Find all real values of the parameter $a$ for which the system \begin{align*} &1+\left(4x^2-12x+9\right)^2+2^{y+2}=a\\ &\log_3\left(x^2-3x+\frac{117}4\right)+32=a+\log_3(2y+3) \end{align*}has a unique real solution. Solve the system for these values of $a$.

Define $T_n =\sum^{n}){i=1} i(n + 1 - i)$. Find $\lim_{n\to \infty} \frac{T_n}{n^3}$.

There're two positive inegers $a_1<a_2$. For every positive integer $n \geq 3$ let $a_n$ be the smallest integer that bigger than $a_{n-1}$ and such that there's unique pair $1\leq i< j\leq n-1$ such that this number equals to $a_i+a_j$. Given that there're finitely many even numbers in this sequence. Prove that sequence $\{a_{n+1}-a_n \}$ is periodic starting from some element.

An urn contains $4$ green balls and $6$ blue balls. A second urn contains $16$ green balls and $N$ blue balls. A single ball is drawn at random from each urn. The probability that both balls are of the same color is $0.58$. Find $N$.

The real numbers $a, b$ satisfy $| a | \ne | b |$ and $$ \frac{a + b}{a - b}+\frac{a - b}{a + b}= -\frac52.$$ Determine the value of the expression $$E= \frac{a^4 - b^4}{a^4 + b^4} - \frac{a^4 + b^4}{a^4- b^4}.$$

Suppose $a_1, a_2, \ldots, a_n$ and $b_1, b_2, \ldots, b_n$ are real numbers such that \[ (a_1 ^ 2 + a_2 ^ 2 + \cdots + a_n ^ 2 -1)(b_1 ^ 2 + b_2 ^ 2 + \cdots + b_n ^ 2 - 1) > (a_1 b_1 + a_2 b_2 + \cdots + a_n b_n - 1)^2. \] Prove that $a_1 ^ 2 + a_2 ^ 2 + \cdots + a_n ^ 2 > 1$ and $b_1 ^ 2 + b_2 ^ 2 + \cdots + b_n ^ 2 > 1$.

Define a sequence $\{a_n\}$ as: $\left\{\begin{aligned}& a_1=1 \\ & a_{n+1}=3-\frac{a_{n}+2}{2^{a_{n}}}\ \ \text{for} \ n\geq 1.\end{aligned}\right.$ Prove that this sequence has a finite limit as $n\to+\infty$ . Also determine the limit.

For all positive integer $n$, define $f_n(x)$ such that $f_n(x) = \sum_{k=1}^n{|x - k|}$. Determine all solution from the inequality $f_n(x) < 41$ for all positive $2$-digit integers $n$ (in decimal notation).

A sequence of real numbers $ a_{0},\ a_{1},\ a_{2},\dots$ is defined by the formula \[ a_{i \plus{} 1} \equal{} \left\lfloor a_{i}\right\rfloor\cdot \left\langle a_{i}\right\rangle\qquad\text{for}\quad i\geq 0; \]here $a_0$ is an arbitrary real number, $\lfloor a_i\rfloor$ denotes the greatest integer not exceeding $a_i$, and $\left\langle a_i\right\rangle=a_i-\lfloor a_i\rfloor$. Prove that $a_i=a_{i+2}$ for $i$ sufficiently large. [i]Proposed by Harmel Nestra, Estionia[/i]

For certain real numbers $a$, $b$, and $c$, the polynomial \[g(x) = x^3 + ax^2 + x + 10\] has three distinct roots, and each root of $g(x)$ is also a root of the polynomial \[f(x) = x^4 + x^3 + bx^2 + 100x + c.\] What is $f(1)$? $\textbf{(A)}\ -9009 \qquad\textbf{(B)}\ -8008 \qquad\textbf{(C)}\ -7007 \qquad\textbf{(D)}\ -6006 \qquad\textbf{(E)}\ -5005$

Consider the sequence $a_{1}, a_{2}, a_{3},\ldots$ defined by $a_{1}=2024^{2024}$ and for each positive integer $n$, $$a_{n+1}=\left|a_{n}-\sqrt{2}\right|.$$ Prove that there exists an integer $k$ such that $a_{k+2}=a_k$. [i]Here [/i]$\left|x\right|$[i] denotes the absolute value of [/i]$x$.

Let $A_1,A_2, \dots A_k$ be points on the unit circle.Prove that: $\sum\limits_{1\le i<j \le k} d(A_i,A_j)^2 \le k^2 $ Where $d(A_i,A_j)$ denotes the distance between $A_i,A_j$.

Prove that the polynomial $ x^3 + x + 1 $ is a factor of the polynomial $ P_n(x) = x^{n + 2} + (x+1)^{2n+1} $ for every integer $ n \geq 0 $.

For every natural number $n$ prove that $$\left( 1+ \frac12 + ...+ \frac1n \right)^2+ \left( \frac12 + ...+ \frac1n \right)^2+...+ \left( \frac{1}{n-1} + \frac12 \right)^2+ \left( \frac1n \right)^2=2n- \left( 1+ \frac12 + ...+ \frac1n \right)$$ (S. Manukian, Yerevan)

For any integer $n\ge 2$ denote by $A_n$ the set of solutions of the equation \[x=\left\lfloor\frac{x}{2}\right\rfloor+\left\lfloor\frac{x}{3}\right\rfloor+\cdots+\left\lfloor\frac{x}{n}\right\rfloor .\] a) Determine the set $A_2\cup A_3$. b) Prove that the set $A=\bigcup_{n\ge 2}A_n$ is finite and find $\max A$. [i]Dan Nedeianu & Mihai Baluna[/i]

Let $$P(x) = \sum^n_{k=1}(x^{3^k}+ x^{-3^k}- 1), Q(x) = \sum^n_{k=1}(x^{3^k}+ x^{-3^k}+ 1).$$ Given that $$P(x)Q(x) =\sum^{2\cdot 3^n}_{k=-2\cdot 3^n} a_kx^k,$$ Compute $\sum^{3^n}_{k=0}a_k$ in terms of $n$.

[u]Round 4[/u] [b]p13.[/b] What is the units digit of the number $(2^1 + 1)(2^2 - 1)(2^3 + 1)(2^4 - 1)...(2^{2010} - 1)$? [b]p14.[/b] Mr. Fat noted that on January $2$, $2010$, the display of the day is $01/02/2010$, and the sequence $01022010$ is a palindrome (a number that reads the same forwards and backwards). How many days does Mr. Fat need to wait between this palindrome day and the last palindrome day of this decade? [b]p15.[/b] Farmer Tim has a $30$-meter by $30$-meter by $30\sqrt2$-meter triangular barn. He ties his goat to the corner where the two shorter sides meet with a 60-meter rope. What is the area, in square meters, of the land where the goat can graze, given that it cannot get inside the barn? [b]p16.[/b] In triangle $ABC$, $AB = 3$, $BC = 4$, and $CA = 5$. Point $P$ lies inside the triangle and the distances from $P$ to two of the sides of the triangle are $ 1$ and $2$. What is the maximum distance from $P$ to the third side of the triangle? [u]Round 5[/u] [b]p17.[/b] Let $Z$ be the answer to the third question on this guts quadruplet. If $x^2 - 2x = Z - 1$, find the positive value of $x$. [b]p18.[/b] Let $X$ be the answer to the first question on this guts quadruplet. To make a FATRON2012, a cubical steel body as large as possible is cut out from a solid sphere of diameter $X$. A TAFTRON2013 is created by cutting a FATRON2012 into $27$ identical cubes, with no material wasted. What is the length of one edge of a TAFTRON2013? [b]p19.[/b] Let $Y$ be the smallest integer greater than the answer to the second question on this guts quadruplet. Fred posts two distinguishable sheets on the wall. Then, $Y$ people walk into the room. Each of the Y people signs up on $0, 1$, or $2$ of the sheets. Given that there are at least two people in the room other than Fred, how many possible pairs of lists can Fred have? [b]p20.[/b] Let $A, B, C$, be the respective answers to the first, second, and third questions on this guts quadruplet. At the Robot Design Convention and Showcase, a series of robots are programmed such that each robot shakes hands exactly once with every other robot of the same height. If the heights of the $16$ robots are $4$, $4$, $4$, $5$, $5$, $7$, $17$, $17$, $17$, $34$, $34$, $42$, $100$, $A$, $B$, and $C$ feet, how many handshakes will take place? [u]Round 6[/u] [b]p21.[/b] Determine the number of ordered triples $(p, q, r)$ of primes with $1 < p < q < r < 100$ such that $q - p = r - q$. [b]p22.[/b] For numbers $a, b, c, d$ such that $0 \le a, b, c, d \le 10$, find the minimum value of $ab + bc + cd + da - 5a - 5b - 5c - 5d$. [b]p23.[/b] Daniel has a task to measure $1$ gram, $2$ grams, $3$ grams, $4$ grams , ... , all the way up to $n$ grams. He goes into a store and buys a scale and six weights of his choosing (so that he knows the value for each weight that he buys). If he can place the weights on either side of the scale, what is the maximum value of $n$? [b]p24.[/b] Given a Rubik’s cube, what is the probability that at least one face will remain unchanged after a random sequence of three moves? (A Rubik’s cube is a $3$ by $3$ by $3$ cube with each face starting as a different color. The faces ($3$ by $3$) can be freely turned. A move is defined in this problem as a $90$ degree rotation of one face either clockwise or counter-clockwise. The center square on each face–six in total–is fixed.) PS. You should use hide for answers. First rounds have been posted [url=https://artofproblemsolving.com/community/c4h2766534p24230616]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $x$ be a solution to the equation $\lfloor x \lfloor x + 2\rfloor + 2\rfloor = 10$. Compute the smallest $C$ such that for any solution $x$, $x < C$. Here, $\lfloor m \rfloor$ is defined as the greatest integer less than or equal to $m$. For example, $\lfloor 3\rfloor = 3$ and $\lfloor -4.25\rfloor = -5$.

Let $\{x_i\}, 1\le i\le 6$ be a given set of six integers, none of which are divisible by $7$. $(a)$ Prove that at least one of the expressions of the form $x_1\pm x_2\pm x_3\pm x_4\pm x_5\pm x_6$ is divisible by $7$, where the $\pm$ signs are independent of each other. $(b)$ Generalize the result to every prime number.

Let $\{x_n\}$ be a sequence of natural numbers such that \[(a) 1 = x_1 < x_2 < x_3 < \ldots; \quad (b) x_{2n+1} \leq 2n \quad \forall n.\] Prove that, for every natural number $k$, there exist terms $x_r$ and $x_s$ such that $x_r - x_s = k.$

[b]p1.[/b] Elsie M. is fixing a watch with three gears. Gear $A$ makes a full rotation every $5$ minutes, gear $B$ makes a full rotation every $8$ minutes, and gear $C$ makes a full rotation every $12$ minutes. The gears continue spinning until all three gears are in their original positions at the same time. How many minutes will it take for the gears to stop spinning? [b]p2.[/b] Optimus has to pick $10$ distinct numbers from the set of positive integers $\{2, 3, 4,..., 29, 30\}$. Denote the numbers he picks by $\{a_1, a_2, ...,a_{10}\}$. What is the least possible value of $$d(a_1 ) + d(a_2) + ... + d(a_{10}),$$ where $d(n)$ denotes the number of positive integer divisors of $n$? For example, $d(33) = 4$ since $1$, $3$, $11$, and $33$ divide $33$. [b]p3.[/b] Michael is given a large supply of both $1\times 3$ and $1\times 5$ dominoes and is asked to arrange some of them to form a $6\times 13$ rectangle with one corner square removed. What is the minimum number of $1\times 3$ dominoes that Michael can use? [img]https://cdn.artofproblemsolving.com/attachments/6/6/c6a3ef7325ecee417e37ec9edb5374aceab9fd.png[/img] [b]p4.[/b] Andy, Ben, and Chime are playing a game. The probabilities that each player wins the game are, respectively, the roots $a$, $b$, and $c$ of the polynomial $x^3 - x^2 + \frac{111}{400}x - \frac{9}{400} = 0$ with $a \le b \le c$. If they play the game twice, what is the probability of the same player winning twice? [b]p5.[/b] TongTong is doodling in class and draws a $3 \times 3$ grid. She then decides to color some (that is, at least one) of the squares blue, such that no two $1 \times 1$ squares that share an edge or a corner are both colored blue. In how many ways may TongTong color some of the squares blue? TongTong cannot rotate or reflect the board. [img]https://cdn.artofproblemsolving.com/attachments/6/0/4b4b95a67d51fda0f155657d8295b0791b3034.png[/img] [b]p6.[/b] Given a positive integer $n$, we define $f(n)$ to be the smallest possible value of the expression $$| \square 1 \square 2 ... \square n|,$$ where we may place a $+$ or a $-$ sign in each box. So, for example, $f(3) = 0$, since $| + 1 + 2 - 3| = 0$. What is $f(1) + f(2) + ... + f(2011)$? [b]p7.[/b] The Duke Men's Basketball team plays $11$ home games this season. For each game, the team has a $\frac34$ probability of winning, except for the UNC game, which Duke has a $\frac{9}{10}$ probability of winning. What is the probability that Duke wins an odd number of home games this season? [b]p8.[/b] What is the sum of all integers $n$ such that $n^2 + 2n + 2$ divides $n^3 + 4n^2 + 4n - 14$? [b]p9.[/b] Let $\{a_n\}^N_{n=1}$ be a finite sequence of increasing positive real numbers with $a_1 < 1$ such that $$a_{n+1} = a_n \sqrt{1 - a^2_1}+ a_1\sqrt{1 - a^2_n}$$ and $a_{10} = 1/2$. What is $a_{20}$? [b]p10.[/b] Three congruent circles are placed inside a unit square such that they do not overlap. What is the largest possible radius of one of these circles? PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

[u]Set 1[/u] [b]p1.[/b] Assume the speed of sound is $343$ m/s. Anastasia and Bananastasia are standing in a field in front of you. When they both yell at the same time, you hear Anastasia’s yell $5$ seconds before Bananastasia’s yell. If Bananastasia yells first, and then Anastasia yells when she hears Bananastasia yell, you hear Anastasia’s yell $5$ seconds after Bananastasia’s yell. What is the distance between Anastasia and Bananastasia in meters? [b]p2.[/b] Michelle picks a five digit number with distinct digits. She then reverses the digits of her number and adds that to her original number. What is the largest possible sum she can get? [b]p3.[/b] Twain is trying to crack a $4$-digit number combination lock. They know that the second digit must be even, the third must be odd, and the fourth must be different from the previous three. If it takes Twain $10$ seconds to enter a combination, how many hours would it take them to try every possible combination that satisfies these rules? PS. You should use hide for answers.

Let $x,y,z$ be positive real numbers satisfying $4x^2 - 2xy + y^2 = 64, y^2 - 3yz +3z^2 = 36,$ and $4x^2 +3z^2 = 49.$ If the maximum possible value of $2xy +yz -4zx$ can be expressed as $\sqrt{n}$ for some positive integer $n,$ find $n.$

[b]p1.[/b] Sometimes one finds in an old park a tetrahedral pile of cannon balls, that is, a pile each layer of which is a tightly packed triangular layer of balls. A. How many cannon balls are in a tetrahedral pile of cannon balls of $N$ layers? B. How high is a tetrahedral pile of cannon balls of $N$ layers? (Assume each cannon ball is a sphere of radius $R$.) [b]p2.[/b] A prime is an integer greater than $1$ whose only positive integer divisors are itself and $1$. A. Find a triple of primes $(p, q, r)$ such that $p = q + 2$ and $q = r + 2$ . B. Prove that there is only one triple $(p, q, r)$ of primes such that $p = q + 2$ and $q = r + 2$ . [b]p3.[/b] The function $g$ is defined recursively on the positive integers by $g(1) =1$, and for $n>1$ , $g(n)= 1+g(n-g(n-1))$ . A. Find $g(1)$ , $g(2)$ , $g(3)$ and $g(4)$ . B. Describe the pattern formed by the entire sequence $g(1) , g(2 ), g(3), ...$ C. Prove your answer to Part B. [b]p4.[/b] Let $x$ , $y$ and $z$ be real numbers such that $x + y + z = 1$ and $xyz = 3$ . A. Prove that none of $x$ , $y$ , nor $z$ can equal $1$. B. Determine all values of $x$ that can occur in a simultaneous solution to these two equations (where $x , y , z$ are real numbers). [b]p5.[/b] A round robin tournament was played among thirteen teams. Each team played every other team exactly once. At the conclusion of the tournament, it happened that each team had won six games and lost six games. A. How many games were played in this tournament? B. Define a [i]circular triangle[/i] in a round robin tournament to be a set of three different teams in which none of the three teams beat both of the other two teams. How many circular triangles are there in this tournament? C. Prove your answer to Part B. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Suppose $r$, $s$, and $t$ are nonzero reals such that the polynomial $x^2 + rx + s$ has $s$ and $t$ as roots, and the polynomial $x^2 + tx + r$ has $5$ as a root. Compute $s$.