A series of lockers, numbered 1 through 100, are all initially closed. Student 1 goes through and opens every locker. Student 3 goes through and "flips" every 3rd locker ("flipping") a locker means changing its state: if the locker is open he closes it, and if the locker is closed he opens it). Thus, Student 3 will close the third locker, open the sixth, close the ninth. . . . Student 5 then goes through and "flips"every 5th locker. This process continues with all students with odd numbers $n<100$ going through and "flipping" every $n$th locker. How many lockers are open after this process?

A continuous function $ f: \mathbb {R} \to \mathbb {R} $ satisfies $ f (x) f (f (x)) = 1 $ for every real $ x $ and $ f (2020) = 2019 $ . What is the value of $ f (2018) $?

Right triangle $ABC$ has right angle at $C$ and $\angle BAC=\theta$; the point $D$ is chosen on $AB$ so that $|AC|=|AD|=1$; the point $E$ is chosen on $BC$ so that $\angle CDE=\theta$. The perpendicular to $BC$ at $E$ meets $AB$ at $F$. Evaluate $\lim_{\theta\to 0}|EF|$.

Find continuous functions $x(t),\ y(t)$ such that $\ \ \ \ \ \ \ \ \ x(t)=1+\int_{0}^{t}e^{-2(t-s)}x(s)ds$ $\ \ \ \ \ \ \ \ \ y(t)=\int_{0}^{t}e^{-2(t-s)}\{2x(s)+3y(s)\}ds$

Let $A_1A_2A_3A_4A_5A_6A_7A_8$ be a cyclic octagon. Let $B_i$ by the intersection of $A_iA_{i+1}$ and $A_{i+3}A_{i+4}$. (Take $A_9 = A_1$, $A_{10} = A_2$, etc.) Prove that $B_1, B_2, \ldots , B_8$ lie on a conic. [i]David Yang.[/i]

Show that there exists a $2 \times 2$ matrix of order 6 with rational entries, such that the sum of its entries is 2018. Note: The order of a matrix (if it exists) is the smallest positive integer $n$ such that $A^n = I$, where $I$ is the identity matrix.

In an exhibition there are $100$ paintings each of which is made with exactly $k$ colors. Find the minimum possible value of $k$ if any $20$ paintings have a common color but there is no color that is used in all paintings.

Let $x_1,x_2,x_3, \dots$ be a sequence of nonzero real numbers satisfying $$x_n=\frac{x_{n-2}x_{n-1}}{2x_{n-2}-x_{n-1}} \text{ for } n=3,4,5, \dots.$$ Establish necessary and sufficient conditions on $x_1$ and $x_2$ for $x_n$ to be an integer for infinitely many values of $n.$

Find all functions $f:\mathbb Z^+\to\mathbb R^+$ that satisfy $f(nk^2)=f(n)f^2(k)$ for all positive integers $n$ and $k$, furthermore $\lim\limits_{n\to\infty}\dfrac{f(n+1)}{f(n)}=1$.

Find the conditions that the functions $M(x, y)$ and $N (x, y)$ must satisfy in order that the differential equation $Mdx + Ndy =0$ shall have an integrating factor of the form $f(xy).$ You may assume that $M$ and $N$ have continuous partial derivatives of all orders.

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*}