Let $ABC$, $AB<AC$ be an acute triangle inscribed in circle $\Gamma$ with center $O$. The altitude from $A$ to $BC$ intersects $\Gamma$ again at $A_1$. $OA_1$ intersects $BC$ at $A_2$ Similarly define $B_1$, $B_2$, $C_1$, and $C_2$. Then $B_2C_2=2\sqrt{2}$. If $B_2C_2$ intersects $AA_2$ at $X$ and $BC$ at $Y$, then $XB_2=2$ and $YB_2=k$. Find $k^2$. [i]2017 CCA Math Bonanza Individual Round #15[/i]

Given a triangle $ABC, O$ is the center of the circumcircle, $M$ is the midpoint of $BC, W$ is the second intersection of the bisector of the angle $C$ with this circle. A line parallel to $BC$ passing through $W$, intersects$ AB$ at the point $K$ so that $BK = BO$. Find the measure of angle $WMB$. (Anton Trygub)

Let $ABC$ be a triangle with given sides $a,b,c$. Determine the minimal possible length of the diagonal of an inscribed rectangle in this triangle. [i]Note: A rectangle is inscribed in the triangle if two of its consecutive vertices lie on one side of the triangle, while the other two vertices lie on the other two sides of the triangle. [/i]

[b]p1.[/b] Two perpendicular chords intersect in a circle. The lengths of the segments of one chord are $3$ and $4$. The lengths of the segments of the other chord are $6$ and $2$. Find the diameter of the circle. [b]p2.[/b] Determine the greatest integer that will divide $13,511$, $13,903$ and $14,589$ and leave the same remainder. [b]p3.[/b] Suppose $A, B$ and $C$ are the angles of the triangle. Show that $\cos^2 A + \cos^2 B + \cos^2 C + 2 \cos A \cos B \cos C = 1$ [b]p4.[/b] Given the linear fractional transformation $f_1(x) =\frac{2x - 1}{x + 1}$. Define $f_{n+1}(x) = f_1(f_n(x))$ for $n = 1, 2, 3,...$ . It can be shown that $f_{35} = f_5$. (a) Find a function $g$ such that $f_1(g(x)) = g(f_1(x)) = x$. (b) Find $f_{28}$. [b]p5.[/b] Suppose $a$ is a complex number such that $a^{10} + a^5 + 1 = 0$. Determine the value of $a^{2005} + \frac{1}{a^{2005}}$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

How many $2 \times 2 \times 2$ cubes must be added to a $8 \times 8 \times 8$ cube to form a $12 \times 12 \times 12$ cube? [i]Proposed by Evan Chen[/i]

Consider $\vartriangle A_0B_0C_0$ and points $C_1, A_1, B_1$ on its sides $A_0B_0, B_0C_0, C_0A_0$, points $C_2, A_2,B_2$ on the sides $A_1B_1, B_1C_1, C_1A_1$ of $\vartriangle A_1B_1C_1$, respectively, etc., so that $$\frac{A_0B_1}{B_1C_0}= \frac{B_0C_1}{C_1A_0}= \frac{C_0A_1}{A_1B_0}= k, \frac{A_1B_2}{B_2C_1}= \frac{B_1C_2}{C_2A_1}= \frac{C_1A_2}{A_2B_1}= \frac{1}{k^2}$$ and, in general, $$\frac{A_nB_{n+1}}{B_{n+1}C_n}= \frac{B_nC_{n+1}}{C_{n+1}A_n}= \frac{C_nA_{n+1}}{A_{n+1}B_n} =k^{2n}$$ for $n$ even , $\frac{1}{k^{2n}}$ for $n$ odd. Prove that $\vartriangle ABC$ formed by lines $A_0A_1, B_0B_1, C_0C_1$ is contained in $\vartriangle A_nB_nC_n$ for any $n$.

The point $D$ is the midpoint of the side $[AB]$ of the triangle $ABC$ . The points $E$ and $F$ belong to $[AC]$ and $[BC]$ respectively. Prove that the area of triangle $DEF$ area does not exceed the sum of the areas of triangles $ADE$ and $BDF$.

Let $ABC$ be an arbitrary scalene triangle. Define $\sum$ to be the set of all circles $y$ that have the following properties: [b](i)[/b] $y$ meets each side of $ABC$ in two (possibly coincident) points; [b](ii)[/b] if the points of intersection of $y$ with the sides of the triangle are labeled by $P, Q, R, S, T , U$, with the points occurring on the sides in orders $\mathcal B(B,P,Q,C), \mathcal B(C, R, S,A), \mathcal B(A, T,U,B)$, then the following relations of parallelism hold: $TS \parallel BC; PU\parallel CA; RQ\parallel AB$. (In the limiting cases, some of the conditions of parallelism will hold vacuously; e.g., if $A$ lies on the circle $y$, then $T$ , $S$ both coincide with $A$ and the relation $TS \parallel BC$ holds vacuously.) [i](a)[/i] Under what circumstances is $\sum$ nonempty? [i](b)[/i] Assuming that Σ is nonempty, show how to construct the locus of centers of the circles in the set $\sum$. [i](c)[/i] Given that the set $\sum$has just one element, deduce the size of the largest angle of $ABC.$ [i](d)[/i] Show how to construct the circles in $\sum$ that have, respectively, the largest and the smallest radii.

In an isosceles triangle $ABC$ with $BC=AC$ and $\angle B<60^\circ$, $I$ is the incenter and $O$ the circumcenter. The circle with center $E$ that passes through $A,O$ and $I$ intersects the circumcircle of $\triangle ABC$ again at point $D$. Prove that the lines $DE$ and $CO$ intersect on the circumcircle of $ABC$.

In triangle $ABC$ let $I_a$ be the $A$-excenter. Let $\omega$ be an arbitrary circle that passes through $A,I_a$ and intersects the extensions of sides $AB,AC$ (extended from $B,C$) at $X,Y$ respectively. Let $S,T$ be points on segments $I_aB,I_aC$ respectively such that $\angle AXI_a=\angle BTI_a$ and $\angle AYI_a=\angle CSI_a$.Lines $BT,CS$ intersect at $K$. Lines $KI_a,TS$ intersect at $Z$. Prove that $X,Y,Z$ are collinear. [i]Proposed by Hooman Fattahi[/i]

Let $ABC$ be an acute angle triangle. Suppose that $D,E,F$ are the feet of perpendicluar lines from $A,B,C$ to $BC,CA,AB$. Let $P,Q,R$ be the feet of perpendicular lines from $A,B,C$ to $EF,FD,DE$. Prove that \[ 2(PQ+QR+RP)\geq DE+EF+FD \]

A square has coordinates at $(0, 0)$, $(4, 0)$, $(0, 4)$, and $(4, 4)$. Rohith is interested in circles of radius $ r$ centered at the point $(1, 2)$. There is a range of radii $a < r < b$ where Rohith’s circle intersects the square at exactly $6$ points, where $a$ and $b$ are positive real numbers. Then $b - a$ can be written in the form $m +\sqrt{n}$, where $m$ and $n$ are integers. Compute $m + n$.

A convex quadrilateral $ABCD$ is constructed out of metal rods with negligible thickness. The side lengths are $AB=BC=CD=5$ and $DA=3$. The figure is then deformed, with the angles between consecutive rods allowed to change but the rods themselves staying the same length. The resulting figure is a convex polygon for which $\angle{ABC}$ is as large as possible. What is the area of this figure? $\text{(A) }6\qquad\text{(B) }8\qquad\text{(C) }9\qquad\text{(D) }10\qquad\text{(E) }12$

Let $ABC$ be an acute triangle. $PQRS$ is a rectangle with $P$ on $AB$, $Q$ and $R$ on $BC$, and $S$ on $AC$ such that $PQRS$ has the largest area among all rectangles $TUVW$ with $T$ on $AB$, $U$ and $V$ on $BC$, and $W$ on $AC$. If $D$ is the point on $BC$ such that $AD\perp BC$, then $PQ$ is the harmonic mean of $\frac{AD}{DB}$ and $\frac{AD}{DC}$. What is $BC$? Note: The harmonic mean of two numbers $a$ and $b$ is the reciprocal of the arithmetic mean of the reciprocals of $a$ and $b$. [i]2017 CCA Math Bonanza Lightning Round #4.4[/i]

Given a scalene acute triangle $ ABC$ with $ AC>BC$ let $ F$ be the foot of the altitude from $ C$. Let $ P$ be a point on $ AB$, different from $ A$ so that $ AF\equal{}PF$. Let $ H,O,M$ be the orthocenter, circumcenter and midpoint of $ [AC]$. Let $ X$ be the intersection point of $ BC$ and $ HP$. Let $ Y$ be the intersection point of $ OM$ and $ FX$ and let $ OF$ intersect $ AC$ at $ Z$. Prove that $ F,M,Y,Z$ are concyclic.

For every $n\geq 3$, determine all the configurations of $n$ distinct points $X_1,X_2,\ldots,X_n$ in the plane, with the property that for any pair of distinct points $X_i$, $X_j$ there exists a permutation $\sigma$ of the integers $\{1,\ldots,n\}$, such that $\textrm{d}(X_i,X_k) = \textrm{d}(X_j,X_{\sigma(k)})$ for all $1\leq k \leq n$. (We write $\textrm{d}(X,Y)$ to denote the distance between points $X$ and $Y$.) [i](United Kingdom) Luke Betts[/i]

The face diagonal of a cube has length $4$. Find the value of $n$ given that $n\sqrt2$ is the $\textit{volume}$ of the cube.

Let $A,B,C,D,E,F,G$ and $H$ be eight pairwise distinct points on the surface of a sphere. The quadruples $(A,B,C,D), (A,B,F,E),(B,C,G,F),(C,D,H,G)$ and $(D,A,E,H)$ of points are coplanar. Prove that the quadruple $(E,F,G,H)$ is coplanar aswell.

Suppose the side lengths of triangle $ABC$ are the roots of polynomial $x^3 - 27x^2 + 222x - 540$. What is the product of its inradius and circumradius?

The quadrilateral $ABCD$ is inscribed in a circle. The point $P$ lies in the interior of $ABCD$, and $\angle P AB = \angle P BC = \angle P CD = \angle P DA$. The lines $AD$ and $BC$ meet at $Q$, and the lines $AB$ and $CD$ meet at $R$. Prove that the lines $P Q$ and $P R$ form the same angle as the diagonals of $ABCD$.

Let $ABC$ be a triangle with $\angle BAC = 60^{\circ}$. Let $AP$ bisect $\angle BAC$ and let $BQ$ bisect $\angle ABC$, with $P$ on $BC$ and $Q$ on $AC$. If $AB + BP = AQ + QB$, what are the angles of the triangle?

The four congruent circles below touch one another and each has radius 1. [asy] unitsize(30); fill(box((-1,-1), (1, 1)), gray); filldraw(circle((1, 1), 1), white); filldraw(circle((1, -1), 1), white); filldraw(circle((-1, 1), 1), white); filldraw(circle((-1, -1), 1), white); [/asy] What is the area of the shaded region?

Rectangle $ABCD$ has perimeter $178$ and area $1848$. What is the length of the diagonal of the rectangle? [i]2016 CCA Math Bonanza Individual Round #2[/i]

Let $ABCD$ be a cyclic quadrilateral. Let $M, N$ be the midpoints of the diagonals $AC$ and $BD$ and let $P$ be the midpoint of $MN$. Let $A',B',C',D'$ be the intersections of the rays $AP$, $BP$, $CP$ and $DP$ respectively with the circumcircle of the quadrilateral $ABCD$. Find, with proof, the value of the sum \[ \sigma = \frac{ AP}{PA'} + \frac{BP}{PB'} + \frac{CP}{PC'} + \frac{DP}{PD'} . \]

The triangle $ABC$ is equilateral. Find the locus of the points $M$ such that the triangles $ABM$ and $ACM$ are both isosceles.