Triangles $MA_2B_2$ and $MA_1B_1$ are similar to each other and have the same orientation. Prove that the circles circumcribed around these triangles and the straight lines $A_1A_2$ , $B_1B_2$ have a common point.

The sequence $\left(a_{n}\right)_{n\in\mathbb{N}}$ is defined recursively as $a_{0}=a_{1}=1$, $a_{n+2}=5a_{n+1}-a_{n}-1$, $\forall n\in\mathbb{N}$ Prove that $$a_{n}\mid a_{n+1}^{2}+a_{n+1}+1$$ for any $n\in\mathbb{N}$

Let $f(x) = \frac{\sin(x)}{x}$ Find $$ \lim_{T\to\infty}\frac{1}{T}\int_0^T\sqrt{1+f'(x)^2}dx.$$

Solve the equation \[ \sqrt {x \plus{} \sqrt {4x \plus{} \sqrt {16x \plus{} \sqrt {\dotsc \plus{} \sqrt {4^{2008}x \plus{} 3}}}}} \minus{} \sqrt {x} \equal{} 1. \] Express your answer as a reduced fraction with the numerator and denominator written in their prime factorization.

Describe a method to convert any triangle into a rectangle with side 1 and area equal to the original triangle by dividing that triangle into finitely many subtriangles.

Given that $a,b,c > 0$ and $a + b + c = 1$. Prove that $\sqrt {\frac{{ab}}{{ab + c}}} + \sqrt {\frac{{bc}}{{bc + a}}} + \sqrt {\frac{{ca}}{{ca + b}}} \leqslant \frac{3}{2}$.

An arithmetic sequence is a sequence of numbers for which the difference between two consecutive numbers applies terms is constant. So this is an arithmetic sequence with difference $\frac56$: $$\frac13,\frac76, 2,\frac{17}{6},\frac{11}{3},\frac92.$$ The sequence of seven natural numbers $60$, $70$, $84$, $105$, $140$, $210$, $420$ has the property that the sequence inverted numbers (i.e. the row $\frac{1}{60}$, $\frac{1}{70}$, $\frac{1}{84}$, $\frac{1}{105}$, $\frac{1}{140}$, $\frac{1}{210}$,$\frac{1}{420}$ ) is an arithmetic sequence. (a) Is there a sequence of eight different natural numbers whose inverse numbers are one form an arithmetic sequence? (b) Is there an infinite sequence of distinct natural numbers whose inverses are form an arithmetic sequence?

Prove that for every given positive integer $k$, there exist infinitely many $n$, such that $2^{n}+3^{n}-1, 2^{n}+3^{n}-2,\ldots, 2^{n}+3^{n}-k$ are all composite numbers.

Evaluate the sum $$\sum_{n=1}^{\infty} \frac{1}{n^2(n+1)}$$

For a given positive integer $n$ and prime number $p$, find the minimum value of positive integer $m$ that satisfies the following property: for any polynomial $$f(x)=(x+a_1)(x+a_2)\ldots(x+a_n)$$ ($a_1,a_2,\ldots,a_n$ are positive integers), and for any non-negative integer $k$, there exists a non-negative integer $k'$ such that $$v_p(f(k))<v_p(f(k'))\leq v_p(f(k))+m.$$ Note: for non-zero integer $N$,$v_p(N)$ is the largest non-zero integer $t$ that satisfies $p^t\mid N$.

Alice and Bob play a number game. Starting with a positive integer $n$ they take turns changing the number with Alice going first. Each player may change the current number $k$ to either $k-1$ or $\lceil k/2\rceil$. The person who changes $1$ to $0$ wins. Determine all $n$ such that Alice has a winning strategy.

Compute the number of functions $f\colon\{1, \dots, 15\} \to \{1, \dots, 15\}$ such that, for all $x \in \{1, \dots, 15\}$, \[ \frac{f(f(x)) - 2f(x) + x}{15} \]is an integer. [i]Proposed by Ankan Bhattacharya[/i]

The minimum of $ \sin \frac{A}{2} \minus{} \sqrt3 \cos \frac{A}{2}$ is attained when $ A$ is $ \textbf{(A)}\ \minus{}180^{\circ} \qquad \textbf{(B)}\ 60^{\circ} \qquad \textbf{(C)}\ 120^{\circ} \qquad \textbf{(D)}\ 0^{\circ} \qquad \textbf{(E)}\ \text{none of these}$

Let $ f(x) \equal{} 1 \minus{} \cos x \minus{} x\sin x$. (1) Show that $ f(x) \equal{} 0$ has a unique solution in $ 0 < x < \pi$. (2) Let $ J \equal{} \int_0^{\pi} |f(x)|dx$. Denote by $ \alpha$ the solution in (1), express $ J$ in terms of $ \sin \alpha$. (3) Compare the size of $ J$ defined in (2) with $ \sqrt {2}$.

Find all functions $f: \mathbb{Z}^+\to \mathbb{R}$, which satisfies $f(n+1)\geq f(n)$ for all $n\geq 1$ and $f(mn)=f(m)f(n)$ for all $(m,n)=1$.

The real numbers $a_0,a_1,a_2,\ldots$ satisfy $1=a_0\le a_1\le a_2\le\ldots. b_1,b_2,b_3,\ldots$ are defined by $b_n=\sum_{k=1}^n{1-{a_{k-1}\over a_k}\over\sqrt a_k}$. [b]a.)[/b] Prove that $0\le b_n<2$. [b]b.)[/b] Given $c$ satisfying $0\le c<2$, prove that we can find $a_n$ so that $b_n>c$ for all sufficiently large $n$.

Le $S$ be the set of positive integers greater than or equal to $2$. A function $f: S\rightarrow S$ is italian if $f$ satifies all the following three conditions: 1) $f$ is surjective 2) $f$ is increasing in the prime numbers(that is, if $p_1<p_2$ are prime numbers, then $f(p_1)<f(p_2)$) 3) For every $n\in S$ the number $f(n)$ is the product of $f(p)$, where $p$ varies among all the primes which divide $n$ (For instance, $f(360)=f(2^3\cdot 3^2\cdot 5)=f(2)\cdot f(3)\cdot f(5)$). Determine the maximum and the minimum possible value of $f(2020)$, when $f$ varies among all italian functions.

$ 1 \le n \le 455$ and $ n^3 \equiv 1 \pmod {455}$. The number of solutions is ? $\textbf{(A)}\ 9 \qquad\textbf{(B)}\ 6 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 1 \qquad\textbf{(E)}\ \text{None}$

Show that there exists $n>2$ such that $1991 | 1999 \ldots 91$ (with $n$ 9's).

Let $ABC$ be a triangle, and $M$ and $N$ variable points on $AB$ and $AC$ respectively, such that both $M$ and $N$ do not lie on the vertices, and also, $AM \times MB = AN \times NC$. Prove that the perpendicular bisector of $MN$ passes through a fixed point.

On a long metal stick, $1000$ red marbles are embedded in the stick so the stick is equally partitioned into $1001$ parts by them. $1001$ blue marbles are embedded in the stick too, so the stick is equally partitioned into $1002$ parts by them. If you cut the metal stick equally into $2003$ smaller parts, how many of the smaller parts would contain at least one embedded marble?

Determine the minimum value of $$(r-1)^2 + \left(\frac{s}{r}-1 \right)^2 + \left(\frac{t}{s}-1 \right)^{2} + \left( \frac{4}{t} -1 \right)^2$$ for all real numbers $1\leq r \leq s \leq t \leq 4.$

Let $M$ be the midpoint of side $AB$ of the triangle $ABC$. Let$P$ be a point on $AB$ between $A$ and $M$, and let $MD$ be drawn parallel to $PC$ and intersecting $BC$ at $D$. If the ratio of the area of the triangle $BPD$ to that of triangle $ABC$ is denoted by $r$, then $\text{(A)}\ \tfrac{1}{2}<r<1\text{ depending upon the position of }P \qquad\\ \text{(B)}\ r=\tfrac{1}{2}\text{ independent of the position of }P\qquad\\ \text{(C)}\ \tfrac{1}{2}\le r<1\text{ depending upon the position of }P \qquad\\ \text{(D)}\ \tfrac{1}{3}<r<\tfrac{2}{3}\text{ depending upon the position of }P \qquad\\ \text{(E)}\ r=\tfrac{1}{3} \text{ independent of the position of }P$

Hannusya, Petrus and Mykolka drew independently one isosceles triangle $ABC$, all angles of which are measured as a integer number of degrees. It turned out that the bases $AC$ of these triangles are equals and for each of them on the ray $BC$ there is a point $E$ such that $BE=AC$, and the angle $AEC$ is also measured by an integer number of degrees. Is it in necessary that: a) all three drawn triangles are equal to each other? b) among them there are at least two equal triangles?

Find all pairs of integers $(x,y)$ satisfying the following condition: [i]each of the numbers $x^3 + y$ and $x + y^3$ is divisible by $x^2 + y^2$ [/i] Tournament of Towns