$2.$Let $ABC$ be an acute-angeled triangle , non-isosceles and with barycentre $G$ (which is , in fact , the intersection of the medians).Let $M$ be the midpoint of $BC$ , and let Ω be the circle with centre $G$ and radius $GM$ , and let $N$ be the point of intersection between Ω and $BC$ that is distinct from $M$.Let $S$ be the symmetric point of $A$ with respect to $N$ , that is , the point on the line $AN$ such that $AN=NS$. Prove that $GS$ is perpendicular to $BC$

Given the monic polynomial \begin{align*} P(x) = x^N +a_{N-1}x^{N-1} + \ldots + a_1 x + a_0 \in \mathbb{R}[x] \end{align*} of even degree $N$ $=$ $2n$ and having all real positive roots $x_i$, for $1 \le i \le N$. Prove, for any $c$ $\in$ $[0, \underset{1 \le i \le N}{\min} \{x_i \} )$, the following inequality \begin{align*} c + \sqrt[N]{P(c)} \le \sqrt[N]{a_0} \end{align*}

Prove that if $X{}$ is an infinite set of cardinality $\kappa$ then there is a collection $\mathcal{F}$ of subsets of $X$ such that[list] [*]For any $A\subseteq X$ with cardinality $\kappa$ there exists $F\in\mathcal{F}$ for which $A\cap F$ has cardinality $\kappa,$ and [*]$X$ cannot be written as the union of less than $\kappa$ sets from $\mathcal{F}$ and a single set of cardinality less than $\kappa$. [/list]

The sequence of nonnegative integers $F_0, F_1, F_2, \dots$ is defined recursively as $F_0 = 0$, $F_1 = 1$, and $F_{n+2}= F_{n+1} + F_{n}$ for all integers $n \geq 0$. Let $d$ be the largest positive integer such that, for all integers $n\geq 0$, $d$ divides $F_{n+2020}-F_n$. Compute the remainder when $d$ is divided by $1001$. [i]Proposed by Ankit Bisain[/i]

[b]p1.[/b] Lisa is playing the piano at a tempo of $80$ beats per minute. If four beats make one measure of her rhythm, how many seconds are in one measure? [b]p2.[/b] Compute the smallest integer $n > 1$ whose base-$2$ and base-$3$ representations both do not contain the digit $0$. [b]p3.[/b] In a room of $24$ people, $5/6$ of the people are old, and $5/8$ of the people are male. At least how many people are both old and male? [b]p4.[/b] Juan chooses a random even integer from $1$ to $15$ inclusive, and Gina chooses a random odd integer from $1$ to $15$ inclusive. What is the probability that Juan’s number is larger than Gina’s number? (They choose all possible integers with equal probability.) [b]p5.[/b] Set $S$ consists of all positive integers less than or equal to $ 2016$. Let $A$ be the subset of $S$ consisting of all multiples of $6$. Let $B$ be the subset of $S$ consisting of all multiples of $7$. Compute the ratio of the number of positive integers in $A$ but not $B$ to the number of integers in $B$ but not $A$. [b]p6.[/b] Three peas form a unit equilateral triangle on a flat table. Sebastian moves one of the peas a distance $d$ along the table to form a right triangle. Determine the minimum possible value of $d$. [b]p7.[/b] Oumar is four times as old as Marta. In $m$ years, Oumar will be three times as old as Marta will be. In another $n$ years after that, Oumar will be twice as old as Marta will be. Compute the ratio $m/n$. [b]p8.[/b] Compute the area of the smallest square in which one can inscribe two non-overlapping equilateral triangles with side length $ 1$. [b]p9.[/b] Teemu, Marcus, and Sander are signing documents. If they all work together, they would finish in $6$ hours. If only Teemu and Sander work together, the work would be finished in 8 hours. If only Marcus and Sander work together, the work would be finished in $10$ hours. How many hours would Sander take to finish signing if he worked alone? [b]p10.[/b]Triangle $ABC$ has a right angle at $B$. A circle centered at $B$ with radius $BA$ intersects side $AC$ at a point $D$ different from $A$. Given that $AD = 20$ and $DC = 16$, find the length of $BA$. [b]p11.[/b] A regular hexagon $H$ with side length $20$ is divided completely into equilateral triangles with side length $ 1$. How many regular hexagons with sides parallel to the sides of $H$ are formed by lines in the grid? [b]p12[/b]. In convex pentagon $PEARL$, quadrilateral $PERL$ is a trapezoid with side $PL$ parallel to side $ER$. The areas of triangle $ERA$, triangle $LAP$, and trapezoid $PERL$ are all equal. Compute the ratio $\frac{PL}{ER}$. [b]p13.[/b] Let $m$ and $n$ be positive integers with $m < n$. The first two digits after the decimal point in the decimal representation of the fraction $m/n$ are $74$. What is the smallest possible value of $n$? [b]p14.[/b] Define functions $f(x, y) = \frac{x + y}{2} - \sqrt{xy}$ and $g(x, y) = \frac{x + y}{2} + \sqrt{xy}$. Compute $g (g (f (1, 3), f (5, 7)), g (f (3, 5), f (7, 9)))$. [b]p15.[/b] Natalia plants two gardens in a $5 \times 5$ grid of points. Each garden is the interior of a rectangle with vertices on grid points and sides parallel to the sides of the grid. How many unordered pairs of two non-overlapping rectangles can Nataliia choose as gardens? (The two rectangles may share an edge or part of an edge but should not share an interior point.) PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

The $7$-digit numbers $\underline{7}$ $ \underline{4}$ $ \underline{A}$ $ \underline{5}$ $ \underline{2}$ $ \underline{B}$ $ \underline{1}$ and $\underline{3}$ $ \underline{2}$ $ \underline{6}$ $ \underline{A}$ $ \underline{B}$ $ \underline{4}$ $ \underline{C}$ are each multiples of $3$. Which of the following could be the value of $C$? $\textbf{(A) }1\qquad\textbf{(B) }2\qquad\textbf{(C) }3\qquad\textbf{(D) }5\qquad \textbf{(E) }8$

Three different points $A$, $B$ and $C$ are placed in the plane. Construct a line $m$ through $C$ so that the product of the distances from $A$ and $B$ to $m$ has the maximal value. Is $m$ unique for every triple $A$, $B$ and $C$? (NB Vassiliev)

Prove that there exist at least six points with rational coordinates on the curve of the equation \[y^3=x^3+x+1370^{1370}\]

Let $ABC$ be a triangle with circumcircle $\omega$, incenter $I$, and $A$-excenter $I_A$. Let the incircle and the $A$-excircle hit $BC$ at $D$ and $E$, respectively, and let $M$ be the midpoint of arc $BC$ without $A$. Consider the circle tangent to $BC$ at $D$ and arc $BAC$ at $T$. If $TI$ intersects $\omega$ again at $S$, prove that $SI_A$ and $ME$ meet on $\omega$. [i]Amol Aggarwal.[/i]

Fifteen pairwise coprime positive integers chosen so that each of them less than 2010. Show that at least one of them is prime.

Let $a, b, c$ be positive real numbers such that $a^3 + b^3 + c^3 = 5abc$. Show that \[ \left( \frac{a + b}{c} \right) \left( \frac{b + c}{a} \right) \left( \frac{c + a}{b} \right) \geq 9. \]

Consider the solid with 4 triangles and 4 regular hexagons as faces, where each triangle borders 3 hexagons, and all the sides are of length 1. Compute the [i]square[/i] of the volume of the solid. Express your result in reduced fraction and concatenate the numerator with the denominator (e.g., if you think that the square is $\frac{1734}{274}$, then you would submit 1734274).

Let $a, b, c$ be positive real numbers such that: $$ab - c = 3$$ $$abc = 18$$ Calculate the numerical value of $\frac{ab}{c}$

Determine all triples (x, y, z) of non-negative real numbers that satisfy the following system of equations $\begin{cases} x^2 - y = (z - 1)^2\\ y^2 - z = (x - 1)^2 \\ z^2 - x = (y -1)^2 \end{cases}$.

Suppose that $u_0 , u_1 ,\ldots$ is a sequence of real numbers such that $$u_n = \sum_{k=1}^{\infty} u_{n+k}^{2}\;\;\; \text{for} \; n=0,1,2,\ldots$$ Prove that if $\sum u_n$ converges, then $u_k=0$ for all $k$.

Brian starts at the point $\left(1,0\right)$ in the plane. Every second, he performs one of two moves: he can move from $\left(a,b\right)$ to $\left(a-b,a+b\right)$ or from $\left(a,b\right)$ to $\left(2a-b,a+2b\right)$. How many different paths can he take to end up at $\left(28,-96\right)$? [i]2018 CCA Math Bonanza Individual Round #14[/i]

We define the sets of lattice points $S_0,S_1,\ldots$ as $S_0=\{(0,0)\}$ and $S_k$ consisting of all lattice points that are exactly one unit away from exactly one point in $S_{k-1}$. Determine the number of points in $S_{2017}$. [i]Proposed by Michael Ren

Point $O$ is interior to triangle $ABC$. Through $O$, draw three lines $DE \parallel BC, FG \parallel CA$, and $HI \parallel AB$, where $D, G$ are on $AB$, $I, F$ are on $BC$ and $E, H$ are on $CA$. Denote by $S_1$ the area of hexagon $DGHEFI$, and $S_2$ the area of triangle $ABC$. Prove that $S_1 \geq \frac 23 S_2.$

Prove that there exists infinitely many positive integers $n$ such that $n^2+1$ has a prime divisor greater than $2n+\sqrt{5n+2011}$.

For each pair $a, \,b$ of coprime natural numbers, let $d_{a,\,b}$ be the greatest common divisor of $51a + b$ and $a + 51b$. Find the maximum possible value of $d_{a,\,b}$.

In triangle $ABC$, on line $CA$ it is given point $D$ such that $CD = 3 \cdot CA$ (point $A$ is between points $C$ and $D$), and on line $BC$ it is given point $E$ ($E \neq B$) such that $CE=BC$. If $BD=AE$, prove that $\angle BAC= 90^{\circ}$

Is it possible that two cross-sections of a tetrahedron by two different cutting planes are two squares, one with a side of length no greater than $1$ and another with a side of length at least $100$? Mikhail Evdokimov

In a convex hexagon $ABCDEF$ equalities $\angle ABC= \angle CDE= \angle EFA$ hold, and the angle bisectors of angles $ABC$, $CDE$ and $EFA$ intersect in one point. Rays $AB$ and $DC$ intersect at $P$, rays $BC$ and $ED$ - at $Q$, rays $CD$ and $FE$ - at $R$, rays $DE$ and $AF$ - at $S$. Prove that $PR=QS$ [i]M. Zorka[/i]

Let $ABC$ be an acute triangle. The tangent at $A,B$ of the circumcircle of $ABC$ intersect at $T$. Line $CT$ meets side $AB$ at $D$. Denote by $\Gamma_1,\Gamma_2$ the circumcircle of triangle $CAD$, and the circumcircle of triangle $CBD$, respectively. Let line $TA$ meet $\Gamma_1$ again at $E$ and line $TB$ meet $\Gamma_2$ again at $F$. Line $EF$ intersects sides $AC,BC$ at $P,Q$, respectively. Prove that $EF=PQ+AB$.

$2019$ people (all of whom are perfect logicians), labeled from $1$ to $2019$, partake in a paintball duel. First, they decide to stand in a circle, in order, so that Person $1$ has Person $2$ to his left and person $2019$ to his right. Then, starting with Person $1$ and moving to the left, every person who has not been eliminated takes a turn shooting. On their turn, each person can choose to either shoot one non-eliminated person of his or her choice (which eliminates that person from the game), or deliberately miss. The last person standing wins. If, at any point, play goes around the circle once with no one getting eliminated (that is, if all the people playing decide to miss), then automatic paint sprayers will turn on, and end the game with everyone losing. Each person will, on his or her turn, always pick a move that leads to a win if possible, and, if there is still a choice in what move to make, will prefer shooting over missing, and shooting a person closer to his or her left over shooting someone farther from their left. What is the number of the person who wins this game? Put “$0$” if no one wins.