A sequence infinity $a_{1}, a_{2},...,$ is $generadora$ if: $a_{1}=1,2$ and $a_{n+1}$ is obtained by placing a digit 1 on the left or a digit 2 on the right for all natural n. Prove that there is an infinite $generadora$ sequence such that it does not contain any multiples of 7.

Let $X$ be a finite set of positive integers. Prove that for every subset $A$ of $X$, there is a subset $B$ of $X$, with the following property: For each element $ e$ of $X$, $e$ divides an odd number of elements of $B$, if and only if $e$ is an element of $A$.

Show that there are an infinity of positive integers $n$ such that $2^{n}+3^{n}$ is divisible by $n^{2}$.

An integer $n$ is said to be [i]good[/i] if $|n|$ is not the square of an integer. Determine all integers $m$ with the following property: $m$ can be represented, in infinitely many ways, as a sum of three distinct good integers whose product is the square of an odd integer. [i]Proposed by Hojoo Lee, Korea[/i]

[b]p1.[/b] Two players play the following game. On the lowest left square of an $8\times 8$ chessboard there is a rook. The first player is allowed to move the rook up or to the right by an arbitrary number of squares. The second player is also allowed to move the rook up or to the right by an arbitrary number of squares. Then the first player is allowed to do this again, and so on. The one who moves the rook to the upper right square wins. Who has a winning strategy? [b]p2.[/b] In Crocodile Country there are banknotes of $1$ dollar, $10$ dollars, $100$ dollars, and $1,000$ dollars. Is it possible to get 1,000,000 dollars by using $250,000$ banknotes? [b]p3.[/b] Fifteen positive numbers (not necessarily whole numbers) are placed around the circle. It is known that the sum of every four consecutive numbers is $30$. Prove that each number is less than $15$. [b]p4.[/b] Donald Duck has $100$ sticks, each of which has length $1$ cm or $3$ cm. Prove that he can break into $2$ pieces no more than one stick, after which he can compose a rectangle using all sticks. [b]p5.[/b] Three consecutive $2$ digit numbers are written next to each other. It turns out that the resulting $6$ digit number is divisible by $17$. Find all such numbers. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find all $n\in \mathbb{N}$, $n>1$ with the following property: All divisors of $n$ can be put in a rectangular table in such way that the sums of the numbers by rows are equal and the sums of the numbers by columns are also equal.

Find all pairs of integers $x,y$ for which \[x^3+x^2+x=y^2+y.\]

Let $p$ be a prime number. Prove that there exists a prime number $q$ such that for every integer $n$, the number $n^p-p$ is not divisible by $q$.

An integer $a$ is a quadratic nonresidue modulo a prime $p$ if there does not exist $x \in Z$ such that $x^2 \equiv a$ (mod $p$). How many ordered pairs $(a, b)$ modulo $29$ exist such that $$a + b\equiv 1 \,\,\, (mod \,\,\, 29)$$ where both $a$ and $b$ are quadratic nonresidues modulo $29$?

Find all prime number $p$ such that there exists an integer-coefficient polynomial $f(x)=x^{p-1}+a_{p-2}x^{p-2}+…+a_1x+a_0$ that has $p-1$ consecutive positive integer roots and $p^2\mid f(i)f(-i)$, where $i$ is the imaginary unit.

[b]p1.[/b] How many real roots does the equation $2x^7 + x^5 + 4x^3 + x + 2 = 0$ have? [b]p2.[/b] Given that $f(n) = 1 +\sum^n_{j=1}(1 +\sum^j_{i=1}(2i + 1))$, find the value of $f(99)-\sum^{99}_{i=1} i^2$. [b]p3.[/b] A rectangular prism with dimensions $1\times a \times b$, where $1 < a < b < 2$, is bisected by a plane bisecting the longest edges of the prism. One of the smaller prisms is bisected in the same way. If all three resulting prisms are similar to each other and to the original box, compute $ab$. Note: Two rectangular prisms of dimensions $p \times q\times r$ and$ x\times y\times z$ are similar if $\frac{p}{x} = \frac{q}{y} = \frac{r}{z}$ . [b]p4.[/b] For fixed real values of $p$, $q$, $r$ and $s$, the polynomial $x^4 + px^3 + qx^2 + rx + s$ has four non real roots. The sum of two of these roots is $4 + 7i$, and the product of the other two roots is $3 - 4i$. Compute $q$. [b]p5.[/b] There are $10$ seats in a row in a theater. Say we have an infinite supply of indistinguishable good kids and bad kids. How many ways can we seat $10$ kids such that no two bad kids are allowed to sit next to each other? [b]p6.[/b] There are an infinite number of people playing a game. They each pick a different positive integer $k$, and they each win the amount they chose with probability $\frac{1}{k^3}$ . What is the expected amount that all of the people win in total? [b]p7.[/b] There are $100$ donuts to be split among $4$ teams. Your team gets to propose a solution about how the donuts are divided amongst the teams. (Donuts may not be split.) After seeing the proposition, every team either votes in favor or against the propisition. The proposition is adopted with a majority vote or a tie. If the proposition is rejected, your team is eliminated and will never receive any donuts. Another remaining team is chosen at random to make a proposition, and the process is repeated until a proposition is adopted, or only one team is left. No promises or deals need to be kept among teams besides official propositions and votes. Given that all teams play optimally to maximize the expected value of the number of donuts they receive, are completely indifferent as to the success of the other teams, but they would rather not eliminate a team than eliminate one (if the number of donuts they receive is the same either way), then how much should your team propose to keep? [b]p8.[/b] Dominic, Mitchell, and Sitharthan are having an argument. Each of them is either credible or not credible – if they are credible then they are telling the truth. Otherwise, it is not known whether they are telling the truth. At least one of Dominic, Mitchell, and Sitharthan is credible. Tim knows whether Dominic is credible, and Ethan knows whether Sitharthan is credible. The following conversation occurs, and Tim and Ethan overhear: Dominic: “Sitharthan is not credible.” Mitchell: “Dominic is not credible.” Sitharthan: “At least one of Dominic or Mitchell is credible.” Then, at the same time, Tim and Ethan both simultaneously exclaim: “I can’t tell exactly who is credible!” They each then think for a moment, and they realize that they can. If Tim and Ethan always tell the truth, then write on your answer sheet exactly which of the other three are credible. [b]p9.[/b] Pick an integer $n$ between $1$ and $10$. If no other team picks the same number, we’ll give you $\frac{n}{10}$ points. [b]p10.[/b] Many quantities in high-school mathematics are left undefined. Propose a definition or value for the following expressions and justify your response for each. We’ll give you $\frac15$ points for each reasonable argument. $$(i) \,\,\,(.5)! \,\,\, \,\,\,(ii) \,\,\,\infty \cdot 0 \,\,\, \,\,\,(iii) \,\,\,0^0 \,\,\, \,\,\,(iv)\,\,\, \prod_{x\in \emptyset}x \,\,\, \,\,\,(v)\,\,\, 1 - 1 + 1 - 1 + ...$$ [b]p11.[/b] On the back of your answer sheet, write the “coolest” math question you know, and include the solution. If the graders like your question the most, then you’ll get a point. (With your permission, we might include your question on the Mixer next year!) PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

For a fixed integer $n\geqslant2$ consider the sequence $a_k=\text{lcm}(k,k+1,\ldots,k+(n-1))$. Find all $n$ for which the sequence $a_k$ increases starting from some number.

Let $X \neq \varnothing$ be a finite set and let $f: X \to X$ be a function such that for every $x \in X$ and a fixed prime $p$ we have $f^p(x)=x.$ Let $Y=\{x \in X | f(x) \neq x\}.$ Prove that the number of the members of the set $Y$ is divisible by $p.$ [i]Note.[/i] ${f^p(x)=x = \underbrace{f(f(f(\cdots ((f}_{ p \text{ times}}(x) ) \cdots )))} .$

Let $a_{0},a_{1},a_{2},\ldots $ be an increasing sequence of nonnegative integers such that every nonnegative integer can be expressed uniquely in the form $a_{i}+2a_{j}+4a_{k}$, where $i,j$ and $k$ are not necessarily distinct. Determine $a_{1998}$.

Find all triples $(a,b,c)$ of positive integers such that the product of any two of them when divided by the third leaves the remainder $1$.

Find all pairs of prime numbers $p$ and $q$ such that $p^3-q^5 = (p+q)^2$.

Prove: If the sum of all positive divisors of $ n \in \mathbb{Z}^{\plus{}}$ is a power of two, then the number/amount of the divisors is a power of two.

Prove that the second last digit of each power of three is even . (V . I . Plachkos)

Let $N_{10}$ be the answer to problem 10. Compute the number of ordered pairs of integers $(m,n)$ that satisfy the equation \[ m^2+n^2=mn+N_{10}. \]

Prove that among $39$ consecutive natural numbers, there is always a number that has sum of its digits divisible by $ 12$. Is it true if we replace $39$ with $38$?

Find all positive integers $n$ with the following property: the $k$ positive divisors of $n$ have a permutation $(d_1,d_2,\ldots,d_k)$ such that for $i=1,2,\ldots,k$, the number $d_1+d_2+\cdots+d_i$ is a perfect square.

Find all positive integers $n$ for which there are distinct integer numbers $a_1, a_2, ... , a_n$ such that $$\frac{1}{a_1}+\frac{2}{a_2}+...+\frac{n}{a_n}=\frac{a_1 + a_2 + ... + a_n}{2}$$

Let $p,q,r$ be primes such that for all positive integer $n$, $$n^{pqr}\equiv n (\mod{pqr})$$ Prove that this happens if and only if $p,q,r$ are pairwise distinct and $LCM(p-1,q-1,r-1)|pqr-1$

Let $p$ a positive prime number and $\mathbb{F}_{p}$ the set of integers $mod \ p$. For $x\in \mathbb{F}_{p}$, define $|x|$ as the cyclic distance of $x$ to $0$, that is, if we represent $x$ as an integer between $0$ and $p-1$, $|x|=x$ if $x<\frac{p}{2}$, and $|x|=p-x$ if $x>\frac{p}{2}$ . Let $f: \mathbb{F}_{p} \rightarrow \mathbb{F}_{p}$ a function such that for every $x,y \in \mathbb{F}_{p}$ \[ |f(x+y)-f(x)-f(y)|<100 \] Prove that exist $m \in \mathbb{F}_{p}$ such that for every $x \in \mathbb{F}_{p}$ \[ |f(x)-mx|<1000 \]

[u]Round 1[/u] [b]1.1.[/b] There are $99$ dogs sitting in a long line. Starting with the third dog in the line, if every third dog barks three times, and all the other dogs each bark once, how many barks are there in total? [b]1.2.[/b] Indigo notices that when she uses her lucky pencil, her test scores are always $66 \frac23 \%$ higher than when she uses normal pencils. What percent lower is her test score when using a normal pencil than her test score when using her lucky pencil? [b]1.3.[/b] Bill has a farm with deer, sheep, and apple trees. He mostly enjoys looking after his apple trees, but somehow, the deer and sheep always want to eat the trees' leaves, so Bill decides to build a fence around his trees. The $60$ trees are arranged in a $5\times 12$ rectangular array with $5$ feet between each pair of adjacent trees. If the rectangular fence is constructed $6$ feet away from the array of trees, what is the area the fence encompasses in feet squared? (Ignore the width of the trees.) [u]Round 2[/u] [b]2.1.[/b] If $x + 3y = 2$, then what is the value of the expression $9^x * 729^y$? [b]2.2.[/b] Lazy Sheep loves sleeping in, but unfortunately, he has school two days a week. If Lazy Sheep wakes up each day before school's starting time with probability $1/8$ independent of previous days, then the probability that Lazy Sheep wakes up late on at least one school day over a given week is $p/q$ for relatively prime positive integers $p, q$. Find $p + q$. [b]2.3.[/b] An integer $n$ leaves remainder $1$ when divided by $4$. Find the sum of the possible remainders $n$ leaves when divided by $20$. [u]Round 3[/u] [b]3.1. [/b]Jake has a circular knob with three settings that can freely rotate. Each minute, he rotates the knob $120^o$ clockwise or counterclockwise at random. The probability that the knob is back in its original state after $4$ minutes is $p/q$ for relatively prime positive integers $p, q$. Find $p + q$. [b]3.2.[/b] Given that $3$ not necessarily distinct primes $p, q, r$ satisfy $p+6q +2r = 60$, find the sum of all possible values of $p + q + r$. [b]3.3.[/b] Dexter's favorite number is the positive integer $x$, If $15x$ has an even number of proper divisors, what is the smallest possible value of $x$? (Note: A proper divisor of a positive integer is a divisor other than itself.) [u]Round 4[/u] [b]4.1.[/b] Three circles of radius $1$ are each tangent to the other two circles. A fourth circle is externally tangent to all three circles. The radius of the fourth circle can be expressed as $\frac{a\sqrt{b}-\sqrt{c}}{d}$ for positive integers $a, b, c, d$ where $b$ is not divisible by the square of any prime and $a$ and $d$ are relatively prime. Find $a + b + c + d$. [b]4.2. [/b]Evaluate $$\frac{\sqrt{15}}{3} \cdot \frac{\sqrt{35}}{5} \cdot \frac{\sqrt{63}}{7}... \cdot \frac{\sqrt{5475}}{73}$$ [b]4.3.[/b] For any positive integer $n$, let $f(n)$ denote the number of digits in its base $10$ representation, and let $g(n)$ denote the number of digits in its base $4$ representation. For how many $n$ is $g(n)$ an integer multiple of $f(n)$? PS. You should use hide for answers. Rounds 5-8 have been posted [url=https://artofproblemsolving.com/community/c3h2784571p24468619]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].