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.

Find all positive integers $m$ and $n$ such that the inequality: \[ [ (m+n) \alpha ] + [ (m+n) \beta ] \geq [ m \alpha ] + [n \beta] + [ n(\alpha+\beta)] \] is true for any real numbers $\alpha$ and $\beta$. Here $[x]$ denote the largest integer no larger than real number $x$.

Let $X$ be a set with $n$ elements. Find the largest number of subsets of $X$, each with $3$ elements, so that no two of them are disjoint.

Point $D$ lies in $\triangle ABC$ and $BD=CD$,$\angle BDC=120$. Point $E$ lies outside $ABC$ and $AE=CE,\angle AEC=60$. Points $B$ and $E$ lies on different sides of $AC$. $F$ is midpoint $BE$. Prove, that $\angle AFD=90$

Two circles intersect at $X$ and $Y$ . The line through the centers of the circles intersect the first circle at $A$ and $C$, and intersect the second circle at $B$ and $D$ so that $A, B, C, D$ lie in this order. The common chord $XY$ cuts $BC$ at $P$, and a point $O$ is arbitrarily chosen on segment $XP$. Lines $CO$ and $BO$ are extended to intersect the first and second circles at $M$ and $N$, respectively. If lines $AM$ and $DN$ intersect at $Z$, prove that $X, Y$ and $Z$ lie on the same line.

Let be distinct points on a plane, four of which form a quadrangle, and three of which are in the interior or boundary of this quadrangle. Show that the diagonals of this quadrangle are longer than the double of the minimum of the distances between any two of these seven points. [i]Paul Erdős[/i] [hide=Side note]If the quadrangle is convex, the constant from the inequality can be improved from $ 2 $ to $ \sqrt{\frac{3\pi}{2}}. $[/hide]

For every positive integer $n\ge 3$, let $\phi_n$ be the set of all positive integers less than and coprime to $n$. Consider the polynomial: $$P_n(x)=\sum_{k\in\phi_n} {x^{k-1}}.$$ a. Prove that $P_n(x)=(x^{r_n}+1)Q_n(x)$ for some positive integer $r_n$ and polynomial $Q_n(x)\in\mathbb{Z}[x]$ (not necessary non-constant polynomial). b. Find all $n$ such that $P_n(x)$ is irreducible over $\mathbb{Z}[x]$.

On the team round, an LMT team of six students wishes to divide itself into two distinct groups of three, one group to work on part $1,$ and one group to work on part $2.$ In addition, a captain of each group is designated. In how many ways can this be done?

Let $ABC$ be a triangle with $BC=a$ $AC=b$ $AB=c$. For each line $\Delta$ we denote $d_{A}, d_{B}, d_{C}$ the distances from $A,B, C$ to $\Delta$ and we consider the expresion $E(\Delta)=ad_{A}^{2}+bd_{B}^{2}+cd_{C}^{2}$. Prove that if $E(\Delta)$ is minimum, then $\Delta$ passes through the incenter of $\Delta ABC$.

Let $d_n$ be the last nonzero digit of the decimal representation of $n!$. Prove that $d_n$ is aperiodic; that is, there do not exist $T$ and $n_0$ such that for all $n \geq n_0, d_{n+T} = d_n.$