For each positive integer $p$, let $b(p)$ denote the unique positive integer $k$ such that $|k-\sqrt{p}|<\frac{1}{2}$. For example, $b(6) = 2$ and $b(23)=5$. If $S = \textstyle\sum_{p=1}^{2007}b(p)$, find the remainder when S is divided by 1000.

Consider the four lines $y = mx-k^2$ for different integer $k$. Let $(x_i,y_i)$, $i = 1,2,3,4$ be four different points , such that each belongs to two different lines and on each line pass through just the two of them. Lat $x_1\leq x_2\leq x_3\leq x_4$. Show that $x_1 + x_4 =x_2+x_3$ and $y_1y_4 =y_2y_3$.

Let $n$ be a positive integer and consider an arrangement of $2n$ blocks in a straight line, where $n$ of them are red and the rest blue. A swap refers to choosing two consecutive blocks and then swapping their positions. Let $A$ be the minimum number of swaps needed to make the first $n$ blocks all red and $B$ be the minimum number of swaps needed to make the first $n$ blocks all blue. Show that $A+B$ is independent of the starting arrangement and determine its value.

In the tetrahedron $ABCD$ three of the faces are right-angled triangles and the other is not an obtuse triangle. Prove that: (a) the fourth wall of the tetrahedron is a right-angled triangle if and only if exactly two of the plane angles having common vertex with the some of vertices of the tetrahedron are equal. (b) its volume is equal to $\frac16$ multiplied by the multiple of two shortest edges and an edge not lying on the same wall.

Find the minimal $k$ such that every set of $k$ different lines in $\mathbb R^3$ contains either $3$ mutually parallel lines or $3$ mutually intersecting lines or $3$ mutually skew lines.

The real numbers $x_1,\ldots ,x_{2011}$ satisfy \[x_1+x_2=2x_1',\ x_2+x_3=2x_2', \ \ldots, \ x_{2011}+x_1=2x_{2011}'\] where $x_1',x_2',\ldots,x_{2011}'$ is a permutation of $x_1,x_2,\ldots,x_{2011}$. Prove that $x_1=x_2=\ldots =x_{2011}$ .

Let $O$ is the center of the circumcircle of the triangle $ABC$. We know that $AB =1$ and $AO = AC = 2$ . Points $D$ and $E$ lie on extensions of sides $AB$ and $AC$ beyond points $B$ and $C$ respectively such that $OD = OE$ and $BD =\sqrt2 EC$. Find $OD^2$.

Let $A_1A_2...A_{99}$ be a regular $99-$gon and point $A_{100}$ be its center. find the smallest possible natural number $n$ , such that Parsa can color all segments $A_iA_j$ ( $1 \le i < j \le 100$ ) with one of $n$ colors in such a way that no two homochromatic segments intersect each other or share a vertex. [i]Proposed by Josef Tkadlec - Czech Republic[/i]

Jiiri and Mari both wish to tile an $n \times n$ chessboard with cards shown in the picture (each card covers exactly one square). Jiiri wants that for each two cards that have a common edge, the neighbouring parts are of different color, and Mari wants that the neighbouring parts are always of the same color. How many possibilities does Jiiri have to tile the chessboard and how many possibilities does Mari have? [img]https://cdn.artofproblemsolving.com/attachments/7/3/9c076eb17ba7ae7c000a2893c83288a94df384.png[/img]

Jacob likes to watchMickeyMouse Clubhouse! One day, he decides to create his own MickeyMouse head shown below, with two circles $\omega_1$ and $\omega_2$ and a circle $\omega$, and centers $O_1$, $O_2$, and $O$, respectively. Let $\omega_1$ and $\omega$ meet at points $P_1$ and $Q_1$, and let $\omega_2$ and $\omega$ meet at points $P_2$ and $Q_2$. Point $P_1$ is closer to $O_2$ than $Q_1$, and point $P_2$ is closer to $O_1$ than $Q_2$. Given that $P_1$ and $P_2$ lie on $O_1O_2$ such that $O_1P_1 = P_1P_2 = P_2O_2 = 2$, and $Q_1O_1 \parallel Q_2O_2$, the area of $\omega$ can be written as $n \pi$. Find $n$. [img]https://cdn.artofproblemsolving.com/attachments/6/d/d98a05ee2218e80fd84d299d47201669736d99.png[/img]

Twenty-seven players are randomly split into three teams of nine. Given that Zack is on a different team from Mihir and Mihir is on a different team from Andrew, what is the probability that Zack and Andrew are on the same team?

Sequence $ \{ f_n(a) \}$ satisfies $ \displaystyle f_{n\plus{}1}(a) \equal{} 2 \minus{} \frac{a}{f_n(a)}$, $ f_1(a) \equal{} 2$, $ n\equal{}1,2, \cdots$. If there exists a natural number $ n$, such that $ f_{n\plus{}k}(a) \equal{} f_{k}(a), k\equal{}1,2, \cdots$, then we call the non-zero real $ a$ a $ \textbf{periodic point}$ of $ f_n(a)$. Prove that the sufficient and necessary condition for $ a$ being a $ \textbf{periodic point}$ of $ f_n(a)$ is $ p_n(a\minus{}1)\equal{}0$, where $ \displaystyle p_n(x)\equal{}\sum_{k\equal{}0}^{\left[ \frac{n\minus{}1}{2} \right]} (\minus{}1)^k C_n^{2k\plus{}1}x^k$, here we define $ \displaystyle \frac{a}{0}\equal{} \infty$ and $ \displaystyle \frac{a}{\infty} \equal{} 0$.

Find $n$ such that $n - 76$ and $n + 76$ are both cubes of positive integers.

Let $\omega_1$ be the circumcircle of triangle $ABC$ and $O$ be its circumcenter. A circle $\omega_2$ touches the sides $AB, AC$, and touches the arc $BC$ of $\omega_1$ at point $K$. Let $I$ be the incenter of $ABC$. Prove that the line $OI$ contains the symmedian of triangle $AIK$.

If $a, b, c$ are real numbers such that $a+b+c=6$ and $ab+bc+ca = 9$, find the sum of all possible values of the expression $\lfloor a \rfloor + \lfloor b \rfloor + \lfloor c \rfloor$.

Let $S(n)$ denote the sum of digits of $n$ (in decimal representation). Do there exist three different natural numbers $n$, $p$ and $q$ such that $$n +S(n) = p + S(p) = q + S(q)?$$ (M Gerver)

For any positive $a$ and $b$, find positive solutions of the system $$\begin{cases} \dfrac{a^2}{x^2}- \dfrac{b^2}{y^2}=8(y^4-x^4) \\ ax-by=x^4-y^4 \end{cases}$$

Consider a square $8 \times 8$ board with its $64$ cells initially white. Determine the minimum number of colors needed to color the cells (each one with only one color) in such a way that if four cells on the board can be covered by an $L$-shaped tile as shown in the figure, then the four cells are of different colors. [asy] size(3cm); draw((1,0)--(1,1)--(2,1)--(2,0)--(1,0)--(0,0)--(0,1)--(0,2)--(1,2)--(1,1)--(0,1)--(1,1)--(2,1)--(3,1)--(3,0)--(2,0)); [/asy] [b]Note:[/b] The $L$-shaped tile can be rotated or flipped.

What is the dimension of the space of solutions continuous on $x^2+y^2\ge1$ of the problem \[\Delta u=0\text{ for }x^2+y^2>1\] \[\partial u/\partial n=0\text{ for }x^2+y^2=1\]

Consider the multiplicative group $A=\{z\in\mathbb{C}|z^{2006^k}=1, 0<k\in\mathbb{Z}\}$ of all the roots of unity of degree $2006^k$ for all positive integers $k$. Find the number of homomorphisms $f:A\to A$ that satisfy $f(f(x))=f(x)$ for all elements $x\in A$.

Let $\mathcal O_{1}$ and $\mathcal O_{2}$ two exterior circles. Let $A$, $B$, $C$ be points on $\mathcal O_{1}$ and $D$, $E$, $F$ points on $\mathcal O_{1}$ such that $AD$ and $BE$ are the common exterior tangents to these two circles and $CF$ is one of the interior tangents to these two circles, and such that $C$, $F$ are in the interior of the quadrilateral $ABED$. If $CO_{1}\cap AB=\{M\}$ and $FO_{2}\cap DE=\{N\}$ then prove that $MN$ passes through the middle of $CF$.

Fred chooses a positive two-digit number with distinct nonzero digits. Laura takes Fred’s number and swaps its digits. She notices that the sum of her number and Fred’s number is a perfect square and the positive difference between them is a perfect cube. Find the greater of the two numbers.

Given an integer $k \geq 2$, determine the largest number of divisors the binomial coefficient $\binom{n}{k}$ may have in the range $n-k+1, \ldots, n$ , as $n$ runs through the integers greater than or equal to $k$.

Let $n \geq 3$ be an integer and $a_1,a_2,...,a_n$ be positive real numbers such that $m$ is the smallest and $M$ is the largest of these numbers. It is known that for any distinct integers $1 \leq i,j,k \leq n$, if $a_i \leq a_j \leq a_k$ then $a_ia_k \leq a_j^2$. Show that \[ a_1a_2 \cdots a_n \geq m^2M^{n-2} \] and determine when equality holds

Ethan Song and Bryan Guo are playing an unfair game of rock-paper-scissors. In any game, Ethan has a 2/5 chance to win, 2/5 chance to tie, and 1/5 chance to lose. How many games is Ethan expected to win before losing? [i]2022 CCA Math Bonanza Lightning Round 4.3[/i]