Daredevil Darren challenges Forgetful Fred to spell "Johns Hopkins." Forgetful Fred will spell it correctly except for the 's's; there is a $\frac{1}{3}$ and $\frac{1}{4}$ chance that he will omit the 's' in the first and last names, respectively, with his mistakes being independent of each other. If Forgetful Fred spells the name correctly, then he is happy; otherwise, Daredevil Darren will present him with a dare, and there is a $\frac{9}{10}$ chance that Forgetful Fred will not be happy. Find the probability that Forgetful Fred will be happy.

In the adjoining figure, two circles of radii 6 and 8 are drawn with their centers 12 units apart. At $P$, one of the points of intersection, a line is drawn in such a way that the chords $QP$ and $PR$ have equal length. Find the square of the length of $QP$. [asy]unitsize(2.5mm); defaultpen(linewidth(.8pt)+fontsize(12pt)); dotfactor=3; pair O1=(0,0), O2=(12,0); path C1=Circle(O1,8), C2=Circle(O2,6); pair P=intersectionpoints(C1,C2)[0]; path C3=Circle(P,sqrt(130)); pair Q=intersectionpoints(C3,C1)[0]; pair R=intersectionpoints(C3,C2)[1]; draw(C1); draw(C2); //draw(O2--O1); //dot(O1); //dot(O2); draw(Q--R); label("$Q$",Q,N); label("$P$",P,dir(80)); label("$R$",R,E); //label("12",waypoint(O1--O2,0.4),S);[/asy]

In triangle $ABC$, we consider the concurrent lines $AA_1$, $BB_1$ and $CC_1$, with $A_1$, $B_1$ and $C_1$ lying on the segments $BC$, $CA$ and respectively $AB$. If the point of intersection of the lines is the incenter of $\triangle A_1B_1C_1$, prove that it is also the orthocenter of $\triangle ABC$.

A regular $17$-gon with vertices $V_1, V_2, . . . , V_{17}$ and sides of length $3$ has a point $ P$ on $V_1V_2$ such that $V_1P = 1$. A chord that stretches from $V_1$ to $V_2$ containing $ P$ is rotated within the interior of the heptadecagon around $V_2$ such that the chord now stretches from $V_2$ to $V_3$. The chord then hinges around $V_3$, then $V_4$, and so on, continuing until $ P$ is back at its original position. Find the total length traced by $ P$.

Let $A_1$ and $C_1$ be the feet of altitudes $AH$ and $CH$ of an acute-angled triangle $ABC$. Points $A_2$ and $C_2$ are the reflections of $A_1$ and $C_1$ about $AC$. Prove that the distance between the circumcenters of triangles $C_2HA_1$ and $C_1HA_2$ equals $AC$.

Albert, Bernard, and Cheryl are playing marbles. At the beginning, each of them brings 5 red marbles, 7 green marbles and 13 blue marbles and in the middle of the table, there is a box of infinitely many red, blue and green marbles. In each turn, each player may choose 2 marbles of different color and replace them with 2 marbles of the third color. After a finite number of steps, this conversation happens. Albert : " I have only red marbles" Bernard : "I have only blue marbles" Cheryl: "I have only green marbles" Which of the three are lying?

Let $ABC $be a right triangle with $\angle C = 90^o$ and let $D$ be the foot of the altitude from $C$. Let $E$ be the centroid of triangle $ACD$ and let $F$ be the centroid of triangle $BCD$. The point $P$ satisfies $\angle CEP = 90^o$ and $|CP| = |AP|$, while point $Q$ satisfies $\angle CFQ = 90^o$ and $|CQ| = |BQ|$. Prove that $PQ$ passes through the centroid of triangle $ABC$.

Let $f$ be a polynomial of degree $n$ with integer coefficients and $p$ a prime for which $f$, considered modulo $p$, is a degree-$k$ irreducible polynomial over $\mathbb{F}_p$. Show that $k$ divides the degree of the splitting field of $f$ over $\mathbb{Q}$.

Say that a positive integer is [i]sweet[/i] if it uses only the digits 0, 1, 2, 4, and 8. For instance, 2014 is sweet. There are sweet integers whose squares are sweet: some examples (not necessarily the smallest) are 1, 2, 11, 12, 20, 100, 202, and 210. There are sweet integers whose cubes are sweet: some examples (not necessarily the smallest) are 1, 2, 10, 20, 200, 202, 281, and 2424. Prove that there exists a sweet positive integer $n$ whose square and cube are both sweet, such that the sum of all the digits of $n$ is 2014.

Let $A=(a_{ij})$ be an $m\times n$ real matrix with at least one non-zero element. For each $i\in\{1,\ldots,m\}$, let $R_i=\sum_{j=1}^na_{ij}$ be the sum of the $i$-th row of the matrix $A$, and for each $j\in\{1,\ldots,n\}$, let $C_j =\sum_{i=1}^ma_{ij}$ be the sum of the $j$-th column of the matrix $A$. Prove that there exist indices $k\in\{1,\ldots,m\}$ and $l\in\{1,\ldots,n\}$ such that $$a_{kl}>0,\qquad R_k\ge0,\qquad C_l\ge0,$$or $$a_{kl}<0,\qquad R_k\le0,\qquad C_l\le0.$$

14. Kelvin the frog is hopping on the coordinate plane $\mathbb{R}^{2}$. He starts at the origin, and every second, he hops one unit to the right, left, up, or down, such that he always remains in the first quadrant $\{(x, y): x \geq 0, y \geq 0\}$. In how many ways can Kelvin make his first 14 jumps such that his 14 th jump lands at the origin?

A set $M$ of points in the plane will be called obtuse, if any 3 points from $M$ are the vertices of an obtuse triangle. a.) Prove: For each finite obtuse set $M$ there is a point in the plane with the following property: $P$ is no element from $M$ and $M \cup \{P\}$ is also obtuse. b.) Determine whether the statement from a.) will remain valid, if it is replaced by infinite.

Let $\{x_{n}\}_{n\ge0}$ and $\{y_{n}\}_{n\ge0}$ be two sequences defined recursively as follows \[x_{0}=1, \; x_{1}=4, \; x_{n+2}=3 x_{n+1}-x_{n},\] \[y_{0}=1, \; y_{1}=2, \; y_{n+2}=3 y_{n+1}-y_{n}.\] [list=a][*] Prove that ${x_{n}}^{2}-5{y_{n}}^{2}+4=0$ for all non-negative integers. [*] Suppose that $a$, $b$ are two positive integers such that $a^{2}-5b^{2}+4=0$. Prove that there exists a non-negative integer $k$ such that $a=x_{k}$ and $b=y_{k}$.[/list]

The sum of four real numbers is $9$ and the sum of their squares is $21$. Prove that these numbers can be denoted by $a, b, c, d$ so that $ab-cd \ge 2$ holds.

Let $n \geq 5$ be a given integer. Determine the greatest integer $k$ for which there exists a polygon with $n$ vertices (convex or not, with non-selfintersecting boundary) having $k$ internal right angles. [i]Proposed by Juozas Juvencijus Macys, Lithuania[/i]

The Fibonacci sequence $F_n$ is defined by $F_1=F_2=1$ and the recurrence relation $F_n=F_{n-1}+F_{n-2}$ for all integers $n\geq3$. Let $m,n\geq1$ be integers. Find the minimal degree $d$ for which there exists a polynomial $f(x)=a_dx^d+a_{d-1}x^{d-1}+\dots+a_1x+a_0$, which satisfies $f(k)=F_{m+k}$ for all $k=0,1,...,n$.