3. Let us call a bi-squared card $2 \times 1$ regular, if two positive integers are written on it and the number in the upper square is less than the number in the lower square. It is allowed at each move to change both numbers in the following manner: either add the same integer (possibly negative) to both numbers, or multiply each number by the same positive integer, or divide each number by the same positive integer. The card must remain regular after any changes made. What minimal number of moves is sufficient to get any regular card from any other regular card? Alexey Glebov

Solve the equation $[x]\{x\} = 1999x$, where $[x]$ denotes the largest integer less than or equal to $x$, and $\{x\} = x -[x] $

Determine the greatest number, who is the product of some positive integers, and the sum of these numbers is $1976.$

The points $P$ and $Q$ lie on the diagonals $AC$ and $BD$, respectively, of a quadrilateral $ABCD$ such that $\frac{AP}{AC} + \frac{BQ}{BD} =1$. The line $PQ$ meets the sides $AD$ and $BC$ at points $M$ and $N$. Prove that the circumcircles of the triangles $AMP$, $BNQ$, $DMQ$, and $CNP$ are concurrent.

There are $100$ people in a room. Some are [i]wise[/i] and some are [i]optimists[/i]. $\quad \bullet~$ A [i]wise[/i] person can look at someone and know if they are wise or if they are an optimist. $\quad \bullet~$ An [i]optimist[/i] thinks everyone is wise (including themselves). Everyone in the room writes down what they think is the number of wise people in the room. What is the smallest possible value for the average? $$ \mathrm a. ~ 10\qquad \mathrm b.~25\qquad \mathrm c. ~50 \qquad \mathrm d. ~75 \qquad \mathrm e. ~100 $$

The quadrilateral $A_1A_2A_3A_4$ is cyclic, and its sides are $a_1 = A_1A_2, a_2 = A_2A_3, a_3 = A_3A_4$ and $a_4 = A_4A_1.$ The respective circles with centres $I_i$ and radii $r_i$ are tangent externally to each side $a_i$ and to the sides $a_{i+1}$ and $a_{i-1}$ extended. ($a_0 = a_4$). Show that \[ \prod^4_{i=1} \frac{a_i}{r_i} = 4 \cdot (\csc (A_1) + \csc (A_2) )^2. \]

Find all real numbers $ x$ such that: $ [x^2\plus{}2x]\equal{}{[x]}^2\plus{}2[x]$ (Here $ [x]$ denotes the largest integer not exceeding $ x$.)

Given a set $X=\{x_1,\ldots,x_n\}$ of natural numbers in which for all $1< i \leq n$ we have $1\leq x_i-x_{i-1}\leq 2$, call a real number $a$ [b]good[/b] if there exists $1\leq j \leq n$ such that $2|x_j-a|\leq 1$. Also a subset of $X$ is called [b]compact[/b] if the average of its elements is a good number. Prove that at least $2^{n-3}$ subsets of $X$ are compact. [i]Proposed by Mahyar Sefidgaran[/i]

Let $ ABC$ be a triangle, and $ I$ its incenter. Let the incircle of $ ABC$ touch side $ BC$ at $ D$, and let lines $ BI$ and $ CI$ meet the circle with diameter $ AI$ at points $ P$ and $ Q$, respectively. Given $ BI \equal{} 6, CI \equal{} 5, DI \equal{} 3$, determine the value of $ \left( DP / DQ \right)^2$.

Find a positive integer $N$ and $a_1, a_2, \cdots, a_N$ where $a_k = 1$ or $a_k = -1$, for each $k=1,2,\cdots,N,$ such that $$a_1 \cdot 1^3 + a_2 \cdot 2^3 + a_3 \cdot 3^3 \cdots + a_N \cdot N^3 = 20162016$$ or show that this is impossible.

Let $A$ and $B$ infinite sets of positive real numbers such that: 1. For any pair of elements $u \ge v$ in $A$, it follows that $u+v$ is an element of $B$. 2. For any pair of elements $s>t$ in $B$, it follows that $s-t$ is an element of $A$. Prove that $A=B$ or there exists a real number $r$ such that $B=\{2r, 3r, 4r, 5r, \dots\}$.

A trapezoid $ABCD$ ($AB ///CD$) has the property that there are points $P$ and $Q$ on sides $AD$ and $BC$ respectively such that $\angle APB = \angle CPD$ and $\angle AQB = \angle CQD$. Show that the points $P$ and $Q$ are equidistant from the intersection point of the diagonals of the trapezoid. (M. Smurov)

How many finite sequences $x_1,x_2,\ldots,x_m$ are there such that $x_i=1$ or $2$ and $\sum_{i=1}^mx_i=10$? $\textbf{(A)}~89$ $\textbf{(B)}~73$ $\textbf{(C)}~107$ $\textbf{(D)}~119$

Find the positive integer $n$ such that $32$ is the product of the real number solutions of $x^{\log_2(x^3)-n} = 13$

A ticket to a school play costs $ x$ dollars, where $ x$ is a whole number. A group of 9th graders buys tickets costing a total of $ \$48$, and a group of 10th graders buys tickets costing a total of $ \$64$. How many values of $ x$ are possible? $ \textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5$

Let $L$ be the number of times the letter $L$ appeared on the Speed Round, $M$ be the number of times the letter $M$ appeared on the Speed Round, and $T$ be the number of times the letter $T$ appeared on the Speed Round. Find the value of $LMT$.

Let $x$, $y$ and $z$ be odd positive integers such that $\gcd \ (x, y, z) = 1$ and the sum $x^2 +y^2 +z^2$ is divisible by $x+y+z$. Prove that $x+y+z- 2$ is not divisible by $3$.

On the hypotenuse $AB$ of a right triangle $ABC$, we denote a point $K$ such that $BK = BC$. Let $P$ be a point on the perpendicular from the point $K$ to line $CK$, equidistant from the points $K$ and $B$. Let $L$ be the midpoint of $CK$. Prove that line $AP$ is tangent to the circumcircle of $\Delta BLP$.

For every $ x,y\in \Re ^{\plus{}}$, the function $ f: \Re ^{\plus{}} \to \Re$ satisfies the condition $ f\left(x\right)\plus{}f\left(y\right)\equal{}f\left(x\right)f\left(y\right)\plus{}1\minus{}\frac{1}{xy}$. If $ f\left(2\right)<1$, then $ f\left(3\right)$ will be $\textbf{(A)}\ 2/3 \\ \textbf{(B)}\ 4/3 \\ \textbf{(C)}\ 1 \\ \textbf{(D)}\ \text{More information needed} \\ \textbf{(E)}\ \text{There is no } f \text{ satisfying the condition above.}$

6. If $ p,q$ are rationals, $r=p+\sqrt{7}q$, then prove there exists a matrix $\left(\begin{array}{cc}a&b\\c&d\end{array}\right) \in M_{2}(Z)- ( \pm I_{2})$ for which $\frac{ar+b}{cr+d}=r$ and $det(A)=1$

On the sides $AB, \, \, BC, \, \, CA$ of the triangle $ABC$ the points ${{C} _ {1}}, \, \, {{A} _ { 1}},\, \, {{B} _ {1}}$ are selected respectively, that are different from the vertices. It turned out that $\Delta {{A} _ {1}} {{B} _ {1}} {{C} _ {1}}$ is equilateral, $\angle B{{C}_{1}}{{A}_{1}}=\angle {{C}_{1}}{{B}_{1}}A$ and $\angle B{{A}_{1}}{{C}_{1}}=\angle {{A}_{1}}{{B}_{1}}C$ . Is $ \Delta ABC$ equilateral?

Each angle of a rectangle is trisected. The intersections of the pairs of trisectors adjacent to the same side always form: $ \textbf{(A)}\ \text{a square} \qquad\textbf{(B)}\ \text{a rectangle} \qquad\textbf{(C)}\ \text{a parallelogram with unequal sides} \\ \textbf{(D)}\ \text{a rhombus} \qquad\textbf{(E)}\ \text{a quadrilateral with no special properties}$

Let $n$ be a positive integer. On a $2n\times 2n$ board, the $2n^2$ squares were painted white and the other $2n^2$ squares were painted black. One operation is to choose a $2\times 2$ subtable and mirror its $4$ cells about the vertical or horizontal axis of symmetry of that subtable. For what values of $n$ is it always possible to obtain a chess-like coloring from any initial coloring?

The following diagram shows an eight-sided polygon $ABCDEFGH$ with side lengths $8,15,8,8,8,6,8,$ and $29$ as shown. All of its angles are right angles. Turn this eight-sided polygon into a six-sided polygon by connecting $B$ to $D$ with an edge and $E$ to $G$ with an edge to form polygon $ABDEGH$. Find the perimeter of $ABDEGH$. [asy] size(200); defaultpen(linewidth(2)); pen qq=font("phvb"); pair rectangle[] = {origin,(0,-8),(15,-8),(15,-16),(23,-16),(23,-8),(29,-8),(29,0)}; string point[] = {"A","B","C","D","E","F","G","H"}; int dirlbl[] = {135,225,225,225,315,315,315,45}; string value[] = {"8","15","8","8","8","6","8","29"}; int direction[] = {0,90,0,90,180,90,180,270}; for(int i=0;i<=7;i=i+1) { draw(rectangle[i]--rectangle[(i+1) % 8]); label(point[i],rectangle[i],dir(dirlbl[i]),qq); label(value[i],(rectangle[i]+rectangle[(i+1) % 8])/2,dir(direction[i]),qq); } [/asy]

Prove that $\frac{5^{125}-1}{5^{25}-1}$ is a composite number.