Let $n \geq 1$ be an odd integer. Determine all functions $f$ from the set of integers to itself, such that for all integers $x$ and $y$ the difference $f(x)-f(y)$ divides $x^n-y^n.$ [i]Proposed by Mihai Baluna, Romania[/i]

Prove, that for every natural $N$ exists $k$, such that $N=a_02^0+a_12^1+...+a_k2^k$, where $a_0,a_1,...a_k$ are $1$ or $2$

A sequence of positive integers $g_1$, $g_2$, $g_3$, $. . . $ is defined as follows: $g_1 = 1$ and for every positive integer $n$, $$g_{n + 1} = g^2_n + g_n + 1.$$ Show that $g^2_{n} + 1$ divides $g^2_{n + 1}+1$ for every positive integer $n$.

[b]p1.[/b] David is taking a $50$-question test, and he needs to answer at least $70\%$ of the questions correctly in order to pass the test. What is the minimum number of questions he must answer correctly in order to pass the test? [b]p2.[/b] You decide to flip a coin some number of times, and record each of the results. You stop flipping the coin once you have recorded either $20$ heads, or $16$ tails. What is the maximum number of times that you could have flipped the coin? [b]p3.[/b] The width of a rectangle is half of its length. Its area is $98$ square meters. What is the length of the rectangle, in meters? [b]p4.[/b] Carol is twice as old as her younger brother, and Carol's mother is $4$ times as old as Carol is. The total age of all three of them is $55$. How old is Carol's mother? [b]p5.[/b] What is the sum of all two-digit multiples of $9$? [b]p6.[/b] The number $2016$ is divisible by its last two digits, meaning that $2016$ is divisible by $16$. What is the smallest integer larger than $2016$ that is also divisible by its last two digits? [b]p7.[/b] Let $Q$ and $R$ both be squares whose perimeters add to $80$. The area of $Q$ to the area of $R$ is in a ratio of $16 : 1$. Find the side length of $Q$. [b]p8.[/b] How many $8$-digit positive integers have the property that the digits are strictly increasing from left to right? For instance, $12356789$ is an example of such a number, while $12337889$ is not. [b]p9.[/b] During a game, Steve Korry attempts $20$ free throws, making 16 of them. How many more free throws does he have to attempt to finish the game with $84\%$ accuracy, assuming he makes them all? [b]p10.[/b] How many dierent ways are there to arrange the letters $MILKTEA$ such that $TEA$ is a contiguous substring? For reference, the term "contiguous substring" means that the letters $TEA$ appear in that order, all next to one another. For example, $MITEALK$ would be such a string, while $TMIELKA$ would not be. [b]p11.[/b] Suppose you roll two fair $20$-sided dice. What is the probability that their sum is divisible by $10$? [b]p12.[/b] Suppose that two of the three sides of an acute triangle have lengths $20$ and $16$, respectively. How many possible integer values are there for the length of the third side? [b]p13.[/b] Suppose that between Beijing and Shanghai, an airplane travels $500$ miles per hour, while a train travels at $300$ miles per hour. You must leave for the airport $2$ hours before your flight, and must leave for the train station $30$ minutes before your train. Suppose that the two methods of transportation will take the same amount of time in total. What is the distance, in miles, between the two cities? [b]p14.[/b] How many nondegenerate triangles (triangles where the three vertices are not collinear) with integer side lengths have a perimeter of $16$? Two triangles are considered distinct if they are not congruent. [b]p15.[/b] John can drive $100$ miles per hour on a paved road and $30$ miles per hour on a gravel road. If it takes John $100$ minutes to drive a road that is $100$ miles long, what fraction of the time does John spend on the paved road? [b]p16.[/b] Alice rolls one pair of $6$-sided dice, and Bob rolls another pair of $6$-sided dice. What is the probability that at least one of Alice's dice shows the same number as at least one of Bob's dice? [b]p17.[/b] When $20^{16}$ is divided by $16^{20}$ and expressed in decimal form, what is the number of digits to the right of the decimal point? Trailing zeroes should not be included. [b]p18.[/b] Suppose you have a $20 \times 16$ bar of chocolate squares. You want to break the bar into smaller chunks, so that after some sequence of breaks, no piece has an area of more than $5$. What is the minimum possible number of times that you must break the bar? For an example of how breaking the chocolate works, suppose we have a $2\times 2$ bar and wish to break it entirely into $1\times 1$ bars. We can break it once to get two $2\times 1$ bars. Then, we would have to break each of these individual bars in half in order to get all the bars to be size $1\times 1$, and we end up using $3$ breaks in total. [b]p19.[/b] A class of $10$ students decides to form two distinguishable committees, each with $3$ students. In how many ways can they do this, if the two committees can have no more than one student in common? [b]p20.[/b] You have been told that you are allowed to draw a convex polygon in the Cartesian plane, with the requirements that each of the vertices has integer coordinates whose values range from $0$ to $10$ inclusive, and that no pair of vertices can share the same $x$ or $y$ coordinate value (so for example, you could not use both $(1, 2)$ and $(1, 4)$ in your polygon, but $(1, 2)$ and $(2, 1)$ is fine). What is the largest possible area that your polygon can have? PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Given a positive integer $N$ (written in base $10$), define its [i]integer substrings[/i] to be integers that are equal to strings of one or more consecutive digits from $N$, including $N$ itself. For example, the integer substrings of $3208$ are $3$, $2$, $0$, $8$, $32$, $20$, $320$, $208$, $3208$. (The substring $08$ is omitted from this list because it is the same integer as the substring $8$, which is already listed.) What is the greatest integer $N$ such that no integer substring of $N$ is a multiple of $9$? (Note: $0$ is a multiple of $9$.)

Let $S(n)=1\varphi(1)+2\varphi(2) \ldots +n\varphi(n)$, where $\varphi(n)$ is the number of positive integers less than or equal to $n$ that are relatively prime to $n$. (For instance $\varphi(12)=4$ and $\varphi(20)=8$.) Prove that for all $n \geq 2018$, the following inequality holds: $$0.17n^3 \leq S(n) \leq 0.23n^3$$

We have $2p-1$ integer numbers, where $p$ is a prime number. Prove that we can choose exactly $p$ numbers (from these $2p-1$ numbers) so that their sum is divisible by $p$.

Let $x_n=\binom{2n}{n}$ for all $n\in\mathbb{Z}^+$. Prove there exist infinitely many finite sets $A,B$ of positive integers, satisfying $A \cap B = \emptyset $, and \[\frac{{\prod\limits_{i \in A} {{x_i}} }}{{\prod\limits_{j\in B}{{x_j}} }}=2012.\]

Let $n>2$ be a positive integer. Masha writes down $n$ natural numbers along a circle. Next, Taya performs the following operation: Between any two adjacent numbers $a$ and $b$, she writes a divisor of the number $a+b$ greater than $1$, then Taya erases the original numbers and obtains a new set of $n$ numbers along the circle. Can Taya always perform these operations in such a way that after some number of operations, all the numbers are equal? [i]Proposed by T. Korotchenko[/i]

Let $n, k, \ell$ be positive integers and $\sigma : \lbrace 1, 2, \ldots, n \rbrace \rightarrow \lbrace 1, 2, \ldots, n \rbrace$ an injection such that $\sigma(x)-x\in \lbrace k, -\ell \rbrace$ for all $x\in \lbrace 1, 2, \ldots, n \rbrace$. Prove that $k+\ell|n$.

Find all $f:\mathbb{N}\to\mathbb{N}$ satisfying that for all $m,n\in\mathbb{N}$, the nonnegative integer $|f(m+n)-f(m)|$ is a divisor of $f(n)$. [i] Proposed by usjl[/i]

Let $\sum_{i=1}^n a_i x_i =0$, $a_i\in \mathbb{Z}$. It is known that however we color $\mathbb{N}$ with finite number of colors, then the upper equation has a solution $x_1,x_2,...,x_n$ in one color. Prove that there is some non-empty sum of its coefficients equal to 0.

Let $n$ ($n\ge 3$) be a natural number. Denote by $f(n)$ the least natural number by which $n$ is not divisible (e.g. $f(12)=5$). If $f(n)\ge 3$, we may have $f(f(n))$ in the same way. Similarly, if $f(f(n))\ge 3$, we may have $f(f(f(n)))$, and so on. If $\underbrace{f(f(\dots f}_{k\text{ times}}(n)\dots ))=2$, we call $k$ the “[i]length[/i]” of $n$ (also we denote by $l_n$ the “[i]length[/i]” of $n$). For arbitrary natural number $n$ ($n\ge 3$), find $l_n$ with proof.

$10$ real numbers are given $a_1,a_2,\ldots ,a_{10} $, and the $45$ sums of two of these numbers are formed $a_i+a_j $, $1\leq i<j\leq 10$ . It is known that not all these sums are integers. Determine the minimum value of $k$ such that it is possible that among the $45$ sums there are $k$ that are not integers and $45-k$ that are integers.

Find the smallest positive integer that is equal to the sum of the product of its digits and the sum of its digits.

Find all odd positive integers $n>1$ such that there is a permutation $a_1, a_2, a_3, \ldots, a_n$ of the numbers $1, 2,3, \ldots, n$ where $n$ divides one of the numbers $a_k^2 - a_{k+1} - 1$ and $a_k^2 - a_{k+1} + 1$ for each $k$, $1 \leq k \leq n$ (we assume $a_{n+1}=a_1$).

Let $k, M$ be positive integers such that $k-1$ is not squarefree. Prove that there exist a positive real $\alpha$, such that $\lfloor \alpha\cdot k^n \rfloor$ and $M$ are coprime for any positive integer $n$.

How many integers from $1$ to $2016$ are divisible by $3$ or $7$, but not $21$?

Do there exist positive integers $m,n$, such that $m^{20}+11^n$ is a square number?

Given a fixed rational number $q$. Let's call a number $x$ [i]charismatic [/i] if we can find a natural number $n$ and integers $a_1, a_2,.., a_n$ such that $$x = (q + 1)^{a_1} \cdot (q + 2)^{a_2} \cdot ... \cdot(q + n)^{a_n} .$$ i) Prove that one can find a $q$ such that all positive rational numbers are charismatic. ii) Is it true that for all $q$, if the number $x$ is charismatic, then $x + 1$ is also charismatic?

Prove the theorem: There is no natural number $ n > 1 $ such that the number $ 2^n - 1 $ is divisible by $ n $.

Let $a_1,a_2, \cdots$ be a sequence of integers that satisfies: $a_1=1$ and $a_{n+1}=a_n+a_{\lfloor \sqrt{n} \rfloor} , \forall n\geq 1 $. Prove that for all positive $k$, there is $m \geq 1$ such that $k \mid a_m$.

[b]p1.[/b] How many nonzero digits are in the number $(5^{94} + 5^{92})(2^{94} + 2^{92})$? [b]p2.[/b] Suppose $A$ is a set of $2013$ distinct positive integers such that the arithmetic mean of any subset of $A$ is also an integer. Find an example of $A$. [b]p3.[/b] How many minutes until the smaller angle formed by the minute and hour hands on the face of a clock is congruent to the smaller angle between the hands at $5:15$ pm? Round your answer to the nearest minute. [b]p4.[/b] Suppose $a$ and $b$ are positive real numbers, $a + b = 1$, and $$1 +\frac{a^2 + 3b^2}{2ab}=\sqrt{4 +\frac{a}{b}+\frac{3b}{a}}.$$ Find $a$. [b]p5.[/b] Suppose $f(x) = \frac{e^x- 12e^{-x}}{ 2}$ . Find all $x$ such that $f(x) = 2$. [b]p6.[/b] Let $P_1$, $P_2$,$...$,$P_n$ be points equally spaced on a unit circle. For how many integer $n \in \{2, 3, ... , 2013\}$ is the product of all pairwise distances: $\prod_{1\le i<j\le n} P_iP_j$ a rational number? Note that $\prod$ means the product. For example, $\prod_{1\le i\le 3} i = 1\cdot 2 \cdot 3 = 6$. [b]p7.[/b] Determine the value $a$ such that the following sum converges if and only if $r \in (-\infty, a)$ : $$\sum^{\infty}_{n=1}(\sqrt{n^4 + n^r} - n^2).$$ Note that $\sum^{\infty}_{n=1}\frac{1}{n^s}$ converges if and only if $s > 1$. [b]p8.[/b] Find two pairs of positive integers $(a, b)$ with $a > b$ such that $a^2 + b^2 = 40501$. [b]p9.[/b] Consider a simplified memory-knowledge model. Suppose your total knowledge level the night before you went to a college was $100$ units. Each day, when you woke up in the morning you forgot $1\%$ of what you had learned. Then, by going to lectures, working on the homework, preparing for presentations, you had learned more and so your knowledge level went up by $10$ units at the end of the day. According to this model, how long do you need to stay in college until you reach the knowledge level of exactly $1000$? [b]p10.[/b] Suppose $P(x) = 2x^8 + x^6 - x^4 +1$, and that $P$ has roots $a_1$, $a_2$, $...$ , $a_8$ (a complex number $z$ is a root of the polynomial $P(x)$ if $P(z) = 0$). Find the value of $$(a^2_1-2)(a^2_2-2)(a^2_3-2)...(a^2_8-2).$$ [b]p11.[/b] Find all values of $x$ satisfying $(x^2 + 2x-5)^2 = -2x^2 - 3x + 15$. [b]p12.[/b] Suppose $x, y$ and $z$ are positive real numbers such that $$x^2 + y^2 + xy = 9,$$ $$y^2 + z^2 + yz = 16,$$ $$x^2 + z^2 + xz = 25.$$ Find $xy + yz + xz$ (the answer is unique). [b]p13.[/b] Suppose that $P(x)$ is a monic polynomial (i.e, the leading coefficient is $1$) with $20$ roots, each distinct and of the form $\frac{1}{3^k}$ for $k = 0,1,2,..., 19$. Find the coefficient of $x^{18}$ in $P(x)$. [b]p14.[/b] Find the sum of the reciprocals of all perfect squares whose prime factorization contains only powers of $3$, $5$, $7$ (i.e. $\frac{1}{1} + \frac{1}{9} + \frac{1}{25} + \frac{1}{419} + \frac{1}{811} + \frac{1}{215} + \frac{1}{441} + \frac{1}{625} + ...$). [b]p15.[/b] Find the number of integer quadruples $(a, b, c, d)$ which also satisfy the following system of equations: $$1+b + c^2 + d^3 =0,$$ $$a + b^2 + c^3 + d^4 =0,$$ $$a^2 + b^3 + c^4 + d^5 =0,$$ $$a^3+b^4+c^5+d^6 =0.$$ PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Prove that if $$\frac{p}{q}=1-\frac{1}{2} + \frac{1}{3}- \frac{1}{4} + ... -\frac{1}{1334} + \frac{1}{1335}$$ where $p, q \in Z_+$ then $p$ is divisible by $2003$.

[u]Round 1[/u] [b]p1.[/b] Find all pairs $(a,b)$ of positive integers with $a > b$ and $a^2 -b^2 =111$. [b]p2.[/b] Alice drives at a constant rate of $2017$ miles per hour. Find all positive values of $x$ such that she can drive a distance of $x^2$ miles in a time of $x$ minutes. [b]p3.[/b] $ABC$ is a right triangle with right angle at $B$ and altitude $BH$ to hypotenuse $AC$. If $AB = 20$ and $BH = 12$, find the area of triangle $\vartriangle ABC$. [u]Round 2[/u] [b]p4.[/b] Regular polygons $P_1$ and $P_2$ have $n_1$ and $n_2$ sides and interior angles $x_1$ and $x_2$, respectively. If $\frac{n_1}{n_2}= \frac75$ and $\frac{x_1}{x_2}=\frac{15}{14}$ , find the ratio of the sum of the interior angles of $P_1$ to the sum of the interior angles of $P_2$. [b]p5.[/b] Joey starts out with a polynomial $f (x) = x^2 +x +1$. Every turn, he either adds or subtracts $1$ from $f$ . What is the probability that after $2017$ turns, $f$ has a real root? [b]p6.[/b] Find the difference between the greatest and least positive integer values $x$ such that $\sqrt[20]{\lfloor \sqrt[17]{x}\rfloor}=1$. [u]Round 3[/u] [b]p7.[/b] Let $ABCD$ be a square and suppose $P$ and $Q$ are points on sides $AB$ and $CD$ respectively such that $\frac{AP}{PB} = \frac{20}{17}$ and $\frac{CQ}{QD}=\frac{17}{20}$ . Suppose that $PQ = 1$. Find the area of square $ABCD$. [b]p8.[/b] If $$\frac{\sum_{n \ge 0} r^n}{\sum_{n \ge 0} r^{2n}}=\frac{1+r +r^2 +r^3 +...}{1+r^2 +r^4 +r^6 +...}=\frac{20}{17},$$ find $r$ . [b]p9.[/b] Let $\overline{abc}$ denote the $3$ digit number with digits $a,b$ and $c$. If $\overline{abc}_{10}$ is divisible by $9$, what is the probability that $\overline{abc}_{40}$ is divisible by $9$? [u]Round 4[/u] [b]p10.[/b] Find the number of factors of $20^{17}$ that are perfect cubes but not perfect squares. [b]p11.[/b] Find the sum of all positive integers $x \le 100$ such that $x^2$ leaves the same remainder as $x$ does upon division by $100$. [b]p12.[/b] Find all $b$ for which the base-$b$ representation of $217$ contains only ones and zeros. PS. You should use hide for answers. Rounds 5-8 have been posted [url=https://artofproblemsolving.com/community/c3h3158514p28715373]here[/url].and 9-12 [url=https://artofproblemsolving.com/community/c3h3162362p28764144]here[/url] Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].