Let $BC=6$, $BX=3$, $CX=5$, and let $F$ be the midpoint of $\overline{BC}$. Let $\overline{AX}\perp\overline{BC}$ and $AF=\sqrt{247}$. If $AC$ is of the form $\sqrt{b}$ and $AB$ is of the form $\sqrt{c}$ where $b$ and $c$ are nonnegative integers, find $2c+3b$.

Determine all integers $k$ such that the equation: $$\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\frac{k}{xyz}$$ has an infinite number of integer solutions $(x,y,z)$ with gcd$(k,xyz)=1$.

Let $\omega$ be a circle. Let $E$ be on $\omega$ and $S$ outside $\omega$ such that line segment $SE$ is tangent to $\omega$. Let $R$ be on $\omega$. Let line $SR$ intersect $\omega$ at $B$ other than $R$, such that $R$ is between $S$ and $B$. Let $I$ be the intersection of the bisector of $\angle ESR$ with the line tangent to $\omega$ at $R$; let $A$ be the intersection of the bisector of $\angle ESR$ with $ER$. If the radius of the circumcircle of $\triangle EIA$ is $10$, the radius of the circumcircle of $\triangle SAB$ is $14$, and $SA = 18$, then $IA$ can be expressed in simplest form as $\frac{m}{n}$. Find $m + n$.

We have a stick of length $2n{}$ and a machine which cuts sticks of length $k\in\mathbb{N}$ with $k>1$ into two sticks with arbitrary positive integer lengths. What is the smallest number of cuts after which we can always find some sticks whose lengths sum up to $n{}$?

$(GDR 2)^{IMO1}$ Prove that there exist infinitely many natural numbers $a$ with the following property: The number $z = n^4 + a$ is not prime for any natural number $n.$

Thirteen black and six white hexagonal tiles were used to create the figure below. If a new figure is created by attaching a border of white tiles with the same size and shape as the others, what will be the difference between the total number of white tiles and the total number of black tiles in the new figure? [asy] defaultpen(linewidth(1)); real x=sqrt(3)/2; path p=rotate(30)*polygon(6); filldraw(p^^shift(0,3)*p^^shift(4x,0)*p^^shift(3x,1.5)*p^^shift(2x,3)*p^^shift(-4x,0)*p^^shift(-3x,1.5)*p^^shift(-2x,3)*p^^shift(3x,-1.5)*p^^shift(-3x,-1.5)*p^^shift(2x,-3)*p^^shift(-2x,-3)*p^^shift(0,-3)*p, black, black); draw(shift(2x,0)*p^^shift(-2x,0)*p^^shift(x,1.5)*p^^shift(-x,1.5)*p^^shift(x,-1.5)*p^^shift(-x,-1.5)*p); [/asy] $ \textbf{(A)}\ 5\qquad\textbf{(B)}\ 7\qquad\textbf{(C)}\ 11\qquad\textbf{(D)}\ 12\qquad\textbf{(E)}\ 18 $

In rectangle $ABCD$, $AB=6$ and $BC=3$. Point $E$ between $B$ and $C$, and point $F$ between $E$ and $C$ are such that $BE=EF=FC$. Segments $\overline{AE}$ and $\overline{AF}$ intersect $\overline{BD}$ at $P$ and $Q$, respectively. The ratio $BP:PQ:QD$ can be written as $r:s:t$, where the greatest common factor of $r,s$ and $t$ is $1$. What is $r+s+t$? $\textbf{(A) } 7 \qquad \textbf{(B) } 9 \qquad \textbf{(C) } 12 \qquad \textbf{(D) } 15 \qquad \textbf{(E) } 20$

Let $ABC$ be a triangle such that $AB = 3$, $BC = 4$, and $AC = 5$. Let $X$ be a point in the triangle. Compute the minimal possible value of $AX^2 + BX^2 + CX^2$

For which positive integer $n$ is the quantity $\frac{n}{3} + \frac{40}{n}$ minimized? [i]Proposed by Eugene Chen[/i]

A regular $n$-gon $P_1P_2...P_n$ satisfies $\angle P_1P_7P_8 = 178^o$. Compute $n$.

Let $ABC$ be a triangle in which $BC<AC$. Let $M$ be the mid-point of $AB$, $AP$ be the altitude from $A$ on $BC$, and $BQ$ be the altitude from $B$ on to $AC$. Suppose that $QP$ produced meets $AB$ (extended) at $T$. If $H$ is the orthocenter of $ABC$, prove that $TH$ is perpendicular to $CM$.

Let $*$ be a commutative and associative binary operation on a set $S.$ Assume that for every $x$ and $y$ in $S,$ there exists $z$ in $S$ such that $x*z=y.$ (This $z$ may depend on $x$ and $y.$) Show that if $a,b,c$ are in $S$ and $a*c=b*c,$ then $a=b.$

Let $(b_{n})_{n\in \mathbb{N}}$ be a sequence of positive real numbers such that $b_{0}=1$, $b_{n}=2+\sqrt{b_{n-1}}-2\sqrt{1+\sqrt{b_{n-1}}}$. Calculate $$\sum_{n=1}^{\infty}b_{n}2^{n}.$$

Let $ABC$ be an acute triangle and let $M$ be the midpoint of $AC$. A circle $\omega$ passing through $B$ and $M$ meets the sides $AB$ and $BC$ at points $P$ and $Q$ respectively. Let $T$ be the point such that $BPTQ$ is a parallelogram. Suppose that $T$ lies on the circumcircle of $ABC$. Determine all possible values of $\frac{BT}{BM}$.

In the triangle$ABC$ ($AC > AB$), point $N$ is the midpoint of $BC$, and $I$ is the intersection point of the angle bisectors. Ray $AI$ intersects the circumscribed circle of triangle $ABC$ at point $W$, a perpendicular $WF$ is drawn from it on side $AC$. Find the length of the segment $CF$ , if the radius of the circle inscribed in the triangle $ABC$ is equal to $r$ and $\angle INB = 45^o$. (Gryhoriy Filippovskyi)

Let $n$ and $k$ be positive integers such that $k\leq 2^n$. Banana and Corona are playing the following variant of the guessing game. First, Banana secretly picks an integer $x$ such that $1\leq x\leq n$. Corona will attempt to determine $x$ by asking some questions, which are described as follows. In each turn, Corona chooses $k$ distinct subsets of $\{1, 2, \ldots, n\}$ and, for each chosen set $S$, asks the question "Is $x$ in the set $S$?''. Banana picks one of these $k$ questions and tells both the question and its answer to Corona, who can then start another turn. Find all pairs $(n,k)$ such that, regardless of Banana's actions, Corona could determine $x$ in finitely many turns with absolute certainty. [i]Pitchayut Saengrungkongka, Thailand[/i]