There exists 4 circles, $ a,b,c,d$, such that $ a$ is tangent to both $ b$ and $ d$, $ b$ is tangent to both $ a$ and $ c$, $ c$ is both tangent to $ b$ and $ d$, and $ d$ is both tangent to $ a$ and $ c$. Show that all these tangent points are located on a circle.

A real number $\alpha$ is given. Find all functions $f : R^+ \to R^+$ satisfying $\alpha x^2f\left(\frac{1}{x}\right) +f(x) =\frac{x}{x+1}$ for all $x > 0$.

Given a regular polygon with $n$ sides. It is known that there are $1200$ ways to choose three of the vertices of the polygon such that they form the vertices of a [b]right triangle[/b]. What is the value of $n$?

Determine all positive integers $n>1$ such that for any divisor $d$ of $n,$ the numbers $d^2-d+1$ and $d^2+d+1$ are prime. [i]Lucian Petrescu[/i]

Let $ABCD$ be a parallelogram with $\angle DAB < 90$ Let $E$ be the point on the line $BC$ such that $AE = AB$ and let $F$ be the point on the line $CD$ such that $AF = AD$. The circumcircle of the triangle $CEF$ intersects the line $AE$ again in $P$ and the line $AF$ again in $Q$. Let $X$ be the reflection of $P$ over the line $DE$ and $Y$ the reflection of $Q$ over the line $BF$. Prove that $A, X, Y$ lie on the same line.

Let $S$ be the sum of all positive real numbers $x$ for which $$x^{2^{\sqrt2}}=\sqrt2^{2^x}.$$ Which of the following statements is true? $\textbf{(A) }S<\sqrt2 \qquad \textbf{(B) }S=\sqrt2 \qquad \textbf{(C) }\sqrt2<S<2\qquad \textbf{(D) }2\le S<6 \qquad \textbf{(E) }S\ge 6$

Let $n \geq 3$ be an integer. A labelling of the $n$ vertices, the $n$ sides and the interior of a regular $n$-gon by $2n + 1$ distinct integers is called [i]memorable[/i] if the following conditions hold: (a) Each side has a label that is the arithmetic mean of the labels of its endpoints. (b) The interior of the $n$-gon has a label that is the arithmetic mean of the labels of all the vertices. Determine all integers $n \geq 3$ for which there exists a memorable labelling of a regular $n$-gon consisting of $2n + 1$ consecutive integers.

Let triangle $ABC$ be perpendicular at $A.$ Let $M$ be the midpoint of segment $\overline{BC}.$ Point $D$ lies on side $\overline{AC}$ and satisfies $|AD|=|AM|.$ Let $P \neq C$ be the intersection of the circumcircle of triangles $AMC$ and $BDC.$ Prove that $CP$ bisects the angle at $C$ of triangle $ABC.$

Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$, such that for arbitrary $x,y \in \mathbb{R}$: \[ (y+1)f(x)+f(xf(y)+f(x+y))=y.\]

Let $M$ and $N$ be two points on different sides of the square $ABCD$. Suppose that segment $MN$ divides the square into two tangential polygons. If $R$ and $r$ are radii of the circles inscribed in these polygons ($R> r$), calculate the length of the segment $MN$ in terms of $R$ and $r$. (Moldova)

Let $f:\mathbb{R}^{\geq 0}\to\mathbb{R}$ be a function such that $f(x)-3x$ and $f(x)-x^3$ are ascendant functions. Prove that $f(x)-x^2-x$ is an ascendant function, too. (We call the function $g(x)$ ascendant, when for every $x\leq{y}$ we have $g(x)\leq{g(y)}$.)

Given a triangle $ABC$, a point $P$ is chosen on side $BC$. Points $M$ and $N$ lie on sides $AB$ and $AC$, respectively, such that $MP \parallel AC$ and $NP \parallel AB$. Point $P$ is reflected across $MN$ to point $Q$. Show that triangle $QMB$ is similar to triangle $CNQ$. [i]Brian Hamrick.[/i]

In a rectangle with sides of length 20 and 25 there are 120 squares of side length 1. Prove that there is a circle with a diameter of 1 contained in this rectangle and having no points in common with any of these squares.

Determine all solutions $(p,n)$ of the equation \[n^3=p^2-p-1\] where $p$ is a prime number and $n$ is an integer

Let $f(n)$ be the smallest prime which divides $n^4+1$. What is the remainder when the sum $f(1)+f(2)+\cdots+f(2014)$ is divided by $8$? $ \textbf{(A)}\ 1 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}\ 7 \qquad\textbf{(E)}\ \text{None of the preceding} $

Can the product of $2$ consecutive natural numbers equal the product of $2$ consecutive even natural numbers? (natural means positive integers)

Let $n\geq 1$ be a positive integer. We say that an integer $k$ is a [i]fan [/i]of $n$ if $0\leq k\leq n-1$ and there exist integers $x,y,z\in\mathbb{Z}$ such that \begin{align*} x^2+y^2+z^2 &\equiv 0 \pmod n;\\ xyz &\equiv k \pmod n. \end{align*} Let $f(n)$ be the number of fans of $n$. Determine $f(2020)$.

Let $\triangle ABC$ be a right triangle with right angle at $C$. Let $D$ and $E$ be points on $\overline{AB}$ with $D$ between $A$ and $E$ such that $\overline{CD}$ and $\overline{CE}$ trisect $\angle C$. If $\frac{DE}{BE} = \frac{8}{15}$, then $\tan B$ can be written as $\frac{m\sqrt{p}}{n}$, where $m$ and $n$ are relatively prime positive integers, and $p$ is a positive integer not divisible by the square of any prime. Find $m+n+p$.

For each polynomial $P(x)$, define $$P_1(x)=P(x), \forall x \in \mathbb{R},$$ $$P_2(x)=P(P_1(x)), \forall x \in \mathbb{R},$$ $$...$$ $$P_{2024}(x)=P(P_{2023}(x)), \forall x \in \mathbb{R}.$$ Let $a>2$ be a real number. Is there a polynomial $P$ with real coefficients such that for all $t \in (-a, a)$, the equation $P_{2024}(x)=t$ has $2^{2024}$ distinct real roots?

There are two values of $ a$ for which the equation $ 4x^2 \plus{} ax \plus{} 8x \plus{} 9 \equal{} 0$ has only one solution for $ x$. What is the sum of these values of $ a$? $ \textbf{(A)}\ \minus{}16\qquad \textbf{(B)}\ \minus{}8\qquad \textbf{(C)}\ 0\qquad \textbf{(D)}\ 8\qquad \textbf{(E)}\ 20$

Let $a_{0}=\sqrt{2}, b_{0}=2,a_{n+1}=\sqrt{2-\sqrt{4-a_{n}^{2}}},b_{n+1}=\frac{2b_{n}}{2+\sqrt{4+b_{n}^{2}}}$. a) Prove that the sequences $(a_{n})$ and $(b_{n})$ are decreasing and converge to $0$. b) Prove that the sequence $(2^{n}a_{n})$ is increasing, the sequence $(2^{n}b_{n})$ is decreasing and both converge to the same limit. c) Prove that there exists a positive constant $C$ such that for all $n$ the following inequality holds: $0 <b_{n}-a_{n} <\frac{C}{8^{n}}$.

$(GDR 5)$ Given a ring $G$ in the plane bounded by two concentric circles with radii $R$ and $\frac{R}{2}$, prove that we can cover this region with $8$ disks of radius $\frac{2R}{5}$. (A region is covered if each of its points is inside or on the border of some disk.)

Under the new AMC 10, 12 scoring method, $6$ points are given for each correct answer, $2.5$ points are given for each unanswered question, and no points are given for an incorrect answer. Some of the possible scores between $0$ and $150$ can be obtained in only one way, for example, the only way to obtain a score of $146.5$ is to have 24 correct answers and one unanswered question. Some scores can be obtained in exactly two ways; for example, a score of $104.5$ can be obtained with $17$ correct answers, $1$ unanswered question, and $7$ incorrect, and also with $12$ correct answers and $13$ unanswered questions. There are three scores that can be obtained in exactly three ways. What is their sum? $\textbf{(A) }175\qquad\textbf{(B) }179.5\qquad\textbf{(C) }182\qquad\textbf{(D) }188.5\qquad\textbf{(E) }201$

There is a $8\times 8$ table, drawn in a plane and painted in a chess board fashion. Peter mentally chooses a square and an interior point in it. Basil can draws any polygon (without self-intersections) in the plane and ask Peter whether the chosen point is inside or outside this polygon. What is the minimal number of questions suffcient to determine whether the chosen point is black or white?

What is the maximal number of elements we can choose form the set $\{1, 2, \ldots, 31\}$, such that the sum of any two of them is not a perfect square.