Determine the smallest positive real number $r$ such that there exist differentiable functions $f: \mathbb{R} \rightarrow \mathbb{R}$ and $g: \mathbb{R} \rightarrow \mathbb{R}$ satisfying (a) $f(0)>0$, (b) $g(0)=0$, (c) $\left|f^{\prime}(x)\right| \leq|g(x)|$ for all $x$, (d) $\left|g^{\prime}(x)\right| \leq|f(x)|$ for all $x$, and (e) $f(r)=0$.

Let $ABC$ be a triangle with $AB > AC$ with incenter $I{}$. The internal bisector of the angle $BAC$ intersects the $BC$ at the point $D{}$. Let $M{}$ the midpoint of the segment $AD{}$, and let $F{}$ be the second intersection point of $MB$ with the circumcircle of the triangle $BIC$. Prove that $AF$ is perpendicular to $FC$.

Prove that for any natural numbers min the inequality holds $$1^m + 2^m + \ldots + n^m \geq n\cdot \left( \frac{n+1}{2}\right)^m$$

Let $a,b,c$ be real numbers such that $ab\not= 0$ and $c>0$. Let $(a_{n})_{n\geq 1}$ be the sequence of real numbers defined by: $a_{1}=a, a_{2}=b$ and \[a_{n+1}=\frac{a_{n}^{2}+c}{a_{n-1}}\] for all $n\geq 2$. Show that all the terms of the sequence are integer numbers if and only if the numbers $a,b$ and $\frac{a^{2}+b^{2}+c}{ab}$ are integers.

Let $\varepsilon$ be positive constant and $u$ satisfies that \[ \begin{cases} (\partial_t-\varepsilon\partial_x^2-\partial_y^2)u=0, & (t,x,y) \in \mathbb{R}_+ \times \mathbb{R} \times \mathbb{R}_+,\\ \partial_y u\vert_{y=0}=\partial_x h, &\\u\vert_{t=0}=0. & \end{cases}\] Here $h(t,x)$ is a smooth Schwartz function. Define the operator $e^{a\langle D\rangle}$ \[\mathcal{F}_x(e^{a\langle D\rangle} f)(k)=e^{a\langle k\rangle} \mathcal{F}_x(f)(k), \quad \langle k\rangle=1+\vert k\vert,\] where $\mathcal{F}_x$ stands for the Fourier transform in $x$. Show that \[\int_0^T \|e^{(1-s)\langle D\rangle} u\|_{L_{x,y}^2}^2 ds \le C \int_0^T \|e^{(1-s)\langle D\rangle} h\|_{H_x^{\frac{1}{4}}}^2 ds\] with constant $C$ independent of $\varepsilon, T$ and $h$.

Determine all functions $f:\mathbb{R}\setminus\{0\}\to \mathbb{R}\setminus\{0\}$ such that $$f(x^2yf(x))+f(1)=x^2f(x)+f(y)$$ holds for all nonzero real numbers $x$ and $y$.

Which of the following functions are differentiable at $x=0$? $\textbf{(A)}~f(x)=\begin{cases}\tan^{-1}\left(\frac1{|x|}\right)&\text{if }x\ne0\\\frac\pi2&\text{if }x=0\end{cases}$ $\textbf{(B)}~f(x)=|x|^{1/2}x$ $\textbf{(C)}~f(x)=\begin{cases}x^2\left|\cos\frac{\pi}x\right|&\text{if }x\ne0\\0&\text{if }x=0\end{cases}$ $\textbf{(D)}~\text{None of the above}$

Consider the sequence $ a_n\equal{}\left|z^n\plus{}\frac{1}{z^n}\right|\ ,\ n\ge 1$, where $ z\in \mathbb{C}^*$ is given. i) Prove that if $ a_1>2$, then: \[ a_{n\plus{}1}<\frac{a_n\plus{}a_{n\plus{}2}}{2}\ ,\ (\forall)n\in \mathbb{N}^*\] ii) Prove that if there is a $ k\in \mathbb{N}^*$ such that $ a_k\le 2$, then $ a_1\le 2$.

Consider the set $$ \mathcal{S}=\{(a, b, c, d, e): 0<a<b<c<d<e<100\} $$ where $a, b, c, d, e$ are integers. If $D$ is the average value of the fourth element of such a tuple in the set, taken over all the elements of $\mathcal{S}$, find the largest integer less than or equal to $D$.

Let $a_n$ be a sequence given by $a_1 = 18$, and $a_n = a_{n-1}^2+6a_{n-1}$, for $n>1$. Prove that this sequence contains no perfect powers.

Calculate the sum $ x^4 + y^4 + z^4 $ knowing that $ x + y + z = 0 $ and $ x^2 + y^2 + z^2 = a $, where $ a $ is a given positive number.

[b]interesting sequence[/b] $n$ is a natural number and $x_1,x_2,...$ is a sequence of numbers $1$ and $-1$ with these properties: it is periodic and its least period number is $2^n-1$. (it means that for every natural number $j$ we have $x_{j+2^n-1}=x_j$ and $2^n-1$ is the least number with this property.) There exist distinct integers $0\le t_1<t_2<...<t_k<n$ such that for every natural number $j$ we have \[x_{j+n}=x_{j+t_1}\times x_{j+t_2}\times ... \times x_{j+t_k}\] Prove that for every natural number $s$ that $s<2^n-1$ we have \[\sum_{i=1}^{2^n-1}x_ix_{i+s}=-1\] Time allowed for this question was 1 hours and 15 minutes.

At a party there are only single women and married men with their wives. The probability that a randomly selected woman is single is $\frac{2}{5}$. What fraction of the people in the room are married men? $\textbf{(A)}\ \frac{1}{3}\qquad \textbf{(B)}\ \frac{3}{8}\qquad \textbf{(C)}\ \frac{2}{5}\qquad \textbf{(D)}\ \frac{5}{12}\qquad \textbf{(E)}\ \frac{3}{5}$

Given a $ \triangle ABC $ with circumcircle $ \Gamma. $ Let $ A' $ be the antipode of $ A $ in $ \Gamma $ and $ D $ be the point s.t. $ \triangle BCD $ is an equilateral triangle ($ A $ and $ D $ are on the opposite side of $ BC $). Let the perpendicular from $ A' $ to $ A'D $ cuts $ CA, $ $ AB $ at $ E, $ $ F, $ resp. and $ T $ be the point s.t. $ \triangle ETF $ is an isosceles triangle with base $ EF $ and base angle $ 30^{\circ} $ ($ A $ and $ T $ are on the opposite side of $ EF $). Prove that $ AT $ passes through the 9-point center of $ \triangle ABC. $ [i]Proposed by Telv Cohl[/i]

a) Find : $A=\{(a,b,c) \in \mathbb{R}^{3} | a+b+c=3 , (6a+b^2+c^2)(6b+c^2+a^2)(6c+a^2+b^2) \neq 0\}$ b) Prove that for any $(a,b,c) \in A$ next inequality hold : \begin{align*} \frac{a}{6a+b^2+c^2}+\frac{b}{6b+c^2+a^2}+\frac{c}{6c+a^2+b^2} \le \frac{3}{8} \end{align*}

In the garden of Wonderland, there are $2016$ apples, $2017$ bananas and $2018$ oranges.Two monkeys Adu and Bakar play the following game: alternatively each of them takes and eats one fruit of any kind except for the one that he took in previous turn (in the first turn, each of them can take a fruit of any kind). Who can not take a fruit is the loser. Which monkey has the winning strategy if Adu plays first?

Find positive integers $a_1 < a_2<... <a_{2010}$ such that $$a_1(1!)^{2010} + a_2(2!)^{2010} + ... + a_{2010}(2010!)^{2010} = (2011 !)^{2010}. $$

It takes Clea $60$ seconds to walk down an escalator when it is not operating and only $24$ seconds to walk down the escalator when it is operating. How many seconds does it take Clea to ride down the operating escalator when she just stands on it? $ \textbf{(A)}\ 36\qquad\textbf{(B)}\ 40\qquad\textbf{(C)}\ 42\qquad\textbf{(D)}\ 48\qquad\textbf{(E)}\ 52 $

A person who left home between 4 p.m. and 5 p.m. returned between 5 p.m. and 6 p.m. and found that the hands of his watch had exactly exchanged place, when did he go out ?

Let $P(x)$ and $Q(x)$ be polynomials with nonnegative coefficients. We denote by $P'(x)$ the derivative of $P(x)$. Suppose that $P(0)=Q(0)=0$ and $Q(1) \leq 1 \leq P'(0)$. $(1)$ Prove that $0 \leq Q(x) \leq x \leq P(x)$ for all $0 \leq x \leq 1$. $(2)$ Prove that $P(Q(x)) \leq Q(P(x))$ for all $0 \leq x \leq 1$. [i]Proposed by Otgonbayar Uuye.[/i]

A coin is flipped three times. What is the probability that there are no instances of two consecutive heads or two consecutive tails? $\textbf{(A) }\frac{1}{8}\qquad\textbf{(B) }\frac{1}{4}\qquad\textbf{(C) }\frac{3}{8}\qquad\textbf{(D) }\frac{5}{8}\qquad\textbf{(E) }\frac{3}{4}$

Given $2021$ distinct positive integers non divisible by $2^{1010}$, show that it's always possible to choose $3$ of them $a$, $b$ and $c$, such that $|b^2-4ac|$ is not a perfect square.

Given a cyclic quadrilateral $ABCD$, let the diagonals $AC$ and $BD$ meet at $E$ and the lines $AD$ and $BC$ meet at $F$. The midpoints of $AB$ and $CD$ are $G$ and $H$, respectively. Show that $EF$ is tangent at $E$ to the circle through the points $E$, $G$ and $H$. [i]Proposed by David Monk, United Kingdom[/i]

Evaluate the integrals as follows. (1) $\int \frac{x^2}{2-x}\ dx$ (2) $\int \sqrt[3]{x^5+x^3}\ dx$ (3) $\int_0^1 (1-x)\cos \pi x\ dx$

The graphs of $ x^2 \plus{} y^2 \equal{} 4 \plus{} 12x \plus{} 6y$ and $ x^2 \plus{} y^2 \equal{} k \plus{} 4x \plus{} 12y$ intersect when $ k$ satisfies $ a \leq k \leq b$, and for no other values of $ k$. Find $ b \minus{} a$. $ \textbf{(A)}\ 5\qquad \textbf{(B)}\ 68\qquad \textbf{(C)}\ 104\qquad \textbf{(D)}\ 140\qquad \textbf{(E)}\ 144$