Let $P(x, y)$ be a non-constant homogeneous polynomial with real coefficients such that $P(\sin t, \cos t) = 1$ for every real number $t$. Prove that there exists a positive integer $k$ such that $P(x, y) = (x^2 + y^2)^k$.

The product $ (1.8)(40.3\plus{}.07)$ is closest to \[ \textbf{(A)}\ 7 \qquad \textbf{(B)}\ 42 \qquad \textbf{(C)}\ 74 \qquad \textbf{(D)}\ 84 \qquad \textbf{(E)}\ 737 \]

$BC$ is a diameter of a circle and the points $X,Y$ are on the circle such that $XY\perp BC$. The points $P,M$ are on $XY,CY$ (or their stretches), respectively, such that $CY||PB$ and $CX||PM$. Let $K$ be the meet point of the lines $XC,BP$. Prove that $PB\perp MK$.

Let $n$ be an integer greater than 2. A positive integer is said to be [i]attainable [/i]if it is 1 or can be obtained from 1 by a sequence of operations with the following properties: 1.) The first operation is either addition or multiplication. 2.) Thereafter, additions and multiplications are used alternately. 3.) In each addition, one can choose independently whether to add 2 or $n$ 4.) In each multiplication, one can choose independently whether to multiply by 2 or by $n$. A positive integer which cannot be so obtained is said to be [i]unattainable[/i]. [b]a.)[/b] Prove that if $n\geq 9$, there are infinitely many unattainable positive integers. [b]b.)[/b] Prove that if $n=3$, all positive integers except 7 are attainable.

For a graph $G$ on $n$ vertices, let $P_G(x)$ be the unique polynomial of degree at most $n$ such that for each $i=0,1,2,\dots,n$, $P_G (i)$ equals the number of ways to color the vertices of the graph $G$ with $i$ distinct colors such that no two vertices connected by an edge have the same color. For each integer $3\le k \le 2017$, define a $k$-[i]tasty[/i] graph to be a connected graph on $2017$ vertices with $2017$ edges and a cycle of length $k$. Let the [i]tastiness[/i] of a $k$-tasty graph $G$ be the number of coefficients in $P_G(x)$ that are odd integers, and let $t$ be the minimal tastiness over all $k$-tasty graphs with $3\le k \le 2017$. Determine the sum of all integers $b$ between $3$ and $2017$ inclusive for which there exists a $b$-tasty graph with tastiness $t$. [i]Proposed by Vincent Huang[/i]

Let $D = \{z\in \mathbb{C}: |z| < 1\}$ be the open unit disk in the complex plane and let $f : D \to D$ be a holomorphic function such that $\lim_{|z|\to 1}|f(z)| = 1$. Let the Taylor series of $f$ be $f(z) = \sum_{n=0}^{\infty} a_nz^n$. Prove that the number of zeroes of $f$ (counted with multiplicities) equals $\sum_{n=0}^{\infty} n|a_n|^2$.

Determine whether it's possible to cover a $K_{2012}$ with a) 1000 $K_{1006}$'s; b) 1000 $K_{1006,1006}$'s. [i]David Yang.[/i]

A positive integer $n$ is said to be a [i]perfect power[/i] if $n=a^b$ for some integers $a,b$ with $b>1$. $(\text{a})$ Find $2004$ perfect powers in arithmetic progression. $(\text{b})$ Prove that perfect powers cannot form an infinite arithmetic progression.

What is the greatest multiple of $9$ that can be formed by using each of the digits in the set $\{1, 3,5, 7, 9\}$ at most once.

Let $ ABCD$ be a trapezoid with $ AB\parallel{}CD$. Let $ a \equal{} AB$ and $ b \equal{} CD$. For $ MN\parallel{}AB$ such that $ M$ lies on $ AD,$ $ N$ lies on $ BC$, and the trapezoids $ ABNM$ and $ MNCD$ have the same area, the length of $ MN$ equals [img]http://i250.photobucket.com/albums/gg265/geometry101/NielsHenrikAbel1997Number6.jpg[/img] A. $ \sqrt{ab}$ B. $ \frac{a\plus{}b}{2}$ C. $ \frac{a^2 \plus{} b^2}{a\plus{}b}$ D. $ \sqrt{\frac{a^2 \plus{} b^2}{2}}$ E. $ \frac{a^2 \plus{} (2 \sqrt{2} \minus{} 2)ab \plus{} b^2}{\sqrt{2} (a\plus{}b)}$

Given the equation $x^2+ax+1=0$, determine: a) The interval of possible values for $a$ where the solutions to the previous equation are not real. b) The loci of the roots of the polynomial, when $a$ is in the previous interval.

A triangle $ABC$ with circumcenter $U$ is given, so that $\angle CBA = 60^o$ and $\angle CBU = 45^o$ apply. The straight lines $BU$ and $AC$ intersect at point $D$. Prove that $AD = DU$. (Karl Czakler)

[b]a)[/b] Prove that $$ k+\frac{1}{2}-\frac{1}{8k}<\sqrt{k^2+k}<k+\frac{1}{2}-\frac{1}{8k}+\frac{1}{16k^2} , $$ for any natural number $ k. $ [b]b)[/b] Prove that there exists four numbers $ \alpha,\beta,\gamma,\delta\in\{0,1,2,3,4,5,6,7,8,9\} $ such that $$ \left\lfloor\sum_{k=1}^{2012} \sqrt{k(k+1)\left( k^2+k+1 \right)}\right\rfloor =\underbrace{\ldots\alpha \beta\gamma\delta}_{\text{decimal form}} $$ and $ \alpha +\delta =\gamma . $

Find the positive integer $n$ such that the least common multiple of $n$ and $n - 30$ is $n + 1320$.

The quadrilateral $ABCD$, in which $\angle BAC < \angle DCB$ , is inscribed in a circle $c$, with center $O$. If $\angle BOD = \angle ADC = \alpha$. Find out which values of $\alpha$ the inequality $AB <AD + CD$ occurs.

Let $ABC$ be a triangle with $AB = 8, AC = 12,$ and $BC = 5.$ Let $M$ be the second intersection of the internal angle bisector of $\angle BAC$ with the circumcircle of $ABC.$ Let $\omega$ be the circle centered at $M$ tangent to $AB$ and $AC.$ The tangents to $\omega$ from $B$ and $C,$ other than $AB$ and $AC$ respectively, intersect at a point $D.$ Compute $AD.$

Let $ABC$ be an acute-angled triangle with $AC>AB$, and $\Omega$ is your circumcircle. Let $P$ be the midpoint of the arc $BC$ of $\Omega$ (not containing $A$) and $Q$ be the midpoint of the arc $BC$ of $\Omega$(containing the point $A$). Let $M$ be the foot of perpendicular of $Q$ on the line $AC$. Prove that the circumcircle of $\triangle AMB$ cut the segment $AP$ in your midpoint.

Let $ABCD$ be a convex quadrilateral which admits an incircle. Let $AB$ produced beyond $B$ meet $DC$ produced towards $C$, at $E$. Let $BC$ produced beyond $C$ meet $AD$ produced towards $D$, at $F$. Let $G$ be the point on line $AB$ so that $FG \parallel CD$, and let $H$ be the point on line $BC$ so that $EH \parallel AD$. Prove that the (concave) quadrilateral $EGFH$ admits an excircle tangent to $\overline{EG}, \overline{EH}, \overrightarrow{FG}, \overrightarrow{FH}$. [i]Proposed by Rijul Saini[/i]

Let $n \ge 3$ be an integer. A frog is to jump along the real axis, starting at the point $0$ and making $n$ jumps: one of length $1$, one of length $2$, $\dots$ , one of length $n$. It may perform these $n$ jumps in any order. If at some point the frog is sitting on a number $a \le 0$, its next jump must be to the right (towards the positive numbers). If at some point the frog is sitting on a number $a > 0$, its next jump must be to the left (towards the negative numbers). Find the largest positive integer $k$ for which the frog can perform its jumps in such an order that it never lands on any of the numbers $1, 2, \dots , k$.

In a quadrilateral ABCD, it is given that AB = AD = 13, BC = CD = 20, BD = 24. If r is the radius of the circle inscribable in the quadrilateral, then what is the integer closest to r?

Given an equilateral triangle, cut it into four polygonal shapes so that, reassembled appropriately, these figures form a square.

Decompose the space of homogeneous polynomials of degree $5$ in $(x, y, z)$ into irreducible subspaces invariant with respect to the rotation group $SO(3)$.

Two non-intersecting circles, not lying inside each other, are drawn in the plane. Two lines pass through a point P which lies outside each circle. The first line intersects the first circle at A and A′ and the second circle at B and B′; here A and B are closer to P than A′ and B′, respectively, and P lies on segment AB. Analogously, the second line intersects the first circle at C and C′ and the second circle at D and D′. Prove that the points A, B, C, D are concyclic if and only if the points A′, B′, C′, D′ are concyclic.

Define a parabola $C$ by $y=x^2+1$ on the coordinate plane. Let $s,\ t$ be real numbers with $t<0$. Denote by $l_1,\ l_2$ the tangent lines drawn from the point $(s,\ t)$ to the parabola $C$. (1) Find the equations of the tangents $l_1,\ l_2$. (2) Let $a$ be positive real number. Find the pairs of $(s,\ t)$ such that the area of the region enclosed by $C,\ l_1,\ l_2$ is $a$.

The least common multiple of a positive integer $n$ and 18 is 180, and the greatest common divisor of $n$ and 45 is 15. What is the sum of the digits of $n$? $\textbf{(A) }3\qquad\textbf{(B) }6\qquad\textbf{(C) }8\qquad\textbf{(D) }9\qquad\textbf{(E) }12$