Suppose a circle passes through the feet of the symmedians of a non-isosceles triangle $ABC$ , and is tangent to one of the sides. Show that $a^2 +b^2, b^2 + c^2 , c^2 + a^2$ are in geometric progression when taken in some order

Find the smallest positive integer solution to the equation $2^{2^k}\equiv k\pmod{29}$.

If $a,b,c,$ and $d$ are non-zero numbers such that $c$ and $d$ are the solutions of $x^2+ax+b=0$ and $a$ and $b$ are the solutions of $x^2+cx+d=0$, then $a+b+c+d$ equals $\textbf{(A) }0\qquad\textbf{(B) }-2\qquad\textbf{(C) }2\qquad\textbf{(D) }4\qquad \textbf{(E) }(-1+\sqrt{5})/2$

In a triangle $ABC$, let $D$, $E$ and $F$ be the feet of the altitudes from $A$, $B$ and $C$ respectively. Let $D'$, $E'$ and $F'$ be the second intersection of lines $AD$, $BE$ and $CF$ with the circumcircle of $ABC$. Show that the triangles $DEF$ and $D'E'F'$ are similar.

Let $a_1, a_2, \dots$ be an arithmetic sequence and $b_1, b_2, \dots$ be a geometric sequence. Suppose that $a_1 b_1 = 20$, $a_2 b_2 = 19$, and $a_3 b_3 = 14$. Find the greatest possible value of $a_4 b_4$.

Let $S = \mathbb Z \times \mathbb Z$. A subset $P$ of $S$ is called [i]nice[/i] if [list] [*] $(a, b) \in P \implies (b, a) \in P$ [*] $(a, b)$, $(c, d)\in P \implies (a + c, b - d) \in P$ [/list] Find all $(p, q) \in S$ so that if $(p, q) \in P$ for some [i]nice[/i] set $P$ then $P = S$.

Find the minimum value attained by $\sum_{m=1}^{100} \gcd(M - m, 400)$ for $M$ an integer in the range $[1746, 2017]$.

$(USS 5)$ Given $5$ points in the plane, no three of which are collinear, prove that we can choose $4$ points among them that form a convex quadrilateral.

We have a polyhedron such that an ant can walk from one vertex to another, traveling only along edges, and traversing every edge exactly once. What is the smallest possible total number of vertices, edges, and faces of this polyhedron?

Let $G$ be a complete directed graph with $2024$ vertices and let $k \leq 10^5$ be a positive integer. Angel and Boris play the following game: Angel colors $k$ of the edges in red and puts a pawn in one of the vertices. After that in each move, first Angel moves the pawn to a neighboring vertex and then Boris has to flip one of the non-colored edges. Boris wins if at some point Angel can't make a move. Find, depending on $G$ and $k$, whether or not Boris has a winning strategy.

Principal Skinner is thinking of two integers $m$ and $n$ and bets Superintendent Chalmers that he will not be able to determine these integers with a single piece of information. Chalmers asks Skinner the numerical value of $mn+13m+13n-m^2-n^2$. From the value of this expression alone, he miraculously determines both $m$ and $n$. What is the value of the above expression.

Let $A$ and $B$ be points on circle $\Gamma$ such that $AB=\sqrt{10}.$ Point $C$ is outside $\Gamma$ such that $\triangle ABC$ is equilateral. Let $D$ be a point on $\Gamma$ and suppose the line through $C$ and $D$ intersects $AB$ and $\Gamma$ again at points $E$ and $F \neq D.$ It is given that points $C, D, E, F$ are collinear in that order and that $CD=DE=EF.$ What is the area of $\Gamma?$ [i]Proposed by Kyle Lee[/i]

Let $f$ be a linear function such that $f(0) = -5$ and $f(f(0)) = -15$. Find the values of $ k \in R$ for which the solutions of the inequality $f(x) \cdot f(k - x) > 0$, lie in an interval of[u][/u] length $2$.

A triangle $\vartriangle ABC$ has $AC = 16$ and $BC = 12$. $E$ and $F$ are points on $AC$ and $BC$, respectively, so that $CE = 3CF$. Let $M$ be the midpoint of $AB$, and let lines $EF$ and $CM$ intersect at $G$. Compute the ratio $EG : GF$.

A paper rectangle $ABCD $ of area $1$ is folded along a straight line so that $C$ coincides with $A$. Prove that the area of the pentagon thus obtained is less than $3/4$.

Let $ABC$ be a right triangle with hypotenuse $BC$ and center $I$. Let bisectors of the angles $\angle B$ and $\angle C$ intersect the sides $AC$ and $AB$ in$ D$ and $E$, respectively. Let $P$ and $Q$ be the feet of the perpendiculars of the points $D$ and $E$ on the side $BC$. Prove that $I$ is the circumcenter of $APQ$.

Find the positive integer $ n$ such that \[\arctan\frac{1}{3}\plus{}\arctan\frac{1}{4}\plus{}\arctan\frac{1}{5}\plus{}\arctan\frac{1}{n}\equal{}\frac{\pi}{4}.\]

Let $\,{\mathbb{R}}\,$ denote the set of all real numbers. Find all functions $\,f: {\mathbb{R}}\rightarrow {\mathbb{R}}\,$ such that \[ f\left( x^{2}+f(y)\right) =y+\left( f(x)\right) ^{2}\hspace{0.2in}\text{for all}\,x,y\in \mathbb{R}. \]

For a set of points in a plane, we construct the perpendicular bisectors of the line segments connecting every pair of those points and we count the number of points in which these perpendicular bisectors intersect each other. If we start with twelve points, the maximum possible number of intersection points is 1705. What is the maximum possible number of intersection points if we start with thirteen points?

Find the sum of the values of $x$ for which $\binom{x}{0}-\binom{x}{1}+\binom{x}{2}-...+\binom{x}{2008}=0$

Circle $\omega$ has radius 10 with center $O$. Let $P$ be a point such that $PO=6$. Let the midpoints of all chords of $\omega$ through $P$ bound a region of area $R$. Find the value of $\lfloor 10R \rfloor$. [i]Proposed by Andrew Zhao[/i]

The polynomial $f(x)=x^3-3x^2-4x+4$ has three real roots $r_1$, $r_2$, and $r_3$. Let $g(x)=x^3+ax^2+bx+c$ be the polynomial which has roots $s_1$, $s_2$, and $s_3$, where $s_1=r_1+r_2z+r_3z^2$, $s_2=r_1z+r_2z^2+r_3$, $s_3=r_1z^2+r_2+r_3z$, and $z=\frac{-1+i\sqrt3}2$. Find the real part of the sum of the coefficients of $g(x)$.

Let \[\begin{array}{ccccccccccc}A&=&5\cdot 6&-&6\cdot 7&+&7\cdot 8&-&\cdots&+&2003\cdot 2004,\\B&=&1\cdot 10&-&2\cdot 11&+&3\cdot 12&-&\cdots&+&1999\cdot 2008.\end{array}\] Find the value of $A-B$.

A toy factory produces several kinds of clay toys. The toys are painted in $k$ colours. [i]Diversity[/i] of a colour is the number of [i]different[/i] toys of that colour. (Thus, if there are $5$ blue cats, $7$ blue mice and nothing else is blue, the diversity of colour blue is $2$.) The painting protocol requires that [i]each colour is used and the diversities of each two colours are different[/i]. The toys in the store could be painted according to the protocol. However, a batch of clay Cheburashkas arrived at the store before painting (there were no Cheburashkas before). The number of Cheburashkas is not less that the number of the toys of any other kind. The total number of all toys, including Cheburashkas, is at least $\frac{(k+1)(k+2)}{2}$. Prove that now the toys can be painted in $k + 1$ colours according to the protocol. [i]Proposed by F. Petrov[/i]

The chords $AC$ and $BD$ of a, circle with centre $O$ intersect at the point $K$. The circumcentres of triangles $AKB$ and $CKD$ are $M$ and $N$ respectively. Prove that $OM = KN$. (A Zaslavsky )