Compute the sum of all positive real numbers $x \le 5$ satisfying $$x =\frac{ \lceil x^2 \rceil + \lceil x \rceil \cdot \lfloor x \rfloor}{ \lceil x\rceil + \lfloor x \rfloor}$$

Let $P$ be an interior point of triangle $ABC$ such that $\angle BPC = \angle CPA =\angle APB = 120^o$. Prove that the Euler lines of triangles $APB,BPC,CPA$ are concurrent.

In a $ 19\times 19$ board, a piece called [i]dragon[/i] moves as follows: It travels by four squares (either horizontally or vertically) and then it moves one square more in a direction perpendicular to its previous direction. It is known that, moving so, a dragon can reach every square of the board. The [i]draconian distance[/i] between two squares is defined as the least number of moves a dragon needs to move from one square to the other. Let $ C$ be a corner square, and $ V$ the square neighbor of $ C$ that has only a point in common with $ C$. Show that there exists a square $ X$ of the board, such that the draconian distance between $ C$ and $ X$ is greater than the draconian distance between $ C$ and $ V$.

Let $a_1, a_2, \ldots, a_n$ be $n$ positive integers that are not divisible by each other, i.e. for any $i \neq j$, $a_i$ is not divisible by $a_j$. Show that \[ a_1+a_2+\cdots+a_n \ge 1.1n^2-2n. \] [i]Note:[/i] A proof of the inequality when $n$ is sufficient large will be awarded points depending on your results.

Prove that there doesn't exist any function $f:\mathbb{N}\rightarrow \mathbb{N}$ such that : $f(f(n-1)=f(n+1)-f(n)$, for every natural $n\geq2$

Let $p$ be a prime number. Find all triples $(a, b, c)$ of integers (not necessarily positive) such that $a^bb^cc^a = p$.

Let $n$ be a positive integer. Let $\mathcal{F}$ be a family of sets that contains more than half of all subsets of an $n$-element set $X$. Prove that from $\mathcal{F}$ we can select $\lceil \log_2 n \rceil + 1$ sets that form a separating family on $X$, i.e., for any two distinct elements of $X$ there is a selected set containing exactly one of the two elements. Moderator says: http://www.artofproblemsolving.com/Forum/viewtopic.php?f=41&t=614827&hilit=Schweitzer+2014+separating

Prove that there are infinitely natural numbers $n$ such that $n$ can't be written as a sum of two positive integers with prime factors less than $1394$.

Points $A, B$ and $C$ lie on the same line so that $CA = AB$. Square $ABDE$ and the equilateral triangle $CFA$, are constructed on the same side of line $CB$. Find the acute angle between straight lines $CE$ and $BF$.

A utility company is building a network to send electricity to fifty houses, with addresses $0, 1, 2, \ldots , 49. $ The power center only connects directly to house $0$, so electricity reaches all other houses through a system of wires that connects specific pairs of houses. To save money, the company only lays wires between as few pairs of distinct houses as possible; additionally, two houses with addresses $a$ and $b$ can only have a wire between them if at least one of the following three conditions is met: [list] [*]$10$ divides both $a$ and $b.$ [*]$\lfloor \tfrac{a}{10} \rfloor \equiv \lfloor \tfrac{b}{10}\rfloor \pmod{5}.$ [*]$\lceil \tfrac{a}{10} \rceil\equiv \lceil \tfrac{b}{10}\rceil\pmod{5}.$ [/list] Letting $N$ be the number of distinct ways such a wire system can be configured so that every house receives electricity , find the remainder when $N$ is divided by $1000.$

A white ball has finitely many disk-shaped black region (closed) painted on its surface. Each black region has area less than half of the surface area of the sphere. No two black regions touch or overlap. Determine, with proof, whether there is always a diameter of the ball with two white endpoints.

How many integers belong to ($a,2008a$), where $a$ ($a > 0$) is given.

Dan the dog spots Cate the cat 50m away. At that instant, Cate begins running away from Dan at 6 m/s, and Dan begins running toward Cate at 8 m/s. Both of them accelerate instantaneously and run in straight lines. Compute the number of seconds it takes for Dan to reach Cate. [i]Proposed by Eugene Chen[/i]

Find all pairs ($x,y$) of integers such that $x^2-2xy+126y^2=2009$.

The incenter of $\triangle A_1A_2A_3$ is $I$, and the center of the $A_i$-excircle is $J_i$ ($i=1,2,3$). Let $B_i$ be the intersection point of side $A_{i+1}A_{i+2}$ and the bisector of $\angle A_{i+1}IA_{i+2}$ ($A_{i+3}:=A_i$ $\forall i$). Prove that the three lines $B_iJ_i$ are concurrent.

Let $p$ and $q$ be integers. Show that there exists an interval $I$ of length $1/q$ and a polynomial $P$ with integral coefficients such that \[ \left|P(x)-\frac pq \right| < \frac{1}{q^2}\]for all $x \in I.$

"If a whole number $n$ is not prime, then the whole number $n-2$ is not prime." A value of $n$ which shows this statement to be false is $\text{(A)}\ 9 \qquad \text{(B)}\ 12 \qquad \text{(C)}\ 13 \qquad \text{(D)}\ 16 \qquad \text{(E)}\ 23$

What is the arithmetic mean of the smallest elements of $r$-subsets of set $\{1, 2, \dots , n\}$ where $1 \leq r \leq n$? $ \textbf{a)}\ \dfrac{n+1}{r+1} \qquad\textbf{b)}\ \dfrac{r(n+1)}{r+1} \qquad\textbf{c)}\ \dfrac{nr}{r+1} \qquad\textbf{d)}\ \dfrac{r(n+1)}{(r+1)n} \qquad\textbf{e)}\ \text{None of above} $

Determine all functions $f:\mathbb{R}\to\mathbb{R}$ such that for every pair of real numbers $x$ and $y$, \[f(x+y^2)=f(x)+|yf(y)|.\]

Determine the number of three digit integers that are equal to $19$ times the sum of its digits.

Source: 1976 Euclid Part B Problem 1 ----- Triangle $ABC$ has $\angle{B}=30^{\circ}$, $AB=150$, and $AC=50\sqrt{3}$. Determine the length of $BC$.

a) Prove that $\lfloor x\rfloor$ is odd iff $\Big\lfloor 2\{\frac{x}{2}\}\Big\rfloor=1$ ($\lfloor x\rfloor$ denotes the largest integer less than or equal to $x$ and $\{x\}=x-\lfloor x\rfloor$). b) Let $n$ be a natural number. Find the number of [i]square free[/i] numbers $a$, such that $\Big\lfloor\frac{n}{\sqrt{a}}\Big\rfloor$ is odd. (A natural number is [i]square free[/i] if it's not divisible by any square of a prime number).

Write using with the floor function: the last, the second last, and the first digit of the number $n$ written in the decimal system.

Four congruent semicircles are drawn on the surface of a sphere with radius $2$, as shown, creating a close curve that divides the surface into two congruent regions. The length of the curcve is $\pi \sqrt{n}$. What is $n$? $\textbf{(A) } 32 \qquad \textbf{(B) } 12 \qquad \textbf{(C) } 48 \qquad \textbf{(D) } 36 \qquad \textbf{(E) } 27$

Find all twice differentiable functions $f : \mathbb{R} \to \mathbb{R}$ such that $$f''(x) \cos(f(x))\geq(f'(x))^2 \sin(f(x)) $$ for every $x\in \mathbb{R}$. [i]Proposed by Orif Ibrogimov (Czech Technical University of Prague), Karim Rakhimov (University of Pisa)[/i]