Find all real numbers $x$ such that $\left\lfloor x^2 \right\rfloor + \left\lceil x^2 \right\rceil = 2003$.

In a triangle $[ABC]$, $\angle C = 2\angle A$. A point $D$ is marked on the side $[AC]$ such that $\angle ABD = \angle DBC$. Knowing that $AB = 10$ and $CD = 3$, what is the length of the side $[BC]$?

Suppose $X, Y, Z$ are collinear points in that order such that $XY = 1$ and $YZ = 3$. Let $W$ be a point such that $YW = 5$, and define $O_1$ and $O_2$ as the circumcenters of triangles $\triangle WXY$ and $\triangle WYZ$, respectively. What is the minimum possible length of segment $\overline{O_1O_2}$?

Find all $n\in\mathbb{Z}$ such that the number $\sqrt{\frac{4n-2}{n+5}}$ is rational.

Determine whether or not there exist two different sets $A,B$, each consisting of at most $2011^2$ positive integers, such that every $x$ with $0 < x < 1$ satisfies the following inequality: \[\left| \sum_{a \in A} x^a - \sum_{b \in B} x^b \right| < (1-x)^{2011}.\]

A point $P$ in the interior of a convex polyhedron in Euclidean space is called a [i]pivot point[/i] of the polyhedron if every line through $P$ contains exactly $0$ or $2$ vertices of the polyhedron. Determine, with proof, the maximum number of pivot points that a polyhedron can contain.

Consider the following two-player game: player $A$ (first mover) and $B$ take turns to write a positive integer less than or equal to $10$ on the blackboard. The integer written at any step cannot be a factor of any existing integer on board. Determine, with proof, who wins.

Point $M$ is chosen in triangle $ABC$ so that the radii of the circumcircles of triangles $AMC, BMC$, and $BMA$ are no smaller than the radius of the circumcircle of $ABC$. Prove that all four radii are equal.

A sleeping rabbit lies in the interior of a convex $2024$-gon. A hunter picks three vertices of the polygon and he lays a trap which covers the interior and the boundary of the triangular region determined by them. Determine the minimum number of times he needs to do this to guarantee that the rabbit will be trapped. [i]Proposed by Anant Mudgal and Rohan Goyal[/i]

Find all positive integers $a, b, c$ such that $a < b$, $a < 4c$, and $b c^3 \le a c^3 + b$.

The line $L$ is parallel to the plane $y=z$ and meets the parabola $2x=y^2 ,z=0$ and the parabola $3x=z^2, y=0$. Prove that if $L$ moves freely subject to these constraints then it generates the surface $x=(y-z)\left(\frac{y}{2}-\frac{z}{3}\right)$.

a) A certain locker room contains $n$ lockers numbered $1,2,\ldots,n$ and all are originally locked. An attendant performs a sequence of operations $T_1, T_2 ,\ldots, T_n$, whereby with the operation $T_k$ the state of those lockers whose number is divisible by $k$ is swapped. After all $n$ operations have been performed, it is observed that all lockers whose number is a perfect square (and only those lockers) are open. Prove this. b) Investigate in a meaningful mathematical way a procedure or set of operations similar to those above which will produce the set of cubes, or the set of numbers of the form $2 m^2 $, or the set of numbers of the form $m^2 +1$, or some nontrivial similar set of your own selection.

Josh and Mike live 13 miles apart. Yesterday, Josh started to ride his bicycle toward Mike's house. A little later Mike started to ride his bicycle toward Josh's house. When they met, Josh had ridden for twice the length of time as Mike and at four-fifths of Mike's rate. How many miles had Mike ridden when they met? $ \textbf{(A)}\ 4\qquad \textbf{(B)}\ 5\qquad \textbf{(C)}\ 6\qquad \textbf{(D)}\ 7\qquad \textbf{(E)}\ 8$

Set $X$ is called [i]fancy[/i] if it satisfies all of the following conditions: [list] [*]The number of elements of $X$ is $2022$. [*]Each element of $X$ is a closed interval contained in $[0, 1]$. [*]For any real number $r \in [0, 1]$, the number of elements of $X$ containing $r$ is less than or equal to $1011$. [/list] For [i]fancy[/i] sets $A, B$, and intervals $I \in A, J \in B$, denote by $n(A, B)$ the number of pairs $(I, J)$ such that $I \cap J \neq \emptyset$. Determine the maximum value of $n(A, B)$.

Let $A$ and $B$ be matrices of size $3\times 2$ and $2\times 3$ respectively. Suppose that $$AB =\begin{pmatrix} 8 & 2 & -2\\ 2 & 5 &4 \\ -2 &4 &5 \end{pmatrix}.$$ Show that the product $BA$ is equal to $\begin{pmatrix} 9 &0\\ 0 &9 \end{pmatrix}.$

Given a circle \( \omega \) and two points \( A \) and \( B \) outside \( \omega \), a quadrilateral \( PQRS \) is defined as[i] "good"[/i] if \( P, Q, R, S \) are four distinct points on \( \omega \) in order, and lines \( PQ \) and \( RS \) intersect at \( A \) and lines \( PS \) and \( QR \) intersect at \( B \). For a quadrilateral \( T \), let \( S_T \) denote its area. If there exists a [i]good[/i] quadrilateral, prove that there exists [i]good[/i] quadrilateral \( T \) such that for any good quadrilateral $T_1 (T_1 \neq T)$, \( S_{T_1} < S_T \).

Two circles $ w_1 $ and $ w_2 $ intersect at two points $ P $ and $ Q $. The common tangent to $ w_1 $ and $ w_2 $, which is closer to the point $ P $ than to $ Q $, touches these circles at $ A $ and $ B $, respectively. The tangent to $ w_1 $ at the point $ P $ intersects $ w_2 $ at the point $ E $ (different from $ P $), and the tangent to $ w_2 $ at the point $ P $ intersects $ w_1 $ at $ F $ (different from $ P $). Let $ H $ and $ K $ be points on the rays $ AF $ and $ BE $, respectively, such that $ AH = AP $ and $ BK = BP $. Prove that the points $ A $, $ H $, $ Q $, $ K $ and $ B $ lie on the same circle.

Find all prime number $p$ such that $p^{2002}+2003^{p-1}-1$ is a multiple of $2003p$.

$a$, $b$, $c$ are three distinct real numbers, given $\lambda >0$. Proof that \[\frac{1+ \lambda ^2a^2b^2}{(a-b)^2}+\frac{1+ \lambda ^2b^2c^2}{(b-c)^2}+\frac{1+ \lambda ^2c^2a^2}{(c-a)^2} \geq \frac 32 \lambda.\] [hide=Remark]Old problem, can be found [url=https://artofproblemsolving.com/community/c6h588854p3487434]here[/url]. Double post to have a cleaner thread for collection (as the original one contains a messy quote)[/hide]

Let $ABC$ be an acute triangle with $\displaystyle{AB<AC<BC}$ inscribed in circle $ \displaystyle{c(O,R)}$.The excircle $\displaystyle{(c_A)}$ has center $\displaystyle{I}$ and touches the sides $\displaystyle{BC,AC,AB}$ of the triangle $ABC$ at $\displaystyle{D,E,Z} $ respectively.$ \displaystyle{AI}$ cuts $\displaystyle{(c)}$ at point $M$ and the circumcircle $\displaystyle{(c_1)}$ of triangle $\displaystyle{AZE}$ cuts $\displaystyle{(c)}$ at $K$.The circumcircle $\displaystyle{(c_2)}$ of the triangle $\displaystyle{OKM}$ cuts $\displaystyle{(c_1)} $ at point $N$.Prove that the point of intersection of the lines $AN,KI$ lies on $ \displaystyle{(c)}$.

Determine whether there exist two reals $x,y$ and a sequence $\{a_n\}_{n=0}^{\infty}$ of nonzero reals such that $a_{n+2}=xa_{n+1}+ya_n$ for all $n\ge0$ and for every positive real number $r$, there exist positive integers $i,j$ such that $|a_i|<r<|a_j|$. [i]Alex Zhu.[/i]

In their base $ 10$ representation, the integer $ a$ consists of a sequence of $ 1985$ eights and the integer $ b$ consists of a sequence of $ 1985$ fives. What is the sum of the digits of the base 10 representation of $ 9ab$? $ \textbf{(A)}\ 15880 \qquad \textbf{(B)}\ 17856 \qquad \textbf{(C)}\ 17865 \qquad \textbf{(D)}\ 17874 \qquad \textbf{(E)}\ 19851$

The incircle of $\triangle ABC$ touch $AB$,$BC$,$CA$ at $K$,$L$,$M$. The common external tangents to the incircles of $\triangle AMK$,$\triangle BKL$,$\triangle CLM$, distinct from the sides of $\triangle ABC$, are drawn. Show that these three lines are concurrent.

Let $n$ be a positive integer with $k$ digits. A number $m$ is called an $alero$ of $n$ if there exist distinct digits $a_1$, $a_2$, $\dotsb$, $a_k$, all different from each other and from zero, such that $m$ is obtained by adding the digit $a_i$ to the $i$-th digit of $n$, and no sum exceeds 9. For example, if $n$ $=$ $2024$ and we choose $a_1$ $=$ $2$, $a_2$ $=$ $1$, $a_3$ $=$ $5$, $a_4$ $=$ $3$, then $m$ $=$ $4177$ is an alero of $n$, but if we choose the digits $a_1$ $=$ $2$, $a_2$ $=$ $1$, $a_3$ $=$ $5$, $a_4$ $=$ $6$, then we don't obtain an alero of $n$, because $4$ $+$ $6$ exceeds $9$. Find the smallest $n$ which is a multiple of $2024$ that has an alero which is also a multiple of $2024$.

Prove that there are infinitely many quadruples of integers $(a,b,c,d)$ such that \begin{align*} a^2 + b^2 + 3 &= 4ab\\ c^2 + d^2 + 3 &= 4cd\\ 4c^3 - 3c &= a \end{align*} [i]Travis Hance.[/i]