Show that a unit square can be covered with three equal disks with radius less than $\frac{1}{\sqrt{2}}$. What is the smallest possible radius?

Let $ABCD$ be a convex quadrilateral with \[\angle ADC = 90^\circ, \ \ \angle BCD = \angle ABC > 90^\circ, \mbox{ and } AB = 2CD.\] The line through \(C\), parallel to \(AD\), intersects the external angle bisector of \(\angle ABC\) at point \(T\). Prove that the angles $\angle ATB$, $\angle TBC$, $\angle BCD$, $\angle CDA$, $\angle DAT$ can be divided into two groups, so that the angles in each group have a sum of $270^{\circ}$. [i] Miroslav Marinov, Bulgaria [/i]

Let $ABC$ be an isosceles triangle with $AB=AC$. Points $D$ and $E$ lie on the sides $AB$ and $AC$, respectively. The line passing through $B$ and parallel to $AC$ meets the line $DE$ at $F$. The line passing through $C$ and parallel to $AB$ meets the line $DE$ at $G$. Prove that \[\frac{[DBCG]}{[FBCE]}=\frac{AD}{DE} \]

On the plane are given four distinct points $A,B,C,D$ on a line $g$ in this order, at the mutual distances $AB = a, BC = b, CD = c$. (a) Construct (if possible) a point $P$ outside line $g$ such that $\angle APB =\angle BPC =\angle CPD$. (b) Prove that such a point $P$ exists if and only if $ (a+b)(b+c) < 4ac$

Let $ a,$ $ b,$ $ c,$ $ d,$ and $ e$ be five consecutive terms in an arithmetic sequence, and suppose that $ a \plus{} b \plus{} c \plus{} d \plus{} e \equal{} 30.$ Which of the following can be found? $ \textbf{(A)}\ a \qquad \textbf{(B)}\ b \qquad \textbf{(C)}\ c \qquad \textbf{(D)}\ d \qquad \textbf{(E)}\ e$

If $x$ is the cube of a positive integer and $d$ is the number of positive integers that are divisors of $x$, then $d$ could be $ \textbf{(A)}\ 200\qquad\textbf{(B)}\ 201\qquad\textbf{(C)}\ 202\qquad\textbf{(D)}\ 203\qquad\textbf{(E)}\ 204 $

$\textbf{a) }$ Let $p \geq 2$ be a natural number and $G_p = \bigcup\limits_{n \in \mathbb{N}} \lbrace z \in \mathbb{C} \mid z^{p^n}=1 \rbrace.$ Prove that $(G_p, \cdot)$ is a subgroup of $(\mathbb{C}^*, \cdot).$ $\textbf{b) }$ Let $(H, \cdot)$ be an infinite subgroup of $(\mathbb{C}^*, \cdot).$ Prove that all proper subgroups of $H$ are finite if and only if $H=G_p$ for some prime $p.$

In a triangle, $AB=BC$ and $M$ is the midpoint of $AC$. $H$ is chosen on $BC$ so that $MH$ is perpendicular to $BC$. $P$ is the midpoint of $MH$. Prove that $AH$ is perpendicular to $BP$.

Each side of triangle $ABC$ is $12$ units. $D$ is the foot of the perpendicular dropped from $A$ on $BC$, and $E$ is the midpoint of $AD$. The length of $BE$, in the same unit, is: ${{ \textbf{(A)}\ \sqrt{18} \qquad\textbf{(B)}\ \sqrt{28} \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}\ \sqrt{63} }\qquad\textbf{(E)}\ \sqrt{98} } $

A trapezoid and an isosceles triangle are inscribed in a circle. The larger base of the trapezoid is the diameter of the circle, and the sides of the triangle are parallel to the sides of the trapezoid. Show that the trapezoid and the triangle have equal areas.

Prove that $5^n-3^n$ is not divisible by $2^n+65$ for any positive integer $n$.

Prove that for all positive integers $m$ and $n$ the following inequality hold: $$\pi(m)-\pi(n)\leq\frac{(m-1)\varphi(n)}{n}$$ When does equality hold? [i]Proposed by Shend Zhjeqi and Dorlir Ahmeti, Kosovo[/i]

Cirlce $\Omega$ is inscribed in triangle $ABC$ with $\angle BAC=40$. Point $D$ is inside the angle $BAC$ and is the intersection of exterior bisectors of angles $B$ and $C$ with the common side $BC$. Tangent form $D$ touches $\Omega$ in $E$. FInd $\angle BEC$.

The area of a circle centered at the origin, which is inscribed in the parabola $y=x^2-25$, can be expressed as $\tfrac ab\pi$, where $a$ and $b$ are coprime positive integers. What is the value of $a+b$?

The three roots of $P(x) = x^3 - 2x^2 - x + 1$ are $a>b>c \in \mathbb{R}$. Find the value of $a^2b+b^2c+c^2a$. :D

Let $ABC$ be a triangle with $\angle{BAC}>90^\circ.$ Let $D$ be the foot of the perpendicular from $A$ to side $BC.$ Let $M$ and $N$ be the midpoints of segments $BC$ and $BD,$ respectively. Suppose that $AC=2, \angle{BAN}=\angle{MAC},$ and $AB \cdot BC = AM.$ Compute the distance from $B$ to line $AM.$

Let $G$ be a finite abelian group. There is a magic box $T$. At any point, an element of $G$ may be added to the box and all elements belonging to the subgroup (of $G$) generated by the elements currently inside $T$ are moved from outside $T$ to inside (unless they are already inside). Initially $ T$ contains only the group identity, $1_G$. Alice and Bob take turns moving an element from outside $T$ to inside it. Alice moves first. Whoever cannot make a move loses. Find all $G$ for which Bob has a winning strategy.

Let $M$ be the set of all positive integers that do not contain the digit $9$ (base $10$). If $x_1, \ldots , x_n$ are arbitrary but distinct elements in $M$, prove that \[\sum_{j=1}^n \frac{1}{x_j} < 80 .\]

Let $n$ be the $200$th smallest positive real solution to the equation $x- \frac{\pi}{2} =\ tan x$. Find the greatest integer that does not exceed $\frac{n}{2}$.

Prove that the ten's digit of any power of 3 is even.

Define $f(x, y)$ to be $\frac{|x|}{|y|}$ if that value is a positive integer, $\frac{|y|}{|x|}$ if that value is a positive integer, and zero otherwise. We say that a sequence of integers $\ell_1$ through $\ell_n$ is [i]good [/i] if $f(\ell_i, \ell_{i+1})$ is nonzero for all $i$ where $1 \le i \le n - 1$, and the score of the sequence is $\sum^{n-1}_{i=1} f(\ell_i, \ell_{i+1})$

All four angles of quadrilateral are greater than $60^o$. Prove that we can choose three sides to make a triangle.

Let $n$ be an integer bigger than $0$. Let $\mathbb{A}= ( a_1,a_2,...,a_n )$ be a set of real numbers. Find the number of functions $f:A \rightarrow A$ such that $f(f(x))-f(f(y)) \ge x-y$ for any $x,y \in \mathbb{A}$, with $x>y$.

The diagram below shows nine points on a circle where $AB=BC=CD=DE=EF=FG=GH$. Given that $\angle GHJ=117^\circ$ and $\overline{BH}$ is perpendicular to $\overline{EJ}$, there are relatively prime positive integers $m$ and $n$ so that the degree measure of $\angle AJB$ is $\textstyle\frac mn$. Find $m+n$. [asy] size(175); defaultpen(linewidth(0.6)); draw(unitcircle,linewidth(0.9)); string labels[] = {"A","B","C","D","E","F","G"}; int start=110,increment=20; pair J=dir(210),x[],H=dir(start-7*increment); for(int i=0;i<=6;i=i+1) { x[i]=dir(start-increment*i); draw(J--x[i]--H); dot(x[i]); label("$"+labels[i]+"$",x[i],dir(origin--x[i])); } draw(J--H); dot(H^^J); label("$H$",H,dir(origin--H)); label("$J$",J,dir(origin--J)); [/asy]

Show that $\tan \left( \frac{\pi}{m} \right)$ is irrational for all positive integers $m \ge 5$.