Let $a, b, c$ be positive real numbers satisfying \[a+b+c = a^2 + b^2 + c^2.\] Let \[M = \max\left(\frac{2a^2}{b} + c, \frac{2b^2}{a} + c \right) \quad \text{ and } \quad N = \min(a^2 + b^2, c^2).\] Find the minimum possible value of $M/N$.

ِDo there exist non-zero reals numbers $a_1, a_2, ....., a_{10}$ for which \[(a_1+\frac{1}{a_1})(a_2+\frac{1}{a_2}) \cdots(a_{10}+\frac{1}{a_{10}})= (a_1-\frac{1}{a_1})(a_2-\frac{1}{a_2})\cdots(a_{10}-\frac{1}{a_{10}}) \ ? \]

$ ABCD$ is a square. Let $ E$ be a point on the segment $ BC$ and $ F$ be a point on the segment $ ED$. If $ DF \equal{} BF$ and $ EF \equal{} BE$, then $ \angle DFA$ is $\textbf{(A)}\ 45^\circ \qquad\textbf{(B)}\ 60^\circ \qquad\textbf{(C)}\ 75^\circ \qquad\textbf{(D)}\ 80^\circ \qquad\textbf{(E)}\ 85^\circ$

In an acute triangle $ABC$, the feet of the perpendiculars from $A$ and $C$ to the opposite sides are $D$ and $E$, respectively. The line passing through $E$ and parallel to $BC$ intersects $AC$ at $F$, the line passing through $D$ and parallel to $AB$ intersects $AC$ at $G$. The feet of the perpendiculars from $F$ to $DG$ and $GE$ are $K$ and $L$, respectively. $KL$ intersects $ED$ at $M$. Prove that $FM \perp ED$.

Let $n\ge 3$ be an integer and $a_1,a_2,\dots ,a_n$ be a finite sequence of positive integers, such that, for $k=2,3,\dots ,n$ $$n(a_k+1)-(n-1)a_{k-1}=1.$$ Prove that $a_n$ is not divisible by $(n-1)^2$.

Let $ABC$ be a triangle with $AB\neq AC$, and let $A_{1}B_{1}C_{1}$ be the image of triangle $ABC$ through a rotation $R$ centered at $C$. Let $M,E , F$ be the midpoints of the segments $BA_{1}, AC, BC_{1}$ respectively Given that $EM = FM$, compute $\angle EMF$.

Inside the convex quadrilateral $ABCD$ there is a point $M$ such that $\angle AMB = \angle ADM + \angle BCM$ and $\angle AMD = \angle ABM + \angle DCM$. Prove that $AM \cdot CM + BM \cdot DM \ge \sqrt{AB \cdot BC\cdot CD \cdot DA}$

Path graph $U$ is given, and a blindfolded player is standing on one of its vertices. The vertices of $U$ are labeled with positive integers between 1 and $n$, not necessarily in the natural order. In each step of the game, the game master tells the player whether he is in a vertex with degree 1 or with degree 2. If he is in a vertex with degree 1, he has to move to its only neighbour, if he is in a vertex with degree 2, he can decide whether he wants to move to the adjacent vertex with the lower or with the higher number. All the information the player has during the game after $k$ steps are the $k$ degrees of the vertices he visited and his choice for each step. Is there a strategy for the player with which he can decide in finitely many steps how many vertices the path has?

Let $N$ be an integer greater than $1$. A deck has $N^3$ cards, each card has one of $N$ colors, has one of $N$ figures and has one of $N$ numbers (there are no two identical cards). A collection of cards of the deck is "complete" if it has cards of every color, or if it has cards of every figure or of all numbers. How many non-complete collections are there such that, if you add any other card from the deck, the collection becomes complete?

Let $\mathcal{S}$ denote the set of positive integer sequences (with at least two terms) whose terms sum to $2019$. For a sequence of positive integers $a_1, a_2, \dots, a_k$, its \emph{value} is defined to be \[V(a_1, a_2, \dots, a_k) = \frac{a_1^{a_2} a_2^{a_3} \cdots a_{k-1}^{a_k}}{a_1! a_2! \cdots a_k!}.\] Then the sum of the values over all sequences in $\mathcal{S}$ is $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Compute the remainder when $m+n$ is divided by $1000$. [i]Proposed by Sean Li[/i]

Given a real number $x>1$, prove that there exists a real number $y >0$ such that \[\lim_{n \to \infty} \underbrace{\sqrt{y+\sqrt {y + \cdots+\sqrt y}}}_{n \text{ roots}}=x.\]

Consider an acute triangle $ABC$. Let $D$, $E$ and $F$ be the feet of the altitudes through vertices $A$, $B$ and $C$. Denote by $A'$, $B'$, $C'$ the projections of $A$, $B$, $C$ onto lines $EF$, $FD$, $DE$, respectively. Further, let $H_D$, $H_E$, $H_F$ be the orthocenters of triangles $DB'C'$, $EC'A'$, $FA'B'$. Show that $$H_DB^2 + H_EC^2 + H_FA^2 = H_DC^2 + H_EA^2 + H_FB^2.$$

Let $f:\mathbb{R}^+\to\mathbb{R}^+$ be such that $$f(x+f(y))^2\geq f(x)\left(f(x+f(y))+f(y)\right)$$ for all $x,y\in\mathbb{R}^+$. Show that $f$ is [i]unbounded[/i], i.e. for each $M\in\mathbb{R}^+$, there exists $x\in\mathbb{R}^+$ such that $f(x)>M$.

How many ways are there to color the vertices of a square green, red, or blue so that no two adjacent vertices have the same color? (Two colorings are considered different even if one coloring can be rotated to product the other coloring.)

Given that $x$ and $y$ are nonzero real numbers such that $x+\frac{1}{y}=10$ and $y+\frac{1}{x}=\frac{5}{12}$, find all possible values of $x$.

Players $1,2,\ldots n$ are seated around a table, and each has a single penny. Player $1$ passes a penny to Player $2$, who then passes two pennies to Player $3$, who then passes one penny to player $4$, who then passes two pennies to Player $5$ and so on, players alternately pass one or two pennies to the next player who still has some pennies. The player who runs out of pennies drops out of the game and leaves the table. Find an infinite set of numbers $n$ for which some player ends up with all the $n$ pennies.

[b]a)[/b] Find an example of an infinite list of numbers of the form $a + n \cdot d$, with $n \geqslant 0$, where $a$ and $d$ are positive integers, such that no number in the list is equal to the $k$-th power of an integer, for all $k = 2, 3, 4, \dots$ [b]b)[/b] Find an example of an infinite list of numbers of the form $a + n \cdot d$, with $n \geqslant 0$, where $a$ and $d$ are positive integers, such that no number in the list is equal to the square of an integer, but the list contains infinitely many numbers that are equal to the cubes of positive integers.

Let $a > 0$ be a real number and let $f : R \to R$ be a function satisfying $$f(x_1) + f(x_2) \ge a f(x_1 + x_2), \forall x_1 ,x_2 \in R.$$ Prove that $$f(x_1) + f(x_2) +(x_3) \ge \frac{3a^2}{a+2} f(x_1+ x_2 + x_3), \forall x_1 ,x_2,x_3 \in R$$.

In the plane $xOy$, a lot of points are considered $$X = \{P (a, b) | (a, b) \in \{1, 2,..., 10\} \times \{1, 2,..., 10 \}\}$$ Determine the number of different lines that can be obtained by joining two of them between the points of the set $X$; so that any two lines are not parallel.

Find all positive real solutions to the equation $x+\left\lfloor\frac x3\right\rfloor=\left\lfloor\frac{2x}3\right\rfloor+\left\lfloor\frac{3x}5\right\rfloor$

The student wrote on the board three natural numbers that are consecutive members of one arithmetic progression. Then he erased the commas separating the numbers, resulting in a seven-digit number. What is the largest number that could result?

The sidelengths of a triangle $ABC$ are not greater than $1$. Prove that $p(1 -2Rr)$ is not greater than $1$, where $p$ is the semiperimeter, $R$ and $r$ are the circumradius and the inradius of $ABC$.

Let $ABC$ be a triangle with $AB=AC$, and let $M$ be the midpoint of $BC$. Let $P$ be a point such that $PB<PC$ and $PA$ is parallel to $BC$. Let $X$ and $Y$ be points on the lines $PB$ and $PC$, respectively, so that $B$ lies on the segment $PX$, $C$ lies on the segment $PY$, and $\angle PXM=\angle PYM$. Prove that the quadrilateral $APXY$ is cyclic.

A robot initially at position $0$ along a number line has a [i]movement function[/i] $f(u, v)$. It rolls a fair $26$-sided die repeatedly, with the $k$-th roll having value $r_k$. For $k \ge 2$, if $r_k > r_{k-1}$, it moves $f(r_k, r_{k-1})$ units in the positive direction. If $r_k < r_{k-1}$, it moves $f(r_k, r_{k-1})$ units in the negative direction. If $r_k = r_{k-1}$, all movement and die-rolling stops and the robot remains at its final position $x$. If $f(u, v) = (u^2 - v^2)^2 + (u - 1)(v + 1)$, compute the expected value of $x$.

Let the sequence of rationals $x_1,x_2,\dots$ be defined such that $x_1=\frac{25}{11}$ and \[x_{k+1}=\frac{1}{3}\left(x_k+\frac{1}{x_k}-1\right).\] $x_{2025}$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Find the remainder when $m+n$ is divided by $1000$.