$ABC$ is an acute triangle and $AD$, $BE$ and $CF$ are the altitudes, with $H$ being the point of intersection of these altitudes. Points $A_1$, $B_1$, $C_1$ are chosen on rays $AD$, $BE$ and $CF$ respectively such that $AA_1 = HD$, $BB_1 = HE$ and $CC_1 =HF$. Let $A_2$, $B_2$ and $C_2$ be midpoints of segments $A_1D$, $B_1E$ and $C_1F$ respectively. Prove that $H$, $A_2$, $B_2$ and $C_2$ are concyclic. (Mykhailo Barkulov)

Consider $ \Lambda = \left\{ \lambda\in\text{Hol} \left[ \mathbb{C}\longrightarrow\mathbb{C} \right] |z\in\mathbb{C}\implies |\lambda (z)|\le e^{|\text{Im}(z)|} \right\} . $ Prove that $ g\in\Lambda $ implies $ g'\in\Lambda . $

Let $ABCD$ be a convex quadrilateral. A circle passing through the points $A$ and $D$ and a circle passing through the points $B$ and $C$ are externally tangent at a point $P$ inside the quadrilateral. Suppose that \[\angle{PAB}+\angle{PDC}\leq 90^\circ\qquad\text{and}\qquad\angle{PBA}+\angle{PCD}\leq 90^\circ.\] Prove that $AB+CD \geq BC+AD$. [i]Proposed by Waldemar Pompe, Poland[/i]

Let $ ABC$ be an acute-angled, non-isosceles triangle with orthocenter $H$, $M$ midpoint of side $AB$ and $w$ bisector of angle $\angle ACB$. Let $S$ be the point of intersection of the perpendicular bisector of side $AB$ with $w$ and $F$ the foot of the perpendicular from $H$ on $w$. Prove that the segments $MS$ and $MF$ are equal. [i](Karl Czakler)[/i]

The dollar is now worth $\frac{1}{980}$ ounce of gold. After the $n^{th}$ 7001 billion dollars bailout package passed by congress, the dollar gains $\frac{1}{2{}^2{}^{n-1}}$ of its $(n-1)^{th}$ value in gold. After four bank bailouts, the dollar is worth $\frac{1}{b}(1-\frac{1}{2^c})$ in gold, where $b, c$ are positive integers. Find $b + c$.

Integers are placed on every cell of an infinite checkerboard. For each cell if it contains integer $a$ then the sum of the numbers in the cell under it and the cell right to it is $2a+1$. Prove that in every infinite diagonal row of direction [i] top-right down-left [/i] all numbers are different.

The diagonals of the inscribed quadrilateral $ABCD$ meet at the point $M$, $\angle AMB = 60^o$. Equilateral triangles $ADK$ and $BCL$ are built outward on sides $AD$ and $BC$. Line $KL$ meets the circle circumscribed ariound $ABCD$ at points $P$ and $Q$. Prove that $PK = LQ$.

Find the number of integer polynomials $P$ such that $P(x)^2 = P(P(x)) \forall x$.

Prove or disprove: For any positive integer $k$ there exists an integer $n > 1$ such that the binomial coeffcient $\binom{n}{i}$ is divisible by $k$ for any $1 \leq i \leq n-1.$

Let $\Omega_1, \Omega_2$ be two intersecting circles with centres $O_1, O_2$ respectively. Let $l$ be a line that intersects $\Omega_1$ at points $A, C$ and $\Omega_2$ at points $B, D$ such that $A, B, C, D$ are collinear in that order. Let the perpendicular bisector of segment $A B$ intersect $\Omega_1$ at points $P, Q$; and the perpendicular bisector of segment $C D$ intersect $\Omega_2$ at points $R, S$ such that $P, R$ are on the same side of $l$. Prove that the midpoints of $P R, Q S$ and $O_1 O_2$ are collinear.

Let $n$ be a positive integer. Answer the following questions. (1) Find the maximum value of $f_n(x)=x^{n}e^{-x}$ for $x\geq 0$. (2) Show that $\lim_{x\to\infty} f_n(x)=0$. (3) Let $I_n=\int_0^x f_n(t)\ dt$. Find $\lim_{x\to\infty} I_n(x)$.

In the circumcircle of triange $\triangle ABC,$ $AA'$ is a diameter. We draw lines $l'$ and $l$ from $A'$ parallel with Internal and external bisector of the vertex $A$. $l'$ Cut out $AB , BC$ at $B_1$ and $B_2$. $l$ Cut out $AC , BC$ at $C_1$ and $C_2$. Prove that the circumcircles of $\triangle ABC$ $\triangle CC_1C_2$ and $\triangle BB_1B_2$ have a common point. (20 points)

Let $ABC$ be a triangle with $|AB|=18$, $|AC|=24$, and $m(\widehat{BAC}) = 150^\circ$. Let $D$, $E$, $F$ be points on sides $[AB]$, $[AC]$, $[BC]$, respectively, such that $|BD|=6$, $|CE|=8$, and $|CF|=2|BF|$. Let $H_1$, $H_2$, $H_3$ be the reflections of the orthocenter of triangle $ABC$ over the points $D$, $E$, $F$, respectively. What is the area of triangle $H_1H_2H_3$? $ \textbf{(A)}\ 70 \qquad\textbf{(B)}\ 72 \qquad\textbf{(C)}\ 84 \qquad\textbf{(D)}\ 96 \qquad\textbf{(E)}\ 108 $

Determine all infinite sequences of nonnegative integers $a_1,a_2,\ldots$ such that: 1. Every positive integer appears in the sequence at least once, and; 2. $a_i$ is the smallest integer $j$ such that $a_{j+2}=i$, for all $i\ge 1$. [i](Proposed by Ho Janson)[/i]

If Percy rolls a fair six-sided die until he rolls a $5$, what is his expected number of rolls, given that all of his rolls are prime?

Let $ABC$ be a triangle, and let $D, E$ and $F$ be the feet of the altitudes from $A, B$ and $C,$ respectively. A circle $\omega_A$ through $B$ and $C$ crosses the line $EF$ at $X$ and $X'$. Similarly, a circle $\omega_B$ through $C$ and $A$ crosses the line $FD$ at $Y$ and $Y',$ and a circle $\omega_C$ through $A$ and $B$ crosses the line $DE$ at $Z$ and $Z'$. Prove that $X, Y$ and $Z$ are collinear if and only if $X', Y'$ and $Z'$ are collinear. [i]Vlad Robu[/i]

Given a (simple) graph $G$ with $n \geq 2$ vertices $v_1, v_2, \dots, v_n$ and $m \geq 1$ edges, Joël and Robert play the following game with $m$ coins: [list=i] [*]Joël first assigns to each vertex $v_i$ a non-negative integer $w_i$ such that $w_1+\cdots+w_n=m$. [*]Robert then chooses a (possibly empty) subset of edges, and for each edge chosen he places a coin on exactly one of its two endpoints, and then removes that edge from the graph. When he is done, the amount of coins on each vertex $v_i$ should not be greater than $w_i$. [*]Joël then does the same for all the remaining edges. [*]Joël wins if the number of coins on each vertex $v_i$ is equal to $w_i$. [/list] Determine all graphs $G$ for which Joël has a winning strategy.

Nine positive integers $a_1,a_2,...,a_9$ have their last $2$-digit part equal to $11,12,13,14,15,16,17,18$ and $19$ respectively. Find the last $2$-digit part of the sum of their squares.

$ABC$ is an isosceles triangle ($AB=AC$). Point $X$ is an arbitrary point on $BC$. $Z \in AC$ and $Y \in AB$ such that $\angle BXY = \angle ZXC$. A line parallel to $YZ$ passes through $B$ and cuts $XZ$ at $T$. Prove that $AT$ bisects $\angle A$.

Define a function $P(n)$ from the set of positive integers to itself, where $P(1)=1$ and if an integer $n > 1$ has prime factorization $$n = p_1^{a_1}p_2^{a_2} \dots p_k^{a_k}$$ then $$P(n) = a_1^{p_1}a_2^{p_2} \dots a_k^{p_k}.$$ Prove that $P(P(n)) \le n$ for all positive integers $n.$ [i]Proposed by Evan Chang (squareman), USA[/i]

Which one of the following integers can be expressed as the sum of $ 100$ consecutive positive integers? $ \textbf{(A)}\ 1,\!627,\!384,\!950\qquad \textbf{(B)}\ 2,\!345,\!678,\!910\qquad \textbf{(C)}\ 3,\!579,\!111,\!300\qquad \textbf{(D)}\ 4,\!692,\!581,\!470\qquad \textbf{(E)}\ 5,\!815,\!937,\!260$

The ratio of Mary's age to Alice's age is $ 3: 5$. Alice is $ 30$ years old. How old is Mary? $ \textbf{(A) } 15\qquad \textbf{(B) } 18\qquad \textbf{(C) } 20\qquad \textbf{(D) } 24\qquad \textbf{(E) } 50$

If in the formula $ C \equal{} \frac {en}{R\plus{}nr}$, $n$ is increased while $ e$, $R$ and $r$ are kept constant, then $C$: $\textbf{(A)}\ \text{Increases} \qquad \textbf{(B)}\ \text{Decreases} \qquad \textbf{(C)}\ \text{Remains constant} \qquad \textbf{(D)}\ \text{Increases and then decreases} \qquad\\ \textbf{(E)}\ \text{Decreases and then increases}$

Let $\mathcal{L}$ be a finite collection of lines in the plane in general position (no two lines in $\mathcal{L}$ are parallel and no three are concurrent). Consider the open circular discs inscribed in the triangles enclosed by each triple of lines in $\mathcal{L}$. Determine the number of such discs intersected by no line in $\mathcal{L}$, in terms of $|\mathcal{L}|$. [i]B. Aronov et al.[/i]

A tetrahedron $ABCD$ is given. The incircles of triangles $ ABC$ and $ABD$ with centers $O_1, O_2$, touch $AB$ at points $T_1, T_2$. The plane $\pi_{AB}$ passing through the midpoint of $T_1T_2$ is perpendicular to $O_1O_2$. The planes $\pi_{AC},\pi_{BC}, \pi_{AD}, \pi_{BD}, \pi_{CD}$ are defined similarly. Prove that these six planes have a common point.