Suppose $P(x,y)$ is a homogenous non-constant polynomial with real coefficients such that $P(\sin t, \cos t) = 1$ for all real $t$. Prove that $P(x,y) = (x^2+y^2)^k$ for some positive integer $k$. (A polynomial $A(x,y)$ with real coefficients and having a degree of $n$ is homogenous if it is the sum of $a_ix^iy^{n-i}$ for some real number $a_i$, for all integer $0 \le i \le n$.)

In the acute triangle $ABC$ point $I$ is the incenter, $O$ is the circumcenter, while $I_a$ is the excenter opposite the vertex $A$. Point $A'$ is the reflection of $A$ across the line $BC$. Prove that angles $\angle IOI_a$ and $\angle IA'I_a$ are equal.

Prove that if every subspace of a Hausdorff space $X$ is $\sigma$-compact, then $X$ is countable.

In a $1024$ person randomly seeded single elimination tournament bracket, each player has a unique skill rating. In any given match, the player with the higher rating has a $\frac34$ chance of winning the match. What is the probability the second lowest rated player wins the tournament?

Find prime numbers $p$, $q$, $r$ and $s$, pairwise distinct, such that their sum is prime number and numbers $p^2+qr$ and $p^2+qs$ are perfect squares

Find all pairs of prime numbers $(p,q)$, such that $\frac{(2p^2-1)^q+1}{p+q}$ and $\frac{(2q^2-1)^p+1}{p+q}$ are both integers.

Let $P(x) = x^3 + (t - 1)x^2 - (t + 3)x + 1$. For what values of real $t$ the sum of the squares and the reciprocals of the roots of $ P(x)$ is minimum?

Points $X$ and $Y$ are chosen inside an acute triangle $ABC$ so that: $$\angle AXB = \angle CYB = 180^\circ - \angle ABC, \text{ } \angle ABX = \angle CBY$$ Show that the points $X$ and $Y$ are equidistant from the center of the circumscribed circle of $\triangle ABC$. [i]Proposed by Anton Trygub[/i]

Let $ABC$ be an acute triangle. Let $P$ be a point on the circle $(ABC)$, and $Q$ be a point on the segment $AC$ such that $AP\perp BC$ and $BQ\perp AC$. Lot $O$ be the circumcenter of triangle $APQ$. Find the angle $OBC$.

Find all pairs of functions $f ,g : Z \rightarrow Z $ that satisfy $f (g(x)+y) = g( f (y)+x) $ for all integers $ x,y$ and such that $g(x) = g(y)$ only if $x = y$.

We say that a square-free positive integer $n$ is [i]almost prime[/i] if \[n \mid x^{d_1}+x^{d_2}+\dots+x^{d_k}-kx\] for all integers $x$, where $1=d_1<d_2<\dots<d_k=n$ are all the positive divisors of $n$. Suppose that $r$ is a Fermat prime (i.e. it is a prime of the form $2^{2^m}+1$ for an integer $m \ge 0$), $p$ is a prime divisor of an almost prime integer $n$, and $p \equiv 1 \pmod{r}$. Show that, with the above notation, $d_i \equiv 1 \pmod{r}$ for all $1 \le i \le k$. (An integer $n$ is called [i]square-free[/i] if it is not divisible by $d^2$ for any integer $d>1$.)

Let $n$ be a positive integer. If $4^n + 2^n + 1$ is a prime, prove that $n$ is a power of three.

Let $n$ be a positive integer and let $k_0,k_1, \dots,k_{2n}$ be nonzero integers such that $k_0+k_1 +\dots+k_{2n}\neq 0$. Is it always possible to a permutation $(a_0,a_1,\dots,a_{2n})$ of $(k_0,k_1,\dots,k_{2n})$ so that the equation \begin{align*} a_{2n}x^{2n}+a_{2n-1}x^{2n-1}+\dots+a_0=0 \end{align*} has not integer roots?

Let $\alpha$ be the radian measure of the smallest angle in a $3{-}4{-}5$ right triangle. Let $\beta$ be the radian measure of the smallest angle in a $7{-}24{-}25$ right triangle. In terms of $\alpha$, what is $\beta$? $ \textbf{(A) }\frac{\alpha}{3}\qquad \textbf{(B) }\alpha - \frac{\pi}{8}\qquad \textbf{(C) }\frac{\pi}{2} - 2\alpha \qquad \textbf{(D) }\frac{\alpha}{2}\qquad \textbf{(E) }\pi - 4\alpha\qquad $

A $2N\times2N$ board is covered by non-overlapping dominos of $1\times2$ size. A lame rook (which can only move one cell at a time, horizontally or vertically) has visited each cell once on its route across the board. Call a move by the rook longitudinal if it is a move from one cell of a domino to another cell of the same domino. What is: [list=a] [*]the maximum possible number of longitudinal moves? [*]the minimum possible number of longitudinal moves? [/list]

It is known that the sequence $\{a_n\}$ satisfies $a_1=2$, $a_n=2^{2n}a_{n-1}+n\cdot 2^{n^2}$, $(n \ge 2)$, find the general term of $a_n$.

Let $X =\{ X_1 , X_2 , ...\}$ be a countable set of points in space. Show that there is a positive sequence $\{a_k\}$ such that for any point $Z\not\in X$ the distance between the point Z and the set $\{X_1,X_2 , ...,X_k\}$ is at least $a_k$ for infinitely many k.

Let $m$ be a convex polygon in a plane, $l$ its perimeter and $S$ its area. Let $M\left( R\right) $ be the locus of all points in the space whose distance to $m$ is $\leq R,$ and $V\left(R\right) $ is the volume of the solid $M\left( R\right) .$ [i]a.)[/i] Prove that \[V (R) = \frac 43 \pi R^3 +\frac{\pi}{2} lR^2 +2SR.\] Hereby, we say that the distance of a point $C$ to a figure $m$ is $\leq R$ if there exists a point $D$ of the figure $m$ such that the distance $CD$ is $\leq R.$ (This point $D$ may lie on the boundary of the figure $m$ and inside the figure.) additional question: [i]b.)[/i] Find the area of the planar $R$-neighborhood of a convex or non-convex polygon $m.$ [i]c.)[/i] Find the volume of the $R$-neighborhood of a convex polyhedron, e. g. of a cube or of a tetrahedron. [b]Note by Darij:[/b] I guess that the ''$R$-neighborhood'' of a figure is defined as the locus of all points whose distance to the figure is $\leq R.$

Two train lines intersect each other at a junction at an acute angle $\theta$. A train is passing along one of the two lines. When the front of the train is at the junction, the train subtends an angle $\alpha$ at a station on the other line. It subtends an angle $\beta (<\alpha)$ at the same station, when its rear is at the junction. Show that \[\tan\theta=\frac{2\sin\alpha\sin\beta}{\sin(\alpha-\beta)}\]

Given that $x_{i}>0$, $i = 1, 2, \cdots, n$, $k \geq 1$. Show that: \[\sum_{i=1}^{n}\frac{1}{1+x_{i}}\cdot \sum_{i=1}^{n}x_{i}\leq \sum_{i=1}^{n}\frac{x_{i}^{k+1}}{1+x_{i}}\cdot \sum_{i=1}^{n}\frac{1}{x_{i}^{k}}\]

Let $a>1$ be a positive integer and $f\in \mathbb{Z}[x]$ with positive leading coefficient. Let $S$ be the set of integers $n$ such that \[n \mid a^{f(n)}-1.\] Prove that $S$ has density $0$; that is, prove that $\lim_{n\rightarrow \infty} \frac{|S\cap \{1,...,n\}|}{n}=0$.

The positive real numbers $a, b, c$ satisfy the equation $a+b+c=1$. Prove the identity: $\sqrt{\frac{(a+bc)(b+ca)}{c+ab}}+\sqrt{\frac{(b+ca)(c+ab)}{a+bc}}+\sqrt{\frac{(c+ab)(a+bc)}{b+ca}} = 2$

In the figure, $ \overline{CD}, \overline{AE}$ and $ \overline{BF}$ are one-third of their respective sides. It follows that $ \overline{AN_2}: \overline{N_2N_1}: \overline{N_1D} \equal{} 3: 3: 1$, and similarly for lines $ BE$ and $ CF.$ Then the area of triangle $ N_1N_2N_3$ is: [asy]unitsize(27); defaultpen(linewidth(.8pt)+fontsize(10pt)); pair A,B,C,D,E,F,X,Y,Z; A=(3,3); B=(0,0); C=(6,0); D=(4,0); E=(4,2); F=(1,1); draw(A--B--C--cycle); draw(A--D); draw(B--E); draw(C--F); X=intersectionpoint(A--D,C--F); Y=intersectionpoint(B--E,A--D); Z=intersectionpoint(B--E,C--F); label("$A$",A,N); label("$B$",B,SW); label("$C$",C,SE); label("$D$",D,S); label("$E$",E,NE); label("$F$",F,NW); label("$N_1$",X,NE); label("$N_2$",Y,WNW); label("$N_3$",Z,S);[/asy]$ \textbf{(A)}\ \frac {1}{10} \triangle ABC \qquad\textbf{(B)}\ \frac {1}{9} \triangle ABC \qquad\textbf{(C)}\ \frac {1}{7} \triangle ABC \qquad\textbf{(D)}\ \frac {1}{6} \triangle ABC \qquad\textbf{(E)}\ \text{none of these}$

Let $ABCD$ be a square of side length 2, and let $M$ and $N$ be points on the sides $AB$ and $CD$ respectively. The lines $CM$ and $BN$ meet at $P$, while the lines $AN$ and $DM$ meet at $Q$. Prove that $\left| PQ \right|\ge 1$.

An integer $ x$, with $ 10 \le x \le 99$, is to be chosen. If all choices are equally likely, what is the probability that at least one digit of $ x$ is a $ 7$? $ \textbf{(A)}\ \frac19\qquad \textbf{(B)}\ \frac15\qquad \textbf{(C)}\ \frac{19}{90}\qquad \textbf{(D)}\ \frac29\qquad \textbf{(E)}\ \frac13$