Let $P$ be an interior point of triangle $ABC$. Let $a,b,c$ be the sidelengths of triangle $ABC$ and let $p$ be it's semiperimeter. Find the maximum possible value of $$ \min\left(\frac{PA}{p-a},\frac{PB}{p-b},\frac{PC}{p-c}\right)$$ taking into consideration all possible choices of triangle $ABC$ and of point $P$. by Elton Bojaxhiu, Albania

A quadrilateral $ABCD$ is inscribed in a circle of radius 1 whose diameter is $AB$. If the quadrilateral $ABCD$ has an incircle, prove that $CD \leq 2 \sqrt{5} - 2$.

Let $W(x) = x^4 - 3x^3 + 5x^2 - 9x$ be a polynomial. Determine all pairs of different integers $a$, $b$ satisfying the equation $W(a) = W(b)$.

Show that one cannot find compact sets $A_1, A_2, A_3, \ldots$ in $\mathbb{R}$ such that (1) All elements of $A_n$ are rational. (2) Any compact set $K\subset \mathbb{R}$ which only contains rational numbers is contained in some $A_{m}$.

Let $ABC$ be an acute triangle.The lines $l_1$ and $l_2$ are perpendicular to $AB$ at the points $A$ and $B$, respectively.The perpendicular lines from the midpoint $M$ of $AB$ to the lines $AC$ and $BC$ intersect $l_1$ and $l_2$ at the points $E$ and $F$, respectively.If $D$ is the intersection point of the lines $EF$ and $MC$, prove that \[\angle ADB = \angle EMF.\]

Find all integers $\,a,b,c\,$ with $\,1<a<b<c\,$ such that \[ (a-1)(b-1)(c-1) \] is a divisor of $abc-1.$

Find the maximum possible length of a sequence consisting of non-zero integers, in which the sum of any seven consecutive terms is positive and that of any eleven consecutive terms is negative.

A tetrahedron has four congruent faces, each of which is a triangle with side lengths $6$, $5$, and $5$. If the volume of the tetrahedron is $V$ , compute $V^2$ .

A regular $1997$-gon is divided into triangles by non-intersecting diagonals. Prove that exactly one of them is acute-angled.

Let $a,b,c \in [1, \infty)$. Prove that: $$\frac{a\sqrt{b}}{a+b}+\frac{b\sqrt{c}}{b+c}+\frac{c\sqrt{b}}{c+a}+\frac32 \le a+b+c$$

Let $S$ be a circle with center $C$ and radius $r$, and let $P \ne C$ be an arbitrary point. A line $\ell$ through $P$ intersects the circle in $X$ and $Y$. Let $Z$ be the midpoint of $XY$. Prove that the points $Z$, as $\ell$ varies, describe a circle. Find the center and radius of this circle.

Two circles have radii $15$ and $95$. If the two external tangents to the circles intersect at $60$ degrees, how far apart are the centers of the circles?

Let a convex polygon $P$ be contained in a square of side one. Show that the sum of the sides of $P$ is less than or equal to $4$.

(V.Protasov, 10-11) Given parallelogram $ ABCD$ such that $ AB \equal{} a$, $ AD \equal{} b$. The first circle has its center at vertex $ A$ and passes through $ D$, and the second circle has its center at $ C$ and passes through $ D$. A circle with center $ B$ meets the first circle at points $ M_1$, $ N_1$, and the second circle at points $ M_2$, $ N_2$. Determine the ratio $ M_1N_1/M_2N_2$.

If $y = 2x$ and $z = 2y$, then $x + y + z$ equals \[ \text{(A)}\ x \qquad \text{(B)}\ 3x \qquad \text{(C)}\ 5x \qquad \text{(D)}\ 7x \qquad \text{(E)}\ 9x \]

Let $f(k)$ denote the number of triples $(a, b, c)$ of positive integers satisfying $a + b + c = 2020$ with $(k - 1)$ not dividing $a, k$ not dividing $b$, and $(k + 1)$ not dividing $c$. Find the product of all integers $k$ in the range 3 \le k \le 20 such that $(k + 1)$ divides $f(k)$.

Is there integer $n$ such that $n!$ begins with $2005$ ?

Initially, the point-like particles $A, B$ and $C{}$ are located respectively at the points $(0,0), (1,0)$ and $(0,1)$ in the coordinate plane. Every minute some two particles repel each other along the straight line connecting their current positions, moving the same (positive) distance. [list=a] [*]Can the particle $A{}$ be at the point $(3,3)$? What about the point $(2,3)$? [*]Can the particles $B{}$ and $C{}$ be at the same time at the points $(0,100)$ and $(100,0)$ respectively? [/list] [i]Proposed by K. Krivosheev[/i]

A point $ P$ inside a circle is such that there are three chords of the same length passing through $ P$. Prove that $ P$ is the center of the circle.

Given $t\in\mathbb C$. Complex numbers $x,y,z$ satisfy that $|x|=|y|=|z|=1$ and $\frac{t}{y}=\frac{1}{x}+\frac{1}{z}$. Calculate $$\left|\frac{2xy+2yz+3zx}{x+y+z}\right|.$$

Let $a, b, c$ be nonnegative real numbers with arithmetic mean $m =\frac{a+b+c}{3}$ . Provethat $$\sqrt{a+\sqrt{b + \sqrt{c}}} +\sqrt{b+\sqrt{c + \sqrt{a}}} +\sqrt{c +\sqrt{a + \sqrt{b}}}\le 3\sqrt{m+\sqrt{m + \sqrt{m}}}.$$

Every point with integer coordinates in the plane is the center of a disk with radius $1/1000$. (1) Prove that there exists an equilateral triangle whose vertices lie in different discs. (2) Prove that every equilateral triangle with vertices in different discs has side-length greater than $96$. [i]Radu Gologan, Romania[/i] [hide="Remark"] The "> 96" in [b](b)[/b] can be strengthened to "> 124". By the way, part [b](a)[/b] of this problem is the place where I used [url=http://mathlinks.ro/viewtopic.php?t=5537]the well-known "Dedekind" theorem[/url]. [/hide]

Let $a, b, c, d$ be positive reals such that $abcd = 1$. Prove that $$\frac{1}{a(b + 1)} +\frac{1}{b(c + 1)} +\frac{1}{c(d + 1)} +\frac{1}{d(a + 1)} \ge 2.$$

A grocer stacks oranges in a pyramid-like stack whose rectangular base is $ 5$ oranges by $ 8$ oranges. Each orange above the first level rests in a pocket formed by four oranges in the level below. The stack is completed by a single row of oranges. How many oranges are in the stack? $ \textbf{(A)}\ 96 \qquad \textbf{(B)}\ 98 \qquad \textbf{(C)}\ 100 \qquad \textbf{(D)}\ 101 \qquad \textbf{(E)}\ 134$

Two circles that share the same center have radii $10$ meters and $20$ meters. An aardvark runs along the path shown, starting at $A$ and ending at $K$. How many meters does the aardvark run? [asy] size((150)); draw((10,0)..(0,10)..(-10,0)..(0,-10)..cycle); draw((20,0)..(0,20)..(-20,0)..(0,-20)..cycle); draw((20,0)--(-20,0)); draw((0,20)--(0,-20)); draw((-2,21.5)..(-15.4, 15.4)..(-22,0), EndArrow); draw((-18,1)--(-12, 1), EndArrow); draw((-12,0)..(-8.3,-8.3)..(0,-12), EndArrow); draw((1,-9)--(1,9), EndArrow); draw((0,12)..(8.3, 8.3)..(12,0), EndArrow); draw((12,-1)--(18,-1), EndArrow); label("$A$", (0,20), N); label("$K$", (20,0), E); [/asy] $ \textbf{(A)}\ 10\pi+20\qquad\textbf{(B)}\ 10\pi+30\qquad\textbf{(C)}\ 10\pi+40\qquad\textbf{(D)}\ 20\pi+20\qquad \textbf{(E)}\ 20\pi+40$