Let $ 0<x_0,x_1, \dots , x_{669}<1$ be pairwise distinct real numbers. Show that there exists a pair $ (x_i,x_j)$ with $ 0<x_ix_j(x_j\minus{}x_i)<\frac{1}{2007}$

Let $a$, $b$ and $c$ positive integers. Three sequences are defined as follows: [list] [*] $a_1=a$, $b_1=b$, $c_1=c$[/*] [*] $a_{n+1}=\lfloor{\sqrt{a_nb_n}}\rfloor$, $\:b_{n+1}=\lfloor{\sqrt{b_nc_n}}\rfloor$, $\:c_{n+1}=\lfloor{\sqrt{c_na_n}}\rfloor$ for $n \ge 1$[/*] [/list] [list = a] [*]Prove that for any $a$, $b$, $c$, there exists a positive integer $N$ such that $a_N=b_N=c_N$.[/*] [*]Find the smallest $N$ such that $a_N=b_N=c_N$ for some choice of $a$, $b$, $c$ such that $a \ge 2$ y $b+c=2a-1$.[/*] [/list]

In the non-convex quadrilateral $ABCD$ shown below, $\angle BCD$ is a right angle, $AB=12$, $BC=4$, $CD=3$, and $AD=13$. [asy]draw((0,0)--(2.4,3.6)--(0,5)--(12,0)--(0,0)); label("$B$", (0, 0), SW); label("$A$", (12, 0), ESE); label("$C$", (2.4, 3.6), SE); label("$D$", (0, 5), N);[/asy] What is the area of quadrilateral $ABCD$? $\textbf{(A) }12\qquad\textbf{(B) }24\qquad\textbf{(C) }26\qquad\textbf{(D) }30\qquad\textbf{(E) }36$

Find the number of ordered pairs of positive integers $(a,b)$ such that $a+b=1000$ and neither $a$ nor $b$ has a zero digit.

Let $b$ and $c$ be random integers from the set $\{1, 2, \ldots, 100\}$, chosen uniformly and independently. What is the probability that the roots of the quadratic $x^2 + bx + c$ are real?

Given that the ratio of $3x-4$ to $y+15$ is constant, and $y=3$ when $x=2$, then, when $y=12$, $x$ equals: $\text{(A)} \ \frac 18 \qquad \text{(B)} \ \frac 73 \qquad \text{(C)} \ \frac78 \qquad \text{(D)} \ \frac72 \qquad \text{(E)} \ 8$

Find the least positive integer $N$ which is both a multiple of 19 and whose digits add to 23.

$$\int_1^2\left(x^5+6x^4+14x^3+16x^2+9x+2\right)dx$$ [i]Proposed by Connor Gordon[/i]

There are given one black box and $n$ white boxes ($n$ is a random natural number). White boxes are numbered with the numbers $1,2,\ldots,n$. In them are put $n$ balls. It is allowed the following rearrangement of the balls: if in the box with number $k$ there are exactly $k$ balls, that box is made empty - one of the balls is put in the black box and the other $k-1$ balls are put in the boxes with numbers: $1,2,\ldots,k-1$. [i](Ivan Tonov)[/i]

The point $P(a,b)$ in the $xy$-plane is first rotated counterclockwise by $90^{\circ}$ around the point $(1,5)$ and then reflected about the line $y=-x$. The image of $P$ after these two transformations is at $(-6,3)$. What is $b-a$? $\textbf{(A) }1 \qquad \textbf{(B) }3 \qquad \textbf{(C) }5 \qquad \textbf{(D) }7 \qquad \textbf{(E) }9$

What number should be removed from the list \[1,2,3,4,5,6,7,8,9,10,11\] so that the average of the remaining numbers is $6.1$? $\text{(A)}\ 4 \qquad \text{(B)}\ 5 \qquad \text{(C)}\ 6 \qquad \text{(D)}\ 7 \qquad \text{(E)}\ 8$

The graph of $ y\equal{}\log x$ $\textbf{(A)}\text{Cuts the }y\text{-axis} \qquad\\ \textbf{(B)}\ \text{Cuts all lines perpendicular to the }x\text{-axis} \qquad\\ \textbf{(C)}\ \text{Cuts the }x\text{-axis} \qquad\\ \textbf{(D)}\ \text{Cuts neither axis} \qquad\\ \textbf{(E)}\ \text{Cuts all circles whose center is at the origin}$

$N$ people have a series of calls. Each call is between two people, and is started by exactly one of them. Each person starts at most $10$ calls. Two people can call at most once. In any group of $3$ people, there are at least two people who have a call. Find the maximum possible value of $N$. [i]Proposed by Serena Xu[/i]

Five points are arbitrarily put inside a given equilateral triangle $ABC$ whose area is equal to $1$. Show that we can draw three equilateral triangles within triangle $ABC$ such that the following conditions are all satisfied: i) the five points are covered by the three equilateral triangles; ii) any side of the three equilateral triangles is parallel to a certain side of the triangle $ABC$; iii) the sum of the areas of the three equilateral triangles is not larger than $0.64$.

Given a scalene triangle $ABC$ with circuncenter $O$ and circumscribed circle $\Gamma$. Let $D, E ,F$ the midpoints of $BC, AC, AB$. Let $M=OE \cap AD$, $N=OF \cap AD$ and $P=CM \cap BN$. Let $X=AO \cap PE$, $Y=AP \cap OF$. Let $r$ the tangent of $\Gamma$ through $A$. Prove that $r, EF, XY$ are concurrent.

Let the equation $a^2 + b^2 + 1=abc$ have answer in $\mathbb{N}$.Prove that $c=3$.

Given triangle $ ABC$ with medians $ AE$, $ BF$, $ CD$; $ FH$ parallel and equal to $ AE$; $ BH$ and $ HE$ are drawn; $ FE$ extended meets $ BH$ in $ G$. Which one of the following statements is not necessarily correct? $ \textbf{(A)}\ AEHF \text{ is a parallelogram} \qquad \textbf{(B)}\ HE\equal{}HG \\ \textbf{(C)}\ BH\equal{}DC \qquad \textbf{(D)}\ FG\equal{}\frac{3}{4}AB \qquad \textbf{(E)}\ FG\text{ is a median of triangle }BFH$

Determine all functions $f : \mathbb{R}^+ \to \mathbb{R}^+$ such that \[ f(2x+f(y))=x+y +f(f(x)) , \ \ \ \forall x,y \in \mathbb{R}^+.\]

How many ways are there to place 8 1s and 8 0s in a $4\times 4$ array such that the sum in every row and column is 2? \begin{tabular}{|c|c|c|c|} \hline 1 & 0 & 0 & 1 \\ \hline 0 & 1 & 1 & 0 \\ \hline 0 & 1 & 1 & 0 \\ \hline 1 & 0 & 0 & 1 \\ \hline \end{tabular} [i]2016 CCA Math Bonanza Individual #11[/i]

Let $ABCDEF $ be a regular hexagon. On the line $AF $ mark the point $X$so that $ \angle DCX = 45^o$ . Find the value of the angle $FXE$. (Vyacheslav Yasinsky)

For each positive integer $m$, be $P(m)$ the product of all the digits of $m$. It defines the succession $a_1,a_2, a_3\cdots, $ as follows: . $a_1$ is a positive integer less than 2018 . $a_{n+1}=a_n+P(a_n)$ for each integer $n\ge 1$ Prove that for every integer $n \ge1$ it is true that $a_n \le 2^{2018}.$

On an infinite triangular lattice, there is a single atom at a lattice point. We allow for four operations as illustrated in Figure 1. In words, one could take an existing atom, split it into three atoms, and place them at adjacent lattice points in one of the two displayed fashions (a “split”). One could also reverse the process, i.e. taking three existing atoms in the displayed configurations, and merge them into a single atom at the center (a “merge”). [center][img]https://cdn.artofproblemsolving.com/attachments/2/5/41abc4dc8fb8235e5eb0c98638f9e4a0896c05.png[/img][/center] Figure 1: The four possible operations on an atom. Assume that, after finitely many operations, there is again only a single atom remaining on the lattice. Show that this is possible if and only if the final atom is contained in the sublattice implied by Figure 2. [center][img]https://cdn.artofproblemsolving.com/attachments/b/4/7a7bd10a1862947c250fa07571c061367a5a71.png[/img][/center] Figure 2: The possible positions for the final atom is the green sublattice. The position of the original atom is marked in purple.

Isosceles triangle $ABC$, with $AB=AC$, is inscribed in circle $\omega$. Let $D$ be an arbitrary point inside $BC$ such that $BD\neq DC$. Ray $AD$ intersects $\omega$ again at $E$ (other than $A$). Point $F$ (other than $E$) is chosen on $\omega$ such that $\angle DFE = 90^\circ$. Line $FE$ intersects rays $AB$ and $AC$ at points $X$ and $Y$, respectively. Prove that $\angle XDE = \angle EDY$. [i]Proposed by Anton Trygub[/i]

A circle with circumference 6n units is given and 3n points divide the circumference in n intervals of 1 unit, n intervals of 2 units, and n intervals of 3 units. Prove that there is at least one pair of points that are diametrically opposite to each other.

$a$ and $b$ are real numbers that satisfy \[a^4+a^2b^2+b^4=900,\] \[a^2+ab+b^2=45.\] Find the value of $2ab.$ [i]Author: Ray Li[/i]