Dirock has a very neat rectangular backyard that can be represented as a $32\times 32$ grid of unit squares. The rows and columns are each numbered $1,2,\ldots, 32$. Dirock is very fond of rocks, and places a rock in every grid square whose row and column number are both divisible by $3$. Dirock would like to build a rectangular fence with vertices at the centers of grid squares and sides parallel to the sides of the yard such that [list] [*] The fence does not pass through any grid squares containing rocks; [*] The interior of the fence contains exactly 5 rocks. [/list] In how many ways can this be done? [i]Ray Li[/i]

Find the second smallest positive integer $n$ such that when $n$ is divided by $5,$ the remainder is $3,$ and when $n$ is divided by $7,$ the remainder is $4.$

Within an equilateral triangle $ABC$, take any point $P$. Let $L, M, N$ be the projections of $P$ on sides $AB, BC, CA$ respectively. Prove that $\frac{AP}{NL}=\frac{BP}{LM}=\frac{CP}{MN}$.

A cat is tied to one corner of the base of a tower. The base forms an equilateral triangle of side length $4$ m, and the cat is tied with a leash of length $8$ m. Let $A$ be the area of the region accessible to the cat. If we write $A = \frac{m}{n} k - \sqrt{\ell}$, where $m,n, k, \ell$ are positive integers such that $m$ and $n$ are relatively prime, and $\ell$ is squarefree, what is the value of $m + n + k + \ell$ ?

Fold the next seven corners into a rectangle. [img]https://cdn.artofproblemsolving.com/attachments/b/b/2b8b9d6d4b72024996a66d41f865afb91bb9b7.png[/img]

[b]p1. [/b](a) Show that for every positive integer $m > 1$, there are positive integers $x$ and $y$ such that $x^2 - y^2 = m^3$. (b) Find all pairs of positive integers $(x, y)$ such that $x^6 = y^2 + 127$. [b]p2.[/b] (a) Let $P(x)$ be a polynomial with integer coefficients. Suppose that $P(0)$ is an odd integer and that $P(1)$ is also an odd integer. Show that if $c$ is an integer then $P(c)$ is not equal to $0$. (b) Let P(x) be a polynomial with integer coefficients. Suppose that $P(1,000) = 1,000$ and $P(2,000) = 2,000.$ Explain why $P(3,000)$ cannot be equal to $1,000$. [b]p3.[/b] Triangle $\vartriangle ABC$ is created from points $A(0, 0)$, $B(1, 0)$ and $C(1/2, 2)$. Let $q, r$, and $s$ be numbers such that $0 < q < 1/2 < s < 1$, and $q < r < s$. Let D be the point on $AC$ which has $x$-coordinate $q$, $E$ be the point on AB which has $x$-coordinate $r$, and $F$ be the point on $BC$ that has $x$-coordinate $s$. (a) Find the area of triangle $\vartriangle DEF$ in terms of $q, r$, and $s$. (b) If $r = 1/2$, prove that at least one of the triangles $\vartriangle ADE$, $\vartriangle CDF$, or $\vartriangle BEF$ has an area of at least $1/4$. [b]p4.[/b] In the Gregorian calendar: (i) years not divisible by $4$ are common years, (ii) years divisible by $4$ but not by $100$ are leap years, (iii) years divisible by $100$ but not by $400$ are common years, (iv) years divisible by $400$ are leap years, (v) a leap year contains $366$ days, a common year $365$ days. From the information above: (a) Find the number of common years and leap years in $400$ consecutive Gregorian years. Show that $400$ consecutive Gregorian years consists of an integral number of weeks. (b) Prove that the probability that Christmas falls on a Wednesday is not equal to $1/7$. [b]p5.[/b] Each of the first $13$ letters of the alphabet is written on the back of a card and the $13$ cards are placed in a row in the order $$A,B,C,D,E, F, G,H, I, J,K, L,M$$ The cards are then turned over so that the letters are face down. The cards are rearranged and again placed in a row, but of course they may be in a different order. They are rearranged and placed in a row a second time and both rearrangements were performed exactly the same way. When the cards are turned over the letters are in the order $$B,M, A,H, G,C, F,E,D, L, I,K, J$$ What was the order of the letters after the cards were rearranged the first time? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $t$ and $n$ be fixed integers each at least $2$. Find the largest positive integer $m$ for which there exists a polynomial $P$, of degree $n$ and with rational coefficients, such that the following property holds: exactly one of \[ \frac{P(k)}{t^k} \text{ and } \frac{P(k)}{t^{k+1}} \] is an integer for each $k = 0,1, ..., m$. [i]Proposed by Michael Kural[/i]

Let $S = \{1, 2, \ldots, n\}$ for some positive integer $n$, and let $A$ be an $n$-by-$n$ matrix having as entries only ones and zeroes. Define an infinite sequence $\{x_i\}_{i \ge 0}$ to be [i]strange[/i] if: [list] [*] $x_i \in S$ for all $i$, [*] $a_{x_kx_{k+1}} = 1$ for all $k$, where $a_{ij}$ denotes the element in the $i^{\text{th}}$ row and $j^{\text{th}}$ column of $A$. [/list] Prove that the set of strange sequences is empty if and only if $A$ is nilpotent, i.e. $A^m = 0$ for some integer $m$.

For a given real number $a$ and a positive integer $n$, prove that: i) there exists exactly one sequence of real numbers $x_0,x_1,\ldots,x_n,x_{n+1}$ such that \[\begin{cases} x_0=x_{n+1}=0,\\ \frac{1}{2}(x_i+x_{i+1})=x_i+x_i^3-a^3,\ i=1,2,\ldots,n.\end{cases}\] ii) the sequence $x_0,x_1,\ldots,x_n,x_{n+1}$ in i) satisfies $|x_i|\le |a|$ where $i=0,1,\ldots,n+1$. [i]Liang Yengde[/i]

Suppose $n$ is an integer $\geq 2$. Determine the first digit after the decimal point in the decimal expansion of the number \[\sqrt[3]{n^{3}+2n^{2}+n}\]

There are $100$ students who praticipate at exam.Also there are $25$ members of jury.Each student is checked by one jury.Known that every student likes $10$ jury $a)$ Prove that we can select $7$ jury such that any student likes at least one jury. $b)$ Prove that we can make this every student will be checked by the jury that he likes and every jury will check at most $10$ students.

Let $X{}$ be a set of integers which can be partitioned into $N{}$ disjoint increasing arithmetic progressions (infinite in both directions), and cannot be partitioned into a smaller number of such progressions. Is such partition into $N{}$ progressions unique for every such $X{}$ if a) $N = 2{}$ and b) $N = 3$? [i]Viktor Kleptsyn[/i]

[b]p1.[/b] Determine the number of ways to place $4$ rooks on a $4 \times 4$ chessboard such that: (a) no two rooks attack one another, and (b) the main diagonal (the set of squares marked $X$ below) does not contain any rooks. [img]https://cdn.artofproblemsolving.com/attachments/e/e/e3aa96de6c8ed468c6ef3837e66a0bce360d36.png[/img] The rooks are indistinguishable and the board cannot be rotated. (Two rooks attack each other if they are in the same row or column.) [b]p2.[/b] Seven students, numbered $1$ to $7$ in counter-clockwise order, are seated in a circle. Fresh Mann has 100 erasers, and he wants to distribute them to the students, albeit unfairly. Starting with person $ 1$ and proceeding counter-clockwise, Fresh Mann gives $i$ erasers to student $i$; for example, he gives $ 1$ eraser to student $ 1$, then $2$ erasers to student $2$, et cetera. He continues around the circle until he does not have enough erasers to give to the next person. At this point, determine the number of erasers that Fresh Mann has. [b]p3.[/b] Let $ABC$ be a triangle with $AB = AC = 17$ and $BC = 24$. Approximate $\angle ABC$ to the nearest multiple of $10$ degrees. [b]p4.[/b] Define a sequence of rational numbers $\{x_n\}$ by $x_1 =\frac35$ and for $n \ge 1$, $x_{n+1} = 2 - \frac{1}{x_n}$ . Compute the product $x_1x_2x_3... x_{2013}$. [b]p5.[/b] In equilateral triangle $ABC$, points $P$ and $R$ lie on segment $AB$, points $I$ and $M$ lie on segment $BC$, and points $E$ and $S$ lie on segment $CA$ such that $PRIMES$ is a equiangular hexagon. Given that $AB = 11$, $PR = 2$, $IM = 3$, and $ES = 5$, compute the area of hexagon $PRIMES$. [b]p6.[/b] Let $f(a, b) = \frac{a^2}{a+b}$ . Let $A$ denote the sum of $f(i, j)$ over all pairs of integers $(i, j)$ with $1 \le i < j \le 10$; that is, $$A = (f(1, 2) + f(1, 3) + ...+ f(1, 10)) + (f(2, 3) + f(2, 4) +... + f(2, 10)) +... + f(9, 10).$$ Similarly, let $B$ denote the sum of $f(i, j)$ over all pairs of integers $(i, j)$ with $1 \le j < i \le 10$, that is, $$B = (f(2, 1) + f(3, 1) + ... + f(10, 1)) + (f(3, 2) + f(4, 2) +... + f(10, 2)) +... + f(10, 9).$$ Compute $B - A$. [b]p7.[/b] Fresh Mann has a pile of seven rocks with weights $1, 1, 2, 4, 8, 16$, and $32$ pounds and some integer X between $1$ and $64$, inclusive. He would like to choose a set of the rocks whose total weight is exactly $X$ pounds. Given that he can do so in more than one way, determine the sum of all possible values of $X$. (The two $1$-pound rocks are indistinguishable.) [b]p8.[/b] Let $ABCD$ be a convex quadrilateral with $AB = BC = CA$. Suppose that point $P$ lies inside the quadrilateral with $AP = PD = DA$ and $\angle PCD = 30^o$. Given that $CP = 2$ and $CD = 3$, compute $CA$. [b]p9.[/b] Define a sequence of rational numbers $\{x_n\}$ by $x_1 = 2$, $x_2 = \frac{13}{2}$ , and for $n \ge 1$, $x_{n+2} = 3 -\frac{3}{x_{n+1}}+\frac{1}{x_nx_{n+1}}$. Compute $x_{100}$. [b]p10.[/b] Ten prisoners are standing in a line. A prison guard wants to place a hat on each prisoner. He has two colors of hats, red and blue, and he has $10$ hats of each color. Determine the number of ways in which the prison guard can place hats such that among any set of consecutive prisoners, the number of prisoners with red hats and the number of prisoners with blue hats differ by at most $2$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find the number of positive integers $n$ less than $2017$ such that \[ 1+n+\frac{n^2}{2!}+\frac{n^3}{3!}+\frac{n^4}{4!}+\frac{n^5}{5!}+\frac{n^6}{6!} \] is an integer.

There are $2014$ students from high schools nationwide communications sit around a round table in arbitrary manner. Then the organizers want to rearrange students from the same school sit next to each other by performing the following swapping: permutation view of two adjacent groups of students (see illustration). Find the smallest $k$ number so that a result can be obtained results as desired by the organizers with no more than $k$ swapping permits. Permission to change places like after $...\underbrace{ABCD}_\text{1}\underbrace{EFG}_\text{2}... \to ...\underbrace{EFG}_\text{2}\underbrace{ABCD}_\text{1}...$ Vu The Khoi, Institute of Mathematics, Vietnam Academy of Science and Technology, Cau Giay, Hanoi.

Given is a prime number $p$. Find the number of positive integer solutions $(a, b, c, d)$ of the system of equations $ac+bd = p(a+c)$ and $bc-ad = p(b-d)$.

If $0 < a \leq b$ then$$\frac{2}{\sqrt{3}}\tan^{-1}\left(\frac{2(b^2 - a^2)}{(a^2+2)(b^2+2)}\right)\leq \int \limits_a^b \frac{(x^2+1)(x^2+x+1)}{(x^3 + x^2 + 1) (x^3 + x + 1)}dx\leq \frac{4}{\sqrt{3}}\tan^{-1}\left(\frac{(b-a)\sqrt{3}}{a+b+2(1+ab)}\right)$$

Determine the smallest rational number $\frac{r}{s}$ such that $\frac{1}{k}+\frac{1}{m}+\frac{1}{n}\leq \frac{r}{s}$ whenever $k, m,$ and $n$ are positive integers that satisfy the inequality $\frac{1}{k}+\frac{1}{m}+\frac{1}{n} < 1$.

Find the maximum and the minimum values of $S = (1 - x_1)(1 -y_1) + (1 - x_2)(1 - y_2)$ for real numbers $x_1, x_2, y_1,y_2$ with $x_1^2 + x_2^2 = y_1^2 + y_2^2 = 2013$.

If five boys and three girls are randomly divided into two four-person teams, what is the probability that all three girls will end up on the same team? $\text{(A) }\frac{1}{7}\qquad\text{(B) }\frac{2}{7}\qquad\text{(C) }\frac{1}{10}\qquad\text{(D) }\frac{1}{14}\qquad\text{(E) }\frac{1}{28}$

Le $S$ be the set of positive integers greater than or equal to $2$. A function $f: S\rightarrow S$ is italian if $f$ satifies all the following three conditions: 1) $f$ is surjective 2) $f$ is increasing in the prime numbers(that is, if $p_1<p_2$ are prime numbers, then $f(p_1)<f(p_2)$) 3) For every $n\in S$ the number $f(n)$ is the product of $f(p)$, where $p$ varies among all the primes which divide $n$ (For instance, $f(360)=f(2^3\cdot 3^2\cdot 5)=f(2)\cdot f(3)\cdot f(5)$). Determine the maximum and the minimum possible value of $f(2020)$, when $f$ varies among all italian functions.

Let $ABC$ be an acute-angled triangle with $|AB| \neq |AC|$. Let $D$ be the foot of the altitude from $A$ to $BC$, and let $E$ be the intersection of the bisector of angle $\angle BAC$ with side $BC$. Let $P$ and $Q$ be the intersection points of the circumcircle of triangle $ADE$ with $AC$ and $AB$, respectively. Prove that the lines $AD$, $BP$, and $CQ$ pass through a common point.

[u]Part 1[/u] You might think this round is broken after solving some of these problems, but everything is intentional. [b]1.1.[/b] The number $n$ can be represented uniquely as the sum of $6$ distinct positive integers. Find $n$. [b]1.2.[/b] Let $ABC$ be a triangle with $AB = BC$. The altitude from $A$ intersects line $BC$ at $D$. Suppose $BD = 5$ and $AC^2 = 1188$. Find $AB$. [b]1.3.[/b] A lemonade stand analyzes its earning and operations. For the previous month it had a \$45 dollar budget to divide between production and advertising. If it spent $k$ dollars on production, it could make $2k - 12$ glasses of lemonade. If it spent $k$ dollars on advertising, it could sell each glass at an average price of $15 + 5k$ cents. The amount it made in sales for the previous month was $\$40.50$. Assuming the stand spent its entire budget on production and advertising, what was the absolute dierence between the amount spent on production and the amount spent on advertising? [b]1.4.[/b] Let $A$ be the number of dierent ways to tile a $1 \times n$ rectangle with tiles of size $1 \times 1$, $1 \times 3$, and $1 \times 6$. Let B be the number of different ways to tile a $1 \times n$ rectangle with tiles of size $1 \times 2$ and $1 \times 5$, where there are 2 different colors available for the $1 \times 2$ tiles. Given that $A = B$, find $n$. (Two tilings that are rotations or reflections of each other are considered distinct.) [b]1.5.[/b] An integer $n \ge 0$ is such that $n$ when represented in base $2$ is written the same way as $2n$ is in base $5$. Find $n$. [b]1.6.[/b] Let $x$ be a positive integer such that $3$, $ \log_6(12x)$, $\log_6(18x)$ form an arithmetic progression in some order. Find $x$. [u]Part 2[/u] Oops, it looks like there were some [i]intentional [/i] printing errors and some of the numbers from these problems got removed. Any $\blacksquare$ that you see was originally some positive integer, but now its value is no longer readable. Still, if things behave like they did for Part 1, maybe you can piece the answers together. [b]2.1.[/b] The number $n$ can be represented uniquely as the sum of $\blacksquare$ distinct positive integers. Find $n$. [b]2.2.[/b] Let $ABC$ be a triangle with $AB = BC$. The altitude from $A$ intersects line $BC$ at $D$. Suppose $BD = \blacksquare$ and $AC^2 = 1536$. Find $AB$. [b]2.3.[/b] A lemonade stand analyzes its earning and operations. For the previous month it had a $\$50$ dollar budget to divide between production and advertising. If it spent k dollars on production, it could make $2k - 2$ glasses of lemonade. If it spent $k$ dollars on advertising, it could sell each glass at an average price of $25 + 5k$ cents. The amount it made in sales for the previous month was $\$\blacksquare$. Assuming the stand spent its entire budget on production and advertising, what was the absolute dierence between the amount spent on production and the amount spent on advertising? [b]2.4.[/b] Let $A$ be the number of different ways to tile a $1 \times n$ rectangle with tiles of size $1 \times \blacksquare$, $1 \times \blacksquare$, and $1 \times \blacksquare$. Let $B$ be the number of different ways to tile a $1\times n$ rectangle with tiles of size $1 \times \blacksquare$ and $1 \times \blacksquare$, where there are $\blacksquare$ different colors available for the $1 \times \blacksquare$ tiles. Given that $A = B$, find $n$. (Two tilings that are rotations or reflections of each other are considered distinct.) [b]2.5.[/b] An integer $n \ge \blacksquare$ is such that $n$ when represented in base $9$ is written the same way as $2n$ is in base $\blacksquare$. Find $n$. [b]2.6.[/b] Let $x$ be a positive integer such that $1$, $\log_{96}(6x)$, $\log_{96}(\blacksquare x)$ form an arithmetic progression in some order. Find $x$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Centers of adjacent faces of a unit cube are joined to form a regular octahedron. What is the volume of this octahedron? $ \textbf{(A) } \frac 18 \qquad \textbf{(B) } \frac 16 \qquad \textbf{(C) } \frac 14 \qquad \textbf{(D) } \frac 13 \qquad \textbf{(E) } \frac 12$

Let $N$ be the product of all odd primes less than $2^4$. What remainder does $N$ leave when divided by $2^4$? $\text{(A) }5\qquad\text{(B) }7\qquad\text{(C) }9\qquad\text{(D) }11\qquad\text{(E) }13$