Two points $A, B$ are randomly chosen on a circle with radius $100.$ For a positive integer $x$, denote $P(x)$ as the probability that the length of $AB$ is less than $x$. Find the minimum possible integer value of $x$ such that $\text{P}(x) > \frac{2}{3}$.

A diabolical combination lock has $n$ dials (each with $c$ possible states), where $n,c>1$. The dials are initially set to states $d_1, d_2, \ldots, d_n$, where $0\le d_i\le c-1$ for each $1\le i\le n$. Unfortunately, the actual states of the dials (the $d_i$'s) are concealed, and the initial settings of the dials are also unknown. On a given turn, one may advance each dial by an integer amount $c_i$ ($0\le c_i\le c-1$), so that every dial is now in a state $d_i '\equiv d_i+c_i \pmod{c}$ with $0\le d_i ' \le c-1$. After each turn, the lock opens if and only if all of the dials are set to the zero state; otherwise, the lock selects a random integer $k$ and cyclically shifts the $d_i$'s by $k$ (so that for every $i$, $d_i$ is replaced by $d_{i-k}$, where indices are taken modulo $n$). Show that the lock can always be opened, regardless of the choices of the initial configuration and the choices of $k$ (which may vary from turn to turn), if and only if $n$ and $c$ are powers of the same prime. [i]Bobby Shen.[/i]

Find all the pairs of real numbers $(x,y)$ that are solutions of the system: $(x^{2}+y^{2})^{2}-xy(x+y)^{2}=19 $ $| x - y | = 1$

Given two triangles and such that the lengths of the sides of the first triangle are the lengths of the medians of the second triangle. Determine the ratio of the areas of these triangles.

Does there exist points $A,B$ on the curve $y=x^3$ and on $y=x^3+|x|+1$ respectively such that distance between $A,B$ is less than $\frac{1}{100}$ ?

Each positive integer $a$ undergoes the following procedure in order to obtain the number $d = d\left(a\right)$: (i) move the last digit of $a$ to the first position to obtain the numb er $b$; (ii) square $b$ to obtain the number $c$; (iii) move the first digit of $c$ to the end to obtain the number $d$. (All the numbers in the problem are considered to be represented in base $10$.) For example, for $a=2003$, we get $b=3200$, $c=10240000$, and $d = 02400001 = 2400001 = d(2003)$.) Find all numbers $a$ for which $d\left( a\right) =a^2$. [i]Proposed by Zoran Sunic, USA[/i]

From the foot $D$ of the height $CD$ in the triangle $ABC,$ perpendiculars to $BC$ and $AC$ are drawn, which they intersect at points $M$ and $N.$ Let $\angle CAB = 60^{\circ} , \angle CBA = 45^{\circ} ,$ and $H$ be the orthocentre of $MNC.$ If $O$ is the midpoint of $CD,$ find $\angle COH.$

On the parabola $y = x^2$ lie three different points $P, Q$ and $R$. Their projections $P', Q'$ and $R'$ on the $x$-axis are equidistant and equal to $s$ , i.e. $| P'Q'| = | Q'R'| = s$. Determine the area of $\vartriangle PQR$ in terms of $s$

Show that there exist integers $j,k,l,m,n$ greater than $100$ such that $j^2 +k^2 +l^2 +m^2 +n^2 = jklmn-12$.

Let $ABCD$ be a convex quadrilateral. Let $E,F,G,H$ be points on segments $AB$, $BC$, $CD$, $DA$, respectively, and let $P$ be the intersection of $EG$ and $FH$. Given that quadrilaterals $HAEP$, $EBFP$, $FCGP$, $GDHP$ all have inscribed circles, prove that $ABCD$ also has an inscribed circle. [i]Evan O'Dorney.[/i]

Let $ABC$ be a triangle. Let $BE$ and $CF$ be internal angle bisectors of $\angle B$ and $\angle C$ respectively with $E$ on $AC$ and $F$ on $AB$. Suppose $X$ is a point on the segment $CF$ such that $AX$ perpendicular $CF$; and $Y$ is a point on the segment $BE$ such that $AY$ perpendicular $BE$. Prove that $XY = (b + c-a)/2$ where $BC = a, CA = b $and $AB = c$.

Given a finite sequence $ S \equal{} (a_1,a_2,\ldots,a_n)$ of $ n$ real numbers, let $ A(S)$ be the sequence \[ \left(\frac {a_1 \plus{} a_2}2,\frac {a_2 \plus{} a_3}2,\ldots,\frac {a_{n \minus{} 1} \plus{} a_n}2\right) \]of $ n \minus{} 1$ real numbers. Define $ A^1(S) \equal{} A(S)$ and, for each integer $ m$, $ 2\le m\le n \minus{} 1$, define $ A^m(S) \equal{} A(A^{m \minus{} 1}(S)).$ Suppose $ x > 0$, and let $ S \equal{} (1,x,x^2,\ldots,x^{100})$. If $ A^{100}(S) \equal{} (1/2^{50})$, then what is $ x$? $ \textbf{(A) } 1 \minus{} \frac {\sqrt {2}}2\qquad \textbf{(B) } \sqrt {2} \minus{} 1\qquad \textbf{(C) } \frac 12\qquad \textbf{(D) } 2 \minus{} \sqrt {2}\qquad \textbf{(E) } \frac {\sqrt {2}}2$

Compute $$\left(20+\frac{1}{23}\right)\cdot\left(23+\frac{1}{20}\right)-\left(20-\frac{1}{23}\right)\cdot\left(23-\frac{1}{20}\right)$$ [i]Proposed by Andy Xu[/i]

Rectangle $ABCD$ and right triangle $DCE$ have the same area. They are joined to form a trapezoid, as shown. What is $DE$? [asy] size(250); defaultpen(linewidth(0.8)); pair A=(0,5),B=origin,C=(6,0),D=(6,5),E=(18,0); draw(A--B--E--D--cycle^^C--D); draw(rightanglemark(D,C,E,30)); label("$A$",A,NW); label("$B$",B,SW); label("$C$",C,S); label("$D$",D,N); label("$E$",E,S); label("$5$",A/2,W); label("$6$",(A+D)/2,N); [/asy] $\textbf{(A) }12\qquad\textbf{(B) }13\qquad\textbf{(C) }14\qquad\textbf{(D) }15\qquad \textbf{(E) }16$

How many ordered triples $(x,y,z)$ of real numbers satisfy the system of equations $$x^2+y^2+z^2=9,$$ $$x^4+y^4+z^4=33,$$ $$xyz=-4?$$

Solve the equation \[ \sin 9^\circ \sin 21^\circ \sin(102^\circ + x^\circ) = \sin 30^\circ \sin 42^\circ \sin x^\circ \] for $x$ where $0 < x < 90$.