Trains $A$ and $B$ are on the same track a distance $100$ miles apart heading towards one another, each at a speed of $50$ miles per hour. A fly starting out at the front of train $A$ flies towards train $B$ at a speed of $75$ miles per hour. Upon reaching train $B$, the fly turns around and flies towards train $A$, again at $75$ miles per hour. The fly continues flying back and forth between the two trains at $75$ miles per hour until the two trains hit each other. How many minutes does the fly spend closer to train $A$ than to train $B$ before getting squashed?

Let $x, y, z$ denote positive real numbers for which $x+y+z = 1$ and $x > yz$, $y > zx$, $z > xy$. Prove that $$\left(\frac{x - yz}{x + yz}\right)^2+ \left(\frac{y - zx}{y + zx}\right)^2+\left(\frac{z - xy}{z + xy}\right)^2< 1.$$

Let $n$ be a positive integer. There are $n$ lamps, each with a switch that changes the lamp from on to off, or from off to on, each time it is pressed. The lamps are initially all off. You are going to press the switches in a series of rounds. In the first round, you are going to press exactly $1$ switch; in the second round, you are going to press exactly $2$ switches; and so on, so that in the $k$th round you are going to press exactly $k$ switches. In each round you will press each switch at most once. Your goal is to finish a round with all of the lamps switched on. Determine for which $n$ you can achieve this goal.

a) Prove that $10201$ is composite in all bases greater than 2. b) Prove that $10101$ is composite in all bases.

It is given the function $f:\left( \mathbb{R} - \{0\} \right) \times \left( \mathbb{R}-\{0\} \right) \to \mathbb{R}$ such that $f(a,b)= \left| \frac{|b-a|}{|ab|}+\frac{b+a}{ab}-1 \right|+ \frac{|b-a|}{|ab|}+ \frac{b+a}{ab}+1$ where $a,b \not=0$. Prove that: \[ f(a,b)=4 \cdot \text{max} \left\{\frac{1}{a},\frac{1}{b},\frac{1}{2} \right\}\]

Calculate $ \inf_{x> 0} \sqrt{(1+x)^2+4/x} . $ [i]Constantin Rusu[/i] and [i]Mihai Neagu[/i]

For each positive integer $k$ find the number of solutions in nonnegative integers $x,y,z$ with $x\le y \le z$ of the equation $$8^k=x^3+y^3+z^3-3xyz$$

A cart rolls down a hill, traveling 5 inches the first second and accelerating so that each successive 1-second time interval, it travels 7 inches more than during the previous 1-second interval. The cart takes 30 seconds to reach the bottom of the hill. How far, in inches, does it travel? $\textbf{(A) }215 \qquad \textbf{(B) }360 \qquad \textbf{(C) }2992 \qquad \textbf{(D) }3195 \qquad \textbf{(E) }3242$

Each corner cube is removed from this $3\text{ cm}\times 3\text{ cm}\times 3\text{ cm}$ cube. The surface area of the remaining figure is [asy]draw((2.7,3.99)--(0,3)--(0,0)); draw((3.7,3.99)--(1,3)--(1,0)); draw((4.7,3.99)--(2,3)--(2,0)); draw((5.7,3.99)--(3,3)--(3,0)); draw((0,0)--(3,0)--(5.7,0.99)); draw((0,1)--(3,1)--(5.7,1.99)); draw((0,2)--(3,2)--(5.7,2.99)); draw((0,3)--(3,3)--(5.7,3.99)); draw((0,3)--(3,3)--(3,0)); draw((0.9,3.33)--(3.9,3.33)--(3.9,0.33)); draw((1.8,3.66)--(4.8,3.66)--(4.8,0.66)); draw((2.7,3.99)--(5.7,3.99)--(5.7,0.99)); [/asy] $\textbf{(A)}\ 19\text{ sq.cm} \qquad \textbf{(B)}\ 24\text{ sq.cm} \qquad \textbf{(C)}\ 30\text{ sq.cm} \qquad \textbf{(D)}\ 54\text{ sq.cm} \qquad \textbf{(E)}\ 72\text{ sq.cm}$

Quadrilateral $ABCD$ has both an inscribed and a circumscribed circle and sidelengths $BC = 4, CD = 5, DA = 6$. Find the area of $ABCD$.

Let $ABC$ be a triangle and $D$ be a point on segment $AC$. The circumscribed circle of the triangle $BDC$ cuts $AB$ again at $E$ and the circumference circle of the triangle $ABD$ cuts $BC$ again at $F$. Prove that $AE = CF$ if and only if $BD$ is the interior bisector of $\angle ABC$.

A rhombus is formed by two radii and two chords of a circle whose radius is $ 16$ feet. The area of the rhombus in square feet is: $ \textbf{(A)}\ 128 \qquad\textbf{(B)}\ 128\sqrt {3} \qquad\textbf{(C)}\ 256 \qquad\textbf{(D)}\ 512 \qquad\textbf{(E)}\ 512\sqrt {3}$

Let $n \geq 3$ be an integer. Consider the set $A=\{1,2,3,\ldots,n\}$, in each move, we replace the numbers $i, j$ by the numbers $i+j$ and $|i-j|$. After doing such moves all of the numbers are equal to $k$. Find all possible values for $k$.

In a triangle $ABC$, it is drawn a circumference with center in the incenter $I$ and that meet twice each of the sides of the triangle: the segment $BC$ on $D$ and $P$ (where $D$ is nearer two $B$); the segment $CA$ on $E$ and $Q$ (where $E$ is nearer to $C$); and the segment $AB$ on $F$ and $R$ ( where $F$ is nearer to $A$). Let $S$ be the point of intersection of the diagonals of the quadrilateral $EQFR$. Let $T$ be the point of intersection of the diagonals of the quadrilateral $FRDP$. Let $U$ be the point of intersection of the diagonals of the quadrilateral $DPEQ$. Show that the circumcircle to the triangle $\triangle{FRT}$, $\triangle{DPU}$ and $\triangle{EQS}$ have a unique point in common.

The incenter of a triangle is the midpoint of the line segment of length $4$ joining the centroid and the orthocenter of the triangle. Determine the maximum possible area of the triangle.

Let $x_{1}$, $x_{2}$, $\ldots$, $x_{n}$ be $n$ real numbers in $\left(\frac{1}{4},\frac{2}{3}\right)$. Find the minimal value of the expression: \[ \log_{\frac 32x_{1}}\left(\frac{1}{2}-\frac{1}{36x_{2}^{2}}\right)+\log_{\frac 32x_{2}}\left(\frac{1}{2}-\frac{1}{36x_{3}^{2}}\right)+\cdots+ \log_{\frac 32x_{n}}\left(\frac{1}{2}-\frac{1}{36x_{1}^{2}}\right). \]

Let consider the following function set $$F=\{f\ |\ f:\{1,\ 2,\ \cdots,\ n\}\to \{1,\ 2,\ \cdots,\ n\} \}$$ [list=1] [*] Find $|F|$ [*] For $n=2k$ prove that $|F|< e{(4k)}^{k}$ [*] Find $n$, if $|F|=540$ and $n=2k$ [/list]

Let $ABCD$ be a convex quadrilateral, and $M$, $N$, and $P$ be the midpoints of diagonals $AC$ and $BD$, and side $AD$, respectively. Also, suppose that $\angle{ABC} + \angle{DCB} = 90$ and that $AB = 6$, $CD = 8$. Calculate the perimeter of triangle $MNP$.

Prove that for any integers $ n> m> 0 $ the number $ 2 ^n-1 $ has a prime divisor not dividing $ 2 ^m-1 $.

Mark the Martian and Bark the Bartian live on planet Blok, in the year $2019$. Mark and Bark decide to play a game on a $10 \times 10$ grid of cells. First, Mark randomly generates a subset $S$ of $\{1, 2, \dots, 2019\}$ with $|S|=100$. Then, Bark writes each of the $100$ integers in a different cell of the $10 \times 10$ grid. Afterwards, Bark constructs a solid out of this grid in the following way: for each grid cell, if the number written on it is $n$, then she stacks $n$ $1 \times 1 \times 1$ blocks on top of one other in that cell. Let $B$ be the largest possible surface area of the resulting solid, including the bottom of the solid, over all possible ways Bark could have inserted the $100$ integers into the grid of cells. Find the expected value of $B$ over all possible sets $S$ Mark could have generated. [i]Proposed by Yang Liu[/i]

The vertices of a $ 3 \minus{} 4 \minus{} 5$ right triangle are the centers of three mutually externally tangent circles, as shown. What is the sum of the areas of the three circles? [asy]unitsize(5mm); defaultpen(fontsize(10pt)+linewidth(.8pt)); pair B=(0,0), C=(5,0); pair A=intersectionpoints(Circle(B,3),Circle(C,4))[0]; draw(A--B--C--cycle); draw(Circle(C,3)); draw(Circle(A,1)); draw(Circle(B,2)); label("$A$",A,N); label("$B$",B,W); label("$C$",C,E); label("3",midpoint(B--A),NW); label("4",midpoint(A--C),NE); label("5",midpoint(B--C),S);[/asy]$ \textbf{(A) } 12\pi\qquad \textbf{(B) } \frac {25\pi}{2}\qquad \textbf{(C) } 13\pi\qquad \textbf{(D) } \frac {27\pi}{2}\qquad \textbf{(E) } 14\pi$

In a triangle $ABC$, point $P$ is the incenter and $A'$, $B'$, $C'$ its orthogonal projections on $BC$, $CA$, $AB$, respectively. Show that $\angle B'A'C'$ is acute.

Prove that there exists no innite sequence of prime numbers $p_0, p_1, p_2,...$ such that for all positive integers $k$: $p_k = 2p_{k-1} + 1$ or $p_k = 2p_{k-1} - 1$.

Show that, if $\sqrt{2}$ is written in decimal notation, there is at least one nonzero digit at the interval of 1,000,000-th and 3,000,000-th digits.

Let $ I$ be the incenter of triangle $ ABC.$ Let $ M,N$ be the midpoints of $ AB,AC,$ respectively. Points $ D,E$ lie on $ AB,AC$ respectively such that $ BD\equal{}CE\equal{}BC.$ The line perpendicular to $ IM$ through $ D$ intersects the line perpendicular to $ IN$ through $ E$ at $ P.$ Prove that $ AP\perp BC.$