Let $ABCD$ be a parallelogram of area $10$ with $AB = 3$ and $BC = 5$. Locate $E$, $F$ and $G$ on segments $\overline{AB}$, $\overline{BC}$ and $\overline{AD}$, respectively, with $AE = BF = AG = 2$. Let the line through $G$ parallel to $\overline{EF}$ intersect $\overline{CD}$ at $H$. The area of the quadrilateral $EFHG$ is $ \textbf{(A)}\ 4\qquad\textbf{(B)}\ 4.5\qquad\textbf{(C)}\ 5\qquad\textbf{(D)}\ 5.5\qquad\textbf{(E)}\ 6 $

Doofy duck buy tangerines in the store. All tangerines have equal weight and are divided into $9$, $10$, $11$, $12$, or $13$ equal wedges, although this cannot be seen without peeling them. How many tangerines does Doofy duck need to buy if he wishes to eat exactly one tangerine’s worth while eating at most one wedge from every tangerine? [i]Doofy duck only peels the tangerines at home.[/i]

Prove that if $x, y, z$ are reals, then $x^2(3y^2+3z^2-2yz)=>yz(2xy+2xz-yz)$

On the diagonals $AC$ and $CE$ of a regular hexagon $ABCDEF$ with side $1$ we mark points$ M$ and $N$ such that $AM = CN = a$. Find $a$ if the points $B, M, N$ lie on the same line.

The phrase "COLORFUL TARTAN'' is spelled out with wooden blocks, where blocks of the same letter are indistinguishable. How many ways are there to distribute the blocks among two bags of different color such that neither bag contains more than one of the same letter?

In right triangle $ ABC$, $ BC \equal{} 5$, $ AC \equal{} 12$, and $ AM \equal{} x$; $ \overline{MN} \perp \overline{AC}$, $ \overline{NP} \perp \overline{BC}$; $ N$ is on $ AB$. If $ y \equal{} MN \plus{} NP$, one-half the perimeter of rectangle $ MCPN$, then: [asy]defaultpen(linewidth(.8pt)); unitsize(2cm); pair A = origin; pair M = (1,0); pair C = (2,0); pair P = (2,0.5); pair B = (2,1); pair Q = (1,0.5); draw(A--B--C--cycle); draw(M--Q--P); label("$A$",A,SW); label("$M$",M,S); label("$C$",C,SE); label("$P$",P,E); label("$B$",B,NE); label("$N$",Q,NW);[/asy]$ \textbf{(A)}\ y \equal{} \frac {1}{2}(5 \plus{} 12) \qquad \textbf{(B)}\ y \equal{} \frac {5x}{12} \plus{} \frac {12}{5}\qquad \textbf{(C)}\ y \equal{} \frac {144 \minus{} 7x}{12}\qquad$ $ \textbf{(D)}\ y \equal{} 12\qquad \qquad\quad\,\, \textbf{(E)}\ y \equal{} \frac {5x}{12} \plus{} 6$

Let $k$ be a fixed positive integer. For each integer $1 \leq i \leq 4$, let $x_i$ and $y_i$ be positive integers such that their least common multiple is $k$. Suppose that the four points $(x_1, y_1)$, $(x_2, y_2)$, $(x_3, y_3)$, $(x_4, y_4)$ are the vertices of a non-degenerate rectangle in the Cartesian plane. Prove that $x_1x_2x_3x_4$ is a perfect square. [i]Andrei Chirita[/i]

Find all real polynomials $P(x)$ of degree $5$ such that $(x-1)^3| P(x)+1$ and $(x+1)^3| P(x)-1$.

A convex $2019$-gon $A_1A_2...A_{2019}$ is cut into smaller pieces along its $2019$ diagonals of the form $A_iA_{i+3}$ for $1 \le i \le2019$, where $A_{2020} = A_1$, $A_{2021} = A_2$, and $A_{2022} = A_3$. What is the least possible number of resulting pieces?

Given a trapezoid $ABCD$ ($AD \parallel BC$) with $\angle ABC > 90^\circ$ . Point $M$ is chosen on the lateral side $AB$. Let $O_1$ and $O_2$ be the circumcenters of the triangles $MAD$ and $MBC$, respectively. The circumcircles of the triangles $MO_1D$ and $MO_2C$ meet again at the point $N$. Prove that the line $O_1O_2$ passes through the point $N$.

In the acute-angled triangle $ABC$, the point $F$ is the foot of the altitude from $A$, and $P$ is a point on the segment $AF$. The lines through $P$ parallel to $AC$ and $AB$ meet $BC$ at $D$ and $E$, respectively. Points $X \ne A$ and $Y \ne A$ lie on the circles $ABD$ and $ACE$, respectively, such that $DA = DX$ and $EA = EY$. Prove that $B, C, X,$ and $Y$ are concyclic.

If $a,b,c$ are the sides of a triangle and $m_a , m_b, m_c$ are the medians prove that \[4(m_a^2+m_b^2+m_c^2)=3(a^2+b^2+c^2)\]

Let $a$,$b$,$c$ be nonnegative reals such that $$a^2+b^2+c^2+ab+\frac{2}{3}ac+\frac{4}{3}bc=1$$ Find the maximum and minimum value of $a+b+c$.

Prove the theorem: the lengths $ a$, $ b $, $ c $ of the sides of a triangle and the arc measures $ \alpha $, $ \beta $, $ \gamma $of its opposite angles satisfy the inequalities $$\frac{\pi}{3}\leq \frac{a \alpha + b \beta +c \gamma}{a+b+c}<\frac{\pi }{ 2}.$$

Let $S$ be the sum of all real $x$ such that $4^x = x^4$. Find the nearest integer to $S$.

Let $P(x) = x^2 - 20x - 11$. If $a$ and $b$ are natural numbers such that $a$ is composite, $\gcd(a, b) = 1$, and $P(a) = P(b)$, compute $ab$. Note: $\gcd(m, n)$ denotes the greatest common divisor of $m$ and $n$. [i]Proposed by Aaron Lin [/i]

Let $a,b,c$ be given positive integers. Prove that there exists some positive integer $N$ such that \[ a\mid Nbc+b+c,\ b\mid Nca+c+a,\ c\mid Nab+a+b \] if and only if, denoting $d=\gcd(a,b,c)$ and $a=dx$, $b=dy$, $c=dz$, the positive integers $x,y,z$ are pairwise coprime, and also $\gcd(d,xyz) \mid x+y+z$. (Dan Schwarz)

In neverland, there are $n$ cities and $n$ airlines. Each airline serves an odd number of cities in a circular way, that is, if it serves cities $c_1,c_2,\dots,c_{2k+1}$, then they fly planes connecting $c_1c_2,c_2c_3,\dots,c_1c_{2k+1}$. Show that we can select an odd number of cities $d_1,d_2,\dots,d_{2m+1}$ such that we can fly $d_1\rightarrow d_2\rightarrow\dots\rightarrow d_{2m+1}\rightarrow d_1$ while using each airline at most once.

Circle $\omega_1$ of radius $1$ and circle $\omega_2$ of radius $2$ are concentric. Godzilla inscribes square $CASH$ in $\omega_1$ and regular pentagon $MONEY$ in $\omega_2$. It then writes down all 20 (not necessarily distinct) distances between a vertex of $CASH$ and a vertex of $MONEY$ and multiplies them all together. What is the maximum possible value of his result?

Determine all pairs $P(x),Q(x)$ of complex polynomials with leading coefficient $1$ such that $P(x)$ divides $Q(x)^2+1$ and $Q(x)$ divides $P(x)^2+1$. [i]Proposed by Rodrigo Angelo, Princeton University and Matheus Secco, PUC, Rio de Janeiro[/i]

What is the sum of real roots of $(2+(2+(2+x)^2)^2)^2=2000$ ? $ \textbf{(A)}\ -4 \qquad\textbf{(B)}\ -2 \qquad\textbf{(C)}\ 0 \qquad\textbf{(D)}\ 2 \qquad\textbf{(E)}\ 4 $

Call a positive integer [b]good[/b] if either $N=1$ or $N$ can be written as product of [i]even[/i] number of prime numbers, not necessarily distinct. Let $P(x)=(x-a)(x-b),$ where $a,b$ are positive integers. (a) Show that there exist distinct positive integers $a,b$ such that $P(1),P(2),\cdots ,P(2010)$ are all good numbers. (b) Suppose $a,b$ are such that $P(n)$ is a good number for all positive integers $n$. Prove that $a=b$.

A polynomial $P(x)$ has unity as the coefficient of its highest power, and has the property that with natural number arguments, it can take all values of form $2^M$ , where $M$ is a natural number. Prove that the polynomial is of degree $1$.

[u]Round 6[/u] [b]p16.[/b] Let $ABC$ be a triangle with $AB = 3$, $BC = 4$, and $CA = 5$. There exist two possible points $X$ on $CA$ such that if $Y$ and $Z$ are the feet of the perpendiculars from $X$ to $AB$ and $BC,$ respectively, then the area of triangle $XY Z$ is $1$. If the distance between those two possible points can be expressed as $\frac{a\sqrt{b}}{c}$ for positive integers $a$, $b$, and $c$ with $b$ squarefree and $gcd(a, c) = 1$, then find $a +b+ c$. [b]p17.[/b] Let $f(n)$ be the number of orderings of $1,2, ... ,n$ such that each number is as most twice the number preceding it. Find the number of integers $k$ between $1$ and $50$, inclusive, such that $f (k)$ is a perfect square. [b]p18.[/b] Suppose that $f$ is a function on the positive integers such that $f(p) = p$ for any prime p, and that $f (xy) = f(x) + f(y)$ for any positive integers $x$ and $y$. Define $g(n) = \sum_{k|n} f (k)$; that is, $g(n)$ is the sum of all $f(k)$ such that $k$ is a factor of $n$. For example, $g(6) = f(1) + 1(2) + f(3) + f(6)$. Find the sum of all composite $n$ between $50$ and $100$, inclusive, such that $g(n) = n$. [u]Round 7[/u] [b]p19.[/b] AJ is standing in the center of an equilateral triangle with vertices labelled $A$, $B$, and $C$. They begin by moving to one of the vertices and recording its label; afterwards, each minute, they move to a different vertex and record its label. Suppose that they record $21$ labels in total, including the initial one. Find the number of distinct possible ordered triples $(a, b, c)$, where a is the number of $A$'s they recorded, b is the number of $B$'s they recorded, and c is the number of $C$'s they recorded. [b]p20.[/b] Let $S = \sum_{n=1}^{\infty} (1- \{(2 + \sqrt3)^n\})$, where $\{x\} = x - \lfloor x\rfloor$ , the fractional part of $x$. If $S =\frac{\sqrt{a} -b}{c}$ for positive integers $a, b, c$ with $a $ squarefree, find $a + b + c$. [b]p21.[/b] Misaka likes coloring. For each square of a $1\times 8$ grid, she flips a fair coin and colors in the square if it lands on heads. Afterwards, Misaka places as many $1 \times 2$ dominos on the grid as possible such that both parts of each domino lie on uncolored squares and no dominos overlap. Given that the expected number of dominos that she places can be written as $\frac{a}{b}$, for positive integers $a$ and $b$ with $gcd(a, b) = 1$, find $a + b$. PS. You should use hide for answers. Rounds 1-3 have been posted [url=https://artofproblemsolving.com/community/c4h3131401p28368159]here [/url] and 4-5 [url=https://artofproblemsolving.com/community/c4h3131422p28368457]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $S$ the set of natural numbers from $1$ up to $1001$ , $S=\{1,2,...,1001\}$. Lisandro thinks of a number $N$ of $S$ , and Carla has to find out that number with the following procedure. She gives Lisandro a list of subsets of $S$, Lisandro reads it and tells Carla how many subsets of her list contain $N$ . If Carla wishes, she can repeat the same thing with a second list, and then with a third, but no more than $3$ are allowed. What is the smallest total number of subsets that allow Carla to find $N$ for sure?