$480$ $1$ cm unit cubes are used to build a block measuring $6$ cm by $8$ cm by $10$ cm. A tiny ant then chews his way in a straight line from one vertex of the block to the furthest vertex. How many cubes does the ant pass through? The ant is so tiny that he does not "pass through" cubes if he is merely passing through where their edges or vertices meet. [i]2017 CCA Math Bonanza Individual Round #11[/i]

[b]p1.[/b] Prove that if $a, b, c, d$ are real numbers, then $$\max \{a + c, b + d\} \le \max \{a, b\} + \max \{c, d\}$$ [b]p2.[/b] Find the smallest positive integer whose digits are all ones which is divisible by $3333333$. [b]p3.[/b] Find all integer solutions of the equation $\sqrt{x} +\sqrt{y} =\sqrt{2560}$. [b]p4.[/b] Find the irrational number: $$A =\sqrt{ \frac12+\frac12 \sqrt{\frac12+\frac12 \sqrt{ \frac12 +...+ \frac12 \sqrt{ \frac12}}}}$$ ($n$ square roots). [b]p5.[/b] The Math country has the shape of a regular polygon with $N$ vertexes. $N$ airports are located on the vertexes of that polygon, one airport on each vertex. The Math Airlines company decided to build $K$ additional new airports inside the polygon. However the company has the following policies: (i) it does not allow three airports to lie on a straight line, (ii) any new airport with any two old airports should form an isosceles triangle. How many airports can be added to the original $N$? [b]p6.[/b] The area of the union of the $n$ circles is greater than $9$ m$^2$(some circles may have non-empty intersections). Is it possible to choose from these $n$ circles some number of non-intersecting circles with total area greater than $1$ m$^2$? PS. You should use hide for answers.

Let $BB_1$, $CC_1$ be the altitudes of triangle $ABC$, and $AD$ be the diameter of its circumcircle. The lines $BB_1$ and $DC_1$ meet at point $E$, the lines $CC_1$ and $DB_1$ meet at point $F$. Prove that $\angle CAE = \angle BAF$.

Find the equation of the circle in the complex plane determined by the roots of the equation $z^3 +(-1+i)z^2+(1-i)z+i= 0$.

Let $M$ be a point on the altitude $CD$ of an acute-angled triangle $ABC$, and $K$ and $L$ the orthogonal projections of $M$ on $AC$ and $BC$. Suppose that the incenter and circumcenter of the triangle lie on the segment $KL$. (a) Prove that $CD=R+r$, where $R$ and $r$ are the circumradius and inradius, respectively. (b) Find the minimum value of the ratio $CM:CD$.

It is known that there is a straight line dividing the perimeter and area of a certain polygon circumscribed around a circle in the same ratio. Prove that this line passes through the center of the indicated circle.

For each positive integer $ n$, let $ c(n)$ be the largest real number such that \[ c(n) \le \left| \frac {f(a) \minus{} f(b)}{a \minus{} b}\right|\] for all triples $ (f, a, b)$ such that --$ f$ is a polynomial of degree $ n$ taking integers to integers, and --$ a, b$ are integers with $ f(a) \neq f(b)$. Find $ c(n)$. [i]Shaunak Kishore.[/i]

The vertex $A$, center $O$ and Euler line $\ell$ of a triangle $ABC$ is given. It is known that $\ell$ intersects $AB,AC$ at two points equidistant from $A$. Restore the triangle.

Let $I$ be the incenter of triangle $ABC$, and $K$ be the common point of $BC$ with the external bisector of angle $A$. The line $KI$ meets the external bisectors of angles $B$ and $C$ at points $X$ and $Y$ . Prove that $\angle BAX = \angle CAY$

Point $M$ is the midpoint of the side $AB$ of an acute triangle $ABC$. Point $P$ lies on the segment $AB$, and points $S_1$ and $S_2$ are the centers of the circumcircles of $APC$ and $BPC$, respectively. Show that the midpoint of segment $S_1S_2$ lies on the perpendicular bisector of segment $CM$.

On a cyclic quadrilateral $ABCD$, $M$ is the midpoint of $AB$ and $N$ is the midpoint of $CD$. Let $E$ be the projection of $C$ onto $AB$ and $F$ the reflection of $N$ about the midpoint of $DE$. If $F$ is inside quadrilateral $ABCD$, show that $\angle BMF = \angle CBD$.

Let $ABCD$ be a tangential quadrilateral with inscribed circle $k(O,r)$ which is tangent to the sides $BC$ and $AD$ at $K$ and $L$, respectively. Show that the circle with diameter $OC$ passes through the intersection point of $KL$ and $OD$. [i]Proposed by Ilija Jovchevski[/i]

Given a triangle $ABC$. The points $A$, $B$, $C$ divide the circumcircle $\Omega$ of the triangle $ABC$ into three arcs $BC$, $CA$, $AB$. Let $X$ be a variable point on the arc $AB$, and let $O_{1}$ and $O_{2}$ be the incenters of the triangles $CAX$ and $CBX$. Prove that the circumcircle of the triangle $XO_{1}O_{2}$ intersects the circle $\Omega$ in a fixed point.

Let O be the circumcenter of a triangle ABC. Points M and N are choosen on the sides AB and BC respectively so that the angle AOC is two times greater than angle MON. Prove that the perimeter of triangle MBN is not less than the lenght of side AC

Points $ A$ and $ B$ lie on a circle centered at $ O$, and $ \angle AOB\equal{}60^\circ$. A second circle is internally tangent to the first and tangent to both $ \overline{OA}$ and $ \overline{OB}$. What is the ratio of the area of the smaller circle to that of the larger circle? $ \textbf{(A)}\ \frac{1}{16} \qquad \textbf{(B)}\ \frac{1}{9} \qquad \textbf{(C)}\ \frac{1}{8} \qquad \textbf{(D)}\ \frac{1}{6} \qquad \textbf{(E)}\ \frac{1}{4}$

In the trapezoid $ABCD$ , the base $AB$ is smaller than the $CD$ base. The point $K$ is chosen such that $AK$ is parallel to BC and $BK$ is parallel to $AD$. The points $P$ and $Q$ are chosen on the $AK$ and $BK$ rays respectively, such that $\angle ADP = \angle BCK$ and $\angle BCQ = \angle ADK$. (a) Show that the lines $AD, BC$ and $PQ$ go through the same point. (b) Assuming that the circumscribed circumferences of the $APD$ and $BCQ$ triangles intersect at two points, show that one of those points belongs to the line $PQ$.

Circles $o_1, o_2$ with equal radii intersect at points $A, B$. Points $C, D, E, F$ lie in this order on one line, with $C, E$ lying on $o_1$ and $D, F$ on $o_2$. Perpendicular bisectors of $CD$ and $EF$ intersect $AB$ at $X, Y$ respectively. Prove that $AX=BY$.

Located inside equilateral triangle $ABC$ is a point $P$ such that $PA=8$, $PB=6$, and $PC=10$. To the nearest integer the area of triangle $ABC$ is: $\textbf{(A)}\ 159\qquad \textbf{(B)}\ 131\qquad \textbf{(C)}\ 95\qquad \textbf{(D)}\ 79\qquad \textbf{(E)}\ 50$

In a trapezoid $ABCD$ with $AD$ parallel to $BC$ points $E, F$ are on sides $AB, CD$ respectively. $A_1, C_1$ are on $AD,BC$ such that $A_1, E, F, A$ lie on a circle and so do $C_1, E, F, C$. Prove that lines $A_1C_1, BD, EF$ are concurrent.

On sides of convex quadrilateral $ABCD$ on external side constructed equilateral triangles $ABK, BCL, CDM, DAN$. Let $P,Q$- midpoints of $BL, AN$ respectively and $X$- circumcenter of $CMD$. Prove, that $PQ$ perpendicular to $KX$

The diagram shows a rectangle that has been dissected into nine non-overlapping squares. Given that the width and the height of the rectangle are relatively prime positive integers, find the perimeter of the rectangle. [asy] defaultpen(linewidth(0.7)); draw((0,0)--(69,0)--(69,61)--(0,61)--(0,0));draw((36,0)--(36,36)--(0,36)); draw((36,33)--(69,33));draw((41,33)--(41,61));draw((25,36)--(25,61)); draw((34,36)--(34,45)--(25,45)); draw((36,36)--(36,38)--(34,38)); draw((36,38)--(41,38)); draw((34,45)--(41,45));[/asy]

Joe has $1729$ randomly oriented and randomly arranged unit cubes, which are initially unpainted. He makes two cubes of sidelengths $9$ and $10$ or of sidelengths $1$ and $12$ (randomly chosen). These cubes are dipped into white paint. Then two more cubes of sidelengths $1$ and $12$ or $9$ and $10$ are formed from the same unit cubes, again randomly oriented and randomly arranged, and dipped into paint. Joe continues this process until every side of every unit cube is painted. After how many times of doing this is the expected number of painted faces closest to half of the total?

In a quadrilateral $ABCD$ with an incircle, $AB = CD; BC < AD$ and $BC$ is parallel to $AD$. Prove that the bisector of $\angle C$ bisects the area of $ABCD$.

An acute triangle $ABC$ is given, in which $AB<AC$. Let $\Omega$ be the circumcircle of $ABC$. Points $M$ and $N$ are the midpoints of the longer arc $BC$ and shorter arc $BC$ of $\Omega$ respectively. Points $X\ne M$ and $Y\ne N$ lie on the line $AM$ and satisfy $BX=BM=CM=CY$. Let $E$ be a point on $AC$ such that $BE$ and $AC$ are perpendicular. Prove that $\angle FNX=\angle YNE$.

Let $ABC$ be a scalene triangle with circumcenter $O$ and orthocenter $H$. Let $AYZ$ be another triangle sharing the vertex $A$ such that its circumcenter is $H$ and its orthocenter is $O$. Show that if $Z$ is on $BC$, then $A,H,O,Y$ are concyclic. [i]Proposed by usjl[/i]