Find the largest constant $K>0$ such that for any $0\le k\le K$ and non-negative reals $a,b,c$ satisfying $a^2+b^2+c^2+kabc=k+3$ we have $a+b+c\le 3$. (Dan Schwarz)

Given a regular $72$-gon. Lenya and Kostya play the game "Make an equilateral triangle." They take turns marking with a pencil on one still unmarked angle of the $72$-gon: Lenya uses red. Kostya uses blue. Lenya starts the game, and the one who marks first wins if its color is three vertices that are the vertices of some equilateral triangle, if all the vertices are marked and no such a triangle exists, the game ends in a draw. Prove that Kostya can play like this so as not to lose.

There are $20$ geese numbered $1-20$ standing in a line. The even numbered geese are standing at the front in the order $2,4,\dots,20,$ where $2$ is at the front of the line. Then the odd numbered geese are standing behind them in the order, $1,3,5,\dots ,19,$ where $19$ is at the end of the line. The geese want to rearrange themselves in order, so that they are ordered $1,2,\dots,20$ (1 is at the front), and they do this by successively swapping two adjacent geese. What is the minimum number of swaps required to achieve this formation? [i]Author: Ray Li[/i]

a) We are given $n$ circles $O_1, O_2, . . . , O_n$, passing through one point $O$. Let $A_1, . . . , A_n$ denote the second intersection points of $O_1$ with $O_2, O_2$ with $O_3$, etc., $O_n$ with $O_1$, respectively. We choose an arbitrary point $B_1$ on $O_1$ and draw a line segment through $A_1$ and $B_1$ to the second intersection with $O_2$ at $B_2$, then draw a line segment through $A_2$ and $B_2$ to the second intersection with $O_3$ at $B_3$, etc., until we get a point $B_n$ on $O_n$. We draw the line segment through $B_n$ and $A_n$ to the second intersection with $O_1$ at $B_{n+1}$. If $B_k$ and $A_k$ coincide for some $k$, we draw the tangent to $O_k$ through $A_k$ until this tangent intersects $O_{k+1}$ at $B_{k+1}$. Prove that $B_{n+1}$ coincides with $B_1$. b) for $n=3$ the same problem.

Let $P$ be an interior point of a triangle $ABC$ whose sidelengths are 26, 65, 78. The line through $P$ parallel to $BC$ meets $AB$ in $K$ and $AC$ in $L$. The line through $P$ parallel to $CA$ meets $BC$ in $M$ and $BA$ in $N$. The line through $P$ parallel to $AB$ meets $CA$ in $S$ and $CB$ in $T$. If $KL,MN,ST$ are of equal lengths, find this common length.

Let $P$ be a variable point on the circumference of a quarter-circle with radii $OA, OB$ and $\angle AOB = 90^\circ$. H is the projection of $P$ on $OA$. Find the locus of the incenter of the right-angled triangle $HPO.$

Let $ABC$ be a triangle with $\angle BAC \ne 45^o$ and $\angle ABC \ne 135^o$. Let $P$ be the point on the line $AB$ with $\angle CPB = 45^o$. Let $O_1$ and $O_2$ be the centers of the circumcircles of the triangles $ACP$ and $BCP$ respectively. Show that the area of the square $CO_1P O_2$ is equal to the area of the triangle $ABC$.

We consider a spherical bowl without any great circle. The distance between points $A$ and $B$ on such a bowl is defined as the length of the arc of the great circle of the sphere with ends at points $A$ and $B$, which is contained in the bowl. Prove that there is no isometry mapping this bowl to a subset of the plane. Attention. A spherical bowl is each of the two parts into which the surface of the sphere is divided by a plane intersecting the sphere.

A male volcano is in the shape of a hollow cone with the point side up, but with everything above a height of 6 meters removed. The resulting shape has a bottom radius of 10 meters and a top radius of 7 meters, with a height of 6 meters. He sat above his bay, watching all the couples play. His lava grew and grew until he was half full of lava. Then, he erupted, lowering the height of the lava to 2 meters. What fraction of the lava remained in the volcano? [i]Proposed by Matthew Weiss

Find, with proof, the maximal length of a non-constant arithmetic progression with all the terms squares of positive integers.

Let $ABCD$ be a square of side length $1$, and let $E$ and $F$ be points on $BC$ and $DC$ such that $\angle{EAF}=30^\circ$ and $CE=CF$. Determine the length of $BD$. [i]2015 CCA Math Bonanza Lightning Round #4.2[/i]

Prove that for infinitely many pairs $(a,b)$ of integers the equation \[x^{2012} = ax + b\] has among its solutions two distinct real numbers whose product is 1.

Given a rectangular grid with some cells containing one letter, we say a row or column is [i]edible [/i] if it has more than one cell with a letter and all such cells contain the same letter. Given such a grid, the hungry, hungry letter monster repeats the following procedure: he nds all edible rows and all edible columns and simultaneously eats all the letters in those rows and columns, removing those letters from the grid and leaving those cells empty. He continues this until no more edible rows and columns remain. Call a grid a [i]meal [/i] if the letter monster can eat all of its letters using this procedure. In the $7$ by $7$ grid to the right, ll each empty space with one letter so that the grid is a meal and there are a total of eight Us, nine Ss, ten As, eleven Ms, and eleven Ts. Some letters have been given to you. You do not need to prove that your answer is the only one possible; you merely need to find an answer that satisfies the constraints above. (Note: In any other USAMTS problem, you need to provide a full proof. Only in this problem is an answer without justication acceptable.) [img]https://cdn.artofproblemsolving.com/attachments/9/a/d1886720796e4befd9d3ce0cbd2868d1b649d1.png[/img]

Let $f(x)$ is such function, that $f(x)=1$ for integer $x$ and $f(x)=0$ for non integer $x$. Build such function using only variable $x$, integer numbers, and operations $+,-,*,/,[.]$(plus, minus, multiply,divide and integer part)

Let $n$ be an integer greater than $1$. A certain school has $1+2+\dots+n$ students and $n$ classrooms, with capacities for $1, 2, \dots, n$ people, respectively. The kids play a game in $k$ rounds as follows: in each round, when the bell rings, the students distribute themselves among the classrooms in such a way that they don't exceed the room capacities, and if two students shared a classroom in a previous round, they cannot do it anymore in the current round. For each $n$, determine the greatest possible value of $k$. [i]Proposed by Victor Domínguez[/i]

There are $4n$ segments of unit length inside a circle radius $n$. Show that given any line $L$ there is a chord of the circle parallel or perpendicular to $L$ which intersects at least two of the $4n$ segments.

Two different points $A,S$ are given in the plane. Furthermore, positive numbers $d,\omega$ are given, $\omega<180^\circ.$ Let $X$ be a point and $X'$ its image under the rotation by the angle $\omega$ (in counter-clockwise direction) with respect to the origin $S.$ Construct all points $X$ such that $XX'=d$ and $A$ is a point of the segment $XX'.$ Discuss conditions of solvability (in terms of $d,\omega,SA$).

$\frac{2 \sqrt 6}{\sqrt 2 + \sqrt 3 + \sqrt 5}$ equals A. $\sqrt 2 + \sqrt 3 - \sqrt 5$ B. $4 - \sqrt 2 - \sqrt 3$ C. $\sqrt 2 + \sqrt 3 + \sqrt 6 - 5$ D. $\frac{1}{2} (\sqrt 2 + \sqrt 5 - \sqrt 3)$ E. $\frac{1}{3} (\sqrt 3 + \sqrt 5 - \sqrt 2)$

Let the $ABCA'B'C'$ be a regular prism. The points $M$ and $N$ are the midpoints of the sides $BB'$, respectively $BC$, and the angle between the lines $AB'$ and $BC'$ is of $60^\circ$. Let $O$ and $P$ be the intersection of the lines $A'C$ and $AC'$, with respectively $B'C$ and $C'N$. a) Prove that $AC' \perp (OPM)$; b) Find the measure of the angle between the line $AP$ and the plane $(OPM)$. [i]Mircea Fianu[/i]

Set $S = \{1, 2, 3, ..., 2005\}$. If among any $n$ pairwise coprime numbers in $S$ there exists at least a prime number, find the minimum of $n$.

A positive integer $n$ is called $special$ if there exist integers $a > 1$ and $b > 1$ such that $n=a^b + b$. Is there a set of $2014$ consecutive positive integers that contains exactly $2012$ $special$ numbers?

Compute the number of nonempty subsets $S$ of $\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$ such that $\frac{\max \,\, S + \min \,\,S}{2}$ is an element of $S$.

$(1)$ Start with $a$ white balls and $b$ black balls. $(2)$ Draw one ball at random. $(3)$ If the ball is white, then stop. Otherwise, add two black balls and go to step $2$. Let $S$ be the number of draws before the process terminates. For the cases $a = b = 1$ and $a = b = 2$ only, find $a_n = P(S = n), b_n = P(S \le n), \lim_{n\to\infty} b_n$, and the expectation value of the number of balls drawn: $E(S) =\displaystyle\sum_{n\ge1} na_n.$

Let $ABC$ be a triangle. Let $X$ be the tangency point of the incircle with $BC$. Let $Y$ be the second intersection point of segment $AX$ with the incircle. Prove that \[AX+AY+BC>AB+AC\]

Side $\overline{AB} = 3$. $\vartriangle ABF$ is an equilateral triangle. Side $\overline{DE} =\overline{ AB} = \overline{AF} = \overline{GE}$, $\angle FED = 60^o$, $FG = 1$. Calculate the area of $ABCDE$. [img]https://cdn.artofproblemsolving.com/attachments/e/9/0ac1a88b4a83cdf3d562af0ce11b5ddbc5b8bc.png[/img]