Let $ABCD$ be a square, $E$ a point on the side $CD$, and $F$ a point inside the square such that that triangle $BFE$ is isosceles and $\angle BFE = 90^o$ . If $DF=DE$, find the measure of angle $\angle FDE$.

Let $VABC$ be a regular triangular pyramid with base $ABC$, of center $O$. Points $I$ and $H$ are the center of the inscribed circle, respectively the orthocenter $\vartriangle VBC$. Knowing that $AH = 3 OI$, determine the measure of the angle between the lateral edge of the pyramid and the plane of the base.

A point $A$ is chosen outside a circle with diameter $BC$ so that $\vartriangle ABC$ is acute. Segments $AB$ and $AC$ intersect the circle at $D$ and $E$, respectively, and $CD$ intersects $BE$ at $F$. Line $AF$ intersects the circle again at $G$ and intersects $BC$ at $H$. Prove that $AH \cdot F H = GH^2$. .

Find a constant $ c > 1$ with the property that, for arbitrary positive integers $ n$ and $ k$ such that $ n>c^k$, the number of distinct prime factors of $ \binom{n}{k}$ is at least $ k$. [i]P. Erdos[/i]

Sarah has five rings (numbered 1 through 5), each with ten rungs labeled $1$ through $10$. Rung $i$ is adjacent to rung $i+1$ for $1 \le i \le 9$, and rung $10$ is adjacent to rung $1$. How many ways can Sarah paint some (possibly none) of the rungs red such that in each ring, the red rungs form a contiguous block, and the total number of red rungs across the five rings is divisible by $11$? (For example, Sarah can paint rungs $8, 9, 10, 1, 2$ on ring $1$, rungs $3, 4, 5$ on ring $2$, no rungs on rings $3$ and $4$, and rungs $1,2,3$ on ring $5$.)

Prove the inequality \[(n + 1)\cos\frac{\pi}{n + 1}- n\cos\frac{\pi}{n}> 1\] for all natural numbers $n \ge 2.$

Find the number of positive integer solutions $(a,b,c,d)$ to the equation \[(a^2+b^2)(c^2-d^2)=2020.\] Note: The solutions $(10,1,6,4)$ and $(1,10,6,4)$ are considered different.

Let $ABCD$ be a fixed convex quadrilateral with $BC=DA$ and $BC$ not parallel with $DA$. Let two variable points $E$ and $F$ lie of the sides $BC$ and $DA$, respectively and satisfy $BE=DF$. The lines $AC$ and $BD$ meet at $P$, the lines $BD$ and $EF$ meet at $Q$, the lines $EF$ and $AC$ meet at $R$. Prove that the circumcircles of the triangles $PQR$, as $E$ and $F$ vary, have a common point other than $P$.

Let $a$, $b$, $c$ be nonnegative reals such that $ab+bc+ca = \frac{1}{3}$. Prove that \[\frac{1}{a^{2}-bc+1}+\frac{1}{b^{2}-ca+1}+\frac{1}{c^{2}-ab+1}\leq 3 \]

We say a finite set $S$ of points in the plane is [i]very[/i] if for every point $X$ in $S$, there exists an inversion with center $X$ mapping every point in $S$ other than $X$ to another point in $S$ (possibly the same point). (a) Fix an integer $n$. Prove that if $n \ge 2$, then any line segment $\overline{AB}$ contains a unique very set $S$ of size $n$ such that $A, B \in S$. (b) Find the largest possible size of a very set not contained in any line. (Here, an [i]inversion[/i] with center $O$ and radius $r$ sends every point $P$ other than $O$ to the point $P'$ along ray $OP$ such that $OP\cdot OP' = r^2$.) [i]Proposed by Sammy Luo[/i]

Compute the number of sequences of real numbers $a_1, a_2, a_3, \dots, a_{16}$ satisfying the condition that for every positive integer $n$, \[ a_1^n + a_2^{2n} + \dots + a_{16}^{16n} = \left \{ \begin{array}{ll} 10^{n+1} + 10^n + 1 & \text{for even } n \\ 10^n - 1 & \text{for odd } n \end{array} \right. . \][i]Proposed by Evan Chen[/i]

Six real numbers $x_1<x_2<x_3<x_4<x_5<x_6$ are given. For each triplet of distinct numbers of those six Vitya calculated their sum. It turned out that the $20$ sums are pairwise distinct; denote those sums by $$s_1<s_2<s_3<\cdots<s_{19}<s_{20}.$$ It is known that $x_2+x_3+x_4=s_{11}$, $x_2+x_3+x_6=s_{15}$ and $x_1+x_2+x_6=s_{m}$. Find all possible values of $m$.

Find out the number of real solutions of $x^2e^{\sin x}=1$ [list=1] [*] 0 [*] 1 [*] 2 [*] 3 [/list]

Indicate all the roots of the equation $x^2+1 = \cos x$.

Determine all sets of real numbers $x,y,z$ which satisfy the system of equations $$\begin{cases} xy = z \\ xz =y \\ yz =x \end{cases}$$

Suppose $a,b$ are positive real numbers such that $a+a^2 = 1$ and $b^2+b^4=1$. Compute $a^2+b^2$. [i]Proposed by Thomas Lam[/i]

A scanning code consists of a $7 \times 7$ grid of squares, with some of its squares colored black and the rest colored white. There must be at least one square of each color in this grid of $49$ squares. A scanning code is called [i]symmetric[/i] if its look does not change when the entire square is rotated by a multiple of $90 ^{\circ}$ counterclockwise around its center, nor when it is reflected across a line joining opposite corners or a line joining midpoints of opposite sides. What is the total number of possible symmetric scanning codes? $\textbf{(A)} \text{ 510} \qquad \textbf{(B)} \text{ 1022} \qquad \textbf{(C)} \text{ 8190} \qquad \textbf{(D)} \text{ 8192} \qquad \textbf{(E)} \text{ 65,534}$

$I$ is the incenter of a given triangle $\triangle ABC$. The angle bisectors of $ABC$ meet the sides at $D,E,F$, and $EF$ meets $(ABC)$ at $L$ and $T$ ($F$ is on segment $LE$.). Suppose $M$ is the midpoint of $BC$. Prove that if $DT$ is tangent to the incircle of $ABC$, then $IL$ bisects $\angle MLT$.

Let $n\geq 1$ be an integer and let $a$ and $b$ be its positive divisors satisfying $a+b+ab=n$. Prove that $a=b$.

Does there exist a sequence $\{a_n\}$ of positive integers satisfying the following conditions: $a)$ every natural number occurs in this sequence and exactly once; $b)$ $a_1 + a_2 +... + a_n$ is divisible by $n^n$ for each $n = 1,2,3, ...$ ?

Let $n,k$ be given natural numbers. Determine all ordered n-tuples of non-negative real numbers $(x_1,x_2,...,x_n)$ that satisfy the system of equations $$x_1^k+x_2^k+...+x_n^k=1$$ $$(1+x_1)(1+x_2)...(1+x_n)=2$$

Let $A$ and $B$ be nonempty subsets of $\mathbb{N}$. The sum of $2$ distinct elements in $A$ is always an element of $B$. Furthermore, the result of the division of $2$ distinct elements in $B$ (where the larger number is divided by the smaller number) is always a member of $A$. Determine the maximum number of elements in $A \cup B$.

Let $ABCD$ be a parallelogram with area $15$. Points $P$ and $Q$ are the projections of $A$ and $C,$ respectively, onto the line $BD;$ and points $R$ and $S$ are the projections of $B$ and $D,$ respectively, onto the line $AC.$ See the figure, which also shows the relative locations of these points. [asy] size(350); defaultpen(linewidth(0.8)+fontsize(11)); real theta = aTan(1.25/2); pair A = 2.5*dir(180+theta), B = (3.35,0), C = -A, D = -B, P = foot(A,B,D), Q = -P, R = foot(B,A,C), S = -R; draw(A--B--C--D--A^^B--D^^R--S^^rightanglemark(A,P,D,6)^^rightanglemark(C,Q,D,6)); draw(B--R^^C--Q^^A--P^^D--S,linetype("4 4")); dot("$A$",A,dir(270)); dot("$B$",B,E); dot("$C$",C,N); dot("$D$",D,W); dot("$P$",P,SE); dot("$Q$",Q,NE); dot("$R$",R,N); dot("$S$",S,dir(270)); [/asy] Suppose $PQ=6$ and $RS=8,$ and let $d$ denote the length of $\overline{BD},$ the longer diagonal of $ABCD.$ Then $d^2$ can be written in the form $m+n\sqrt p,$ where $m,n,$ and $p$ are positive integers and $p$ is not divisible by the square of any prime. What is $m+n+p?$ $\textbf{(A) }81 \qquad \textbf{(B) }89 \qquad \textbf{(C) }97\qquad \textbf{(D) }105 \qquad \textbf{(E) }113$

In the collection of all right circular cylinders of fixed volume $c$, what is the ratio $\frac hr$ of the cylinder which has the least total surface area?

Let $x = 1 - 2^{-2009}$. Show that $x + x^2 + x^4 + x^8 +... + x^{2^m}< 2010$ for all positive integers $m$.