A frog makes $ 3$ jumps, each exactly $ 1$ meter long. The directions of the jumps are chosen independently and at random. What is the probability the frog's final position is no more than $ 1$ meter from its starting position? $ \textbf{(A)}\ \frac {1}{6} \qquad \textbf{(B)}\ \frac {1}{5} \qquad \textbf{(C)}\ \frac {1}{4} \qquad \textbf{(D)}\ \frac {1}{3} \qquad \textbf{(E)}\ \frac {1}{2}$

Compute $\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1}}}}...}$

The centers of two circles are $ 41$ inches apart. The smaller circle has a radius of $ 4$ inches and the larger one has a radius of $ 5$ inches. The length of the common internal tangent is: $ \textbf{(A)}\ 41\text{ inches} \qquad\textbf{(B)}\ 39\text{ inches} \qquad\textbf{(C)}\ 39.8\text{ inches} \qquad\textbf{(D)}\ 40.1\text{ inches}\\ \textbf{(E)}\ 40\text{ inches}$

In triangle $ ABC$ we have $ a \geq b$ and $ a \geq c.$ Prove that the ratio of circumcircle radius to incircle diameter is at least as big as the length of the centroidal axis $ s_a$ to the altitude $ a_a.$ When do we have equality?

In the figure, AQPB and ASRC are squares, and AQS is an equilateral triangle. If QS  = 4 and BC  = x, what is the value of x? [asy] unitsize(16); pair A,B,C,P,Q,R,T; A=(3.4641016151377544, 2); B=(0, 0); C=(6.928203230275509, 0); P=(-1.9999999999999991, 3.464101615137755); Q=(1.4641016151377544, 5.464101615137754); R=(8.928203230275509, 3.4641016151377544); T=(5.464101615137754, 5.464101615137754); dot(A);dot(B);dot(C);dot(P); dot(Q);dot(R);dot(T); label("$A$", (3.4641016151377544, 2),E); label("$B$", (0, 0),S); label("$C$", (6.928203230275509, 0),S); label("$P$", (-1.9999999999999991, 3.464101615137755), W); label("$Q$", (1.4641016151377544, 5.464101615137754),N); label("$R$", (8.928203230275509, 3.4641016151377544),E); label("$S$", (5.464101615137754, 5.464101615137754),N); draw(B--C--A--B); draw(B--P--Q--A--B); draw(A--C--R--T--A); draw(Q--T--A--Q); label("$x$", (3.4641016151377544, 0), S); label("$4$", (Q+T)/2, N);[/asy]

For every triangle $ ABC$, let $ D,E,F$ be a point located on segment $ BC,CA,AB$, respectively. Let $ P$ be the intersection of $ AD$ and $ EF$. Prove that: \[ \frac{AB}{AF}\times DC\plus{}\frac{AC}{AE}\times DB\equal{}\frac{AD}{AP}\times BC\]

We are given an isosceles triangle $ABC$ such that $BC=a$ and $AB=BC=b$. The variable points $M\in (AC)$ and $N\in (AB)$ satisfy $a^2\cdot AM \cdot AN = b^2 \cdot BN \cdot CM$. The straight lines $BM$ and $CN$ intersect in $P$. Find the locus of the variable point $P$. [i]Dan Branzei[/i]

Three circles $k_1, k_2$, and $k_3$ intersect in point $O$. Let $A, B$, and $C$ be the second intersection points (other than $O$) of $k_2$ and $k_3, k_1$ and $k_3$, and $k_1$ and $k_2$, respectively. Assume that $O$ lies inside of the triangle $ABC$. Let lines $AO,BO$, and $CO$ intersect circles $k_1, k_2$, and $k_3$ for a second time at points $A', B'$, and $C'$, respectively. If $|XY|$ denotes the length of segment $XY$, prove that $\frac{|AO|}{|AA'|}+\frac{|BO|}{|BB'|}+\frac{|CO|}{|CC'|}= 1$

Ana and Banana play a game. First, Ana picks a real number $p$ with $0 \le p \le 1$. Then, Banana picks an integer $h$ greater than $1$ and creates a spaceship with $h$ hit points. Now every minute, Ana decreases the spaceship's hit points by $2$ with probability $1-p$, and by $3$ with probability $p$. Ana wins if and only if the number of hit points is reduced to exactly $0$ at some point (in particular, if the spaceship has a negative number of hit points at any time then Ana loses). Given that Ana and Banana select $p$ and $h$ optimally, compute the integer closest to $1000p$. [i]Proposed by Lewis Chen[/i]

In the triangle $ABC$ , the angle bisector $AD$ divides the side $BC$ into the ratio $BD: DC = 2: 1$. In what ratio, does the median $CE$ divide this bisector?

In $\triangle ABC$, points $E$ and $F$ lie on $\overline{AC}, \overline{AB}$, respectively. Denote by $P$ the intersection of $\overline{BE}$ and $\overline{CF}$. Compute the maximum possible area of $\triangle ABC$ if $PB = 14$, $PC = 4$, $PE = 7$, $PF = 2$. [i]Proposed by Eugene Chen[/i]

$BE$ and $CF$ are the altitudes of the acute scalene $\triangle ABC$, $O$ is its circumcenter and $M$ is the midpoint of the side $BC$. If point, which is symmetric to $M$ with respect to $O$, lies on the line $EF$, find all possible values of the ratio $\dfrac{AM}{AO}$. [i]Proposed by Fedir Yudin[/i]

For a positive number such as 3.27, 3 is referred to as the integral part of the number and .27 as the decimal part. Find a positive number such that its decimal part, its integral part, and the number itself form a geometric progression.

In an acute triangle $ ABC$, let $ D$ be a point on $ \left[AC\right]$ and $ E$ be a point on $ \left[AB\right]$ such that $ \angle ADB\equal{}\angle AEC\equal{}90{}^\circ$. If perimeter of triangle $ AED$ is 9, circumradius of $ AED$ is $ \frac{9}{5}$ and perimeter of triangle $ ABC$ is 15, then $ \left|BC\right|$ is $\textbf{(A)}\ 5 \qquad\textbf{(B)}\ \frac{24}{5} \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}\ 8 \qquad\textbf{(E)}\ \frac{48}{5}$

The average (arithmetic mean) age of a group consisting of doctors and lawyers in 40. If the doctors average 35 and the lawyers 50 years old, then the ratio of the numbers of doctors to the number of lawyers is $ \textbf{(A)}\ 3: 2 \qquad \textbf{(B)}\ 3: 1 \qquad \textbf{(C)}\ 2: 3 \qquad \textbf{(D)}\ 2: 1 \qquad \textbf{(E)}\ 1: 2$