Consider the rectangular strip of length $12$ below, divided into three rectangles. The distance between the centers of two of the rectangles is $4$. What is the length of the other rectangle? [asy] size(120); draw((0,0)--(12,0)--(12,1)--(0,1)--cycle); draw((8,1)--(8,0)); draw((3,1)--(3,0)); dot((1.5,0.5)); dot((5.5,0.5)); draw((1.5,0.5)--(5.5,0.5)); [/asy] $\textbf{(A) }2.5\qquad\textbf{(B) }3\qquad\textbf{(C) }3.5\qquad\textbf{(D) }4\qquad\textbf{(E) }4.5$

Let $a$, $b$, and $c$ be positive real numbers. Prove that $$(a^5 - a^2 +3)(b^5 - b^2 +3)(c^5 - c^2 +3)\ge (a+b+c)^3$$

Let $p$ be a prime. Define $f_n(k)$ to be the number of positive integers $1\leq x\leq p-1$ such that $$\left(\left\{\frac{x}{p}\right\}-\left\{\frac{k}{p}\right\}\right)\left(\left\{\frac{nx}{p}\right\}-\left\{\frac{k}{p}\right\}\right)<0.$$ Let $a_n=f_n\left(\frac 12\right)+f_n\left(\frac 32\right)+\dots+f_n\left(\frac{2p-1}{2}\right)$, find $\min\{a_2, a_3, \dots, a_{p-1}\}$.

There are 100 students taking an exam. The professor calls them one by one and asks each student a single person question: “How many of 100 students will have a “passed” mark by the end of this exam?” The students answer must be an integer. Upon receiving the answer, the professor immediately publicly announces the student’s mark which is either “passed” or “failed.” After all the students have got their marks, an inspector comes and checks if there is any student who gave the correct answer but got a “failed” mark. If at least one such student exists, then the professor is suspended and all the marks are replaced with “passed.” Otherwise no changes are made. Can the students come up with a strategy that guarantees a “passed” mark to each of them? [i] Denis Afrizonov [/i]

Show that any triangle has two sides whose lengths $a$ and $b$ satisfy $\frac{\sqrt{5}-1}{2}<\frac{a}{b}<\frac{\sqrt{5}+1}{2}$.

Let $ ABC$ be an acute triangle, let $ M,N$ be the midpoints of minor arcs $ \widehat{CA},\widehat{AB}$ of the circumcircle of triangle $ ABC,$ point $ D$ is the midpoint of segment $ MN,$ point $ G$ lies on minor arc $ \widehat{BC}.$ Denote by $ I,I_{1},I_{2}$ the incenters of triangle $ ABC,ABG,ACG$ respectively.Let $ P$ be the second intersection of the circumcircle of triangle $ GI_{1}I_{2}$ with the circumcircle of triangle $ ABC.$ Prove that three points $ D,I,P$ are collinear.

Solve the system of equations in real numbers: \[ \frac{x}{y} + \frac{y}{z} + \frac{z}{x} = \frac{x}{z} + \frac{z}{y} + \frac{y}{x} \] \[ x^2 + y^2 + z^2 = 294 \] \[ x + y + z = 0 \]

Let $\{u_n\}_ {n\ge 1}$ be given sequence satisfying the conditions: $u_1 = 0$, $u_2 = 1$, $u_{n+1} = u_{n-1} + 2n - 1$ for $n \ge 2$. 1) Calculate $u_5$. 2) Calculate $u_{100} + u_{101}$.

An urn contains one red ball and one blue ball. A box of extra red and blue balls lie nearby. George performs the following operation four times: he draws a ball from the urn at random and then takes a ball of the same color from the box and returns those two matching balls to the urn. After the four iterations the urn contains six balls. What is the probability that the urn contains three balls of each color? $\textbf{(A) } \frac16 \qquad \textbf{(B) }\frac15 \qquad \textbf{(C) } \frac14 \qquad \textbf{(D) } \frac13 \qquad \textbf{(E) } \frac12$

Is it true that there are $130$ consecutive natural numbers, such that each of them has exactly $900$ natural divisors?

Let $a\# b$ be defined as $ab-a-3$. For example, $4\#5=20-4-3=13$ Compute $(2\#0)\#(1\#4)$.

Suppose that $x_1 < x_2 < \dots < x_n$ is a sequence of positive integers such that $x_k$ divides $x_{k+2}$ for each $k = 1, 2, \dots, n-2$. Given that $x_n = 1000$, what is the largest possible value of $n$? [i]Proposed by Evan Chen[/i]

Triangle $ABC$ is inscribed in a circle of radius 2 with $\angle ABC \geq 90^\circ$, and $x$ is a real number satisfying the equation $x^4 + ax^3 + bx^2 + cx + 1 = 0$, where $a=BC$, $b=CA$, $c=AB$. Find all possible values of $x$.

In the figure, there is a circle of radius $1$ such that the segment $AG$ is diameter and that line $AF$ is perpendicular to line $DC$. There are also two squares $ABDC$ and $DEGF$, where $B$ and $E$ are points on the circle, and the points $A$, $D$ and $E$ are collinear. What is the area of square $DEGF$? [img]https://cdn.artofproblemsolving.com/attachments/1/e/794da3bc38096ef5d5daaa01d9c0f8c41a6f84.png[/img]

Let $a_1, a_2,...,a_{2005}, b_1, b_2,...,b_{2005}$ be real numbers such that $(a_ix - b_i)^2 \ge \sum_{j\ne i,j=1}^{2005} (a_jx - b_j)$ for all real numbers x and every integer $i$ with $1 \le i \le 2005$. What is maximal number of positive $a_i$'s and $b_i$'s?

Let $ H$ be a set of real numbers that does not consist of $ 0$ alone and is closed under addition. Further, let $ f(x)$ be a real-valued function defined on $ H$ and satisfying the following conditions: \[ \;f(x)\leq f(y)\ \mathrm{if} \;x \leq y\] and \[ f(x\plus{}y)\equal{}f(x)\plus{}f(y) \;(x,y \in H)\ .\] Prove that $ f(x)\equal{}cx$ on $ H$, where $ c$ is a nonnegative number. [M. Hosszu, R. Borges]

A museum wants to protect its most valuable piece by maintaining constant surveillance. To do this, he wants to place guards to watch the place, in shifts of $7$ consecutive hours. Each guard starts his shift at the same time every day. A guard is essential if there is any time during the day when you are alone to watch the item. Indicates all possibilities for the number of guards guarding the piece, so that everyone is indispensable.