Suppose that $n \ge 2$. Prove that \[\sum_{k=2}^{n}\left\lfloor \frac{n^{2}}{k}\right\rfloor = \sum_{k=n+1}^{n^{2}}\left\lfloor \frac{n^{2}}{k}\right\rfloor.\]

Compute $ \sum_{n \equal{} 1}^\infty\sum_{k \equal{} 1}^{n \minus{} 1}\frac {k}{2^{n \plus{} k}}$.

For fixed positive integers $s, t$, define $a_n$ as the following. $a_1 = s, a_2 = t$, and $\forall n \ge 1$, $a_{n+2} = \lfloor \sqrt{a_n+(n+2)a_{n+1}+2008} \rfloor$. Prove that the solution set of $a_n \not= n$, $n \in \mathbb{N}$ is finite.

Proof that $$ \sum_{m=1}^n5^{\omega (m)} \le \sum_{k=1}^n\lfloor \frac{n}{k} \rfloor \tau (k)^2 \le \sum_{m=1}^n5^{\Omega (m)} .$$

Find all functions $f:\mathbb{N}\rightarrow \mathbb{N}$ such that for all $m,n\in\mathbb{N}$, \[(2^m+1)f(n)f(2^mn)=2^mf(n)^2+f(2^mn)^2+(2^m-1)^2n. \]

Find the number of ordered triples $(x,y,z)$ of non-negative integers satisfying (i) $x \leq y \leq z$ (ii) $x + y + z \leq 100.$

Participants to an olympiad worked on $n$ problems. Each problem was worth a [color=#FF0000]positive [/color]integer number of points, determined by the jury. A contestant gets $0$ points for a wrong answer, and all points for a correct answer to a problem. It turned out after the olympiad that the jury could impose worths of the problems, so as to obtain any (strict) final ranking of the contestants. Find the greatest possible number of contestants.

Determine all positive integers $a$ for which there exist exactly $2014$ positive integers $b$ such that $\displaystyle2\leq\frac{a}{b}\leq5$.

Let $\{x\} = x- \lfloor x \rfloor$ . Consider a function f from the set $\{1, 2, . . . , 2020\}$ to the half-open interval $[0, 1)$. Suppose that for all $x, y,$ there exists a $z$ so that $\{f(x) + f(y)\} = f(z)$. We say that a pair of integers $m, n$ is valid if $1 \le m, n \le 2020$ and there exists a function $f$ satisfying the above so $f(1) = \frac{m}{n}$. Determine the sum over all valid pairs $m, n$ of ${m}{n}$.

Solve in the real numbers the equation $ \lfloor ax \rfloor -\lfloor (1+a)x \rfloor = (1+a)(1-x) . $ [i]Dumitru Acu[/i]

Let $x,y,z>0$ such that $$(x+y+z)\left(\frac1x+\frac1y+\frac1z\right)=\frac{91}{10}$$ Compute $$\left[(x^3+y^3+z^3)\left(\frac1{x^3}+\frac1{y^3}+\frac1{z^3}\right)\right]$$ where $[.]$ represents the integer part. [i]Proposed by Marian Cucoanoeş and Marius Drăgan[/i]

Consider $ n$ points in a plane which are vertices of a convex polygon. Prove that the set of the lengths of the sides and the diagonals of the polygon has at least $ \lfloor n/2\rfloor$ elements.

Let two natural number $n > 1$ and $m$ be given. Find the least positive integer $k$ which has the following property: Among $k$ arbitrary integers $a_1, a_2, \ldots, a_k$ satisfying the condition $a_i - a_j$ ( $1 \leq i < j \leq k$) is not divided by $n$, there exist two numbers $a_p, a_s$ ($p \neq s$) such that $m + a_p - a_s$ is divided by $n$.

Some of $n$ towns are connected by two-way airlines. There are $m$ airlines in total. For $i = 1, 2, \cdots, n$, let $d_i$ be the number of airlines going from town $i$. If $1\le d_i \le 2010$ for each $i = 1, 2,\cdots, 2010$, prove that \[\displaystyle\sum_{i=1}^n d_i^2\le 4022m- 2010n\] Find all $n$ for which equality can be attained. [i]Proposed by Aleksandar Ilic[/i]

Solve the equation $\left[\frac{x}{3}\right]+\left [\frac{2x}{3}\right]=x $

A right triangle has perimeter $2008$, and the area of a circle inscribed in the triangle is $100\pi^3$. Let $A$ be the area of the triangle. Compute $\lfloor A\rfloor$.

Find the number of positive integers $n$ such that $1\leq n\leq 1000$ and $n$ is divisible by $\lfloor \sqrt[3]{n} \rfloor$.

[b]p1.[/b] You are on a flat planet. There are $100$ cities at points $x = 1, ..., 100$ along the line $y = -1$, and another $100$ cities at points $x = 1, ... , 100$ along the line $y = 1$. The planet’s terrain is scalding hot, and you cannot walk over it directly. Instead, you must cross archways from city to city. There are archways between all pairs of cities with different $y$ coordinates, but no other pairs: for instance, there is an archway from $(1, -1)$ to $(50, 1)$, but not from $(1, -1)$ to $(50, -1)$. The amount of “effort” necessary to cross an archway equals the square of the distance between the cities it connects. You are at $(1, -1)$, and you want to get to $(100, -1)$. What is the least amount of effort this journey can take? [b]p2.[/b] Let $f(x) = x^4 + ax^3 + bx^2 + cx + 25$. Suppose $a, b, c$ are integers and $f(x)$ has $4$ distinct integer roots. Find $f(3)$. [b]p3.[/b] Frankenstein starts at the point $(0, 0, 0)$ and walks to the point $(3, 3, 3)$. At each step he walks either one unit in the positive $x$-direction, one unit in the positive $y$-direction, or one unit in the positive $z$-direction. How many distinct paths can Frankenstein take to reach his destination? [b]p4.[/b] Let $ABCD$ be a rectangle with $AB = 20$, $BC = 15$. Let $X$ and $Y$ be on the diagonal $\overline{BD}$ of $ABCD$ such that $BX > BY$ . Suppose $A$ and $X$ are two vertices of a square which has two sides on lines $\overline{AB}$ and $\overline{AD}$, and suppose that $C$ and $Y$ are vertices of a square which has sides on $\overline{CB}$ and $\overline{CD}$. Find the length $XY$ . [img]https://cdn.artofproblemsolving.com/attachments/2/8/a3f7706171ff3c93389ff80a45886e306476d1.png[/img] [b]p5.[/b] $n \ge 2$ kids are trick-or-treating. They enter a haunted house in a single-file line such that each kid is friends with precisely the kids (or kid) adjacent to him. Inside the haunted house, they get mixed up and out of order. They meet up again at the exit, and leave in single file. After leaving, they realize that each kid (except the first to leave) is friends with at least one kid who left before him. In how many possible orders could they have left the haunted house? [b]p6.[/b] Call a set $S$ sparse if every pair of distinct elements of S differ by more than $1$. Find the number of sparse subsets (possibly empty) of $\{1, 2,... , 10\}$. [b]p7.[/b] How many ordered triples of integers $(a, b, c)$ are there such that $1 \le a, b, c \le 70$ and $a^2 + b^2 + c^2$ is divisible by $28$? [b]p8.[/b] Let $C_1$, $C_2$ be circles with centers $O_1$, $O_2$, respectively. Line $\ell$ is an external tangent to $C_1$ and $C_2$, it touches $C_1$ at $A$ and $C_2$ at $B$. Line segment $\overline{O_1O_2}$ meets $C_1$ at $X$. Let $C$ be the circle through $A, X, B$ with center $O$. Let $\overline{OO_1}$ and $\overline{OO_2}$ intersect circle $C$ at $D$ and $E$, respectively. Suppose the radii of $C_1$ and $C_2$ are $16$ and $9$, respectively, and suppose the area of the quadrilateral $O_1O_2BA$ is $300$. Find the length of segment $DE$. [b]p9.[/b] What is the remainder when $5^{5^{5^5}}$ is divided by $13$? [b]p10.[/b] Let $\alpha$ and $\beta$ be the smallest and largest real numbers satisfying $$x^2 = 13 + \lfloor x \rfloor + \left\lfloor \frac{x}{2} \right\rfloor +\left\lfloor \frac{x}{3} \right\rfloor + \left\lfloor \frac{x}{4} \right\rfloor .$$ Find $\beta - \alpha$ . ($\lfloor a \rfloor$ is defined as the largest integer that is not larger than $a$.) PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

[b]Problem 3.[/b] Let $n\geq 3$ is given natural number, and $M$ is the set of the first $n$ primes. For any nonempty subset $X$ of $M$ with $P(X)$ denote the product of its elements. Let $N$ be a set of the kind $\ds\frac{P(A)}{P(B)}$, $A\subset M, B\subset M, A\cap B=\emptyset$ such that the product of any 7 elements of $N$ is integer. What is the maximal number of elements of $N$? [i]Alexandar Ivanov[/i]

Let $\mathbb{N}_0$ and $\mathbb{Z}$ be the set of all non-negative integers and the set of all integers, respectively. Let $f:\mathbb{N}_0\rightarrow\mathbb{Z}$ be a function defined as \[f(n)=-f\left(\left\lfloor\frac{n}{3}\right\rfloor \right)-3\left\{\frac{n}{3}\right\} \] where $\lfloor x \rfloor$ is the greatest integer smaller than or equal to $x$ and $\{ x\}=x-\lfloor x \rfloor$. Find the smallest integer $n$ such that $f(n)=2010$.

Let $a_1, a_2, . . . , a_n$ be positive integers, with $n \ge 2$, such that $$ \lfloor \sqrt{a_1 \cdot a_2\cdot\cdot\cdot a_n} \rfloor = \lfloor \sqrt{a_1} \rfloor \cdot \lfloor \sqrt{a_2} \rfloor \cdot\cdot\cdot \lfloor \sqrt{a_n} \rfloor.$$ Prove that at least $n - 1$ of these numbers are perfect squares. Clarification: Given a real number $x$, $\lfloor x\rfloor$ denotes the largest integer that is less than or equal to $x$. For example $\lfloor \sqrt2\rfloor$ and $\lfloor 3\rfloor =3$.

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]

What is the sum of the positive solutions to $2x^2 -\lfloor x \rfloor = 5$, where $\lfloor x \rfloor$ is the largest integer less than or equal to $x$?

Let $a_1, a_2, \dots, a_{1000}$ be positive integers whose sum is $S$. If $a_n!$ divides $n$ for each $n = 1, 2, \dots, 1000$, compute the maximum possible value of $S$. [i]Proposed by Michael Ren[/i]

Let $M$ be the set of palindromic integers of the form $5n+4$ where $n\ge 0$ is an integer. [list=a] [*]If we write the elements of $M$ in increasing order, what is the $50^{\text{th}}$ number? [*]Among all numbers in $M$ with nonzero digits which sum up to $2014$ which is the largest and smallest one?[/list]