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$

$f(x)$ is a periodic even function defined on $\mathbb{R}$, with period of $2$. When $x\in[2,3]$, $f(x)=x$. Then what's $f(x)$ if $x\in[-2,0]$? $\text{(A)}f(x)=x+4\qquad\text{(B)}f(x)=2-x\qquad\text{(C)}f(x)=3-|x+1|\qquad\text{(D)}f(x)=2+|x+1|$

A quadrilateral is given, in which the lengths of some two sides are equal to $1$ and $4$. Also, the diagonal of length $2$ divides it into two isosceles triangles. Find the perimeter of this quadrilateral.

A regular $n$-gonal truncated pyramid is circumscribed around a sphere. Denote the areas of the base and the lateral surfaces of the pyramid by $S_1, S_2$, and $S$, respectively. Let $\sigma$ be the area of the polygon whose vertices are the tangential points of the sphere and the lateral faces of the pyramid. Prove that \[\sigma S = 4S_1S_2 \cos^2 \frac{\pi}{n}.\]

Father moves $3$ steps forward just as son moves $5$ steps, but this while the father takes $6$ steps, the son does $7$ steps. At first, father and son are together, then the son begins to walk away from his father in a straight line. When the son has done $30$ steps, the father starts to follow him. In a few steps, Dad arrives after the son?

Given $n$ a positive integer, define $T_n$ the number of quadruples of positive integers $(a,b,x,y)$ such that $a>b$ and $n= ax+by$. Prove that $T_{2023}$ is odd.