The functions $f,g:[0,1]\to\mathbb R$ are given with the formulas $$f(x)=\sqrt[4]{1-x},\enspace g(x)=f(f(x)),$$ and $c$ denotes any solution of $x=f(x)$. (a) i. Analyze the function $f(x)$ and draw its graph. Prove that the equation $f(x)=x$ has the unique root $c$ satisfying $c\in[0.72,0.73]$. ii. Analyze the function $f'(x)$. Let $M_1$ and $M_2$ be the points of the graph of $f(x)$ with different $x$ coordinates. What is the position of the arc $M_1M_2$ of the graph with respect to the segment $M_1M_2$? iii. Analyze the function $g(x)$ and draw its graph. What is the position of that graph with respect to the line $y=x$? Find the tangents to the graph at points with $x$ coordinates $0$ and $1$. iv. Prove that every sequence $\{a_n\}$ with the conditions $a_1\in(0,1)$ and $a_{n+1}=f(a_n)$ for $n\in\mathbb N$ converges. [hide=Official Hint]Consider the sequences $\{a_{2n-1}\},\{a_{2n}\}~(n\in\mathbb N)$ and the function $g(x)$ associated with the graph.[/hide] (b) On the graph of the function $f(x)$ consider the points $M$ and $M'$ with $x$ coordinates $x$ and $f(x)$, where $x\ne c$. i. Prove that the line $MM'$ intersects with the line $y=x$ at the point with $x$ coordinate $$h(x)=x-\frac{(f(x)-x)^2}{g(x)+x-2f(x)}.$$ ii. Prove that if $x\in(0,c)$ then $h(x)\in(x,c)$. iii. Analyze whether the sequence $\{a_n\}$ satisfying $a_1\in(0,c),a_{n+1}=h(a_n)$ for $n\in\mathbb N$ converges. Prove that the sequence $\{\tfrac{a_{n+1}-c}{a_n-c}\}$ converges and find its limit. (c) Assume that the calculator approximates every number $b\in[-2,2]$ by number $\overline b$ having $p$ decimal digits after the decimal point. We are performing the following sequence of operations on that calculator: 1) Set $a=0.72$; 2) Calculate $\delta(a)=\overline{f(a)}-a$; 3) If $|\delta(a)|>0.5\cdot10^{-p}$, then calculate $\overline{h(a)}$ and go to the operation $2)$ using $\overline{h(a)}$ instead of $a$; 4) If $|\delta(a)|\le0.5\cdot10^{-p}$, finish the calculation. Let $\overleftrightarrow c$ be the last of calculated values for $\overline{h(a)}$. Assuming that for each $x\in[0.72,0.73]$ we have $\left|\overline{f(x)}-f(x)\right|<\epsilon$, determine $\delta(\overleftrightarrow c)$, the accuracy (depending on $\epsilon$) of the approximation of $c$ with $\overleftrightarrow c$. (d) Assume that the sequence $\{a_n\}$ satisfies $a_1=0.72$ and $a_{n+1}=f(a_n)$ for $n\in\mathbb N$. Find the smallest $n_0\in\mathbb N$, such that for every $n\ge n_0$ we have $|a_n-c|<10^{-6}$.

How many rearrangements of $abcd$ are there in which no two adjacent letters are also adjacent letters in the alphabet? For example, no such rearrangements could include either $ab$ or $ba$. $ \textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }3\qquad\textbf{(E) }4 $

The “flying rook” moves as the usual chess rook but can’t move to a neighbouring square in one move. Is it possible for the flying rook on a $4 \times 4$ chess-board to visit every square once and return to the initial square in $16$ moves? (A. Tolpygo, Kiev)

Call an ordered triple $(a, b, c)$ of integers feral if $b -a, c - a$ and $c - b$ are all prime. Find the number of feral triples where $1 \le a < b < c \le 20$.

Solve the system $\begin{cases} x+ y +z = a \\ x^2 + y^2 + z^2 = a^2 \\ x^3 + y^3 +z^3 = a^3 \end{cases}$

Let $C_1, C_2$ and $C_3$ be three circles, of radii $2, 4$ and $6$ respectively. It is known that each of them are tangent exteriorly with the other two circles. Let $\Omega_1$ and $\Omega_2$ be two more circles, each of them tangent to all of the $3$ circles above, of radius $\omega_1$ and $\omega_2$ respectively. Prove that $\omega_1 + \omega_2 = 2\omega_1\omega_2$.

Let $R^+$ be the set of positive real numbers. Determine all functions $f:R^+$ $\rightarrow$ $R^+$ such that for all positive real numbers $x$ and $y:$ \[f(x+f(xy))+y=f(x)f(y)+1\] [i]Ukraine[/i]

Sum of all coefficients of polynomial $P(x)$ is equal with $2$ . Also the sum of coefficients which are at odd exponential in $x^k$ are equal to sum of coefficients which are at even exponential in $x^k$ . Find the residue of polynomial $P(x)$ when it is divide by $x^2-1$ .

Angle bisectors from vertices $B$ and $C$ and the perpendicular bisector of side $BC$ are drawn in a non-isosceles triangle $ABC$. Next, three points of pairwise intersection of these three lines were marked (remembering which point is which), and the triangle itself was erased. Restore it according to the marked points using a compass and ruler. (Yu. Blinkov)

For a positive integer $n,$ let $f(n)$ be the number of integers $m$ satisfying $0 \le m \le n - 1$ such that there exists an integer solution to the congruence $x^2 \equiv m \pmod{n}.$ It is given that as $k$ goes to $\infty,$ the value of $f(225^k)/225^k$ converges to some rational number $p/q,$ where $p,q$ are relatively prime positive integers. Find $p + q.$

There are cities in country, and some cities are connected by roads. Not more than $100$ roads go from every city. Set of roads is called as ideal if all roads in set have not common ends, and we can not add one more road in set without breaking this rule. Every day minister destroy one ideal set of roads. Prove, that he need not more than $199$ days to destroy all roads in country.

Let $a_1,a_2,a_3,\dots$ be an infinite sequence of non-zero reals satisfying \[a_{i} = \frac{a_{i-1}a_{i-2}-2}{a_{i-3}}\]for all $i\geq 4$. Determine all positive integers $n$ such that if $a_1,a_2,\dots,a_n$ are integers, then all elements of the sequence are integers.

A positive integer will be called [i]typical[/i] if the sum of its decimal digits is a multiple of $2011$. a) Show that there are infinitely many [i]typical[/i] numbers, each having at least $2011$ multiples which are also typical numbers. b) Does there exist a positive integer such that each of its multiples is typical?

Prove that $$\frac{2}{3}+\frac{4}{5}+\dots +\frac{2010}{2011}$$ is not an integer.

[b]p1.[/b] Is there an integer such that the product of all whose digits equals $99$ ? [b]p2.[/b] An elevator in a $100$ store building has only two buttons: UP and DOWN. The UP button makes the elevator go $13$ floors up, and the DOWN button makes it go $8$ floors down. Is it possible to go from the $13$th floor to the $8$th floor? [b]p3.[/b] Cut the triangle shown in the picture into three pieces and rearrange them into a rectangle. (Pieces can not overlap.) [img]https://cdn.artofproblemsolving.com/attachments/9/f/359d3b987012de1f3318c3f06710daabe66f28.png[/img] [b]p4.[/b] Two players Tom and Sid play the following game. There are two piles of rocks, $5$ rocks in the first pile and $6$ rocks in the second pile. Each of the players in his turn can take either any amount of rocks from one pile or the same amount of rocks from both piles. The winner is the player who takes the last rock. Who does win in this game if Tom starts the game? [b]p5.[/b] In the next long multiplication example each letter encodes its own digit. Find these digits. $\begin{tabular}{ccccc} & & & a & b \\ * & & & c & d \\ \hline & & c & e & f \\ + & & a & b & \\ \hline & c & f & d & f \\ \end{tabular}$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Francesca uses 100 grams of lemon juice, 100 grams of sugar, and 400 grams of water to make lemonade. There are 25 calories in 100 grams of lemon juice and 386 calories in 100 grams of sugar. Water contains no calories. How many calories are in 200 grams of her lemonade? $ \textbf{(A) } 129 \qquad \textbf{(B) } 137 \qquad \textbf{(C) } 174 \qquad \textbf{(D) } 223 \qquad \textbf{(E) } 411$

Find all $x \in R$ such that $$ x - \left[ \frac{x}{2016} \right]= 2016$$, where $[k]$ represents the largest smallest integer or equal to $k$.

Let $(a_n )_{n \ge o}$ and $(b_n )_{n \ge o}$ be sequences defined by $a_{n+2} = a_{n+1}+ a_n$ , $n = 0 , 1 , . .. $, $a_0 = 1$, $a_1 = 2$, and $b_{n+2} = b_{n+1} + b_n$ , $n = 0 , 1 , . . .$, $b_0 = 2$, $b_1 = 1$. How many integers do the sequences have in common?

Determine all positive integers $ n$, for which $ 2^{n\minus{}1}n\plus{}1$ is a perfect square.

The eighth grade class at Lincoln Middle School has $93$ students. Each student takes a math class or a foreign language class or both. There are $70$ eighth graders taking a math class, and there are $54$ eighth graders taking a foreign language class. How many eighth graders take [i]only[/i] a math class and [i]not[/i] a foreign language class? $\textbf{(A) }16\qquad \textbf{(B) }23\qquad \textbf{(C) }31\qquad \textbf{(D) }39\qquad \textbf{(E) }70\qquad$

Suppose that the $n$ times differentiable real function $f(x)$ has at least $n+1$ distinct zeros in the closed interval $[a,b]$ and that the polynomial $P(z)=z^n +c_{n-1}z^{n-1}+\ldots+c_1 x +c_0$ has only real zeroes. Show that $f^{(n)}(x)+ c_{n-1} f^{(n-1)}(x) +\ldots +c_1 f'(x)+ c_0 f(x)$ has at least one zero in $[a,b]$, where $f^{(n)}$ denotes the $n$-th derivative of $f.$

Prove that $$AB + PQ + QR + RP \le AP + AQ + AR + BP + BQ + BR$$ where $A, B, P, Q$ and $R $ are any five points in a plane.

Solve the equation, $$\sqrt{x+5}+\sqrt{16-x^2}=x^2-25$$

Collinear points $A$, $B$, and $C$ are given in the Cartesian plane such that $A= (a, 0)$ lies along the x-axis, $B$ lies along the line $y=x$, $C$ lies along the line $y=2x$, and $\frac{AB}{BC}=2$. If $D= (a, a)$, and the circumcircle of triangle $ADC$ intersects the line $y=x$ again at $E$, and ray $AE$ intersects $y=2x$ at $F$, evaluate $\frac{AE}{EF}$.

Let $n\in\mathbb{N}, x_0=0$ and $x_1,x_2,\ldots,x_n$ be postive real numbers such that $x_1+x_2+\ldots+x_n=1$. Show that $$1\leq\sum_{i=1}^{n}\frac{x_i}{\sqrt{1+x_0+x_1+\ldots+x_{i-1}}\cdot\sqrt{x_i+x_{i+1}+\ldots+x_n}}<\frac{\pi}{2}.$$