Let $f(x, y) = \left\lfloor \frac{5x}{2y} \right\rfloor + \left\lceil \frac{5y}{2x} \right\rceil$. Suppose $x, y$ are chosen independently uniformly at random from the interval $(0, 1]$. Let $p$ be the probability that $f(x, y) < 6$. If $p$ can be expressed in the form $m/n$ for relatively prime positive integers $m$ and $n$, compute $m + n$. (Note: $\lfloor x\rfloor $ is defined as the greatest integer less than or equal to $x$ and $\lceil x \rceil$ is defined as the least integer greater than or equal to$ x$.)

Let $\triangle ABC$ be a triangle with side length $BC= 4\sqrt{6}$. Denote $\omega$ as the circumcircle of $\triangle{ABC}$. Point $D$ lies on $\omega$ such that $AD$ is the diameter of $\omega$. Let $N$ be the midpoint of arc $BC$ that contains $A$. $H$ is the intersection of the altitudes in $\triangle{ABC}$ and it is given that $HN = HD= 6$. If the area of $\triangle{ABC}$ can be expressed as $\frac{a\sqrt{b}}{c}$, where $a,b,c$ are positive integers with $a$ and $c$ relatively prime and $b$ not divisible by the square of any prime, compute $a+b+c$. [i]Proposed by Andy Xu[/i]

For every integer $ k \geq 2,$ prove that $ 2^{3k}$ divides the number \[ \binom{2^{k \plus{} 1}}{2^{k}} \minus{} \binom{2^{k}}{2^{k \minus{} 1}} \] but $ 2^{3k \plus{} 1}$ does not. [i]Author: Waldemar Pompe, Poland[/i]

Let $ABCD$ be a quadrilateral with $AC=20$, $AD=16$. We take point $P$ on segment $CD$ so that triangle $ABP$ and $ACD$ are congruent. If the area of triangle $APD$ is $28$, find the area of triangle $BCP$. Note that $XY$ expresses the length of segment $XY$.

The numbers $1, 2, . . . , 10$ are randomly arranged in a circle. Let $p$ be the probability that for every positive integer $k < 10$, there exists an integer $k' > k$ such that there is at most one number between $k$ and $k'$ in the circle. If $p$ can be expressed as $\frac{a}{b}$ for relatively prime positive integers $a$ and $b$, compute $100a + b$.

Let $a,b,c$ be positive integers such that $a|b^3, b|c^3$ and $c|a^3$. Prove that $abc|(a+b+c)^{13}$

Find the smallest $\lambda \in \mathbb{R}$ such that for all $n \in \mathbb{N}_+$, there exists $x_1, x_2, \ldots, x_n$ satisfying $n = x_1 x_2 \ldots x_{2023}$, where $x_i$ is either a prime or a positive integer not exceeding $n^\lambda$ for all $i \in \left\{ 1,2, \ldots, 2023 \right\}$. [i]Proposed by Yinghua Ai[/i]

Is it possible to cut a rectangle $2001\times2003$ into pieces of the form [img]https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvNS82L2RjZTZjNzc0M2YxMzM1ZDIzZTY2Zjc2NGJlMWJlMWUwMmU2ZWRlLnBuZw==&rn=U2NyZWVuIFNob3QgMjAyMS0wNS0xMyBhdCAzLjQ2LjQ2IFBNLnBuZw==[/img] each consisting of three unit squares?

For integers $a,b$ we define $f((a,b))=(2a,b-a)$ if $a<b$ and $f((a,b))=(a-b,2b)$ if $a\geq b$. Given a natural number $n>1$ show that there exist natural numbers $m,k$ with $m<n$ such that $f^{k}((n,m))=(m,n)$,where $f^{k}(x)=f(f(f(...f(x))))$,$f$ being composed with itself $k$ times.

Let $ABCD$ be a tetrahedron such that $AB = CD = \sqrt{41}$, $AC = BD = \sqrt{80}$, and $BC = AD = \sqrt{89}$. There exists a point $I$ inside the tetrahedron such that the distances from $I$ to each of the faces of the tetrahedron are all equal. This distance can be written in the form $\frac{m \sqrt{n}}{p}$, when $m$, $n$, and $p$ are positive integers, $m$ and $p$ are relatively prime, and $n$ is not divisible by the square of any prime. Find $m+n+p$.

In a meeting, there are $2011$ scientists attending. We know that, every scientist know at least $1509$ other ones. Prove that a group of five scientists can be formed so that each one in this group knows $4$ people in his group.

Juku built a robot that moves along the border of a regular octagon, passing each side in exactly $1$ minute. The robot starts in some vertex $A$ and upon reaching each vertex can either continue in the same direction, or turn around and continue in the opposite direction. In how many different ways can the robot move so that after $n$ minutes it will be in the vertex $B$ opposite to $A$?

Determine the number of 2010 letter words, formed by the letters $a$, $b$, and $c$, such that at least one of the three letters is odd number of times in the word.

Prove that the sum of an odd number of vectors of length 1, of common origin $O$ and all situated in the same semi-plane determined by a straight line which goes through $O,$ is at least 1.

For all positive integers $n$ the function $f$ satisfies $f(1) = 1, f(2n + 1) = 2f(n),$ and $f(2n) = 3f(n) + 2$. For how many positive integers $x \leq 100$ is the value of $f(x)$ odd? $\mathrm{(A) \ } 4 \qquad \mathrm{(B) \ } 5 \qquad \mathrm {(C) \ } 6 \qquad \mathrm{(D) \ } 7 \qquad \mathrm{(E) \ } 10$

Let $X = \{-5,-4,-3,-2,-1,0,1,2,3,4,5\}$ and $S = \{(a,b)\in X\times X:x^2+ax+b \text{ and }x^3+bx+a \text{ have at least a common real zero .}\}$ How many elements are there in $S$?

Let $\tau(n)$ be the number of distinct positive divisors of $n$ (including $1$ and itself). Find the sum of all positive integers $n$ satisfying $n=\tau(n)^3.$

$A$ and $B$ are distinct points on a circle $T$. $C$ is a point distinct from $B$ such that $|AB|=|AC|$, and such that $BC$ is tangent to $T$ at $B$. Suppose that the bisector of $\angle ABC$ meets $AC$ at a point $D$ inside $T$. Show that $\angle ABC>72^\circ$.

Prove that a circle centered at point $(\sqrt{2},\sqrt{3})$ in the cartesian plane passes through at most one point with integer coordinates. I tried to prove that any circle with center at $(0,0)$ has at most one point with coordinates $(a-\sqrt{2},b-\sqrt{3})$;$a,b \in \mathbb{Z}$. So that when we translate the center to $(\sqrt{2},\sqrt{3})$ we have what we wanted to show.

Solve the system of equations in integers: $$\begin{cases} a^3=abc+2a+2c \\ b^3=abc-c \\ c^3=abc-a+b \end{cases}$$

Consider the L-shaped tromino below with 3 attached unit squares. It is cut into exactly two pieces of equal area by a line segment whose endpoints lie on the perimeter of the tromino. What is the longest possible length of the line segment? [asy] size(1cm); draw((0,0) -- (2,0) -- (2,1) -- (1,1)--(1,2)--(0,2)--(0,0)); [/asy] [i]Proposed by James Lin.[/i]

Vukasin, Dimitrije, Dusan, Stefan and Filip asked their teacher to guess three consecutive positive integers, after these true statements: Vukasin: " The sum of the digits of one number is prime number. The sum of the digits of another of the other two is, an even perfect number.($n$ is perfect if $\sigma\left(n\right)=2n$). The sum of the digits of the third number equals to the number of it's positive divisors". Dimitrije:"Everyone of those three numbers has at most two digits equal to $1$ in their decimal representation". Dusan:"If we add $11$ to exactly one of them, then we have a perfect square of an integer" Stefan:"Everyone of them has exactly one prime divisor less than $10$". Filip:"The three numbers are square free". Professor found the right answer. Which numbers did he mention?

Determine all composite integers $n>1$ that satisfy the following property: if $d_1$, $d_2$, $\ldots$, $d_k$ are all the positive divisors of $n$ with $1 = d_1 < d_2 < \cdots < d_k = n$, then $d_i$ divides $d_{i+1} + d_{i+2}$ for every $1 \leq i \leq k - 2$.

Triangle $ABC$ satisfying $AB<AC$ has circumcircle $\Omega$. $E, F$ lies on $AC, AB$, respectively, such that $BCEF$ is cyclic. $T$ lies on $EF$ such that $\odot(TEF)$ is tangent to $BC$ at $T$. $A'$ is the antipode of $A$ on $\Omega$. $TA', TA$ intersects $\Omega$ again at $X, Y$, respectively, and $EF$ intersects $\odot(TXY)$ again at $W$. Prove that $\measuredangle WBA=\measuredangle ACW$ [i]Proposed by BlessingOfHeaven[/i]

In the Mobius Planet (a plane and infinite planet!, in a similar manner to the $N \times N$ lattice), the Supreme King Mobius is planning to construct a water reservoir. There are some restrictions to this project: 1. There exists only $k < \infty$ bricks. 2. These bricks will delimit a closed finite area. What is the maximum area of this resevoir in function of $k$?