[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].

If $ a$ and $ b$ are natural numbers such that $ a\plus{}13b$ is divisible by $ 11$ and $ a\plus{}11b$ is divisible by $ 13$, then find the least possible value of $ a\plus{}b$.

Let the numbers $x_i \in \{-1, 1\}$ be given for $i = 1, 2,..., n$, satisfying $$x_1x_2 + x_2x_3 +... + x_{n-1}x_n + x_nx_1 = 0.$$ Prove that $n$ is divisible by $4$.

Let $r$ and $b$ be positive integers. The game of [i]Monis[/i], a variant of Tetris, consists of a single column of red and blue blocks. If two blocks of the same color ever touch each other, they both vanish immediately. A red block falls onto the top of the column exactly once every $r$ years, while a blue block falls exactly once every $b$ years. (a) Suppose that $r$ and $b$ are odd, and moreover the cycles are offset in such a way that no two blocks ever fall at exactly the same time. Consider a period of $rb$ years in which the column is initially empty. Determine, in terms of $r$ and $b$, the number of blocks in the column at the end. (b) Now suppose $r$ and $b$ are relatively prime and $r+b$ is odd. At time $t=0$, the column is initially empty. Suppose a red block falls at times $t = r, 2r, \dots, (b-1)r$ years, while a blue block falls at times $t = b, 2b, \dots, (r-1)b$ years. Prove that at time $t=rb$, the number of blocks in the column is $\left\lvert 1+2(r-1)(b+r)-8S \right\rvert$, where \[ S = \left\lfloor \frac{2r}{r+b} \right\rfloor + \left\lfloor \frac{4r}{r+b} \right\rfloor + ... + \left\lfloor \frac{(r+b-1)r}{r+b} \right\rfloor . \] [i]Proposed by Sammy Luo[/i]

Find all positive integers $n$ for which number $A = n^3-n^2+n-1$ is prime

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 $x$.

Determine the largest positive integer $n$ which cannot be written as the sum of three numbers bigger than $1$ which are pairwise coprime.

Prove that for every positive integer $m$ there exists positive integer $n$ such that $m+n+1$ is perfect square and $mn+1$ is perfect cube of some positive integers

Determine all pairs $(p,q)$ of prime numbers with the following property: There are positive integers $a,b,c$ satisfying \[\frac{p}{a}+\frac{p}{b}+\frac{p}{c}=1 \quad \text{and} \quad \frac{a}{p}+\frac{b}{p}+\frac{c}{p}=q+1.\]

The sequence $a_1, a_2, a_3, ...$ is defined by $a_1 = a_2 = a_3 = 1$, $a_{n+3} = a_{n+2}a_{n+1} + a_n$. Show that for any positive integer $r$ we can find $s$ such that $a_s$ is a multiple of $r$.

Determine the positive integers $n$ such that the set of all positive divisors of $30^n$ can be divided into groups of three so that the product of the three numbers in each group is the same.

$k>2$ is an integer. $a_0,a_1,...$ is an integer sequence such that $a_0=0$, $a_{n+1}=ka_n-a_{n-1}$. Prove that for any positive integer $m$, $(2m)!|a_1a_2...a_{3m}$.

Let $a, b$, and $n$ be integers such that $a + b$ is divisible by $n$ and $a^2 + b^2$ is divisible by $n^2$. Prove that $a^m + b^m$ is divisible by $n^m$ for all positive integers $m$.

Determine if there are positive integers $a, b$ such that all terms of the sequence defined by \[ x_{1}= 2010,x_{2}= 2011\\ x_{n+2}= x_{n}+ x_{n+1}+a\sqrt{x_{n}x_{n+1}+b}\quad (n\ge 1) \] are integers.

Given five $n$-digit binary numbers. For each two numbers their digits coincide exactly on $m$ places. There is no place with the common digit for all the five numbers. Prove that $$2/5 \le m/n \le 3/5$$

Let $ x$ and $ y$ be positive integers such that $ 7x^5 \equal{} 11y^{13}$. The minimum possible value of $ x$ has a prime factorization $ a^cb^d$. What is $ a \plus{} b \plus{} c \plus{} d$? $ \textbf{(A)}\ 30 \qquad \textbf{(B)}\ 31 \qquad \textbf{(C)}\ 32 \qquad \textbf{(D)}\ 33 \qquad \textbf{(E)}\ 34$

Given an odd $n$, prove that there exist $2n$ integers $a_1,a_2,\cdots ,a_n$; $b_1,b_2,\cdots ,b_n$, such that for any integer $k$ ($0<k<n$), the following $3n$ integers: $a_i+a_{i+1}, a_i+b_i, b_i+b_{i+k}$ ($i=1,2,\cdots ,n; a_{n+1}=a_1, b_{n+j}=b_j, 0<j<n$) are of different remainders on division by $3n$.

Let $a_1, a_2, \ldots, a_n$ be $n$ positive integers, and let $b_1, b_2, \ldots, b_m$ be $m$ positive integers such that $a_1 a_2 \cdots a_n = b_1 b_2 \cdots b_m$. Prove that a rectangular table with $n$ rows and $m$ columns can be filled with positive integer entries in such a way that * the product of the entries in the $i$-th row is $a_i$ (for each $i \in \left\{1,2,\ldots,n\right\}$); * the product of the entries in the $j$-th row is $b_j$ (for each $i \in \left\{1,2,\ldots,m\right\}$).

Let \( x \), \( y \), \( p \) be positive integers that satisfy the equation \( x^4 = p + 9y^4 \), where \( p \) is a prime number. Show that \( \frac{p^2 - 1}{3} \) is a perfect square and a multiple of 16.

Consider an odd prime $p$ and a positive integer $N < 50p$. Let $a_1, a_2, \ldots , a_N$ be a list of positive integers less than $p$ such that any specific value occurs at most $\frac{51}{100}N$ times and $a_1 + a_2 + \cdots· + a_N$ is not divisible by $p$. Prove that there exists a permutation $b_1, b_2, \ldots , b_N$ of the $a_i$ such that, for all $k = 1, 2, \ldots , N$, the sum $b_1 + b_2 + \cdots + b_k$ is not divisible by $p$. [i]Will Steinberg, United Kingdom[/i]

Positive integers $a,b,z$ satisfy the equation $ab=z^2+1$. Prove that there exist positive integers $x,y$ such that $$\frac{a}{b}=\frac{x^2+1}{y^2+1}$$

Let $p$ and $q$ be different prime numbers. Solve the following system in integers: \[\frac{z+ p}x+\frac{z-p}y= q,\\ \frac{z+ p}y -\frac{z-p}x= q.\]

Let us call the [i]distance [/i] between the numbers $\overline{a_1a_2a_3a_4a_5}$ and $\overline{b_1b_2b_3b_4b_5}$ the maximum $i$ for which $a_i \ne b_i$. All five-digit numbers are written out one after another in some order. What is the minimum possible sum of distances between adjacent numbers?

A representation of $\frac{17}{20}$ as a sum of reciprocals $$ \frac{17}{20} = \frac{1}{a_1} + \frac{1}{a_2} + \cdots + \frac{1}{a_k} $$ is called a [i]calm representation[/i] with $k$ terms if the $a_i$ are distinct positive integers and at most one of them is not a power of two. (a) Find the smallest value of $k$ for which $\frac{17}{20}$ has a calm representation with $k$ terms. (b) Prove that there are infinitely many calm representations of $\frac{17}{20}$.

The sequence $a_n$ is given as $$a_1=1, a_2=2 \;\;\; \text{and} \;\;\;\; a_{n+2}=a_n(a_{n+1}+1) \quad \forall n\geq 1$$ Prove that $a_{a_n}$ is divisible by $(a_n)^n$ for $n\geq 100$.