Circles $\omega_1 , \omega_2$ intersect at points $X,Y$ and they are internally tangent to circle $\Omega$ at points $A,B$,respectively.$AB$ intersect with $\omega_1 , \omega_2$ at points $A_1,B_1$ ,respectively.Another circle is internally tangent to $\omega_1 , \omega_2$ and $A_1B_1$ at $Z$.Prove that $\angle AXZ =\angle BXZ$.(C.Ilyasov)

Let $\mathcal U = \left\{ \left( x,y \right) | x,y \in \mathbb Z, \ 0 \leq x,y < 4 \right\}$. (a) Prove that we can choose $6$ points from $\mathcal U$ such that there are no isosceles triangles with the vertices among the chosen points. (b) Prove that no matter how we choose $7$ points from $\mathcal U$, there are always three which form an isosceles triangle. [i]Radu Gologan, Dinu Serbanescu[/i]

Let $a_{i,j}\enspace(\forall\enspace 1\leq i\leq n, 1\leq j\leq n)$ be $n^2$ real numbers such that $a_{i,j}+a_{j,i}=0\enspace\forall i, j$ (in particular, $a_{i,i}=0\enspace\forall i$). Prove that $$ {1\over n}\sum_{i=1}^{n}\left(\sum_{j=1}^{n} a_{i,j}\right)^2\leq{1\over2}\sum_{i=1}^{n}\sum_{j=1}^{n} a_{i,j}^2. $$

In the country Máxico are two islands, the island "Mayor" and island "Menor". The island "Mayor" has $k>3$ states, with exactly $n>3$ cities each one. The island "Menor" has only one state with $31$ cities. "Aeropapantla" and "Aerocenzontle" are the airlines that offer flights in Máxico. "Aeropapantla" offer direct flights from every city in Máxico to any other city in Máxico. "Aerocenzontle" only offers direct flights from every city of the island "Mayor" to any other city of the island "Mayor". If the percentage of flights that connect two cities in the same state it´s the same for the flights of each airline, What is the least number of cities that can be in the island "Mayor"?

Let $O$ be the circumcenter of an acute-angled triangle $ABC$. Let $AH$ be the altitude of this triangle, $M,N,P,Q$ be the midpoints of the segments $AB, AC, BH, CH$, respectively. Let $\omega_1$ and $\omega_2$ be the circumferences of the triangles $AMN$ and $POQ$. Prove that one of the intersection points of $\omega_1$ and $\omega_2$ belongs to the altitude $AH$. (A. Voidelevich)

Let $ABCD$ be a convex quadrilateral with $BA = BC$ and $DA = DC$. Let $E$ and $F$ be the midpoints of $BC$ and $CD$ respectively, and let$ BF$ and $DE$ intersect at $G$. If the area of $CEGF$ is $50$, what is the area of $ABGD$?

Tom is searching for the $6$ books he needs in a random pile of $30$ books. What is the expected number of books must he examine before finding all $6$ books he needs?

Find all complex polynomial $ P(x)$ such that for any three integers $ a,b,c$ satisfying $ a \plus{} b \plus{} c\not \equal{} 0, \frac{P(a) \plus{} P(b) \plus{} P(c)}{a \plus{} b \plus{} c}$ is an integer.

Everyday at school, Jo climbs a flight of $6$ stairs. Joe can take the stairs $1,2$, or $3$ at a time. For example, Jo could climb $3$, then $1$, then $2$. In how many ways can Jo climb the stairs? $ \textbf{(A)}\ 13 \qquad\textbf{(B)}\ 18\qquad\textbf{(C)}\ 20\qquad\textbf{(D)}\ 22\qquad\textbf{(E)}\ 24 $

Let $\Delta ABC$ be an acute triangle with $a>b$, center $O$ of its circumscribed circle and middle point $M$ of $AC$. Let $K$ be the reflection of $O$ in $M$. Point $E\in BC$ is such that $EO\perp AB$. Point $F\in MK$ is such that $FK=OE$ and $K$ lies between $F$ and $M$. The altitude through $C$ and the angle bisector of $\angle CAB$ intersect in $D$. Let $BD$ intersect the circumscribed circle of $\Delta ABC$ for a second time in $P$. Prove that $AP\perp CF$.

Let $6$ pairwise different digits are given and all of them are different from $0$. Prove that there exist $2$ six-digit integers, such that their difference is equal to $9$ and each of them contains all given $6$ digits.

The Tasty Candy Company always puts the same number of pieces of candy into each one-pound bag of candy they sell. Mike bought 4 one-pound bags and gave each person in his class 15 pieces of candy. Mike had 23 pieces of candy left over. Betsy bought 5 one-pound bags and gave 23 pieces of candy to each teacher in her school. Betsy had 15 pieces of candy left over. Find the least number of pieces of candy the Tasty Candy Company could have placed in each one-pound bag.

We write $\{a,b,c\}$ for the set of three different positive integers $a, b$, and $c$. By choosing some or all of the numbers a, b and c, we can form seven nonempty subsets of $\{a,b,c\}$. We can then calculate the sum of the elements of each subset. For example, for the set $\{4,7,42\}$ we will find sums of $4, 7, 42,11, 46, 49$, and $53$ for its seven subsets. Since $7, 11$, and $53$ are prime, the set $\{4,7,42\}$ has exactly three subsets whose sums are prime. (Recall that prime numbers are numbers with exactly two different factors, $1$ and themselves. In particular, the number $1$ is not prime.) What is the largest possible number of subsets with prime sums that a set of three different positive integers can have? Give an example of a set $\{a,b,c\}$ that has that number of subsets with prime sums, and explain why no other three-element set could have more.

Moor made a lopsided ice cream cone. It turned out to be an oblique circular cone with the vertex directly above the perimeter of the base (see diagram below). The height and base radius are both of length $1$. Compute the radius of the largest spherical scoop of ice cream that it can hold such that at least $50\%$ of the scoop’s volume lies inside the cone. [center]<see attached>[/center]

Find all possible triples of positive integers, $a, b, c$ so that $\frac{a+1}{b}$, $\frac{b+1}{c}$ and $\frac{c+1}{a}$ are also integers.

Let $a,b,c$ be positive real numbers such that $a^2+b^2+c^2=3. $ Prove that $$\frac{a^4+3ab^3}{a^3+2b^3}+\frac{b^4+3bc^3}{b^3+2c^3}+\frac{c^4+3ca^3}{c^3+2a^3}\leq4.$$

An isosceles trapezoid $PQRS$, with $\overline{PQ} = \overline{QR}= \overline{RS}$ and $\angle PQR = 120^o$, is inscribed in the graph of $y = x^2$ such that $QR$ is parallel to the $x$-axis and $R$ is in the first quadrant. The $x$-coordinate of point $R$ can be written as $\frac{\sqrt{A}}{B}$ for positive integers $A$ and $B$ such that $A$ is square-free. Find $1000A +B$.

Let $\mathbb N$ be the set of positive integers. Let $f: \mathbb N \to \mathbb N$ be a function satisfying the following two conditions: (a) $f(m)$ and $f(n)$ are relatively prime whenever $m$ and $n$ are relatively prime. (b) $n \le f(n) \le n+2012$ for all $n$. Prove that for any natural number $n$ and any prime $p$, if $p$ divides $f(n)$ then $p$ divides $n$.

A positive integer $n$ is [i]Danish[/i] if a regular hexagon can be partitioned into $n$ congruent polygons. Prove that there are infinitely many positive integers $n$ such that both $n$ and $2^n+n$ are Danish.

In a test, $3n$ students participate, who are located in three rows of $n$ students in each. The students leave the test room one by one. If $N_1(t), N_2(t), N_3(t)$ denote the numbers of students in the first, second, and third row respectively at time $t$, find the probability that for each t during the test, \[|N_i(t) - N_j(t)| < 2, i \neq j, i, j = 1, 2, \dots .\]

Let $X$ be the set of natural numbers whose all digits in the decimal representation are different. For $n \in N$, denote by $A_n$ the set of numbers whose digits are a permutation of the digits of $n$, and $d_n$ be the greatest common divisor of the numbers in $A_n$. (For example, $A_{1120} =\{112,121,...,2101,2110\}$, so $d_{1120} = 1$.) Find the maximum possible value of $d_n$.

The following diagram shows equilateral triangle $\vartriangle ABC$ and three other triangles congruent to it. The other three triangles are obtained by sliding copies of $\vartriangle ABC$ a distance $\frac18 AB$ along a side of $\vartriangle ABC$ in the directions from $A$ to $B$, from $B$ to $C$, and from $C$ to $A$. The shaded region inside all four of the triangles has area $300$. Find the area of $\vartriangle ABC$. [img]https://cdn.artofproblemsolving.com/attachments/3/a/8d724563c7411547d3161076015b247e882122.png[/img]

In a round-robin tournament with $6$ teams, each team plays one game against each other team, and each game results in one team winning and one team losing. At the end of the tournament, the teams are ranked by the number of games won. What is the maximum number of teams that could be tied for the most wins at the end of the tournament? $ \textbf{(A)}\ 2\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ 5\qquad\textbf{(E)}\ 6 $

A triangle $\vartriangle A_0A_1A_2$ in the plane has sidelengths $A_0A_1 = 7$,$A_1A_2 = 8$,$A_2A_0 = 9$. For $i \ge 0$, given $\vartriangle A_iA_{i+1}A_{i+2}$, let $A_{i+3}$ be the midpoint of $A_iA_{i+1}$ and let Gi be the centroid of $\vartriangle A_iA_{i+1}A_{i+2}$. Let point $G$ be the limit of the sequence of points $\{G_i\}^{\infty}_{i=0}$. If the distance between $G$ and $G_0$ can be written as $\frac{a\sqrt{b}}{c}$ , where $a, b, c$ are positive integers such that $a$ and $c$ are relatively prime and $b$ is not divisible by the square of any prime, find $a^2 + b^2 + c^2$.

Let $ABC$ be an isosceles right triangle with $\angle BCA=90^{\circ}, BC=AC=10$. Let $P$ be a point on $AB$ that is a distance $x$ from $A$, $Q$ be a point on $AC$ such that $PQ$ is parallel to $BC$. Let $R$ and $S$ be points on $BC$ such that $QR$ is parallel to $AB$ and $PS$ is parallel to $AC$. The union of the quadrilaterals $PBRQ$ and $PSCQ$ determine a shaded area $f(x)$. Evaluate $f(2)$