[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].

Factorise $ 5^{1985}\minus{}1$ as a product of three integers, each greater than $ 5^{100}$.

Call a positive integer [b]good[/b] if either $N=1$ or $N$ can be written as product of [i]even[/i] number of prime numbers, not necessarily distinct. Let $P(x)=(x-a)(x-b),$ where $a,b$ are positive integers. (a) Show that there exist distinct positive integers $a,b$ such that $P(1),P(2),\cdots ,P(2010)$ are all good numbers. (b) Suppose $a,b$ are such that $P(n)$ is a good number for all positive integers $n$. Prove that $a=b$.

Suppose that $a$ is an even positive integer and $A=a^{n}+a^{n-1}+\ldots +a+1,n\in \mathbb{N^{*}}$ is a perfect square.Prove that $8\mid a$.

Given $N = 2^ap_1p_2...p_m$, $m \ge 1$, $a \in N$ with $p_1, p_2,..., p_m$ are different primes. It is known that $\sigma (N) = 3N $ where $\sigma (N)$ is the sum of all positive integers which are factors of $N$. Show that there exists a prime number $p$ such that $2^p- 1$ is also a prime, and $2^p - 1|N$.

For lunch, Lamy, Botan, Nene, and Polka each choose one of three options: a hot dog, a slice of pizza, or a hamburger. Lamy and Botan choose different items, and Nene and Polka choose the same item. In how many ways could they choose their items?

Does there exist an integer such that its cube is equal to $3n^2 + 3n + 7,$ where $n$ is an integer.

Find all positive integers $n, k$ such that $(n-1)!=n^{k}-1$.

An odd positive integer $n$ can be expressed as the sum of two or more consecutive integers in exactly $2023$ ways. Find the greatest possible nonnegative integer $k$ such that $3^k$ is a factor of the least possible value of $n$.

Find all functions $f : \mathbb{N} \rightarrow \mathbb{N}$ such that for any two positive integers $m,n$ we have : $$f(n)+1400m^2|n^2+f(f(m))$$

Prove that for each positive integer $ n$ there exist $ n$ consecutive positive integers none of which is an integral power of a prime number.

Determine the biggest possible value of $ m$ for which the equation $ 2005x \plus{} 2007y \equal{} m$ has unique solution in natural numbers.

[b]p1.[/b] Calculate $\left( \frac12 + \frac13 + \frac14 \right)^2$. [b]p2.[/b] Find the $2010^{th}$ digit after the decimal point in the expansion of $\frac17$. [b]p3.[/b] If you add $1$ liter of water to a solution consisting of acid and water, the new solutions will contain of $30\%$ water. If you add another $5$ liters of water to the new solution, it will contain $36\frac{4}{11}\%$ water. Find the number of liters of acid in the original solution. [b]p4.[/b] John places $5$ indistinguishable blue marbles and $5$ indistinguishable red marbles into two distinguishable buckets such that each bucket has at least one blue marble and one red marble. How many distinguishable marble distributions are possible after the process is completed? [b]p5.[/b] In quadrilateral $PEAR$, $PE = 21$, $EA = 20$, $AR = 15$, $RE = 25$, and $AP = 29$. Find the area of the quadrilateral. [b]p6.[/b] Four congruent semicircles are drawn within the boundary of a square with side length $1$. The center of each semicircle is the midpoint of a side of the square. Each semicircle is tangent to two other semicircles. Region $R$ consists of points lying inside the square but outside of the semicircles. The area of $R$ can be written in the form $a - b\pi$, where $a$ and $b$ are positive rational numbers. Compute $a + b$. [b]p7.[/b] Let $x$ and $y$ be two numbers satisfying the relations $x\ge 0$, $y\ge 0$, and $3x + 5y = 7$. What is the maximum possible value of $9x^2 + 25y^2$? [b]p8.[/b] In the Senate office in Exie-land, there are $6$ distinguishable senators and $6$ distinguishable interns. Some senators and an equal number of interns will attend a convention. If at least one senator must attend, how many combinations of senators and interns can attend the convention? [b]p9.[/b] Evaluate $(1^2 - 3^2 + 5^2 - 7^2 + 9^2 - ... + 2009^2) -(2^2 - 4^2 + 6^2 - 8^2 + 10^2- ... + 2010^2)$. [b]p10.[/b] Segment $EA$ has length $1$. Region $R$ consists of points $P$ in the plane such that $\angle PEA \ge 120^o$ and $PE <\sqrt3$. If point $X$ is picked randomly from the region$ R$, the probability that $AX <\sqrt3$ can be written in the form $a - \frac{\sqrt{b}}{c\pi}$ , where $a$ is a rational number, $b$ and $c$ are positive integers, and $b$ is not divisible by the square of a prime. Find the ordered triple $(a, b, c)$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $a$ and $b$ be two positive integers. Prove that the integer \[a^2+\left\lceil\frac{4a^2}b\right\rceil\] is not a square. (Here $\lceil z\rceil$ denotes the least integer greater than or equal to $z$.) [i]Russia[/i]

Given $10$ positive integers with a sum equal to $1000$. The product of their factorials is a $10$-th power of an integer. Prove that all these numbers are equal.

Find the number of positive integers $n < 2018$ such that $25^n + 9^n$ is divisible by $13$.

Find all pairs $(x,y)$ of positive integers such that $y^{x^2}=x^{y+2}$.

Let $h_n$ and $k_n$ be the unique relatively prime positive integers such that \[\frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n} = \frac{h_n}{k_n}.\] Let $L_n$ denote the least common multiple of the numbers $1, 2, 3,\cdots, n$. For how many integers $n$ with $1 \le n \le 22$ is $k_n<L_n$? $\textbf{(A)} ~0 \qquad\textbf{(B)} ~3 \qquad\textbf{(C)} ~7 \qquad\textbf{(D)} ~8 \qquad\textbf{(E)} ~10 $

Let the function $f:N^*\to N^*$ such that [b](1)[/b] $(f(m),f(n))\le (m,n)^{2014} , \forall m,n\in N^*$; [b](2)[/b] $n\le f(n)\le n+2014 , \forall n\in N^*$ Show that: there exists the positive integers $N$ such that $ f(n)=n $, for each integer $n \ge N$. (High School Affiliated to Nanjing Normal University )

[b]p1.[/b] At a conference a mathematician and a chemist were talking. They were amazed to find that they graduated from the same high school. One of them, the chemist, mentioned that he had three sons and asked the other to calculate the ages of his sons given the following facts: (a) their ages are integers, (b) the product of their ages is $36$, (c) the sum of their ages is equal to the number of windows in the high school of the chemist and the mathematician. The mathematician considered this problem and noted that there was not enough information to obtain a unique solution. The chemist then noted that his oldest son had red hair. The mathematician then announced that he had determined the ages of the three sons. Please (aspiring mathematicians) determine the ages of the chemists three sons and explain your solution. [b]p2.[/b] A square is inscribed in a triangle. Two vertices of this square are on the base of the triangle and two others are on the lateral sides. Prove that the length of the side of the square is greater than and less than $2r$, where $r$ is a radius of the circle inscribed in the triangle. [b]p3.[/b] You are given any set of $100$ integers in which none of the integers is divisible by $100$. Prove that it is possible to select a subset of this set of $100$ integers such that their sum is a multiple of $100$. [b]p4.[/b] Find all prime numbers $a$ and $b$ such that $a^b + b^a$ is a prime number. [b]p5.[/b] $N$ airports are connected by airlines. Some airports are directly connected and some are not. It is always possible to travel from one airport to another by changing planes as needed. The board of directors decided to close one of the airports. Prove that it is possible to select an airport to close so that the remaining airports remain connected. [b]p6.[/b] (A simplified version of the Fermat’s Last Theorem). Prove that there are no positive integers $x, y, z$ and $z \le n$ satisfying the following equation: $x^n + y^n = z^n$. PS. You should use hide for answers.

In every vertex of a regular $n$ -gon exactly one chip is placed. At each $step$ one can exchange any two neighbouring chips. Find the least number of steps necessary to reach the arrangement where every chip is moved by $[\frac{n}{2}]$ positions clockwise from its initial position.

Consider the sequence $(3^{2^n} + 1)_{n\geq 1}$. i) Prove there exist infinitely many primes, none dividing any term of the sequence. ii) Prove there exist infinitely many primes, each dividing some term of the sequence. [i](Dan Schwarz)[/i]

Let $ f$ be a polynomial with integer coefficients. Assume that there exists integers $ a$ and $ b$ such that $ f(a)\equal{}41$ and $ f(b)\equal{}49$. Prove that there exists an integer $ c$ such that $ 2009$ divides $ f(c)$. [i]Nanang Susyanto, Jogjakarta[/i]

1 - Prove that $55 < (1+\sqrt{3})^4 < 56$ . 2 - Find the largest power of $2$ that divides $\lceil(1+\sqrt{3})^{2n}\rceil$ for the positive integer $n$

The set $M$ consists of all $7$-digit positive integer numbers that contain (in decimal notation) each of the digits $1, 3, 4, 6, 7, 8$ and $9$ exactly once. (a) Find the smallest positive difference $d$ of two numbers from $M$. (b) How many pairs $(x, y)$ with $x$ and $y$ from M are there for which $x - y = d$? (Gerhard Kirchner)