Prove that $(n!)!$ is a multiple of $(n!)^{(n-1)!}$

A nondegenerate rhombus has side length $l$, and its area is twice that of its inscribed circle. Find the radius of the inscribed circle.

A cone with semivertical angle $30^{\circ}$ is half filled with water. What is the angle it must be tilted by so that water starts spilling?

Prove that, for any real numbers $a_1, a_2, \dots , a_n$, \[ \sum_{i,j=1}^n \frac{a_ia_j}{i+j-1}\ge 0.\]

Find all real functions $f$ defined on the real numbers except zero, satisfying $f(2019) = 1$ and $f(x)f(y)+ f\left(\frac{2019}{x}\right) f\left(\frac{2019}{y}\right) =2f(xy)$ for all $x,y \ne 0$

Walter rolls four standard six-sided dice and finds that the product of the numbers on the upper face is 144. Which of the following could NOT be on the sum of the upper four faces? $ \textbf{(A)}\ 14 \qquad \textbf{(B)}\ 15 \qquad \textbf{(C)}\ 16 \qquad \textbf{(D)}\ 17 \qquad \textbf{(E)}\ 18$

A [i]permutation[/i] of a finite set $S$ is a one-to-one function from $S$ to $S$. Given a permutation $f$ of the set $\{ 1, 2, \ldots, 100 \}$, define the [i]displacement[/i] of $f$ to be the sum \[ \sum_{i = 1}^{100} \left\lvert f(i) - i \right\rvert . \] How many permutations of $\{ 1, 2, \ldots, 100 \}$ have displacement 4?

A line is drawn through a vertex of a triangle and cuts two of its middle lines (i.e. lines connecting the midpoints of two sides) in the same ratio. Determine this ratio.

Let $p$ be an odd prime. Determine the positive integers $x$ and $y$ with $x\leq y$ for which the number $\sqrt{2p}-\sqrt{x}-\sqrt{y}$ is non-negative and as small as possible.

Let $S$ be the set of fractions of the form $\frac{{\text {lcm}}(A, B)}{A+B}$, where $A$ and $B$ are positive integers and ${\text{lcm}}(A, B)$ is the least common multiple of $A$ and $B$. What is the smallest number exceeding 3 in $S$?

Let $a,b,c$ be three positive real numbers with $a+b+c=3$. Prove that \[ (3-2a)(3-2b)(3-2c) \leq a^2b^2c^2 . \] [i]Robert Szasz[/i]

What quadratic polynomial whose coefficient of $x^2$ is $1$ has roots which are the complex conjugates of the solutions of $x^2 -6x+ 11 = 2xi-10i$? (Note that the complex conjugate of $a+bi$ is $a-bi$, where a and b are real numbers.)

The cells of an $n\times n$ table ($n\ge 3$) are painted black and white in the chess-like manner. Per move one can choose any $2\times 2$ square and reverse the color of the cells inside it. Find all $n$ for which one can obtain a table with all cells of the same color after finitely many such moves.

Rectangle $ABCD$ has sides $CD=3$ and $DA=5$. A circle of radius $1$ is centered at $A$, a circle of radius $2$ is centered at $B$, and a circle of radius $3$ is centered at $C$. Which of the following is closest to the area of the region inside the rectangle but outside all three circles? [asy] draw((0,0)--(5,0)--(5,3)--(0,3)--(0,0)); draw(Circle((0,0),1)); draw(Circle((0,3),2)); draw(Circle((5,3),3)); label("A",(0.2,0),W); label("B",(0.2,2.8),NW); label("C",(4.8,2.8),NE); label("D",(5,0),SE); label("5",(2.5,0),N); label("3",(5,1.5),E); [/asy] $\textbf{(A) }3.5\qquad\textbf{(B) }4.0\qquad\textbf{(C) }4.5\qquad\textbf{(D) }5.0\qquad \textbf{(E) }5.5$

You and your friend play a game on a $ 7 \times 7$ grid of buckets. Your friend chooses $5$ “lucky” buckets by marking an “$X$” on the bottom that you cannot see. However, he tells you that they either form a vertical, or horizontal line of length $5$. To clarify, he will select either of the following sets of buckets: either $\{(a, b),(a, b + 1),(a, b + 2),(a, b + 3),(a, b + 4)\}$, or $\{(b, a),(b + 1, a),(b + 2, a),(b + 3, a),(b + 4, a)\}$, with $1\le a \le 7$, and $1 \le b \le 3$. Your friend lets you pick up at most $n$ buckets, and you win if one of the buckets you picked was a “lucky” bucket. What is the minimum possible value of $n$ such that, if you pick your buckets optimally, you can guarantee that at least one is “lucky”?

Let $ (b_0,b_1,b_2,b_3)$ be a permutation of the set $ \{54,72,36,108\}$. Prove that $ x^5\plus{}b_3x^3\plus{}b_2x^2\plus{}b_1x\plus{}b_0$ is irreducible in $ \mathbb Z[x]$.

Find all real numbers $ x$ such that \[ \sqrt{x\plus{}1}\plus{}\sqrt{x\plus{}3} \equal{} \sqrt{2x\minus{}1}\plus{}\sqrt{2x\plus{}1}.\]

Let $ABC$ be a triangle with $\angle B = 90^o$. Given that there exists a point $D$ on $AC$ such that $AD = DC$ and $BD = BC$, compute the value of the ratio $\frac{AB}{BC}$ .

Find all pairs of natural numbers $(m,n)$ bigger than $1$ for which $2^m+3^n$ is the square of whole number. [i]I. Tonov[/i]

Let $ \Delta ABC$ with incircle $ \Gamma$, let $ D, E$ and $ F$ the tangency points of $ \Gamma$ with sides $ BC, AC$ and $ AB$, respectively and let $ P$ the intersection point of $ AD$ with $ \Gamma$. $ a)$ Prove that $ BC, EF$ and the straight line tangent to $ \Gamma$ for $ P$ concur at a point $ A'$. $ b)$ Define $ B'$ and $ C'$ in an anologous way than $ A'$. Prove that $ A'\minus{}B'\minus{}C'$

Let $a,b, c,d$ be four integers. Prove that the product of the six differences $$b - a,c - a,d - a,d - c,d - b, c - b$$ is divisible by $12$.

Let $ABC$ be an acute, non-isosceles triangle with orthocenter $H$. Let $D, E, F$ be the reflections of $H$ over $BC, CA, AB$, respectively, and let $A', B', C'$ be the reflections of $A, B, C$ over $BC, CA, AB$, respectively. Let $S$ be the circumcenter of triangle $A'B'C'$, and let $H'$ be the orthocenter of triangle $DEF$. Define $J$ as the center of the circle passing through the three projections of $H$ onto the lines $B'C', C'A', A'B'$. Prove that $HJ$ is parallel to $H'S$.

Let be a number $ a\in \left[ 1,\infty \right) $ and a function $ f\in\mathcal{C}^2(-a,a) . $ Show that the sequence $$ \left( \sum_{k=1}^n f\left( \frac{k}{n^2} \right) \right)_{n\ge 1} $$ is convergent, and determine its limit.

Let $ABC$ be a right-angled triangle with $\angle A = 90^{\circ}$ and $\angle B = 30^{\circ}$. The perpendicular at the midpoint $M$ of $BC$ meets the bisector $BK$ of the angle $B$ at the point $E$. The perpendicular bisector of $EK$ meets $AB$ at $D$. Prove that $KD$ is perpendicular to $DE$. [i]Proposed by Greece[/i]

There is an infinite set of points $S$ on the plane, and any $1\times 1$ square contains a finite number of points from the set $S$. Prove that there are two different points $A$ and $B$ from $S$ such that for any other point $X$ from $S$ the following inequalities hold: $$|XA|, |XB| \ge 0.999|AB|.$$