Fill in numbers in the boxes below so that the sum of the entries in each three consecutive boxes is $2005$. What is the number that goes into the leftmost box? [asy] size(300); label("999",(2.5,.5)); label("888",(7.5,.5)); draw((0,0)--(9,0)); draw((0,1)--(9,1)); for (int i=0; i<=9; ++i) { draw((i,0)--(i,1)); } [/asy]

If the (convex) area bounded by the x-axis and the lines $y=mx+4$, $x=1$, and $x=4$ is $7$, then $m$ equals: $\textbf{(A)}\ -\frac{1}{2}\qquad \textbf{(B)}\ -\frac{2}{3}\qquad \textbf{(C)}\ -\frac{3}{2} \qquad \textbf{(D)}\ -2 \qquad \textbf{(E)}\ \text{none of these}$

A positive integer n is called [i]primary divisor [/i] if for every positive divisor $d$ of $n$ at least one of the numbers $d - 1$ and $d + 1$ is prime. For example, $8$ is divisor primary, because its positive divisors $1$, $2$, $4$, and $8$ each differ by $1$ from a prime number ($2$, $3$, $5$, and $7$, respectively), while $9$ is not divisor primary, because the divisor $9$ does not differ by $1$ from a prime number (both $8$ and $10$ are composite). Determine the largest primary divisor number.

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

In the Rainbow Nation there are two airways: Red Rockets and Blue Boeings. For any two cities in the Rainbow Nation it is possible to travel from the one to the other using either or both of the airways. It is known, however, that it is impossible to travel from Beanville to Mieliestad using only Red Rockets - not directly nor by travelling via other cities. Show that, using only Blue Boeings, one can travel from any city to any other city by stopping at at most one city along the way.

The two circles $\Gamma_1$ and $\Gamma_2$ with the midpoints $O_1$ resp. $O_2$ intersect in the two distinct points $A$ and $B$. A line through $A$ meets $\Gamma_1$ in $C \neq A$ and $\Gamma_2$ in $D \neq A$. The lines $CO_1$ and $DO_2$ intersect in $X$. Prove that the four points $O_1,O_2,B$ and $X$ are concyclic.

A sequence $(a_n)_{n\ge 1}$ of integers is defined by the recurrence \[a_1=1,\ a_2=5,\ a_n=\frac{a_{n-1}^2+4}{a_{n-2}}\ \text{for}\ n\ge 2.\] Prove that all terms of the sequence are integers and find an explicit formula for $a_n$.

Prove that none of the digits $2$, $4$, $7$, $9$ can be the last digit of a number $$ 1 + 2 + 3 + \ldots + n,$$ where $n$ is a natural number.

Let $ABCD$ be a convex quadrilateral.Let diagonals $AC$ and $BD$ intersect at $P$. Let $PE,PF,PG$ and $PH$ are altitudes from $P$ on the side $AB,BC,CD$ and $DA$ respectively. Show that $ABCD$ has a incircle if and only if $\frac{1}{PE}+\frac{1}{PG}=\frac{1}{PF}+\frac{1}{PH}.$

Let $ABC$ be an acute, scalene triangle with circumcenter $O$, circumcircle $(O)$, orthocenter $H$. Line $AH$ meets $(O)$ again at $D \neq A$. Let $E, F$ be the midpoint of segments $AB, AC$ respectively. The line through $H$ and perpendicular to $HF$ meets line $BC$ at $K$. a) Line $DK$ meets $(O)$ again at $Y \neq D$. Prove that the intersection of line $BY$ and the perpendicular bisector of $BK$ lies on the circumcircle of triangle $OFY$. b) The line through $H$ and perpendicular to $HE$ meets line $BC$ at $L$. Line $DL$ meets $(O)$ again at $Z \neq D$. Let $M$ be the intersection of lines $BZ, OE$; $N$ be the intersection of lines $CY, OF$; $P$ be the intersection of lines $BY, CZ$. Let $T$ be the intersection of lines $YZ, MN$ and $d$ be the line through $T$ and perpendicular to $OA$. Prove that $d$ bisects $AP$.

How many ways can you color the squares of a $ 2 \times 2008$ grid in 3 colors such that no two squares of the same color share an edge?

Prove that the number $a =\overline{{1...1}{5...5}6}$ is a perfect square (where $1$s are $2012$ in total and $5$s are $2011$ in total)

Let $ABCDEF$ be a convex hexagon all of whose sides are tangent to a circle $\omega$ with centre $O$. Suppose that the circumcircle of triangle $ACE$ is concentric with $\omega$. Let $J$ be the foot of the perpendicular from $B$ to $CD$. Suppose that the perpendicular from $B$ to $DF$ intersects the line $EO$ at a point $K$. Let $L$ be the foot of the perpendicular from $K$ to $DE$. Prove that $DJ=DL$. [i]Proposed by Japan[/i]

Doug constructs a square window using $8$ equal-size panes of glass, as shown. The ratio of the height to width for each pane is $5:2$, and the borders around and between the panes are $2$ inches wide. In inches, what is the side length o the square window? [asy] fill((0,0)--(25,0)--(25,25)--(0,25)--cycle,grey); for(int i = 0; i < 4; ++i){ for(int j = 0; j < 2; ++j){ fill((6*i+2,11*j+3)--(6*i+5,11*j+3)--(6*i+5,11*j+11)--(6*i+2,11*j+11)--cycle,white); } }[/asy] $\textbf{(A) }26\qquad\textbf{(B) }28\qquad\textbf{(C) }30\qquad\textbf{(D) }32\qquad\textbf{(E) }34$

For any continuous real-valued function $f$ defined on the interval $[0,1],$ let \[\mu(f)=\int_0^1f(x)\,dx,\text{Var}(f)=\int_0^1(f(x)-\mu(f))^2\,dx, M(f)=\max_{0\le x\le 1}|f(x)|.\] Show that if $f$ and $g$ are continuous real-valued functions defined on the interval $[0,1],$ then \[\text{Var}(fg)\le 2\text{Var}(f)M(g)^2+2\text{Var}(g)M(f)^2.\]

Let $H$ be the orthocenter of an acute-angled triangle $ABC$ with $AC\ne BC$. The line through the midpoints of the segments $AB$ and $HC$ intersects the bisector of $\angle ACB$ at $D$. Suppose that the line $HD$ contains the circumcenter of $\triangle ABC$. Determine $\angle ACB$.

Differentiable functions $f_1,\ldots,f_n:\mathbb R\to\mathbb R$ are linearly independent. Prove that there exist at least $n-1$ linearly independent functions among $f_1',\ldots,f_n'$.

The number of real solutions to the equation \[ \frac{x}{100} = \sin x \] is $\text{(A)} \ 61 \qquad \text{(B)} \ 62 \qquad \text{(C)} \ 63 \qquad \text{(D)} \ 64 \qquad \text{(E)} \ 65$

Ryan is learning number theory. He reads about the [i]Möbius function[/i] $\mu : \mathbb N \to \mathbb Z$, defined by $\mu(1)=1$ and \[ \mu(n) = -\sum_{\substack{d\mid n \\ d \neq n}} \mu(d) \] for $n>1$ (here $\mathbb N$ is the set of positive integers). However, Ryan doesn't like negative numbers, so he invents his own function: the [i]dubious function[/i] $\delta : \mathbb N \to \mathbb N$, defined by the relations $\delta(1)=1$ and \[ \delta(n) = \sum_{\substack{d\mid n \\ d \neq n}} \delta(d) \] for $n > 1$. Help Ryan determine the value of $1000p+q$, where $p,q$ are relatively prime positive integers satisfying \[ \frac{p}{q}=\sum_{k=0}^{\infty} \frac{\delta(15^k)}{15^k}. \] [i]Proposed by Michael Kural[/i]

Find the maximum value of $$\int^1_0|f'(x)|^2|f(x)|\frac1{\sqrt x}dx$$over all continuously differentiable functions $f:[0,1]\to\mathbb R$ with $f(0)=0$ and $$\int^1_0|f'(x)|^2dx\le1.$$

What is the smallest non-square positive integer that is the product of four prime numbers (not necessarily distinct)?

A polynomial $P(x)$ with integer coefficients satisfies the following: if $F(x)$, $G(x)$, and $Q(x)$ are polynomials with integer coefficients satisfying $P\Big(Q(x)\Big)=F(x)\cdot G(x)$, then $F(x)$ or $G(x)$ is a constant polynomial. Prove that $P(x)$ is a constant polynomial.

Let $ a_1,a_2,...,a_6$ be real numbers such that: $ a_1 \not \equal{} 0, a_1a_6 \plus{} a_3 \plus{} a_4 \equal{} 2a_2a_5 \ \mathrm{and}\ a_1a_3 \ge a_2^2$ Prove that $ a_4a_6\le a_5^2$. When does equality holds?

Assume that $ a$, $ b$, $ c$ and $ d$ are the sides of a quadrilateral inscribed in a given circle. Prove that the product $ (ab \plus{} cd)(ac \plus{} bd)(ad \plus{} bc)$ acquires its maximum when the quadrilateral is a square.

[b]p7[/b] Cindy cuts regular hexagon $ABCDEF$ out of a sheet of paper. She folds $B$ over $AC$, resulting in a pentagon. Then, she folds $A$ over $CF$, resulting in a quadrilateral. The area of $ABCDEF$ is $k$ times the area of the resulting folded shape. Find $k$. [b]p8[/b] Call a sequence $\{a_n\} = a_1, a_2, a_3, . . .$ of positive integers [i]Fib-o’nacci[/i] if it satisfies $a_n = a_{n-1}+a_{n-2}$ for all $n \ge 3$. Suppose that $m$ is the largest even positive integer such that exactly one [i]Fib-o’nacci[/i] sequence satisfies $a_5 = m$, and suppose that $n$ is the largest odd positive integer such that exactly one [i]Fib-o’nacci[/i] sequence satisfies $a_5 = n$. Find $mn$. [b]p9[/b] Compute the number of ways there are to pick three non-empty subsets $A$, $B$, and $C$ of $\{1, 2, 3, 4, 5, 6\}$, such that $|A| = |B| = |C|$ and the following property holds: $$A \cap B \cap C = A \cap B = B \cap C = C \cap A.$$