If in a quadrilateral $ABCD$ whose vertices lie on a circle of radius $1$, holds $$|AB| \cdot |BC| \cdot |CD|\cdot |DA| \ge 4$$, then $ABCD$ is a square. Prove it. [hide=Hint given in contest] You can use Ptolemy's formula $|AB| \cdot |CD| + |BC|\cdot |AD|= |AC| \cdot|BD|$[/hide]

Find the value of $\left\lfloor \frac{1}{6}\right\rfloor+\left\lfloor\frac{4}{6}\right\rfloor+\left\lfloor\frac{9}{6}\right\rfloor+\dots+\left\lfloor\frac{1296}{6}\right\rfloor$. [i]Proposed by Zachary Perry[/i]

In a certain Zoom meeting, there are $4$ students. How many ways are there to split them into any number of distinguishable breakout rooms, each with at least $ 1$ student?

[b]Problem 6.[/b] Let $p>2$ be prime. Find the number of the subsets $B$ of the set $A=\{1,2,\ldots,p-1\}$ such that, the sum of the elements of $B$ is divisible by $p.$ [i] Ivan Landgev[/i]

Let $AB$ be a diameter of a circle $\Gamma$ with center $O$. Let $CD$ be a chord where $CD$ is perpendicular to $AB$, and $E$ is the midpoint of $CO$. The line $AE$ cuts $\Gamma$ in the point $F$, the segment $BC$ cuts $AF$ and $DF$ in $M$ and $N$, respectively. The circumcircle of $DMN$ intersects $\Gamma$ in the point $K$. Prove that $KM=MB$.

There are $a$ straight lines in a plane, no two of which are parallel to each other and no three intersect in one point. a) Prove that there exist a straight line for which each of the two Half-Planes defined by it contains at least $\lfloor \frac{(a-1)(a-2)}{10} \rfloor$ intersection points. b) Find all $a$ for which the evaluation in a) is the best possible.

The bisector $AD$ is drawn in the triangle $ABC$. Circle $k$ passes through the vertex $A$ and touches the side $BC$ at point $D$. Prove that the circle circumscribed around $ABC$ touches the circle $k$ at point $A$.

How many positive integers not exceeding $ 2001$ are multiples of $ 3$ or $ 4$ but not $ 5$? $ \textbf{(A)}\ 768 \qquad \textbf{(B)}\ 801 \qquad \textbf{(C)}\ 934 \qquad \textbf{(D)}\ 1067 \qquad \textbf{(E)}\ 1167$

Let $ p(x)$ be a cubic polynomial with rational coefficients. $ q_1$, $ q_2$, $ q_3$, ... is a sequence of rationals such that $ q_n \equal{} p(q_{n \plus{} 1})$ for all positive $ n$. Show that for some $ k$, we have $ q_{n \plus{} k} \equal{} q_n$ for all positive $ n$.

In each square of an $n\times{n}$ grid there is a lamp. If the lamp is touched it changes its state every lamp in the same row and every lamp in the same column (the one that are on are turned off and viceversa). At the begin, all the lamps are off. Show that lways is possible, with an appropriated sequence of touches, that all the the lamps on the board end on and find, in function of $n$ the minimal number of touches that are necessary to turn on every lamp.

$ABC$ is a triangle $E, F$ are points in $AB$, such that $AE = EF = FB$ $D$ is a point at the line $BC$ such that $ED$ is perpendiculat to $BC$ $AD$ is perpendicular to $CF$. The angle CFA is the triple of angle BDF. ($3\angle BDF = \angle CFA$) Determine the ratio $\frac{DB}{DC}$. %Edited!%

(a) Given a point $O$ inside an equilateral triangle $\vartriangle ABC$. Line $OG$ connects $O$ with the center of mass $G$ of the triangle and intersects the sides of the triangle, or their extensions, at points $A', B', C'$ . Prove that $$\frac{A'O}{A'G} + \frac{B'O}{B'G} + \frac{C'O}{C'G} = 3.$$ (b) Point $G$ is the center of the sphere inscribed in a regular tetrahedron $ABCD$. Straight line $OG$ connecting $G$ with a point $O$ inside the tetrahedron intersects the faces at points $A', B', C', D'$. Prove that $$\frac{A'O}{A'G} + \frac{B'O}{B'G} + \frac{C'O}{C'G}+ \frac{D'O}{D'G} = 4.$$

Let $n \geq 2$ be an integer. Positive reals satisfy $a_1\geq a_2\geq \cdots\geq a_n.$ Prove that $$\left(\sum_{i=1}^n\frac{a_i}{a_{i+1}}\right)-n \leq \frac{1}{2a_1a_n}\sum_{i=1}^n(a_i-a_{i+1})^2,$$ where $a_{n+1}=a_1.$

Find all functions $f:\mathbb R^{+} \longrightarrow \mathbb R^{+}$ so that $f(xy + f(x^y)) = x^y + xf(y)$ for all positive reals $x,y$.

For which integer $n$ is $N = 20^n + 16^n - 3^n - 1$ divisible by $323$?

Prove that in the decimal notation of the number $(5+\sqrt{26})^{-1973}$ immediately after the decimal point there are at least $1973$ zeros.

The difference of two angles of a triangle is greater than $90^{\circ}$. Prove that the ratio of its circumradius and inradius is greater than $4$.

On an infinite white checkered sheet, a square $Q$ of size $12$ × $12$ is selected. Petya wants to paint some (not necessarily all!) cells of the square with seven colors of the rainbow (each cell is just one color) so that no two of the $288$ three-cell rectangles whose centers lie in $Q$ are the same color. Will he succeed in doing this? (Two three-celled rectangles are painted the same if one of them can be moved and possibly rotated so that each cell of it is overlaid on the cell of the second rectangle having the same color.) (Bogdanov. I)

Consider an equilateral triangle and square, both with area $1$. What is the product of their perimeters?

Find all functions $f$ that is defined on all reals but $\tfrac13$ and $- \tfrac13$ and satisfies \[ f \left(\frac{x+1}{1-3x} \right) + f(x) = x \] for all $x \in \mathbb{R} \setminus \{ \pm \tfrac13 \}$.

Let $\mathcal{S}$ be the set of all possible $9$-digit numbers that use $1,2,3,\dots,9$ each exactly once as a digit. What is the probability that a randomly selected number $n$ from $\mathcal{S}$ is divisible by $27$?

a) Given any positive integer $n$, show that there exist distint positive integers $x$ and $y$ such that $x + j$ divides $y + j$ for $j = 1 , 2, 3, \ldots, n$; b) If for some positive integers $x$ and $y$, $x+j$ divides $y+j$ for all positive integers $j$, prove that $x = y$.

How many ordered four-tuples of integers $(a,b,c,d)$ with $0 < a < b < c < d < 500$ satisfy $a + d = b + c$ and $bc - ad = 93$?

Divide the grid shown to the right into more than one region so that the following rules are satisfied. 1. Each unit square lies entirely within exactly 1 region. 2. Each region is a single piece connected by the edges of its unit squares. 3. Each region contains the same number of whole unit squares. 4. Each region contains the same sum of numbers. You do not need to prove that your configuration is the only one possible; you merely need to find a configuration that works. (Note: In any other USAMTS problem, you need to provide a full proof. Only in this problem is an answer without justification acceptable.) [asy] size(6cm); for (int i=0; i<=8; ++i) draw((i,0)--(i,7), linewidth(.8)); for (int j=0; j<=7; ++j) draw((0,j)--(8,j), linewidth(.8)); void draw_num(pair ll_corner, int num) { label(string(num), ll_corner + (0.5, 0.5), p = fontsize(18pt)); } draw_num((0, 0), 1); draw_num((1, 0), 1); draw_num((2, 0), 1); draw_num((0, 5), 4); draw_num((1, 1), 4); draw_num((1, 4), 3); draw_num((2, 2), 4); draw_num((3, 4), 3); draw_num((3, 5), 2); draw_num((4, 1), 4); draw_num((4, 3), 4); draw_num((5, 4), 4); draw_num((5, 6), 6); draw_num((6, 2), 3); draw_num((6, 5), 4); draw_num((6, 6), 5); draw_num((7, 1), 4); draw_num((7, 6), 6);[/asy]

Given is a right triangle $ABC$ with perimeter $2$, with $\angle B=90^o$ . Point $S$ is the center of the excircle to the side $AB$ of the triangle and $H$ is the intersection of the heights of the triangle $ABS$ . Determine the smallest possible length of the segment $HS $.