If two points are selected at random on a fixed circle and the chord between the two points is drawn, what is the probability that its length exceeds the radius of the circle?

Let $n =123456789101112 ... 998999$ be the natural number where is obtained by writing the natural numbers from $1$ to $999$ one after the other. What is the $1997$-th digit number in $n$?

Elmo calls a monic polynomial with real coefficients [i]tasty[/i] if all of its coefficients are in the range $[-1,1]$. A monic polynomial $P$ with real coefficients and complex roots $\chi_1,\cdots,\chi_m$ (counted with multiplicity) is given to Elmo, and he discovers that there does not exist a monic polynomial $Q$ with real coefficients such that $PQ$ is tasty. Find all possible values of $\max\left(|\chi_1|,\cdots,|\chi_m|\right)$. [i]Proposed by Carl Schildkraut[/i]

The figure shown can be folded into the shape of a cube. In the resulting cube, which of the lettered faces is opposite the face marked $x$? [asy] defaultpen(linewidth(0.7)); path p=origin--(0,1)--(1,1)--(1,2)--(2,2)--(2,3); draw(p^^(2,3)--(4,3)^^shift(2,0)*p^^(2,0)--origin); draw(shift(1,0)*p, dashed); label("$x$", (0.3,0.5), E); label("$A$", (1.3,0.5), E); label("$B$", (1.3,1.5), E); label("$C$", (2.3,1.5), E); label("$D$", (2.3,2.5), E); label("$E$", (3.3,2.5), E);[/asy] $ \mathbf{(A)}\; A\qquad \mathbf{(B)}\; B\qquad \mathbf{(C)}\; C\qquad \mathbf{(D)}\; D\qquad \mathbf{(E)}\; E$

In a triangle $ABC$, let $AB$ be the shortest side. Points $X$ and $Y$ are given on the circumcircle of $\triangle ABC$ such that $CX=AX+BX$ and $CY=AY+BY$. Prove that $\measuredangle XCY<60^{o}$.

Find the range of the function $f(x)=\frac{\sqrt{x^2+1}}{x-1}$.

Let $C$ be a continuous closed curve in the plane which does not cross itself and let $Q$ be a point inside $C$. Show that there exists points $P_1$ and $P_2$ on $C$ such that $Q$ is the midpoint of the line segment $P_1P_2.$

Determine the value of $a,b$ for which $\int_0^1 (\sqrt{1-x}-ax-b)^2 dx$ is minimized.

Aaliyah rolls two standard 6-sided dice. She notices that the product of the two numbers rolled is a multiple of 6. Which of the following integers [i]cannot[/i] be the sum of the two numbers? $\textbf{(A) } 5\qquad\textbf{(B) } 6\qquad\textbf{(C) } 7\qquad\textbf{(D) } 8\qquad\textbf{(E) } 9$

Let be three nonnegative integers $ m,n,p $ and three real numbers $ x,y,z $ such that $ 2^mx+2^ny+2^pz\ge 0. $ Prove: $$ 2^m\left( 2^x-1 \right)+2^n\left( 2^y-1 \right)+2^p\left( 2^z-1 \right)\ge 0 $$ [i]Cristinel Mortici[/i]

Positive real numbers $x,y,z$ have the sum $1$. Prove that $\sqrt{7x+3}+ \sqrt{7y+3}+\sqrt{7z+3} \le 7$. Can number $7$ on the right hand side be replaced with a smaller constant?

Three lines $k, l, m$ are drawn through a point $P$ inside a triangle $ABC$ such that $k$ meets $AB$ at $A_1$ and $AC$ at $A_2 \ne A_1$ and $PA_1 = PA_2$, $l $ meets $BC$ at $B_1$ and $BA$ at $B_2 \ne B_1$ and $PB_1 = PB_2$, $m$ meets $CA$ at $C_1$ and $CB$ at $C_2\ne C_1$ and $PC_1=PC_2$. Prove that the lines $k,l,m$ are uniquely determined by these conditions. Find point $P$ for which the triangles $AA_1A_2, BB_1B_2, CC_1C_2$ have the same area and show that this point is unique.

The product $N$ of three positive integers is $6$ times their sum, and one of the integers is the sum of the other two. Find the sum of all possible values of $N.$

Let $a,b$, and $x$ be positive integers such that $x^{a+b}=a^b{b}$. Prove that $a=x$ and $b=x^{x}$.

For which $m > 1$ is $(m -1)!$ divisible by $m$?

A circle with a circumscribed and an inscribed square centered at the origin $ O$ of a rectangular coordinate system with positive $ x$ and $ y$ axes $ OX$ and $ OY$ is shown in each figure $ I$ to $ IV$ below. [asy] size((400)); draw((0,0)--(22,0), EndArrow); draw((10,-10)--(10,12), EndArrow); draw((25,0)--(47,0), EndArrow); draw((35,-10)--(35,12), EndArrow); draw((-25,0)--(-3,0), EndArrow); draw((-15,-10)--(-15,12), EndArrow); draw((-50,0)--(-28,0), EndArrow); draw((-40,-10)--(-40,12), EndArrow); draw(Circle((-40,0),6)); draw(Circle((-15,0),6)); draw(Circle((10,0),6)); draw(Circle((35,0),6)); draw((-34,0)--(-40,6)--(-46,0)--(-40,-6)--(-34,0)--(-34,6)--(-46,6)--(-46,-6)--(-34,-6)--cycle); draw((-6.5,0)--(-15,8.5)--(-23.5,0)--(-15,-8.5)--cycle); draw((-10.8,4.2)--(-19.2,4.2)--(-19.2,-4.2)--(-10.8,-4.2)--cycle); draw((14.2,4.2)--(5.8,4.2)--(5.8,-4.2)--(14.2,-4.2)--cycle); draw((16,6)--(4,6)--(4,-6)--(16,-6)--cycle); draw((41,0)--(35,6)--(29,0)--(35,-6)--cycle); draw((43.5,0)--(35,8.5)--(26.5,0)--(35,-8.5)--cycle); label("I", (-49,9)); label("II", (-24,9)); label("III", (1,9)); label("IV", (26,9)); label("X", (-28,0), S); label("X", (-3,0), S); label("X", (22,0), S); label("X", (47,0), S); label("Y", (-40,12), E); label("Y", (-15,12), E); label("Y", (10,12), E); label("Y", (35,12), E);[/asy] The inequalities \[ |x| \plus{} |y| \leq \sqrt {2(x^2 \plus{} y^2)} \leq 2\mbox{Max}(|x|, |y|)\] are represented geometrically* by the figure numbered $ \textbf{(A)}\ I \qquad \textbf{(B)}\ II \qquad \textbf{(C)}\ III \qquad \textbf{(D)}\ IV \qquad \textbf{(E)}\ \mbox{none of these}$ *An inequality of the form $ f(x, y) \leq g(x, y)$, for all $ x$ and $ y$ is represented geometrically by a figure showing the containment \[ \{\mbox{The set of points }(x, y)\mbox{ such that }g(x, y) \leq a\} \subset\\ \{\mbox{The set of points }(x, y)\mbox{ such that }f(x, y) \leq a\}\] for a typical real number $ a$.

The points $A, B, C, D, P$ lie on an circle as shown in the figure such that $\angle AP B = \angle BPC = \angle CPD$. Prove that the lengths of the segments are denoted by $a, b, c, d$ by $\frac{a + c}{b + d} =\frac{b}{c}$. [img]https://cdn.artofproblemsolving.com/attachments/a/2/ba8965f5d7d180426db26e8f7dd5c7ad02c440.png[/img]

Find all pairs $(x,y)$ of positive integers that satisfy the equation \[y^{2}=x^{3}+16.\]

Find all functions $f : R \to R$ such that $x[f(x + y) - f (x - y)] = 4y f (x)$ for any real numbers $x, y$.

Find all triplets $(x,y,p)$ of positive integers such that $p$ be a prime number and $\frac{xy^3}{x+y}=p$

Consider two quadratic equations \begin{align*}x^2+ax+b&=0, \\ x^2+cx+d&=0,\end{align*} with real coefficients. Find necessary and sufficient conditions such that the first equation has (real) roots $x,x_1,$ the second $x,x_2$ and $x>0,x_1>x_2$.

Let $S$ be the set of all the odd positive integers that are not multiples of $5$ and that are less than $30m$, $m$ being an arbitrary positive integer. What is the smallest integer $k$ such that in any subset of $k$ integers from $S$ there must be two different integers, one of which divides the other?

Given a triangle $ ABC $ and a point $ M $ inside it. Prove that $$ \min \{MA, MB, MC\} + MA + MB + MC <AB + BC + AC. $$

Segment $\overline{PQ}$ is drawn and squares $ABPQ$ and $CDQP$ are constructed in the plane such that they lie on opposite sides of segment $\overline{PQ}.$ If $PQ=1,$ find $BD.$

A finite collection of positive real numbers (not necessarily distinct) is [i]balanced [/i]if each number is less than the sum of the others. Find all $m \ge 3$ such that every balanced finite collection of $m$ numbers can be split into three parts with the property that the sum of the numbers in each part is less than the sum of the numbers in the two other parts.