[b]5.[/b] Define the sequence $\{c_n\}_{n=1}^{\infty}$ as follows: $c_1= \frac {1}{2}$, $c_{n+1}= c_{n}-c_{n}^2$($n\geq 1$). Prove that $\lim_{n \to \infty} nc_n= 1$ [b](S.12)[/b]

Let $\triangle PQR$ be a triangle with $\angle P = 75^\circ$ and $\angle Q = 60^\circ$. A regular hexagon $ABCDEF$ with side length 1 is drawn inside $\triangle PQR$ so that side $\overline{AB}$ lies on $\overline{PQ}$, side $\overline{CD}$ lies on $\overline{QR}$, and one of the remaining vertices lies on $\overline{RP}$. There are positive integers $a$, $b$, $c$, and $d$ such that the area of $\triangle PQR$ can be expressed in the form $\tfrac{a+b\sqrt c}d$, where $a$ and $d$ are relatively prime and $c$ is not divisible by the square of any prime. Find $a+b+c+d$.

How many zeros are at the end of the product \[25\times 25\times 25\times 25\times 25\times 25\times 25\times 8\times 8\times 8?\] $\text{(A)}\ 3 \qquad \text{(B)}\ 6 \qquad \text{(C)}\ 9 \qquad \text{(D)}\ 10 \qquad \text{(E)}\ 12$

For an integer $n>3$ denote by $n?$ the product of all primes less than $n$. Solve the equation $n?=2n+16$. [i]V. Senderov [/i]

What is the average (mean) of all $ 5$-digit numbers that can be formed by using each of the digits $ 1$, $ 3$, $ 5$, $ 7$, and $ 8$ exactly once? $ \textbf{(A)}\ 48000\qquad \textbf{(B)}\ 49999.5\qquad \textbf{(C)}\ 53332.8\qquad \textbf{(D)}\ 55555\qquad \textbf{(E)}\ 56432.8$

Let $M$ be the midpoint of side $BC$ of an equilateral triangle $ABC$. The point $D$ is on $CA$ extended such that $A$ is between $D$ and $C$. The point $E$ is on $AB$ extended such that $B$ is between $A$ and $E$, and $|MD| = |ME|$. The point $F$ is the intersection of $MD$ and $AB$. Prove that $\angle BFM = \angle BME$.

Find the smallest positive $x$ for which holds the inequality $$\sin x \le \sin (x+1)\le \sin (x+2)\le sin (x+3)\le \sin (x+4) .$$

Each person stands on a whole number on the number line from $0$ to $2022$ . In each turn, two people are selected by a distance of at least $2$. These go towards each other by $1$. When no more such moves are possible, the process ends. Show that this process always ends after a finite number of moves, and determine all possible configurations where people can end up standing. (whereby is for each configuration is only of interest how many people stand at each number.) [i](Birgit Vera Schmidt)[/i] [hide=original wording]Bei jeder ganzen Zahl auf dem Zahlenstrahl von 0 bis 2022 steht zu Beginn eine Person. In jedem Zug werden zwei Personen mit Abstand mindestens 2 ausgewählt. Diese gehen jeweils um 1 aufeinander zu. Wenn kein solcher Zug mehr möglich ist, endet der Vorgang. Man zeige, dass dieser Vorgang immer nach endlich vielen Zügen endet, und bestimme alle möglichen Konfigurationen, wo die Personen am Ende stehen können. (Dabei ist für jede Konfiguration nur von Interesse, wie viele Personen bei jeder Zahl stehen.)[/hide]

Let be a natural number $ n\ge 2 $ and a real number $ r>1. $ Determine the natural numbers $ k $ having the property that the affixes of $ r^ke^{\pi ki/n} ,r^{k+1}e^{\pi (k+1)i/n} ,r^{k+n}e^{\pi (k+n)i/n} ,r^{k+n+1}e^{\pi (k+n+1) i/n} $ in the complex plane represent the vertices of a trapezoid.

In a table-tennis tournament of $10$ contestants, any $2$ contestants meet only once. We say that there is a winning triangle if the following situation occurs: $i$-th contestant defeated the $j$-th contestant, $j$-th contestant defeated the $k$-th contestant, and, $k$-th contestant defeated the $i$-th contestant. Let, $W_i$ and $L_i $ be respectively the number of games won and lost by the $i$-th contestant. Suppose, $L_i+W_j\geq 8$ whenever the $j$-th contestant defeats the $i$-th contestant. Prove that, there are exactly $40$ winning triangles in this tournament.

Consider an acute triangle $\Delta ABC$. Let $D$ and $E$ be the feet of the altitudes from $A$ to $BC$ and from $B$ to $AC$ respectively. Define $D_1$ and $D_2$ as the reflections of $D$ across lines $AB$ and $AC$, respectively. Let $\Gamma$ be the circumcircle of $\Delta AD_1D_2$. Denote by $P$ the second intersection of line $D_1B$ with $\Gamma$, and by $Q$ the intersection of ray $EB$ with $\Gamma$. If $O$ is the circumcenter of $\Delta ABC$, prove that $O$, $D$, and $Q$ are collinear if and only if quadrilateral $BCQP$ can be inscribed within a circle. $\textbf{Proposed by Kritesh Dhakal, Nepal.}$

$\forall n\in \mathbb{Z_{+}}-\{1,2\}$ find the maximal length of a sequence with elements from a set $\{1,2,\ldots,n\}$, such that any two consecutive elements of this sequence are different and after removing all elements except for the four we do not receive a sequence in form $x,y,x,y$ ($x\neq y$).

Circles $\Omega $ and $\omega $ are tangent at a point $P$ ($\omega $ lies inside $\Omega $). A chord $AB$ of $\Omega $ is tangent to $\omega $ at $C;$ the line $PC$ meets $\Omega $ again at $Q.$ Chords $QR$ and $QS$ of $ \Omega $ are tangent to $\omega .$ Let $I,X,$ and $Y$ be the incenters of the triangles $APB,$ $ARB,$ and $ASB,$ respectively. Prove that $\angle PXI+\angle PYI=90^{\circ }.$

Line segment $AB$ has length $4$ and midpoint $M$. Let circle $C_1$ have diameter $AB$, and let circle $C_2$ have diameter $AM$. Suppose a tangent of circle $C_2$ goes through point $ B$ to intersect circle $C_1$ at $N$. Determine the area of triangle $AMN$.

The area of the region bounded by the graph of $$x^2 + y^2 = 3|x-y| + 3|x+y|$$ is $m + n \pi,$ where $m$ and $n$ are integers. What is $m+n$? $\textbf{(A)} 18\qquad\textbf{(B)} 27\qquad\textbf{(C)} 36\qquad\textbf{(D)} 45\qquad\textbf{(E)} 54$

Find all triples of primes $(p,q,r)$ satisfying $3p^{4}-5q^{4}-4r^{2}=26$.

Let $ n\geq 3 $ be an integer and let $ p\geq 2n-3 $ be a prime number. For a set $ M $ of $ n $ points in the plane, no 3 collinear, let $ f: M\to \{0,1,\ldots, p-1\} $ be a function such that (i) exactly one point of $ M $ maps to 0, (ii) if a circle $ \mathcal{C} $ passes through 3 distinct points of $ A,B,C\in M $ then $ \sum_{P\in M\cap \mathcal{C}} f(P) \equiv 0 \pmod p $. Prove that all the points in $ M $ lie on a circle.

Does there exist a pair $ (f; g)$ of strictly monotonic functions, both from $ \mathbb{N}$ to $ \mathbb{N}$, such that \[ f(g(g(n))) < g(f(n))\] for every $ n \in\mathbb{N}$?

For a polynomial $P(x)$ with integer coefficients and a prime $p$, if there is no $n \in \mathbb{Z}$ such that $p|P(n)$, we say that polynomial $P$ [i]excludes[/i] $p$. Is there a polynomial with integer coefficients such that having degree of 5, excluding exactly one prime and not having a rational root?

Find all polynomials $ P(x)$ with real coefficients such that for every positive integer $ n$ there exists a rational $ r$ with $ P(r)=n$.

The rhombuses $ABDK$ and $CBEL$ are arranged so that $B$ lies on the segment $AC$ and $E$ lies on the segment $BD$. Point $M$ is the midpoint of $KL$. Prove that $\angle DME=90^{\circ}$.

Prove that $$x\cos x \le \frac{\pi^2}{16}$$ for $0 \le x \le \frac{\pi}{2}$

The quadratic trinomial $x^2 + bx + c$ has two roots belonging to the interval $(2, 3)$. Prove that $5b+2c+12 < 0$.

Let $p_i$ denote the $i^{\text{th}}$ prime number. For any positive integer $k$ let $a_k$ denote the number of positive integers $t$ such that $p_tp_{t+1}$ divides $k.$ Let $n$ be an arbitrary positive integer. Prove that \[a_1+a_2+\cdots+a_n<\frac{n}{3}.\]

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be the function as \[ f(x) = \begin{cases} \frac{1}{x-1}& (x > 1)\\ 1& (x=1)\\ \frac{x}{1-x} & (x<1) \end{cases} \] Let $x_1$ be a positive irrational number which is a zero of a quadratic polynomial with integer coefficients. For every positive integer $n$, let $x_{n+1} = f(x_n)$. Prove that there exists different positive integers $k$ and $\ell$ such that $x_k = x_\ell$.