Let \(a_0, a_1, a_2, \dots\) be an infinite sequence of positive integers with the following properties: - \(a_0\) is a given positive integer; - For each integer \(n \geq 1\), \(a_n\) is the smallest integer greater than \(a_{n-1}\) such that \(a_n + a_{n-1}\) is a perfect square. For example, if \(a_0 = 3\), then \(a_1 = 6\), \(a_2 = 10\), \(a_3 = 15\), and so on. (a) Let \(T\) be the set of numbers of the form \(a_k - a_l\), with \(k \geq l \geq 0\) integers. Prove that, regardless of the value of \(a_0\), the number of positive integers not in \(T\) is finite. (b) Calculate, as a function of \(a_0\), the number of positive integers that are not in \(T\).

Consider the second-degree polynomial \(P(x) = 4x^2+12x-3015\). Define the sequence of polynomials \(P_1(x)=\frac{P(x)}{2016}\) and \(P_{n+1}(x)=\frac{P(P_n(x))}{2016}\) for every integer \(n \geq 1\). [list='a'] [*]Show that exists a real number \(r\) such that \(P_n(r) < 0\) for every positive integer \(n\). [*]Find how many integers \(m\) are such that \(P_n(m)<0\) for infinite positive integers \(n\). [/list]

Let $x,y\in\mathbb{R}$ be such that $x = y(3-y)^2$ and $y = x(3-x)^2$. Find all possible values of $x+y$.

The World Cup, featuring $17$ teams from Europe and South America, as well as $15$ other teams that honestly don’t have a chance, is a soccer tournament that is held once every four years. As we speak, Croatia andMorocco are locked in a battle that has no significance whatsoever on the winner, but if you would like live score updates nonetheless, feel free to ask your proctor, who has no obligation whatsoever to provide them. [b]p1.[/b] During the group stage of theWorld Cup, groups of $4$ teams are formed. Every pair of teams in a group play each other once. Each team earns $3$ points for each win and $1$ point for each tie. Find the greatest possible sum of the points of each team in a group. [b]p2.[/b] In the semi-finals of theWorld Cup, the ref is bad and lets $11^2 = 121$ players per team go on the field at once. For a given team, one player is a goalie, and every other player is either a defender, midfielder, or forward. There is at least one player in each position. The product of the number of defenders, midfielders, and forwards is a mulitple of $121$. Find the number of ordered triples (number of defenders, number of midfielders, number of forwards) that satisfy these conditions. [b]p3.[/b] Messi is playing in a game during the Round of $16$. On rectangular soccer field $ABCD$ with $AB = 11$, $BC = 8$, points $E$ and $F$ are on segment $BC$ such that $BE = 3$, $EF = 2$, and $FC = 3$. If the distance betweenMessi and segment $EF$ is less than $6$, he can score a goal. The area of the region on the field whereMessi can score a goal is $a\pi +\sqrt{b} +c$, where $a$, $b$, and $c$ are integers. Find $10000a +100b +c$. [b]p4.[/b] The workers are building theWorld Cup stadium for the $2022$ World Cup in Qatar. It would take 1 worker working alone $4212$ days to build the stadium. Before construction started, there were 256 workers. However, each day after construction, $7$ workers disappear. Find the number of days it will take to finish building the stadium. [b]p5.[/b] In the penalty kick shootout, $2$ teams each get $5$ attempts to score. The teams alternate shots and the team that scores a greater number of times wins. At any point, if it’s impossible for one team to win, even before both teams have taken all $5$ shots, the shootout ends and nomore shots are taken. If each team does take all $5$ shots and afterwards the score is tied, the shootout enters sudden death, where teams alternate taking shots until one team has a higher score while both teams have taken the same number of shots. If each shot has a $\frac12$ chance of scoring, the expected number of times that any team scores can be written as $\frac{A}{B}$ for relatively prime positive integers $A$ and $B$. Find $1000A+B$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

[b]p1.[/b] Alan leaves home when the clock in his cardboard box says $7:35$ AM and his watch says $7:41$ AM. When he arrives at school, his watch says $7:47$ AM and the $7:45$ AM bell rings. Assuming the school clock, the watch, and the home clock all go at the same rate, how many minutes behind the school clock is the home clock? [b]p2.[/b] Compute $$\left( \frac{2012^{2012-2013} + 2013}{2013} \right) \times 2012.$$ Express your answer as a mixed number. [b]p3.[/b] What is the last digit of $$2^{3^{4^{5^{6^{7^{8^{9^{...^{2013}}}}}}}}} ?$$ [b]p4.[/b] Let $f(x)$ be a function such that $f(ab) = f(a)f(b)$ for all positive integers $a$ and $b$. If $f(2) = 3$ and $f(3) = 4$, find $f(12)$. [b]p5.[/b] Circle $X$ with radius $3$ is internally tangent to circle $O$ with radius $9$. Two distinct points $P_1$ and $P_2$ are chosen on $O$ such that rays $\overrightarrow{OP_1}$ and $\overrightarrow{OP_2}$ are tangent to circle $X$. What is the length of line segment $P_1P_2$? [b]p6.[/b] Zerglings were recently discovered to use the same $24$-hour cycle that we use. However, instead of making $12$-hour analog clocks like humans, Zerglings make $24$-hour analog clocks. On these special analog clocks, how many times during $ 1$ Zergling day will the hour and minute hands be exactly opposite each other? [b]p7.[/b] Three Small Children would like to split up $9$ different flavored Sweet Candies evenly, so that each one of the Small Children gets $3$ Sweet Candies. However, three blind mice steal one of the Sweet Candies, so one of the Small Children can only get two pieces. How many fewer ways are there to split up the candies now than there were before, assuming every Sweet Candy is different? [b]p8.[/b] Ronny has a piece of paper in the shape of a right triangle $ABC$, where $\angle ABC = 90^o$, $\angle BAC = 30^o$, and $AC = 3$. Holding the paper fixed at $A$, Ronny folds the paper twice such that after the first fold, $\overline{BC}$ coincides with $\overline{AC}$, and after the second fold, $C$ coincides with $A$. If Ronny initially marked $P$ at the midpoint of $\overline{BC}$, and then marked $P'$ as the end location of $P$ after the two folds, find the length of $\overline{PP'}$ once Ronny unfolds the paper. [b]p9.[/b] How many positive integers have the same number of digits when expressed in base $3$ as when expressed in base $4$? [b]p10.[/b] On a $2 \times 4$ grid, a bug starts at the top left square and arbitrarily moves north, south, east, or west to an adjacent square that it has not already visited, with an equal probability of moving in any permitted direction. It continues to move in this way until there are no more places for it to go. Find the expected number of squares that it will travel on. Express your answer as a mixed number. PS. You had better use hide for answers.

For certain ordered pairs $(a,b)$ of real numbers, the system of equations \begin{eqnarray*} && ax+by =1\\ &&x^2+y^2=50\end{eqnarray*} has at least one solution, and each solution is an ordered pair $(x,y)$ of integers. How many such ordered pairs $(a,b)$ are there?

Let $a_0,a_1,a_2,\dots $ be a sequence of real numbers such that $a_0=0, a_1=1,$ and for every $n\geq 2$ there exists $1 \leq k \leq n$ satisfying \[ a_n=\frac{a_{n-1}+\dots + a_{n-k}}{k}. \]Find the maximum possible value of $a_{2018}-a_{2017}$.

Find all non-zero real numbers $ x, y, z$ which satisfy the system of equations: \[ (x^2 \plus{} xy \plus{} y^2)(y^2 \plus{} yz \plus{} z^2)(z^2 \plus{} zx \plus{} x^2) \equal{} xyz\] \[ (x^4 \plus{} x^2y^2 \plus{} y^4)(y^4 \plus{} y^2z^2 \plus{} z^4)(z^4 \plus{} z^2x^2 \plus{} x^4) \equal{} x^3y^3z^3\]

Let $x,y,z&gt;0$ Prove that $$\frac{x^2+2}{\sqrt{z^2+xy}}+\dfrac{y^2+2}{\sqrt{x ^2+yz}}+\dfrac{z^2+2}{\sqrt{y^2+zx}}\geq 6$$. [i](nooonuii)[/i]

Let $\varphi$ be the positive solution to the equation $$x^2=x+1.$$ For $n\ge 0$, let $a_n$ be the unique integer such that $\varphi^n-a_n\varphi$ is also an integer. Compute $$\sum_{n=0}^{10}a_n.$$

Let $k\geq 2$,$n_1,n_2,\cdots ,n_k\in \mathbb{N}_+$,satisfied $n_2|2^{n_1}-1,n_3|2^{n_2}-1,\cdots ,n_k|2^{n_{k-1}}-1,n_1|2^{n_k}-1$. Prove:$n_ 1=n_ 2=\cdots=n_k=1$.

Prove that the polynomial $x^n+4$ can be expressed as a product of two polynomials (each with degree less than $n$) with integer coefficients, if and only if $n$ is divisible by $4$.

Demonstrate that if $ z_1,z_2\in\mathbb{C}^* $ satisfy the relation: $$ z_1\cdot 2^{\big| z_1\big|} +z_2\cdot 2^{\big| z_2\big|} =\left( z_1+z_2\right)\cdot 2^{\big| z_1 +z_2\big|} , $$ then $ z_1^6=z_2^6 $

Do there exist $2023$ nonzero reals, not necessarily distinct, such that the fractional part of each number is equal to the sum of the rest $2022$ numbers?

Given a quadratic trinomial $f\left(x\right)=x^2+ax+b$. Assume that the equation $f\left(f\left(x\right)\right)=0$ has four different real solutions, and that the sum of two of these solutions is $-1$. Prove that $b\leq -\frac14$.

[u]Round 1[/u] [b]p1.[/b] Solve the maze [img]https://cdn.artofproblemsolving.com/attachments/8/c/6439816a52b5f32c3cb415e2058556edb77c80.png[/img] [b]p2.[/b] Billiam can write a problem in $30$ minutes, Jerry can write a problem in $10$ minutes, and Evin can write a problem in $20$ minutes. Billiam begins writing problems alone at $3:00$ PM until Jerry joins himat $4:00$ PM, and Evin joins both of them at $4:30$ PM. Given that they write problems until the end of math team at $5:00$ PM, how many full problems have they written in total? [b]p3.[/b] How many pairs of positive integers $(n,k)$ are there such that ${n \choose k}= 6$? [u]Round 2 [/u] [b]p4.[/b] Find the sumof all integers $b > 1$ such that the expression of $143$ in base $b$ has an even number of digits and all digits are the same. [b]p5.[/b] Ιni thinks that $a \# b = a^2 - b$ and $a \& b = b^2 - a$, while Mimi thinks that $a \# b = b^2 - a$ and $a \& b = a^2 - b$. Both Ini and Mimi try to evaluate $6 \& (3 \# 4)$, each using what they think the operations $\&$ and $\#$ mean. What is the positive difference between their answers? [b]p6.[/b] A unit square sheet of paper lies on an infinite grid of unit squares. What is the maximum number of grid squares that the sheet of paper can partially cover at once? A grid square is partially covered if the area of the grid square under the sheet of paper is nonzero - i.e., lying on the edge only does not count. [u]Round 3[/u] [b]p7.[/b] Maya wants to buy lots of burgers. A burger without toppings costs $\$4$, and every added topping increases the price by 50 cents. There are 5 different toppings for Maya to choose from, and she can put any combination of toppings on each burger. How much would it cost forMaya to buy $1$ burger for each distinct set of toppings? Assume that the order in which the toppings are stacked onto the burger does not matter. [b]p8.[/b] Consider square $ABCD$ and right triangle $PQR$ in the plane. Given that both shapes have area $1$, $PQ =QR$, $PA = RB$, and $P$, $A$, $B$ and $R$ are collinear, find the area of the region inside both square $ABCD$ and $\vartriangle PQR$, given that it is not $0$. [b]p9.[/b] Find the sum of all $n$ such that $n$ is a $3$-digit perfect square that has the same tens digit as $\sqrt{n}$, but that has a different ones digit than $\sqrt{n}$. [u]Round 4[/u] [b]p10.[/b] Jeremy writes the string: $$LMTLMTLMTLMTLMTLMT$$ on a whiteboard (“$LMT$” written $6$ times). Find the number of ways to underline $3$ letters such that from left to right the underlined letters spell LMT. [b]p11.[/b] Compute the remainder when $12^{2022}$ is divided by $1331$. [b]p12.[/b] What is the greatest integer that cannot be expressed as the sum of $5$s, $23$s, and $29$s? [u]Round 5 [/u] [b]p13.[/b] Square $ABCD$ has point $E$ on side $BC$, and point $F$ on side $CD$, such that $\angle EAF = 45^o$. Let $BE = 3$, and $DF = 4$. Find the length of $FE$. [b]p14.[/b] Find the sum of all positive integers $k$ such that $\bullet$ $k$ is the power of some prime. $\bullet$ $k$ can be written as $5654_b$ for some $b > 6$. [b]p15.[/b] If $\sqrt[3]{x} + \sqrt[3]{y} = 2$ and $x + y = 20$, compute $\max \,(x, y)$. PS. You should use hide for answers. Rounds 6-9 have been posted [url=https://artofproblemsolving.com/community/c3h3167372p28825861]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find the largest real number $\lambda$ with the following property: for any positive real numbers $p,q,r,s$ there exists a complex number $z=a+bi$($a,b\in \mathbb{R})$ such that $$ |b|\ge \lambda |a| \quad \text{and} \quad (pz^3+2qz^2+2rz+s) \cdot (qz^3+2pz^2+2sz+r) =0.$$

Let $f(x), g(x)$ be real polynomials of degrees $2$ and $3$, respectively. Could it happen that $f(g(x))$ has $6$ distinct roots, which are powers of $2$?

Find all polynomials $f$ with real coefficients such that for all reals $a,b,c$ such that $ab+bc+ca = 0$ we have the following relations \[ f(a-b) + f(b-c) + f(c-a) = 2f(a+b+c). \]

Find all functions $f\colon\mathbb R\to\mathbb R$ such that for all real numbers $x$ and $y$, \[f(xf(y))+f(y)=f(x+y)+f(xy).\] [i]Milan Haiman[/i]

Prove that $\frac{5^{125}-1}{5^{25}-1}$ is a composite number.

Find all real $(x,y)$ such that $x + {y^2} = {y^3}$ $y + {x^2} = {x^3}$

The first term of a sequence is 2005. Each succeeding term is the sum of the cubes of the digits of the previous terms. What is the 2005th term of the sequence? $ \textbf{(A)}\ 29\qquad \textbf{(B)}\ 55\qquad \textbf{(C)}\ 85\qquad \textbf{(D)}\ 133\qquad \textbf{(E)}\ 250$

Given infinite sequence $a_n$. It is known that the limit of $$b_n=a_{n+1}-a_n/2$$ equals zero. Prove that the limit of $a_n$ equals zero.

Find all real numbers $x$ such that $x\lfloor x\lfloor x\lfloor x\rfloor\rfloor\rfloor=88$. The notation $\lfloor x\rfloor$ means the greatest integer less than or equal to $x$.