Let $f(X, Y, Z)=X^5Y-XY^5+Y^5Z-YZ^5+Z^5X-ZX^5$. Find $\frac{f(2009, 2010, 2011)+f(2010, 2011, 2009)-f(2011, 2010, 2009)}{f(2009, 2010, 2011)}$

For three sequences $(x_n),(y_n),(z_n)$ with positive starting elements $x_1,y_1,z_1$ we have the following formulae: \[ x_{n+1} = y_n + \frac{1}{z_n} \quad y_{n+1} = z_n + \frac{1}{x_n} \quad z_{n+1} = x_n + \frac{1}{y_n} \quad (n = 1,2,3, \ldots)\] a.) Prove that none of the three sequences is bounded from above. b.) At least one of the numbers $x_{200},y_{200},z_{200}$ is greater than 20.

A particle $P$ moves in the plane in such a way that the angle between the two tangents drawn from $P$ to the curve $y^2=4ax$ is always $90^\circ$. The locus of $P$ is $\textbf{(A)}~\text{a parabola}$ $\textbf{(B)}~\text{a circle}$ $\textbf{(C)}~\text{an ellipse}$ $\textbf{(D)}~\text{a straight line}$

Prove that there is no polynomial $P$ with integer coefficients such that $P\left(\sqrt[3]5+\sqrt[3]{25}\right)=5+\sqrt[3]5$.

In a $N \times N$ array of numbers, all rows are different (two rows are said to be different even if they differ only in one entry). Prove that there is a column which can be deleted in such a way that the resulting rows will still be different. (A Andjans, Riga)

Circles $ \mathcal{C}_1$ and $ \mathcal{C}_2$ intersect at $ A$ and $ B$. Let $ M\in AB$. A line through $ M$ (different from $ AB$) cuts circles $ \mathcal{C}_1$ and $ \mathcal{C}_2$ at $ Z,D,E,C$ respectively such that $ D,E\in ZC$. Perpendiculars at $ B$ to the lines $ EB,ZB$ and $ AD$ respectively cut circle $ \mathcal{C}_2$ in $ F,K$ and $ N$. Prove that $ KF\equal{}NC$.

Vijay chooses three distinct integers $a,b,c$ from the set $\{1,2,3,4,5,6,7,8,9,10,11\}.$ If $k$ is the minimum value taken on by the polynomial $a(x - b)(x - c)$ over all real numbers $x,$ and $l$ is the minimum value taken on by the polynomial $a(x-b)(x+c)$ over all real numbers $x,$ compute the maximum possible value of $k -l.$

Solve the equation $$ tg 7x - \sin 6x=\cos 4x - ctg 7x.$$

Given is a positive integer $n \geq 2$ and three pairwise disjoint sets $A, B, C$, each of $n$ distinct real numbers. Denote by $a$ the number of triples $(x, y, z) \in A \times B \times C$ satisfying $x<y<z$ and let $b$ denote the number of triples $(x, y, z) \in A \times B \times C$ such that $x>y>z$. Prove that $n$ divides $a-b$.

For a positive integer $n$, let function $f(n)$ denote the number of positive integers $a\leq n$ such that $\gcd(a,n) = \gcd(a+1,n) = 1$. Find the sum of all $n$ such that $f(n)=15$. [i]Proposed by Harry Kim[/i]

Consider the sequence $(a^n + 1)_{n\geq 1}$, with $a>1$ a fixed integer. i) Prove there exist infinitely many primes, each dividing some term of the sequence. ii) Prove there exist infinitely many primes, none dividing any term of the sequence. [i](Dan Schwarz)[/i]

Let $p=4k+3$ be a prime number. Find the number of different residues mod p of $(x^{2}+y^{2})^{2}$ where $(x,p)=(y,p)=1.$

Let $M=\{1,2,\cdots,n\}$, each element of $M$ is colored in either red, blue or yellow. Set $A=\{(x,y,z)\in M\times M\times M|x+y+z\equiv 0\mod n$, $x,y,z$ are of same color$\},$ $B=\{(x,y,z)\in M\times M\times M|x+y+z\equiv 0\mod n,$ $x,y,z$ are of pairwise distinct color$\}.$ Prove that $2|A|\geq |B|$.

Let the circles $k_1$ and $k_2$ intersect at two points $A$ and $B$, and let $t$ be a common tangent of $k_1$ and $k_2$ that touches $k_1$ and $k_2$ at $M$ and $N$ respectively. If $t\perp AM$ and $MN=2AM$, evaluate the angle $NMB$.

Prove that a square of side 2.1 units can be completely covered by seven squares of side 1 unit. Extra: Try to prove that 7 is the minimal amount.

[b]p1.[/b] A rectangular pool has diagonal $17$ units and area $120$ units$^2$. Joey and Rachel start on opposite sides of the pool when Rachel starts chasing Joey. If Rachel runs $5$ units/sec faster than Joey, how long does it take for her to catch him? [b]p2. [/b] Alice plays a game with her standard deck of $52$ cards. She gives all of the cards number values where Aces are $1$’s, royal cards are $10$’s and all other cards are assigned their face value. Every turn she flips over the top card from her deck and creates a new pile. If the flipped card has value $v$, she places $12 - v$ cards on top of the flipped card. For example: if she flips the $3$ of diamonds then she places $9$ cards on top. Alice continues creating piles until she can no longer create a new pile. If the number of leftover cards is $4$ and there are $5$ piles, what is the sum of the flipped over cards? [b]p3.[/b] There are $5$ people standing at $(0, 0)$, $(3, 0)$, $(0, 3)$, $(-3, 0)$, and $(-3, 0)$ on a coordinate grid at a time $t = 0$ seconds. Each second, every person on the grid moves exactly $1$ unit up, down, left, or right. The person at the origin is infected with covid-$19$, and if someone who is not infected is at the same lattice point as a person who is infected, at any point in time, they will be infected from that point in time onwards. (Note that this means that if two people run into each other at a non-lattice point, such as $(0, 1.5)$, they will not infect each other.) What is the maximum possible number of infected people after $t = 7$ seconds? [b]p4.[/b] Kara gives Kaylie a ring with a circular diamond inscribed in a gold hexagon. The diameter of the diamond is $2$ mm. If diamonds cost $\$100/ mm ^2$ and gold costs $\$50 /mm ^2$ , what is the cost of the ring? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

A circle$\Gamma$ with radius $R$ and center $\omega$, and a line $d$ are drawn on a plane, such that the distance of $\omega$ from $d$ is greater than $R$. Two points $M$ and $N$ vary on $d$ so that the circle with diameter $MN$ is tangent to $\Gamma$. Prove that there is a point $P$ in the plane from which all the segments $MN$ are visible at a constant angle.

Jingoistic James wants to teach his kindergarten class how to add in Chimese. The Chimese also use base-10 number system, but have replaced the digits 0-9 with ten of its own characters. For example, the two-digit number 十六 represents $16$. What is the sum of the two-digit numbers 十六 and 六十? [i]Proposed by James Lin

In each field of a table there is a real number. We call such $n \times n$ table [i]silly[/i] if each entry equals the product of all the numbers in the neighbouring fields. a) Find all $2 \times 2$ silly tables. b) Find all $3 \times 3$ silly tables.

Find all real solutions of the equation $\sin^{5}x+\cos^{3}x=1$.

We call a positive integer $n$ $\textit{sixish}$ if $n=p(p+6)$, where $p$ and $p+6$ are prime numbers. For example, $187=11\cdot17$ is sixish, but $475=19\cdot25$ is not sixish. Define a function $f$ on the positive integers such that $f(n)$ is the sum of the squares of the positive divisors of $n$. For example, $f(10)=1^2+2^2+5^2+10^2=130$. (a) Find, with proof, an irreducible polynomial function $g(x)$ with integer coefficients such that $f(n)=g(n)$ for all sixish $n$. ("Irreducible" means that $g(x)$ cannot be factored as the product of two polynomials of smaller degree with integer coefficients.) (b) We call a positive integer $n$ $\textit{pseudo-sixish}$ if $n$ is not sixish but nonetheless $f(n)=g(n)$, where $g(n)$ is the polynomial function that you found in part (a). Find, with proof, all pseudo-sixish positive integers.

Enzymes convert glucose $(M=180.2)$ to ethanol $(M=46.1)$ according to the equation \[ \text{C}_6\text{H}_{12}\text{O}_6 \rightarrow 2\text{C}_2\text{H}_2\text{OH} + 2\text{CO}_2 \] What is the maximum mass of ethanol that can be made from $15.5$ kg of glucose? $ \textbf{(A)}\hspace{.05in}0.256 \text{kg}\qquad\textbf{(B)}\hspace{.05in}0.512 \text{kg} \qquad\textbf{(C)}\hspace{.05in}3.96 \text{kg}\qquad\textbf{(D)}\hspace{.05in}7.93 \text{kg}\qquad$

$\dfrac{1}{167}$ has a period of length $166$, and its first $83$ digits after the decimal point are: \begin{align*} &.0059880239 \hspace{0.1 cm} 5209580838 \hspace{0.1 cm} 3233532934 \hspace{0.1 cm} 1317365269 \\ &\hspace{0.1 cm} 4610778443 \hspace{0.1 cm} 1137724550 \hspace{0.1 cm} 8982035928 \hspace{0.1 cm} 1437125748\\ &\hspace{0.1 cm} 502 \end{align*} Let $a$, $b$, $c$ be the $120$th, $121$st, $122$nd digits after the decimal point of $\dfrac{1}{167}$, respectively. Find $100a + 10b + c$. [i]Lightning 4.2[/i]

The incircle of triangle $ABC$ touches sides $BC$, $AC$, and $AB$ at $D, E$, and $F$ respectively. Let $\omega_a, \omega_b$ and $\omega_c$ be the circumcircles of triangles $EAF, DBF$, and $DCE$, respectively. The lines $DE$ and $DF$ cut $\omega_a$ at $E_a\neq{E}$ and $F_a\neq{F}$, respectively. Let $r_A$ be the line $E_{a}F_a$. Let $r_B$ and $r_C$ be defined analogously. Show that the lines $r_A$, $r_B$, and $r_C$ determine a triangle with its vertices on the sides of triangle $ABC$.

Let $ABC$ to be a triangle and $\Gamma$ its circumcircle. Also, let $D, F, G$ and $E$, in this order, on the arc $BC$ which does not contain $A$ satisfying $\angle BAD = \angle CAE$ and $\angle BAF = \angle CAG$. Let $D`, F`, G`$ and $E`$ to be the intersections of $AD, AF, AG$ and $AE$ with $BC$, respectively. Moreover, $X$ is the intersection of $DF`$ with $EG`$, $Y$ is the intersection of $D`F$ with $E`G$, $Z$ is the intersection of $D`G$ with $E`F$ and $W$ is the intersection of $EF`$ with $DG`$. Prove that $X, Y$ and $A$ are collinear, such as $W, Z$ and $A$. Moreover, prove that $\angle BAX = \angle CAZ$.