There are $10$ birds on the ground. For any $5$ of them, there are at least $4$ birds on a circle. Determine the least possible number of birds on the circle with the most birds.

Let $f\left(x\right)=10^{10x}, g\left(x\right)=\log_{10}\left(\frac{x}{10}\right), h_1\left(x\right)=g\left(f\left(x\right)\right),$ and $h_n\left(x\right)=h_1\left(h_{n-1}\left(x\right)\right)$ for integers $n \ge 2$. What is the sum of the digits of $h_{2011}\left(1\right)$? $ \textbf{(A)}\ 16,081 \qquad \textbf{(B)}\ 16,089 \qquad \textbf{(C)}\ 18,089 \qquad \textbf{(D)}\ 18,098 \qquad \textbf{(E)}\ 18,099 $

In the plane, there are $n \geqslant 6$ pairwise disjoint disks $D_{1}, D_{2}, \ldots, D_{n}$ with radii $R_{1} \geqslant R_{2} \geqslant \ldots \geqslant R_{n}$. For every $i=1,2, \ldots, n$, a point $P_{i}$ is chosen in disk $D_{i}$. Let $O$ be an arbitrary point in the plane. Prove that \[O P_{1}+O P_{2}+\ldots+O P_{n} \geqslant R_{6}+R_{7}+\ldots+R_{n}.\] (A disk is assumed to contain its boundary.)

Given triangle $ABC$ . From the $P$ point three lines $(PA),(PB),(PC)$ are drawn. They cross the circumscribed circle at $A_1, B_1,C_1$ points respectively. It comes out that the $A_1B_1C_1$ triangle equals to the initial one. Prove that there are not more than eight such a points $P$ in a plane.

Find all prime numbers $p$ such that the number $$3^p+4^p+5^p+9^p-98$$ has at most $6$ positive divisors.

Let $m$ and $n$ be positive integers. Fuming Zeng gives James a rectangle, such that $m-1$ lines are drawn parallel to one pair of sides and $n-1$ lines are drawn parallel to the other pair of sides (with each line distinct and intersecting the interior of the rectangle), thus dividing the rectangle into an $m\times n$ grid of smaller rectangles. Fuming Zeng chooses $m+n-1$ of the $mn$ smaller rectangles and then tells James the area of each of the smaller rectangles. Of the $\dbinom{mn}{m+n-1}$ possible combinations of rectangles and their areas Fuming Zeng could have given, let $C_{m,n}$ be the number of combinations which would allow James to determine the area of the whole rectangle. Given that \[A=\sum_{m=1}^\infty \sum_{n=1}^\infty \frac{C_{m,n}\binom{m+n}{m}}{(m+n)^{m+n}},\] then find the greatest integer less than $1000A$. [i]Proposed by James Lin

Consider a $2018 \times 2019$ board with integers in each unit square. Two unit squares are said to be neighbours if they share a common edge. In each turn, you choose some unit squares. Then for each chosen unit square the average of all its neighbours is calculated. Finally, after these calculations are done, the number in each chosen unit square is replaced by the corresponding average. Is it always possible to make the numbers in all squares become the same after finitely many turns?

Let $AA', BB'$ and $CC'$ be the altitudes of an acute-angled triangle $ABC$. Points $C_a, C_b$ are symmetric to $C' $ wrt $AA'$ and $BB'$. Points $A_b, A_c, B_c, B_a$ are defined similarly. Prove that lines $A_bB_a, B_cC_b$ and $C_aA_c$ are parallel.

Three spheres $s_1$, $s_2$, $s_3$ intersect along one circle $\omega$. Let $A $be an arbitrary point lying on the circle $\omega$. Ray $AB$ intersects spheres $s_1$, $s_2$, $s_3$ at points $B_1$, $B_2$, $B_3$, respectively, ray $AC$ intersects spheres $s_1$, $s_2$, $s_3$ at points $C_1$, $C_2$, $C_3$, respectively ($B_i \ne A_i$, $C_i \ne A_i$, $i=1,2,3$). It is known that $B_2$ is the midpoint of the segment $B_1B_3$. Prove that $C_2$ is the midpoint of the segment $C_1C_3$.

Let $p_n$ be a bounded sequence of integers which satisfies the recursion $$p_n =\frac{p_{n-1} +p_{n-2} + p_{n-3}p _{n-4}}{p_{n-1} p_{n-2}+ p_{n-3} +p_{n-4}}.$$ Show that the sequence eventually becomes periodic.

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.