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]

Suppose that $|ax^2+bx+c| \geq |x^2-1|$ for all real numbers x. Prove that $|b^2-4ac|\geq 4$.

A bored student walks down a hall that contains a row of closed lockers, numbered 1 to 1024. He opens the locker numbered 1, and then alternates between skipping and opening each closed locker thereafter. When he reaches the end of the hall, the student turns around and starts back. He opens the first closed locker he encounters, and then alternates between skipping and opening each closed locker thereafter. The student continues wandering back and forth in this manner until every locker is open. What is the number of the last locker he opens?

An integer sequence $\{a_{n}\}_{n \ge 1}$ is defined by \[a_{0}=0, \; a_{1}=1, \; a_{n+2}=2a_{n+1}+a_{n}\] Show that $2^{k}$ divides $a_{n}$ if and only if $2^{k}$ divides $n$.

Let be a real number $ c $ and a differentiable function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ such that $$ f(c)\neq \frac{1}{b-a}\int_a^b f(x)dx, $$ for any real numbers $ a\neq b. $ Prove that $ f'(c)=0. $ [i]Florin Rotaru[/i]

A square with side $1$ is cut from the paper. Construct a segment with length $1/2024$ using at most $20$ folds. No instruments are available. It is allowed only to fold the paper and to mark the common points of folding lines.

Let $D$ be the point different from $B$ on the hypotenuse $AB$ of a right triangle $ABC$ such that $|CB| = |CD|$. Let $O$ be the circumcenter of triangle $ACD$. Rays $OD$ and $CB$ intersect at point $P$, and the line through point $O$ perpendicular to side AB and ray $CD$ intersect at point $Q$. Points $A, C, P, Q$ are concyclic. Does this imply that $ACPQ$ is a square?

Let $ ABC$ be a triangle with area $ S$ and points $ D,E,F$ on the sides $ BC,CA,AB$. Perpendiculars at points $ D,E,F$ to the $ BC,CA,AB$ cut circumcircle of the triangle $ ABC$ at points $ (D_1,D_2), (E_1,E_2), (F_1,F_2)$. Prove that: $ |D_1B\cdot D_1C \minus{} D_2B\cdot D_2C| \plus{} |E_1A\cdot E_1C \minus{} E_2A\cdot E_2C| \plus{} |F_1B\cdot F_1A \minus{} F_2B\cdot F_2A| > 4S$

Let $A = \{3 + 10k, 6 + 26k, 5 + 29k, k = 1, 2, 3, 4, ...\}$. Determine the smallest positive integer $r$ such that there exists an integer $b$ with the property that the set $B = \{b + rk, k = 1, 2, 3, 4, ...\}$ is disjoint from $A$.