If the numbers $2a + 2$ and $2b + 2$ add up to $2004$, find the sum of the numbers $\frac{a}{2} - 2$ and $\frac{b}{2} - 2$

For each positive integer $n$, let $f(n)$ denote the number of ways of representing $n$ as a sum of powers of 2 with nonnegative integer exponents. Representations which differ only in the ordering of their summands are considered to be the same. For instance, $f(4)=4$, because the number $4$ can be represented in the following four ways: \[4, 2+2, 2+1+1, 1+1+1+1.\] Prove that, for any integer $n \geq 3$, \[2^{n^{2}/4}< f(2^{n}) < 2^{n^{2}/2}.\]

Let $n>1$ be an integer, and let $k$ be the number of distinct prime divisors of $n$. Prove that there exists an integer $a$, $1<a<\frac{n}{k}+1$, such that $n \mid a^2-a$.

For a certain positive integer $n$ less than $1000$, the decimal equivalent of $\frac{1}{n}$ is $0.\overline{abcdef}$, a repeating decimal of period $6$, and the decimal equivalent of $\frac{1}{n+6}$ is $0.\overline{wxyz}$, a repeating decimal of period $4$. In which interval does $n$ lie? $\textbf{(A)}\ [1,200] \qquad \textbf{(B)}\ [201,400] \qquad \textbf{(C)}\ [401,600] \qquad \textbf{(D)}\ [601,800] \qquad \textbf{(E)}\ [801,999] $

Find all triples $(\alpha, \beta, \gamma)$ of positive real numbers for which the expression $$K = \frac{\alpha+3 \gamma}{\alpha + 2\beta + \gamma} + \frac{4\beta}{\alpha+\beta+2\gamma} - \frac{8 \gamma}{\alpha+ \beta + 3\gamma}$$ obtains its minimum value.

Let $ABC$ be a triangle with $AC > BC,$ let $\omega$ be the circumcircle of $\triangle ABC,$ and let $r$ be its radius. Point $P$ is chosen on $\overline{AC}$ such taht $BC=CP,$ and point $S$ is the foot of the perpendicular from $P$ to $\overline{AB}$. Ray $BP$ mets $\omega$ again at $D$. Point $Q$ is chosen on line $SP$ such that $PQ = r$ and $S,P,Q$ lie on a line in that order. Finally, let $E$ be a point satisfying $\overline{AE} \perp \overline{CQ}$ and $\overline{BE} \perp \overline{DQ}$. Prove that $E$ lies on $\omega$.

$x,y,z$ are positive real numbers such that: $$xyz+x+y+z=6$$ $$xyz+2xy+yz+zx+z=10$$ Find the maximum value of: $$(xy+1)(yz+1)(zx+1)$$

Al has the cards $1,2,\dots,10$ in a row in increasing order. He first chooses the cards labeled $1$, $2$, and $3$, and rearranges them among their positions in the row in one of six ways (he can leave the positions unchanged). He then chooses the cards labeled $2$, $3$, and $4$, and rearranges them among their positions in the row in one of six ways. (For example, his first move could have made the sequence $3,2,1,4,5,\dots,$ and his second move could have rearranged that to $2,4,1,3,5,\dots$.) He continues this process until he has rearranged the cards with labels $8$, $9$, $10$. Determine the number of possible orderings of cards he can end up with. [i]Proposed by Ray Li[/i]

Find all real numbers $x,y$ greater than $1$, satisfying the condition that the numbers $\sqrt{x-1}+\sqrt{y-1}$ and $\sqrt{x+1}+\sqrt{y+1}$ are nonconsecutive integers.

Let be a natural number $ n\ge 2 $ and a real number $ r>1. $ Determine the natural numbers $ k $ having the property that the affixes of $ r^ke^{\pi ki/n} ,r^{k+1}e^{\pi (k+1)i/n} ,r^{k+n}e^{\pi (k+n)i/n} ,r^{k+n+1}e^{\pi (k+n+1) i/n} $ in the complex plane represent the vertices of a trapezoid.

On the bisector $AL$ of triangle $ABC$ a point $K$ is chosen such that $\angle BKL=\angle KBL=30^{\circ}$. Lines $AB$ and $CK$ intersect at point $M$, lines $AC$ and $BK$ intersect at point $N$. FInd the measure of angle $\angle AMN$ [I]Proposed by D. Shiryaev, S. Berlov[/i]

Prove that with $ n\ge 1 $ distinct numbers we can form an arithmetic progression if and only if there are exactly $ n-1 $ distinct elements in the set of positive differences between any two of these numbers.

Let $A=\{1,2,4,5,7,8,\dots\}$ the set with naturals not divisible by three. Find all values of $n$ such that exist $2n$ consecutive elements of $A$ which sum it´s $300$.

[u]Round 5[/u] [b]p13.[/b] Pieck the Frog hops on Pascal’s Triangle, where she starts at the number $1$ at the top. In a hop, Pieck can hop to one of the two numbers directly below the number she is currently on with equal probability. Given that the expected value of the number she is on after $7$ hops is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers, find $m+n$. [b]p14.[/b] Maisy chooses a random set $(x, y)$ that satisfies $$x^2 + y^2 -26x -10y \le 482.$$ The probability that $y>0$ can be expressed as $\frac{A\pi -B\sqrt{C}}{D \pi}$. Find $A+B +C +D$. [color=#f00]Due to the problem having a typo, all teams who inputted answers received points[/color] [b]p15.[/b] $6$ points are located on a circle. How many ways are there to draw any number of line segments between the points such that none of the line segments overlap and none of the points are on more than one line segment? (It is possible to draw no line segments). [u]Round 6[/u] [b]p16.[/b] Find the number of $3$ by $3$ grids such that each square in the grid is colored white or black and no two black squares share an edge. [b]p17.[/b] Let $ABC$ be a triangle with side lengths $AB = 20$, $BC = 25$, and $AC = 15$. Let $D$ be the point on BC such that $CD = 4$. Let $E$ be the foot of the altitude from $A$ to $BC$. Let $F$ be the intersection of $AE$ with the circle of radius $7$ centered at $A$ such that $F$ is outside of triangle $ABC$. $DF$ can be expressed as $\sqrt{m}$, where $m$ is a positive integer. Find $m$. [b]p18.[/b] Bill and Frank were arrested under suspicion for committing a crime and face the classic Prisoner’s Dilemma. They are both given the choice whether to rat out the other and walk away, leaving their partner to face a $9$ year prison sentence. Given that neither of them talk, they both face a $3$ year sentence. If both of them talk, they both will serve a $6$ year sentence. Both Bill and Frank talk or do not talk with the same probabilities. Given the probability that at least one of them talks is $\frac{11}{36}$ , find the expected duration of Bill’s sentence in months. [u]Round 7[/u] [b]p19.[/b] Rectangle $ABCD$ has point $E$ on side $\overline{CD}$. Point $F$ is the intersection of $\overline{AC}$ and $\overline{BE}$. Given that the area of $\vartriangle AFB$ is $175$ and the area of $\vartriangle CFE$ is $28$, find the area of $ADEF$. [b]p20.[/b] Real numbers $x, y$, and $z$ satisfy the system of equations $$5x+ 13y -z = 100,$$ $$25x^2 +169y^2 -z2 +130x y= 16000,$$ $$80x +208y-2z = 2020.$$ Find the value of $x yz$. [color=#f00]Due to the problem having infinitely many solutions, all teams who inputted answers received points. [/color] [b]p21.[/b] Bob is standing at the number $1$ on the number line. If Bob is standing at the number $n$, he can move to $n +1$, $n +2$, or $n +4$. In howmany different ways can he move to the number $10$? [u]Round 8[/u] [b]p22.[/b] A sequence $a_1,a_2,a_3, ...$ of positive integers is defined such that $a_1 = 4$, and for each integer $k \ge 2$, $$2(a_{k-1} +a_k +a_{k+1}) = a_ka_{k-1} +8.$$ Given that $a_6 = 488$, find $a_2 +a_3 +a_4 +a_5$. [b]p23.[/b] $\overline{PQ}$ is a diameter of circle $\omega$ with radius $1$ and center $O$. Let $A$ be a point such that $AP$ is tangent to $\omega$. Let $\gamma$ be a circle with diameter $AP$. Let $A'$ be where $AQ$ hits the circle with diameter $AP$ and $A''$ be where $AO$ hits the circle with diameter $OP$. Let $A'A''$ hit $PQ$ at $R$. Given that the value of the length $RA'$ is is always less than $k$ and $k$ is minimized, find the greatest integer less than or equal to $1000k$. [b]p24.[/b] You have cards numbered $1,2,3, ... ,100$ all in a line, in that order. You may swap any two adjacent cards at any time. Given that you make ${100 \choose 2}$ total swaps, where you swap each distinct pair of cards exactly once, and do not do any swaps simultaneously, find the total number of distinct possible final orderings of the cards. PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h3166472p28814057]here [/url] and 9-12 [url=https://artofproblemsolving.com/community/c3h3166480p28814155]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Points $X_1$ and $X_2$ move along fixed circles with centers $O_1$ and $O_2$, respectively, so that $O_1X_1 \parallel O_2X_2$. Find the locus of the intersection point of lines $O_1X_2$ and $O_2X_1$.

Prove for positive reals $a,b,c$ that $(ab+bc+ca+1)(a+b)(b+c)(c+a) \ge 2abc(a+b+c+1)^2$