There exist primes $p$ and $q$ such that \[ pq = 1208925819614629174706176 \times 2^{4404} - 4503599560261633 \times 134217730 \times 2^{2202} + 1. \] Find the remainder when $p+q$ is divided by $1000$. [i]Proposed by Evan Chen[/i]

Let $\Gamma$ be a circle of center $O$, and $\delta$. be a line in the plane of $\Gamma$, not intersecting it. Denote by $A$ the foot of the perpendicular from $O$ onto $\delta$., and let $M$ be a (variable) point on $\Gamma$. Denote by $\gamma$ the circle of diameter $AM$ , by $X$ the (other than M ) intersection point of $\gamma$ and $\Gamma$, and by $Y$ the (other than $A$) intersection point of $\gamma$ and $\delta$. Prove that the line $XY$ passes through a fixed point.

Is $\log_{10}{8}$ rational?

Numbers $m$ and $n$ are given positive integers. There are $mn$ people in a party, standing in the shape of an $m\times n$ grid. Some of these people are police officers and the rest are the guests. Some of the guests may be criminals. The goal is to determine whether there is a criminal between the guests or not.\\ Two people are considered \textit{adjacent} if they have a common side. Any police officer can see their adjacent people and for every one of them, know that they're criminal or not. On the other hand, any criminal will threaten exactly one of their adjacent people (which is likely an officer!) to murder. A threatened officer will be too scared, that they deny the existence of any criminal between their adjacent people.\\ Find the least possible number of officers such that they can take position in the party, in a way that the goal is achievable. (Note that the number of criminals is unknown and it is possible to have zero criminals.) [i]Proposed by Abolfazl Asadi[/i]

Given an acute triangle $T$ with sides $a,b,c$, find the tetrahedra with base $T$ whose all faces are acute triangles of the same area.

Let $f:\mathbb{R}\to\mathbb{R}$ be a function such that \[f(f(x))=x^2-x+1\] for all real numbers $x$. Determine $f(0)$.

Find all functions $f: R \to R$ such that $f(x)f(yf(x)-1)=x^2f(y)-f(x)$ for all real $x ,y$

There are n cars waiting at distinct points of a circular race track. At the starting signal each car starts. Each car may choose arbitrarily which of the two possible directions to go. Each car has the same constant speed. Whenever two cars meet they both change direction (but not speed). Show that at some time each car is back at its starting point.

Suppose that $728$ coins are set on a table, all facing heads up at first. For each iteration, we randomly choose $314$ coins and flip them (from heads to tails or vice versa). Let $a/b$ be the expected number of heads after we finish $4001$ iterations, where $a$ and $b$ are relatively prime. Find $a + b$ mod $10000$.

A grocer makes a display of cans in which the top row has one can and each lower row has two more cans than the row above it. If the display contains $ 100$ cans, how many rows does it contain? $ \textbf{(A)}\ 5\qquad \textbf{(B)}\ 8\qquad \textbf{(C)}\ 9\qquad \textbf{(D)}\ 10\qquad \textbf{(E)}\ 11$

What is the sum of the real roots of the equation $4x^4-3x^2+7x-3=0$? $ \textbf{(A)}\ -1 \qquad\textbf{(B)}\ -2 \qquad\textbf{(C)}\ -3 \qquad\textbf{(D)}\ -4 \qquad\textbf{(E)}\ \text {None of above} $

Prove that for every $n\in \mathbb N$, there exists a set $S$ of $n$ positive integers such that for any two distinct $a,b\in S$, $a-b$ divides $a$ and $b$ but none of the other elements of $S$. [i]Proposed by Iurie Boreico[/i]

Prove that in an arbitrary convex hexagon there is a diagonal that cuts off from it a triangle whose area does not exceed $\frac16$ of the area of the hexagon. What are the properties of a convex hexagon, each diagonal of which is cut off from it is a triangle whose area is not less than $\frac16$ the area of the hexagon?

Find the sum of the $2$ smallest prime factors of $2^{1024} - 1$. $\textbf{(A) } 4\qquad\textbf{(B) } 6\qquad\textbf{(C) } 8\qquad\textbf{(D) } 10\qquad\textbf{(E) } 12$

For a convex polygon $P$ in the plane let $P'$ denote the convex polygon with vertices at the midpoints of the sides of $P.$ Given an integer $n \geq 3,$ determine sharp bounds for the ratio \[ \frac{\text{area } P'}{\text{area } P}, \] over all convex $n$-gons $P.$

Let $ABC$ be an acute-angled triangle with $AB < AC$. Let $\Omega$ be the circumcircle of $ABC$. Let $S$ be the midpoint of the arc $CB$ of $\Omega$ containing $A$. The perpendicular from $A$ to $BC$ meets $BS$ at $D$ and meets $\Omega$ again at $E \neq A$. The line through $D$ parallel to $BC$ meets line $BE$ at $L$. Denote the circumcircle of triangle $BDL$ by $\omega$. Let $\omega$ meet $\Omega$ again at $P \neq B$. Prove that the line tangent to $\omega$ at $P$ meets line $BS$ on the internal angle bisector of $\angle BAC$.

Determine whether there exist infinitely many triples $(a, b, c)$ of positive integers such that every prime $p$ divides \[\left\lfloor\left(a+b\sqrt{2024}\right)^p\right\rfloor-c.\]

Show that \[\cos(56^{\circ}) \cdot \cos(2 \cdot 56^{\circ}) \cdot \cos(2^2\cdot 56^{\circ})\cdot . . . \cdot \cos(2^{23}\cdot 56^{\circ}) = \frac{1}{2^{24}} .\]

The rectangular faces of rectangular prism $A$ have perimeters $12$, $16$, and $24$. The rectangular faces of rectangular prism $B$ have perimeters $12$, $16$, and $20$. Let $V_A$ denote the volume of $A$ and $V_B$ denote the volume of $B$. Find $V_A-V_B$.

Suppose that $ a$ and $ b$ are digits, not both nine and not both zero, and the repeating decimal $ 0.\overline{ab}$ is expressed as a fraction in lowest terms. How many different denominators are possible? $ \textbf{(A)}\ 3 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 8 \qquad \textbf{(E)}\ 9$

[u]Round 1[/u] [b]p1.[/b] Today, the date $4/9/16$ has the property that it is written with three perfect squares in strictly increasing order. What is the next date with this property? [b]p2.[/b] What is the greatest integer less than $100$ whose digit sumis equal to its greatest prime factor? [b]p3.[/b] In chess, a bishop can only move diagonally any number of squares. Find the number of possible squares a bishop starting in a corner of a $20\times 16$ chessboard can visit in finitely many moves, including the square it stars on. [u]Round 2 [/u] [b]p4.[/b] What is the fifth smallest positive integer with at least $5$ distinct prime divisors? [b]p5.[/b] Let $\tau (n)$ be the number of divisors of a positive integer $n$, including $1$ and $n$. Howmany positive integers $n \le 1000$ are there such that $\tau (n) > 2$ and $\tau (\tau (n)) = 2$? [b]p6.[/b] How many distinct quadratic polynomials $P(x)$ with leading coefficient $1$ exist whose roots are positive integers and whose coefficients sum to $2016$? [u]Round 3[/u] [b]p7.[/b] Find the largest prime factor of $112221$. [b]p8.[/b] Find all ordered pairs of positive integers $(a,b)$ such that $\frac{a^2b^2+1}{ab-1}$ is an integer. [b]p9.[/b] Suppose $f : Z \to Z$ is a function such that $f (2x)= f (1-x)+ f (1-x)$ for all integers $x$. Find the value of $f (2) f (0) +f (1) f (6)$. [u]Round 4[/u] [b]p10.[/b] For any six points in the plane, what is the maximum number of isosceles triangles that have three of the points as vertices? [b]p11.[/b] Find the sum of all positive integers $n$ such that $\sqrt{n+ \sqrt{n -25}}$ is also a positive integer. [b]p12.[/b] Distinct positive real numbers are written at the vertices of a regular $2016$-gon. On each diagonal and edge of the $2016$-gon, the sum of the numbers at its endpoints is written. Find the minimum number of distinct numbers that are now written, including the ones at the vertices. PS. You should use hide for answers. Rounds 5-8 have been posted [url=https://artofproblemsolving.com/community/c3h3158474p28715078]here[/url]. and 9-12 [url=https://artofproblemsolving.com/community/c3h3162282p28763571]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

A right-angled triangle has an angle equal to $30^\circ.$ Prove that one of the bisectors of the triangle is twice as short as another one. [i]Egor Bakaev[/i]

A function $f_n(x)\ (n=1,\ 2,\ \cdots)$ is defined by $f_1(x)=x$ and \[f_n(x)=x+\frac{e}{2}\int_0^1 f_{n-1}(t)e^{x-t}dt\ (n=2,\ 3,\ \cdots)\]. Find $f_n(x)$.

In a cyclic quadrilateral $ ABCD$ the sides $ AB$ and $ AD$ are equal, $ CD>AB\plus{}BC$. Prove that $ \angle ABC>120^\circ$.

A sequence is defined as $a_1=2025$ and for all $n\ge 2$, $$a_n=\frac{a_{n-1}+1}{n}$$ Determine the smallest $k$ such that $\displaystyle a_k<\frac{1}{2025}$. [i]Proposed by Ivan Chan Kai Chin[/i]