Justify the following construction of the bisector of an angle with an inaccessible vertex: [img]https://cdn.artofproblemsolving.com/attachments/9/d/be4f7799d58a28cab3b4c515633b0e021c1502.png[/img] $M \in a$ and $N \in b$ are taken, the $4$ bisectors of the $4$ internal angles formed by $MN$ are traced with $a$ and $ b$. Said bisectors intersect at $P$ and $Q$, then $PQ$ is the bisector sought.

The entries of a $7 \times 5$ table are filled with numbers so that in each $2 \times 3$ rectangle (vertical or horizontal) the sum of numbers is $0$. For $100$ dollars Peter may choose any single entry and learn the number in it. What is the least amount of dollars he should spend in order to learn the total sum of numbers in the table for sure?

Rectangle $EFGH$ with side lengths $8$, $9$ lies inside rectangle $ABCD$ with side lengths $13$, $14$, with their corresponding sides parallel. Let $\ell_A, \ell_B, \ell_C, \ell_D$ be the lines connecting $A,B,C,D$, respectively, with the vertex of $EFGH$ closest to them. Let $P = \ell_A \cap \ell_B$, $Q = \ell_B \cap \ell_C$, $R = \ell_C \cap \ell_D$, and $S = \ell_D \cap \ell_A$. Suppose that the greatest possible area of quadrilateral $PQRS$ is $\frac{m}{n}$, for relatively prime positive integers $m$ and $n$. Find $100m+n$. [i]Proposed by Yannick Yao[/i]

Let be given $n$ points, no three of which are on a line. All the segments with endpoints in these points are colored so that two segments with a common endpoint are of different colors. Determine the least number of colors for which this is possible

Let \( ABCDEF \) be a cyclic hexagon such that \( AD \parallel EF \). Points \( X \) and \( Y \) are marked on diagonals \( AE \) and \( DF \), respectively, such that \( CX = EX \) and \( BY = FY \). Let \( O \) be the intersection point of \( AE \) and \( FD \), \( P \) the intersection point of \( CX \) and \( BY \), and \( Q \) the intersection point of \( BF \) and \( CE \). Prove that points \( O, P, \) and \( Q \) are collinear. [i]Proposed by Matthew Kurskyi[/i]

On a $999\times 999$ board a [i]limp rook[/i] can move in the following way: From any square it can move to any of its adjacent squares, i.e. a square having a common side with it, and every move must be a turn, i.e. the directions of any two consecutive moves must be perpendicular. A [i]non-intersecting route[/i] of the limp rook consists of a sequence of pairwise different squares that the limp rook can visit in that order by an admissible sequence of moves. Such a non-intersecting route is called [i]cyclic[/i], if the limp rook can, after reaching the last square of the route, move directly to the first square of the route and start over. How many squares does the longest possible cyclic, non-intersecting route of a limp rook visit? [i]Proposed by Nikolay Beluhov, Bulgaria[/i]

Let $ABC$ be a triangle with $AB = AC$. The incircle touches $BC$, $AC$ and $AB$ at $D$, $E$ and $F$ respectively. Let $P$ be a point on the arc $\overarc{EF}$ that does not contain $D$. Let $Q$ be the second point of intersection of $BP$ and the incircle of $ABC$. The lines $EP$ and $EQ$ meet the line $BC$ at $M$ and $N$, respectively. Prove that the four points $P, F, B, M$ lie on a circle and $\frac{EM}{EN} = \frac{BF}{BP}$.

Find all function $f:\mathbb{R}\rightarrow \mathbb{R}$ such that for any three real number $a,b,c$ , if $ a + f(b) + f(f(c)) = 0$ : $$ f(a)^3 + bf(b)^2 + c^2f(c) = 3abc $$. [i]Proposed by Amirhossein Zolfaghari [/i]

A tripod has three legs each of length 5 feet. When the tripod is set up, the angle between any pair of legs is equal to the angle between any other pair, and the top of the tripod is 4 feet from the ground. In setting up the tripod, the lower 1 foot of one leg breaks off. Let $h$ be the height in feet of the top of the tripod from the ground when the broken tripod is set up. Then $h$ can be written in the form $\frac m{\sqrt{n}},$ where $m$ and $n$ are positive integers and $n$ is not divisible by the square of any prime. Find $\lfloor m+\sqrt{n}\rfloor.$ (The notation $\lfloor x\rfloor$ denotes the greatest integer that is less than or equal to $x$.)

Let $ABC (AC < BC)$ be an acute triangle with circumcircle $k$ and midpoint $P$ of $AB$. The altitudes $AM$ and $BN$ ($M\in BC$, $N\in AC$) intersect at $H$. The point $E$ on $k$ is such that the segments $CE$ and $AB$ are perpendicular. The line $EP$ intersects $k$ again at point $K$ and the point $Q$ on $k$ is such that $KQ$ and $AB$ are parallel. The circumcircle of $AHB$ intersects the segment $CP$ at an interior point $R$. Prove that the points $C$, $M$, $R$, $H$, $N$ and $Q$ are concyclic.

$1-2-3+4+5-6-7+8+9-10-11+\cdots + 1992+1993-1994-1995+1996=$ $\text{(A)}\ -998 \qquad \text{(B)}\ -1 \qquad \text{(C)}\ 0 \qquad \text{(D)}\ 1 \qquad \text{(E)}\ 998$

Call a point in the Cartesian plane with integer coordinates a $lattice$ $point$. Given a finite set $\mathcal{S}$ of lattice points we repeatedly perform the following operation: given two distinct lattice points $A, B$ in $\mathcal{S}$ and two distinct lattice points $C, D$ not in $\mathcal{S}$ such that $ACBD$ is a parallelogram with $AB > CD$, we replace $A, B$ by $C, D$. Show that only finitely many such operations can be performed. [I]Proposed by Joe Benton, United Kingdom.[/i]

Let $S$ be a finite set of positive integers which has the following property:if $x$ is a member of $S$,then so are all positive divisors of $x$. A non-empty subset $T$ of $S$ is [i]good[/i] if whenever $x,y\in T$ and $x<y$, the ratio $y/x$ is a power of a prime number. A non-empty subset $T$ of $S$ is [i]bad[/i] if whenever $x,y\in T$ and $x<y$, the ratio $y/x$ is not a power of a prime number. A set of an element is considered both [i]good[/i] and [i]bad[/i]. Let $k$ be the largest possible size of a [i]good[/i] subset of $S$. Prove that $k$ is also the smallest number of pairwise-disjoint [i]bad[/i] subsets whose union is $S$.

Let $\triangle ABC$ be inscribed in circle $O$ with $\angle ABC = 36^\circ$. $D$ and $E$ are on the circle such that $\overline{AD}$ and $\overline{CE}$ are diameters of circle $O$. List all possible positive values of $\angle DBE$ in degrees in order from least to greatest. [i]Proposed by Ambrose Yang[/i]

Each of the equations $ 3x^2\minus{}2\equal{}25$, $ (2x\minus{}1)^2\equal{}(x\minus{}1)^2$, $ \sqrt{x^2\minus{}7}\equal{}\sqrt{x\minus{}1}$ has: $ \textbf{(A)}\ \text{two integral roots} \qquad \textbf{(B)}\ \text{no root greater than 3} \qquad \textbf{(C)}\ \text{no root zero} \\ \textbf{(D)}\ \text{only one root} \qquad \textbf{(E)}\ \text{one negative root and one positive root}$

For arbitrary pairwise distinct positive real numbers $a, b, c$, prove the inequality $$\frac{(a^2- b^2)^3 + (b^2-c^2)^3+(c^2-a^2)^3}{(a- b)^3 + (b-c)^3+(c-a)^3}> 8abc$$

A dog owns $4$ different color shoes and $4$ identical green socks. He can fit every shoe and sock on each of his four distinguishable paws. In how many different orders can he put on the shoes and socks, provided that on each paw he must put on the sock before the shoe? [i]2015 CCA Math Bonanza Individual Round #11[/i]

Let $I$ be the incenter of triangle $ABC$. Let $BI$ and $AC$ intersect at $E$, and $CI$ and $AB$ intersect at $F$. Suppose that $R$ is another intersection of $\odot (ABC)$ and $\odot (AEF)$. Let $M$ be the midpoint of $BC$, and $P, Q$ are the intersections of $AI, MI$ and $EF$, respectively. Show that $A, P, Q, R$ are concyclic. (ltf0501).

Let $ f(x)\equal{}\sqrt{ax^2\plus{}bx}$. For how many real values of $ a$ is there at least one positive value of $ b$ for which the domain of $ f$ and the range of $ f$ are the same set? $ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 3 \qquad \textbf{(E)}\ \text{infinitely many}$

Prove that all real numbers $x \ne -1$, $y \ne -1$ with $xy = 1$ satisfy the following inequality: $$\left(\frac{2+x}{1+x}\right)^2 + \left(\frac{2+y}{1+y}\right)^2 \ge \frac92$$ (Karl Czakler)

The system of inequalities $$a-b^2\ge\frac14$$$$b-c^2\ge\frac14$$$$c-d^2\ge\frac14$$$$d-a^2\ge\frac14$$where $a,b,c,d$ are real numbers has $\textbf{(A)}~\text{no solutions}$ $\textbf{(B)}~\text{exactly one solution}$ $\textbf{(C)}~\text{exactly two solutions}$ $\textbf{(D)}~\text{infinitely many solutions}$

John and Mary play the following game. First they choose integers $n > m > 0$ and put $n$ sweets on an empty table. Then they start to make moves alternately. A move consists of choosing a nonnegative integer $k\le m$ and taking $k$ sweets away from the table (if $k = 0$ , nothing happens in fact). In doing so no value for $k$ can be chosen more than once (by none of the players) or can be greater than the number of sweets at the table at the moment of choice. The game is over when one of the players can make no more moves. John and Mary decided that at the beginning Mary chooses the numbers $m$ and $n$ and then John determines whether the performer of the last move wins or looses. Can Mary choose $m$ and $n$ in such way that independently of John’s decision she will be able to win?

In triangle $ABC$, $AC=24$ inches, $BC=10$ inches, $AB=26$ inches. The radius of the inscribed circle is: $\textbf{(A)}\ 26\text{ in} \qquad \textbf{(B)}\ 4\text{ in} \qquad \textbf{(C)}\ 13\text{ in} \qquad \textbf{(D)}\ 8\text{ in} \qquad \textbf{(E)}\ \text{None of these}$

On a table there are $1999$ counters, red on one side and black on the other side, arranged arbitrarily. Two people alternately make moves, where each move is of one of the following two types: (1) Remove several counters which all have the same color up; (2) Reverse several counters which all have the same color up. The player who takes the last counter wins. Decide which of the two players (the one playing first or the other one) has a wining strategy.

Alex, Bertha, Cameron, Dylan, and Ellen each have a different toy. Each kid puts each of their own toys into a large bag. The toys are then randomly distributed such that each kid receives a toy. How many ways are there for exactly one kid to get the same toy that they put in? [i]2018 CCA Math Bonanza Lightning Round #2.4[/i]