Amy and Bob play the game. At the beginning, Amy writes down a positive integer on the board. Then the players take moves in turn, Bob moves first. On any move of his, Bob replaces the number $n$ on the blackboard with a number of the form $n-a^2$, where $a$ is a positive integer. On any move of hers, Amy replaces the number $n$ on the blackboard with a number of the form $n^k$, where $k$ is a positive integer. Bob wins if the number on the board becomes zero. Can Amy prevent Bob’s win? [i]Maxim Didin, Russia[/i]

Assume that $a_1, a_2, a_3$ are three given positive integers consider the following sequence: $a_{n+1}=\text{lcm}[a_n, a_{n-1}]-\text{lcm}[a_{n-1}, a_{n-2}]$ for $n\ge 3$ Prove that there exist a positive integer $k$ such that $k\le a_3+4$ and $a_k\le 0$. ($[a, b]$ means the least positive integer such that$ a\mid[a,b], b\mid[a, b]$ also because $\text{lcm}[a, b]$ takes only nonzero integers this sequence is defined until we find a zero number in the sequence)

A jar has 10 red candies and 10 blue candies. Terry picks two candies at random, then Mary picks two of the remaining candies at random. Given that the probability that they get the same color combination, irrespective of order, is $m/n$, where $m$ and $n$ are relatively prime positive integers, find $m+n$.

Given $k \in \mathbb{Z}$ prove that there exist infinite pairs of distinct natural numbers such that \begin{align*} n+s(2n)=m+s(2m) \\ kn+s(n^2)=km+s(m^2). \end{align*} ($s(n)$ denotes the sum of digits of $n$.) [i]Proposed by Mohammadamin Sharifi[/i]

Solve in positive integers the following equation: \[{1\over n^2}-{3\over 2n^3}={1\over m^2}\] [i]Proposed by A. Golovanov[/i]

Prove that for every positive integer $n$ coprime to $10$ there exists a multiple of $n$ that does not contain the digit $1$ in its decimal representation.

Find all positive integers $A,B$ satisfying the following properties: (i) $A$ and $B$ has same digit in decimal. (ii) $2\cdot A\cdot B=\overline{AB}$ (Here $\cdot$ denotes multiplication, $\overline{AB}$ denotes we write $A$ and $B$ in turn. For example, if $A=12,B=34$, then $\overline{AB}=1234$)

Find the largest integer such that every number after the first is one less than the previous one and is divisible by each of its own numbers.

For a natural number $n$, let $n'$ denote the number obtained by deleting zero digits, if any. (For example, if $n = 260$, $n' = 26$, if $n = 2020$, $n' = 22$.),Find the number of $3$-digit numbers $n$ for which $n'$ is a divisor of $n$, different from $n$.

If ri are integers such that $0 \le r_i < 31$ and $r_i$ satisfies the polynomial $x^4 + x^3 + x^2 + x \equiv 30$ (mod $31$), find $$\sum^4_{i=1}(r^2_i + 1)^{-1} \,\,\,\, (mod \,\,\,\, 31)$$ where $x^{-1}$ is the modulo inverse of $x$, that is, it is the unique integer $y$ such that $0 < y < 31$ and $xy -1$ is divisible by $31$.

How many ways are there to permute the first $n$ positive integers such that in the permutation, for each value of $k \le n$, the first $k$ elements of the permutation have distinct remainder mod $k$?

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]

[u]Round 1 [/u] [b]p1.[/b] Ephram was born in May $2005$. How old will he turn in the first year where the product of the digits of the year number is a nonzero perfect square? [b]p2.[/b] Zhao is studying for his upcoming calculus test by reviewing each of the $13$ lectures, numbered Lecture $1$, Lecture $2$, ..., Lecture $13$. For each $n$, he spends $5n$ minutes on Lecture $n$. Which lecture is he reviewing after $4$ hours? [b]p3.[/b] Compute $$\dfrac{3^3 \div 3(3)+3}{\frac{3}{3}}+3!.$$ [u]Round 2 [/u] [b]p4.[/b] At Ingo’s shop, train tickets normally cost $\$2$, but every $5$th ticket costs only $\$1$. At Emmet’s shop, train tickets normally cost $\$3$, but every $5$th ticket is free. Both Ingo and Emmett sold $1000$ tickets. Find the absolute difference between their sales, in dollars. [b]p5.[/b] Ephram paddles his boat in a river with a $4$-mph current. Ephram travels at $10$ mph in still water. He paddles downstream and then turns around and paddles upstream back to his starting position. Find the proportion of time he spends traveling upstream, as a percentage. [b]p6.[/b] The average angle measure of a $13-14-15$ triangle is $m^o$ and the average angle measure of a $5-6-7$ triangle is $n^o$. Find $m-n$. [u]Round 3[/u] [b]p7.[/b] Let $p(x) = x^2 -10x +31$. Find the minimum value of $p(p(x))$ over all real $x$. [b]p8.[/b] Michael H. andMichael Y. are playing a game with $4$ jellybeans. Michael H starts with $3$ of the jellybeans, and Michael Y starts with the remaining $1$. Every minute, a Michael flips a coin, and if heads, Michael H takes a jellybean from Michael Y. If tails, Michael Y takes a jellybean from Michael H. WhicheverMichael gathers all $4$ jellybeans wins. The probability Michael H wins can be written as $\frac{A}{B}$ for relatively prime positive integers $A$ and $B$. Find $1000A+B$. [b]p9.[/b] Define the digit-product of a positive integer to be the product of its non-zero digits. Let $M$ denote the greatest five-digit number with a digit-product of $360$, and let $N$ denote the least five-digit number with a digit-product of $360$. Find the digit-product of $M-N$. [u]Round 4 [/u] [b]p10.[/b] Hannah is attending one of the three IdeaMath classes running at LHS, while Alex decides to randomly visit some combination of classes. He won’t visit all three classes, but he’s equally likely to visit any other combination. The probability Alex visits Hannah’s class can be expressed as $\frac{A}{B}$ for relatively prime positive integers $A$ and $B$. Find $1000A+B$. [b]p11.[/b] In rectangle $ABCD$, let $E$ be the intersection of diagonal $AC$ and the circle centered at $A$ passing through $D$. Angle $\angle ACD = 24^o$. Find the measure of $\angle CED$ in degrees. [b]p12.[/b] During his IdeaMath class, Zach writes the numbers $2, 3, 4, 5, 6, 7$, and $8$ on a whiteboard. Every minute, he chooses two numbers $a$ and $b$ from the board, erases them, and writes the number $ab +a +b$ on the board. He repeats this process until there’s only one number left. Find the sum of all possible remaining numbers. [u]Round 5[/u] [b]p13.[/b] In isosceles right $\vartriangle ABC$ with hypotenuse $AC$, Let $A'$ be the point on the extension of $AB$ past $A$ such that $AA' = 1$. Let $C'$ be the point on the extension of $BC$ past vertex $C$ such that $CC' = 2$. Given that the difference of the areas of triangle $A'BC'$ and $ABC$ is $10$, find the area of $ABC$. [b]p14.[/b] Compute the sumof the greatest and least values of $x$ such that $(x^2 -4x +4)^2 +x^2 -4x \le 16$. [b]p15.[/b] Ephram is starting a fan club. At the fan club’s first meeting, everyone shakes hands with everyone else exactly once, except for Ephram, who is extremely sociable and shakes hands with everyone else twice. Given that a total of $2015$ handshakes took place, how many people attended the club’s first meeting? PS. You should use hide for answers. Rounds 6-9 have been posted [url=https://artofproblemsolving.com/community/c3h3167139p28823346]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

[u]Round 5 [/u] [b]p13.[/b] A palindrome is a number that reads the same forward as backwards; for example, $121$ and $36463$ are palindromes. Suppose that $N$ is the maximal possible difference between two consecutive three-digit palindromes. Find the number of pairs of consecutive palindromes $(A, B)$ satisfying $A < B$ and $B - A = N$. [b]p14.[/b] Suppose that $x, y$, and $z$ are complex numbers satisfying $x +\frac{1}{yz} = 5$, $y +\frac{1}{zx} = 8$, and $z +\frac{1}{xy} = 6$. Find the sum of all possible values of $xyz$. [b]p15.[/b] Let $\Omega$ be a circle with radius $25\sqrt2$ centered at $O$, and let $C$ and $J$ be points on $\Omega$ such that the circle with diameter $\overline{CJ}$ passes through $O$. Let $Q$ be a point on the circle with diameter $\overline{CJ}$ satisfying $OQ = 5\sqrt2$. If the area of the region bounded by $\overline{CQ}$, $\overline{QJ}$, and minor arc $JC$ on $\Omega$ can be expressed as $\frac{a\pi-b}{c}$ for integers $a, b$, and $c$ with $gcd \,\,(a, c) = 1$, then find $a + b + c$. [u]Round 6[/u] [b]p16.[/b] Veronica writes $N$ integers between $2$ and $2020$ (inclusive) on a blackboard, and she notices that no number on the board is an integer power of another number on the board. What is the largest possible value of $N$? [b]p17.[/b] Let $ABC$ be a triangle with $AB = 12$, $AC = 16$, and $BC = 20$. Let $D$ be a point on $AC$, and suppose that $I$ and $J$ are the incenters of triangles $ABD$ and $CBD$, respectively. Suppose that $DI = DJ$. Find $IJ^2$. [b]p18.[/b] For each positive integer $a$, let $P_a = \{2a, 3a, 5a, 7a, . . .\}$ be the set of all prime multiples of $a$. Let $f(m, n) = 1$ if $P_m$ and $P_n$ have elements in common, and let $f(m, n) = 0$ if $P_m$ and $P_n$ have no elements in common. Compute $$\sum_{1\le i<j\le 50} f(i, j)$$ (i.e. compute $f(1, 2) + f(1, 3) + ,,, + f(1, 50) + f(2, 3) + f(2, 4) + ,,, + f(49, 50)$.) [u]Round 7[/u] [b]p19.[/b] How many ways are there to put the six letters in “$MMATHS$” in a two-by-three grid such that the two “$M$”s do not occupy adjacent squares and such that the letter “$A$” is not directly above the letter “$T$” in the grid? (Squares are said to be adjacent if they share a side.) [b]p20.[/b] Luke is shooting basketballs into a hoop. He makes any given shot with fixed probability $p$ with $p < 1$, and he shoots n shots in total with $n \ge 2$. Miraculously, in $n$ shots, the probability that Luke makes exactly two shots in is twice the probability that Luke makes exactly one shot in! If $p$ can be expressed as $\frac{k}{100}$ for some integer $k$ (not necessarily in lowest terms), find the sum of all possible values for $k$. [b]p21.[/b] Let $ABCD$ be a rectangle with $AB = 24$ and $BC = 72$. Call a point $P$ [i]goofy [/i] if it satisfies the following conditions: $\bullet$ $P$ lies within $ABCD$, $\bullet$ for some points $F$ and $G$ lying on sides $BC$ and $DA$ such that the circles with diameter $BF$ and $DG$ are tangent to one another, $P$ lies on their common internal tangent. Find the smallest possible area of a polygon that contains every single goofy point inside it. PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c4h2800971p24674988]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $f : \{ 1, 2, 3, \dots \} \to \{ 2, 3, \dots \}$ be a function such that $f(m + n) | f(m) + f(n) $ for all pairs $m,n$ of positive integers. Prove that there exists a positive integer $c > 1$ which divides all values of $f$.

Let $x, y$ be distinct positive integers. Prove that the number $$\frac{(x+y)^2}{ x^3 + xy^2 - x^2y - y^3}$$ is not an integer

[b]Problem.[/b] Let $f:\mathbb{R}_0^+\rightarrow\mathbb{R}_0^+$ be a function defined as $$f(n)=n+\lfloor\sqrt{n}\rfloor~\forall~n\in\mathbb{R}_0^+.$$ Prove that for any positive integer $m,$ the sequence $$m,f(m),f(f(m)),f(f(f(m))),\ldots$$ contains a perfect square.

A positive integer $k$ is called [i]powerful [/i] if there are distinct positive integers $p, q, r, s, t$ such that $p^2$, $q^3$, $r^5$, $s^7$, $t^{11}$ all divide k. Find the smallest powerful integer.

Let $ a$ and $ b$ be two positive integers such that $ a \cdot b \plus{} 1$ divides $ a^{2} \plus{} b^{2}$. Show that $ \frac {a^{2} \plus{} b^{2}}{a \cdot b \plus{} 1}$ is a perfect square.

Given a polynomial with integer coefficients, which has at least one integer root. The greatest common divisor of all its integer roots equals $1$. Prove that if the leading coefficient of the polynomial equals $1$ then the greatest common divisor of the other coefficients also equals $1$.

Let $ m$, $ n$ be natural numbers such that $ m\plus{}n\plus{}1$ is prime and divides $ 2(m^2\plus{}n^2)\minus{}1$. Prove that $ m\equal{}n$.

For given positive integers $a$ and $b$, let us consider the equation$$a + \gcd(b, x) = b + \gcd(a, x).$$ [list=a] [*]For $a = 20$ and $b = 25$, find the least positive integer $x$ satisfying this equation. [*]Prove that for any positive integers $a$ and $b$, there exist infinitely many positive integers $x$ satisfying this equation. [/list] [i](Here, $\gcd(m, n)$ denotes the greatest common divisor of positive integers $m$ and $n$.)[/i]

Find the addition of all positive integers n that follows the following: $ \frac{\sqrt{n}}{2} + \frac{30}{\sqrt{n}} $ is an integer.

If $ a,n $ are two natural numbers corelated by the equation $ \left\{ \sqrt{n+\sqrt n}\right\} =\left\{ \sqrt a\right\} , $ then $ 1+4a $ is a perfect square. Justify this statement. Here, $ \{\} $ is the usual fractionary part.

Find all three digit numbers $\overline{abc}$ such that $2 \cdot \overline{abc} = \overline{bca} + \overline{cab}$.