\cdots$. Suppose there exist relatively prime positive integers $p>1$ and $q>1$ such that $a_1^2, a_{p+1}^2$ and $a_{q+1}^2$ are also the terms of the same arithmetic progression. Prove that the terms of the arithmetic progression are all integers.

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?

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$.

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? \\

Determine all integers $n$ for which \[\sqrt{\frac{25}{2}+\sqrt{\frac{625}{4}-n}}+\sqrt{\frac{25}{2}-\sqrt{\frac{625}{4}-n}}\] is an integer.

Find all functions $f : Z_{>0} \to Z_{>0}$ with $a - f(b) | af(a) - bf(b)$ for all $a, b \in Z_{>0}$. [i](Theresia Eisenkoelbl)[/i]

Determine the integers $x$ such that $2^x + x^2 + 25$ is the cube of a prime number