Let $\mathcal{C}$ be the family of circumferences in $\mathbb{R}^2$ that satisfy the following properties: (i) if $C_n$ is the circumference with center $(n,1/2)$ and radius $1/2$, then $C_n\in \mathcal{C}$, for all $n\in \mathbb{Z}$. (ii) if $C$ and $C'$, both in $\mathcal{C}$, are externally tangent, then the circunference externally tangent to $C$ and $C'$ and tanget to $x$-axis also belongs to $\mathcal{C}$. (iii) $\mathcal{C}$ is the least family which these properties. Determine the set of the real numbers which are obtained as the first coordinate of the points of intersection between the elements of $\mathcal{C}$ and the $x$-axis.

The quadrilateral $ABCD$ is divided into cyclic quadrilaterals with pairwise disjoint interiors. None of the vertices of the cyclic quadrilaterals in the decomposition is an interior point of a side of any cyclic quadrilateral in the decomposition or of a side of the quadrilateral $ABCD$. Prove that $ABCD$ is also a cyclic quadrilateral.

A list with $2007$ positive integers is written on a board, such that the arithmetic mean of all the numbers is $12$. Then, seven consecutive numbers are erased from the board. The arithmetic mean of the remaining numbers is $11.915$. The seven erased numbers have this property: the sixth number is half of the seventh, the fifth number is half of the sixth, and so on. Determine the $7$ erased numbers.

[b]p1.[/b] a) Given four points in the plane, no three of which lie on the same line, each subset of three points determines the vertices of a triangle. Can all these triangles have equal areas? If so, give an example of four points (in the plane) with this property, and then describe all arrangements of four joints (in the plane) which permit this. If no such arrangement exists, prove this. b) Repeat part a) with "five" replacing "four" throughout. [b]p2.[/b] Three people at the base of a long stairway begin a race up the stairs. Person A leaps five steps with each stride (landing on steps $5$, $10$, $15$, etc.). Person B leaps a little more slowly but covers six steps with each stride. Person C leaps seven steps with each stride. A picture taken near the end of the race shows all three landing simultaneously, with Person A twenty-one steps from the top, person B seven steps from the top, and Person C one step from the top. How many steps are there in the stairway? If you can find more than one answer, do so. Justify your answer. [b]p3. [/b]Let $S$ denote the sum of an infinite geometric series. Suppose the sum of the squares of the terms is $2S$, and that df the cubes is $64S/13$. Find the first three terms of the original series. [b]p4.[/b] $A$, $B$ and $C$ are three equally spaced points on a circular hoop. Prove that as the hoop rolls along the horizontal line $\ell$, the sum of the distances of the points $A, B$, and $C$ above line $\ell$ is constant. [img]https://cdn.artofproblemsolving.com/attachments/3/e/a1efd0975cf8ff3cf6acb1da56da1dce35d81e.png[/img] [b]p5.[/b] A set of $n$ numbers $x_1,x_2,x_3,...,x_n$ (where $n>1$) has the property that the $k^{th}$ number (that is, $x_k$ ) is removed from the set, the remaining $(n-1)$ numbers have a sum equal to $k$ (the subscript o $x_k$ ), and this is true for each $k = 1,2,3,...,n$. a) SoIve for these $n$ numbers b) Find whether at least one of these $n$ numbers can be an integer. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Two right circular cylinders have the same volume. The radius of the second cylinder is $10\%$ more than the radius of the first. What is the relationship between the heights of the two cylinders? $\textbf{(A) }\text{The second height is 10\% less than the first.}$ $\textbf{(B) }\text{The first height is 10\% more than the second.}$ $\textbf{(C) }\text{The second height is 21\% less than the first.}$ $\textbf{(D) }\text{The first height is 21\% more than the second.}$ $\textbf{(E) }\text{The second height is 80\% of the first.}$

Find all funtions $f : Z_{>0} \to Z_{>0}$ that satisfy $f(f(f(n))) + f(f(n)) + f(n) = 3n$ for all $n \in Z_{>0}$ .

For positive integers $i = 2, 3, \ldots, 2020$, let \[ a_i = \frac{\sqrt{3i^2+2i-1}}{i^3-i}. \]Let $x_2$, $\ldots$, $x_{2020}$ be positive reals such that $x_2^4 + x_3^4 + \cdots + x_{2020}^4 = 1-\frac{1}{1010\cdot 2020\cdot 2021}$. Let $S$ be the maximum possible value of \[ \sum_{i=2}^{2020} a_i x_i (\sqrt{a_i} - 2^{-2.25} x_i) \] and let $m$ be the smallest positive integer such that $S^m$ is rational. When $S^m$ is written as a fraction in lowest terms, let its denominator be $p_1^{\alpha_1} p_2^{\alpha_2}\cdots p_k^{\alpha_k}$ for prime numbers $p_1 < \cdots < p_k$ and positive integers $\alpha_i$. Compute $p_1\alpha_1+p_2\alpha_2 + \cdots + p_k\alpha_k$. [i]Proposed by Edward Wan and Brandon Wang[/i]

In base $10$ there exist two-digit natural numbers that can be factorized into two natural factors such that the two digits and the two factors form a sequence of four consecutive integers (for example, $12 = 3 \cdot 4$). Determine all such numbers in all bases.

Let $ABCDEF$ be a convex hexagon such that all of its vertices are on a circle. Prove that $AD$, $BE$ and $CF$ are concurrent if and only if $\frac {AB}{BC}\cdot\frac {CD}{DE}\cdot\frac {EF}{FA}= 1$.

Jamal wants to store $ 30$ computer files on floppy disks, each of which has a capacity of $ 1.44$ megabytes (MB). Three of his files require $ 0.8$ MB of memory each, $ 12$ more require $ 0.7$ MB each, and the remaining $ 15$ require $ 0.4$ MB each. No file can be split between floppy disks. What is the minimal number of floppy disks that will hold all the files? $ \textbf{(A)}\ 12 \qquad \textbf{(B)}\ 13 \qquad \textbf{(C)}\ 14 \qquad \textbf{(D)}\ 15 \qquad \textbf{(E)}\ 16$

[b]2.[/b] Show that threre exist a compact set $K \subset \mathbb{R}$ and a set $A \subset \mathbb{R}$ of type $F_{\sigma}$ such that the set $\{ x\in \mathbb{R} : K+x \subset A\}$ is not Borel-measurable (here $K+x = \{y+x : y \in K\}$). ([b]M.16[/b]) [M. Laczkovich]

Let $I$ be the incenter of triangle $ABC$ and let $D$ be the midpoint of side $AB$. Prove that if the angle $\angle AOD$ is right, then $AB+BC=3AC$. [I]Proposed by S. Ivanov[/i]

Show that there exists a positive constant $C$ such that, for all positive reals $a$ and $b$ with $a + b$ being an integer, we have $$\left\{a^3\right\} + \left\{b^3\right\} + \frac{C}{(a+b)^6} \le 2. $$ Here $\{x\} = x - \lfloor x\rfloor$ is the fractional part of $x$. [i]Proposed by Li4 and Untro368.[/i]

Find all functions $f : Z^+ \to Z^+$ such that $n^3 - n^2 \le f(n) \cdot (f(f(n)))^2 \le n^3 + n^2$ for every $n$ in positive integers

Let $n$ be a positive integer $\geq 3.$ There are $n$ real numbers $x_1,x_2,\cdots x_n$ that satisfy: \[\left\{\begin{aligned}&\ x_1\ge x_2\ge\cdots \ge x_n;\\& \ x_1+x_2+\cdots+x_n=0;\\& \ x_1^2+x_2^2+\cdots+x_n^2=n(n-1).\end{aligned}\right.\] Find the maximum and minimum value of the sum $S=x_1+x_2.$

Kim earned scores of 87,83, and 88 on her first three mathematics examinations. If Kim receives a score of 90 on the fourth exam, then her average will $\textbf{(A)}\ \text{remain the same} \qquad \textbf{(B)}\ \text{increase by 1} \qquad \textbf{(C)}\ \text{increase by 2} \qquad \textbf{(D)}\ \text{increase by 3} \qquad \textbf{(E)}\ \text{increase by 4}$

Let $$F(x)=\frac{x^4}{\exp(x^3)}\int^x_0\int^{x-u}_0\exp(u^3+v^3)dvdu.$$Find $\lim_{x\to\infty}F(x)$ or prove that it does not exist.

In a trapezium $ABCD$, the sides $AB$ and $CD$ are parallel and the angles $\angle ABC$ and $\angle BAD$ are acute. Show that it is possible to divide the triangle $ABC$ into 4 disjoint triangle $X_1. . . , X_4$ and the triangle $ABD$ into 4 disjoint triangles $Y_1,. . . , Y_4$ such that the triangles $X_i$ and $Y_i$ are congruent for all $i$.

An integer $n$ is called $k$-special, with $k$ a positive integer, if it's the sum of the squares of $k$ consecutive integers. For example, $13$ is $2$-special, since $13=2^2+3^2$, and $2$ is $3$-special, since $2=(-1)^2+0^2+1^2$. a) Prove that there's no perfect square that is $4$-special. b) Find a perfect square that is $I^2$-special, for some odd positive integer $I$ with $I\ge 3$.

Find all nondecreasing functions $f:\mathbb R\to \mathbb R$ such that, for all $x,y\in \mathbb R$, $$f(f(x))+f(y)=f(x+f(y))+1.$$ [i]Proposed by Carl Schildkraut[/i]

In circle $\theta_1$ with radius $1$, circles $\phi_1, \phi_2, \dots, \phi_8$, with equal radii, are drawn such that for $1 \le i \le 8$, $\phi_i$ is tangent to $\omega_1$, $\phi_{i-1}$, and $\phi_{i+1}$, where $\phi_0 = \phi_8$ and $\phi_1 = \phi_9$. There exists a circle $\omega_2$ such that $\omega_1 \neq \omega_2$ and $\omega_2$ is tangent to $\phi_i$ for $1 \le i \le 8$. The radius of $\omega_2$ can be expressed in the form $a - b\sqrt{c} -d\sqrt{e - \sqrt{f}} + g \sqrt{h - j \sqrt{k}}$ such that $a, b, \dots, k$ are positive integers and the numbers $e, f, k, \gcd(h, j)$ are squarefree. What is $a+b+c+d+e+f+g+h+j+k$. [i]Proposed by Eugene Chen [/i]

In $\triangle ABC$, let point $D$ be on $\overline{BC}$ such that the perimeters of $\triangle ADB$ and $\triangle ADC$ are equal. Let point $E$ be on $\overline{AC}$ such that the perimeters of $\triangle BEA$ and $\triangle BEC$ are equal. Let point $F$ be the intersection of $\overline{AB}$ with the line that passes through $C$ and the intersection of $\overline{AD}$ and $\overline{BE}$. Given that $BD = 10$, $CD = 2$, and $BF/FA = 3$, what is the perimeter of $\triangle ABC$?

Let $0\leq x_1\leq x_2\leq \cdots \leq x_n\leq 1 $ $(n\geq 2).$ Prove that $$\sqrt[n]{x_1x_2 \cdots x_n}+ \sqrt[n]{(1-x_1)(1-x_2)\cdots (1-x_n)}\leq \sqrt[n]{1-(x_1- x_n)^2}.$$

Let $ABC$ be a triangle with $\angle C=90^\circ$, and $D$ be a point outside $ABC$, such that $\angle ADC=\angle BAC$. The segments $CD$ and $AB$ meet at point $E$. It is known that the distance from $E$ to $AC$ is equal to the circumradius of triangle $ADE$. Find the angles of triangle $ABC$.

Show that the sequence \begin{align*} a_n = \left \lfloor \left( \sqrt[3]{n-2} + \sqrt[3]{n+3} \right)^3 \right \rfloor \end{align*} contains infinitely many terms of the form $a_n^{a_n}$