The hostess takes a piece of meat from the fridge; kittens gather around her. Each minute, the hostess cuts a part from the piece and feeds it to one of the kittens (on her choice). Each time, the cut part is in the same proportion to the current piece. At some moment, the hostess puts the rest of the meat into the fridge. Can the hostess give the same amount of meat in total to each kitten if a) the number of kittens equals two; (3 marks) b) the number of kittens equals three? (7 marks)

Patrick started walking at a constant rate along a straight road from school to the park. One hour after Patrick left, Tanya started running along the same road from school to the park. One hour after Tanya left, Jose started bicycling along the same road from school to the park. Tanya ran at a constant rate of $2$ miles per hour faster than Patrick walked, Jose bicycled at a constant rate of $7$ miles per hour faster than Tanya ran, and all three arrived at the park at the same time. The distance from the school to the park is $\frac{m}{n}$ miles, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

Suppose that $n$ polygons of area $s = (n - 1)^2$ are placed on a polygon of area $S = \frac{n(n - 1)^2}{2}$. Prove that there exist two of the $n$ smaller polygons whose intersection has the area at least $1$.

Prove that all positive rational numbers can be written as a fraction, which numerator and denominator are products of factorials of not necessarily different prime numbers. For example $\frac{10}{9}=\frac{2!5!}{3!3!3!}$.

Points $G$ and $N$ are chosen on the interiors of sides $ED$ and $DO$ of unit square $DOME$, so that pentagon $GNOME$ has only two distinct side lengths. The sum of all possible areas of quadrilateral $NOME$ can be expressed as $\frac{a-b\sqrt{c}}{d}$, where $a,b,c,d$ are positive integers such that $\gcd(a,b,d) = 1$ and $c$ is square-free (i.e. no perfect square greater than $1$ divides $c$). Compute $1000a+100b+10c+d$.

Find largest possible constant $M$ such that, for any sequence $a_n$, $n=0,1,2,...$ of real numbers, that satisfies the conditions : i) $a_0=1$, $a_1=3$ ii) $a_0+a_1+...+a_{n-1} \ge 3 a_n - a_{n+1}$ for any integer $n\ge 1$ to be true that $$\frac{a_{n+1}}{a_n} >M$$ for any integer $n\ge 0$.

Given a positive integer $n$, find all $n$-tuples of real number $(x_1,x_2,\ldots,x_n)$ such that \[ f(x_1,x_2,\cdots,x_n)=\sum_{k_1=0}^{2} \sum_{k_2=0}^{2} \cdots \sum_{k_n=0}^{2} \big| k_1x_1+k_2x_2+\cdots+k_nx_n-1 \big| \] attains its minimum.

We have $100$ cards with two sides, the [i]even[/i] and the [i]odd[/i]. In each side there are written two succesive integers, in the [i]odd[/i] side and odd integer and at the back in the [i]even[/i] side the even number that follows the odd number of the [i]odd[/i] side, such that all intgers from $1$ to $200$ are used. Student $A$ randomly choses $21$ cards and sums all the numbers of boths sides and announces as their sum the number $913$. Student $B$ randomly choses from the remaining cards $20$ cards and sums all the numbers of boths sides and announces as their sum the number $2400$. a) Explain why student $A$ has done an error in the addition. b) If the correct result for student $A$ is $903$, explain why also student $B$ has done an error in the addition.

Buzz Bunny is hopping up and down a set of stairs, one step at a time. In how many ways can Buzz start on the ground, make a sequence of $6$ hops, and end up back on the ground? (For example, one sequence of hops is up-up-down-down-up-down.) $\textbf{(A) }4\qquad\textbf{(B) }5\qquad\textbf{(C) }6\qquad\textbf{(D) }8\qquad\textbf{(E) }12$

Niki and Kyle play a triangle game. Niki first draws $\triangle ABC$ with area $1$, and Kyle picks a point $X$ inside $\triangle ABC$. Niki then draws segments $\overline{DG}$, $\overline{EH}$, and $\overline{FI}$, all through $X$, such that $D$ and $E$ are on $\overline{BC}$, $F$ and $G$ are on $\overline{AC}$, and $H$ and $I$ are on $\overline{AB}$. The ten points must all be distinct. Finally, let $S$ be the sum of the areas of triangles $DEX$, $FGX$, and $HIX$. Kyle earns $S$ points, and Niki earns $1-S$ points. If both players play optimally to maximize the amount of points they get, who will win and by how much?

Find all integer solutions $(m,n)$ to $m^{n}=n^{m}.$

Let $ABC$ be a triangle such that $AC=2AB$. Let $D$ be the point of intersection of the angle bisector of the angle $CAB$ with $BC$. Let $F$ be the point of intersection of the line parallel to $AB$ passing through $C$ with the perpendicular line to $AD$ passing through $A$. Prove that $FD$ passes through the midpoint of $AC$.

Given a $\triangle ABC$ with the edges $a,b$ and $c$ and the area $S$: (a) Prove that there exists $\triangle A_1B_1C_1$ with the sides $\sqrt a,\sqrt b$ and $\sqrt c$. (b) If $S_1$ is the area of $\triangle A_1B_1C_1$, prove that $S_1^2\ge\frac{S\sqrt3}4$.

Given a parallelogram $ABCD$, join $A$ to the midpoints $E$ and $F$ of the opposite sides $BC$ and $CD$. $AE$ and $AF$ intersect the diagonal $BD$ in $M$ and $N$. Prove that $M$ and $N$ divide $BD$ into three equal parts.

Find the self-intersection index of the surface $x^4+y^4=1$ in the projective plane $\text{CP}^2$.

[u]Round 1[/u] [b]p1.[/b] Five children, Aisha, Baesha, Cosha, Dasha, and Erisha, competed in running, jumping, and throwing. In each event, first place was won by someone from Renton, second place by someone from Seattle, and third place by someone from Tacoma. Aisha was last in running, Cosha was last in jumping, and Erisha was last in throwing. Could Baesha and Dasha be from the same city? [b]p2.[/b] Fifty-five Brits and Italians met in a coffee shop, and each of them ordered either coffee or tea. Brits tell the truth when they drink tea and lie when they drink coffee; Italians do it the other way around. A reporter ran a quick survey: Forty-four people answered “yes” to the question, “Are you drinking coffee?” Thirty-three people answered “yes” to the question, “Are you Italian?” Twenty-two people agreed with the statement, “It is raining outside.” How many Brits in the coffee shop are drinking tea? [b]p3.[/b] Doctor Strange is lost in a strange house with a large number of identical rooms, connected to each other in a loop. Each room has a light and a switch that could be turned on and off. The lights might initially be on in some rooms and off in others. How can Dr. Strange determine the number of rooms in the house if he is only allowed to switch lights on and off? [b]p4.[/b] Fifty street artists are scheduled to give solo shows with three consecutive acts: juggling, drumming, and gymnastics, in that order. Each artist will spend equal time on each of the three activities, but the lengths may be different for different artists. At least one artist will be drumming at every moment from dawn to dusk. A new law was just passed that says two artists may not drum at the same time. Show that it is possible to cancel some of the artists' complete shows, without rescheduling the rest, so that at least one show is going on at every moment from dawn to dusk, and the schedule complies with the new law. [b]p5.[/b] Alice and Bob split the numbers from $1$ to $12$ into two piles with six numbers in each pile. Alice lists the numbers in the first pile in increasing order as $a_1 < a_2 < a_3 < a_4 < a_5 < a_6$ and Bob lists the numbers in the second pile in decreasing order $b_1 > b_1 > b_3 > b_4 > b_5 > b_6$. Show that no matter how they split the numbers, $$|a_1 -b_1| + |a_2 -b_2| + |a_3 -b_3| + |a_4 -b_4| + |a_5 -b_5| + |a_6 -b_6| = 36.$$ [u]Round 2[/u] [b]p6.[/b] The Martian alphabet has ? letters. Marvin writes down a word and notices that within every sub-word (a contiguous stretch of letters) at least one letter occurs an odd number of times. What is the length of the longest possible word he could have written? [b]p7.[/b] For a long space journey, two astronauts with compatible personalities are to be selected from $24$ candidates. To find a good fit, each candidate was asked $24$ questions that required a simple yes or no answer. Two astronauts are compatible if exactly $12$ of their answers matched (that is, both answered yes or both answered no). Miraculously, every pair of these $24$ astronauts was compatible! Show that there were exactly $12$ astronauts whose answer to the question “Can you repair a flux capacitor?” was exactly the same as their answer to the question “Are you afraid of heights?” (that is, yes to both or no to both). PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $D,E,F$ be points on the sides $BC,CA,AB$ of a triangle $ABC$, respectively. Suppose $AD, BE,CF$ are concurrent at $P$. If $PF/PC =2/3, PE/PB = 2/7$ and $PD/PA = m/n$, where $m, n$ are positive integers with $gcd(m, n) = 1$, find $m + n$.

David, Hikmet, Jack, Marta, Rand, and Todd were in a $12$-person race with $6$ other people. Rand finished $6$ places ahead of Hikmet. Marta finished $1$ place behind Jack. David finished $2$ places behind Hikmet. Jack finished $2$ places behind Todd. Todd finished $1$ place behind Rand. Marta finished in $6$th place. Who finished in $8$th place? $\textbf{(A) } \text{David} \qquad\textbf{(B) } \text{Hikmet} \qquad\textbf{(C) } \text{Jack} \qquad\textbf{(D) } \text{Rand} \qquad\textbf{(E) } \text{Todd} $

Each corner of a rectangular prism is cut off. Two (of the eight) cuts are shown. How many edges does the new figure have? [asy] draw((0,0)--(3,0)--(3,3)--(0,3)--cycle); draw((3,0)--(5,2)--(5,5)--(2,5)--(0,3)); draw((3,3)--(5,5)); draw((2,0)--(3,1.8)--(4,1)--cycle,linewidth(1)); draw((2,3)--(4,4)--(3,2)--cycle,linewidth(1));[/asy] $ \text{(A)}\ 24\qquad\text{(B)}\ 30\qquad\text{(C)}\ 36\qquad\text{(D)}\ 42\qquad\text{(E)}\ 48 $ [i]Assume that the planes cutting the prism do not intersect anywhere in or on the prism.[/i]

In the figure, the chord $[CD]$ is perpendicular to the diameter $[AB]$ and intersects it at $H$. Length of $AB$ is a two-digit natural number. Changing the order of these two digits gives length of $CD$. Knowing that distance from $H$ to the center $O$ is a positive rational number, calculate $AB$. [img]https://cdn.artofproblemsolving.com/attachments/5/f/eb9c61579a38118b4f753bbc19a9a50e0732dc.png[/img]

Let $\left[\sqrt{(n+1)^2 + n^2} \right], n = 1,2, \ldots,$ where $[x]$ denotes the integer part of $x.$ Prove that [b]i.)[/b] there are infinitely many positive integers $m$ such that $a_{m+1} - a_m > 1;$ [b]ii.)[/b] there are infinitely many positive integers $m$ such that $a_{m+1} - a_m = 1.$

Given is a triangle $ABC$ with its circumscribed circle and $| AC | <| AB |$. On the short arc $AC$, there is a variable point $D\ne A$. Let $E$ be the reflection of $A$ wrt the inner bisector of $\angle BDC$. Prove that the line $DE$ passes through a fixed point, regardless of point $D$.

Prove that if the real numbers $ a $, $ b $, $ c $ satisfy the inequalities $$a + b + c> 0,$$ $$ ab + bc + ca > 0$$ $$ abc > 0$$ then $a > 0, b > 0, c > 0$.

Natural numbers are written in the cells of of a $2014\times2014$ regular square grid such that every number is the average of the numbers in the adjacent cells. Describe and prove how the number distribution in the grid can be.

Andrew and Barry play the following game: there are two heaps with $a$ and $b$ pebbles, respectively. In the first round Barry chooses a positive integer $k,$ and Andrew takes away $k$ pebbles from one of the two heaps (if $k$ is bigger than the number of pebbles in the heap, he takes away the complete heap). In the second round, the roles are reversed: Andrew chooses a positive integer and Barry takes away the pebbles from one of the two heaps. This goes on, in each round the two players are reversing the roles. The player that takes the last pebble loses the game. Which player has a winning strategy? [i]Submitted by András Imolay, Budapest[/i]