What is the minimum number of digits to the right of the decimal point needed to express the fraction $\dfrac{123\,456\,789}{2^{26}\cdot 5^4}$ as a decimal? $\textbf{(A) }4\qquad\textbf{(B) }22\qquad\textbf{(C) }26\qquad\textbf{(D) }30\qquad\textbf{(E) }104$

Let $O$ be a point in the plane of the triangle $ABC$. A circle $\mathcal{C}$ which passes through $O$ intersects the second time the lines $OA,OB,OC$ in $P,Q,R$ respectively. The circle $\mathcal{C}$ also intersects for the second time the circumcircles of the triangles $BOC$, $COA$ and $AOB$ respectively in $K,L,M$. Prove that the lines $PK,QL$ and $RM$ are concurrent.

Betty has a $4$-by-$4$ square box of chocolates. Every time Betty eats a chocolate, she picks one from a row with the greatest number of remaining chocolates. In how many ways can Betty eat $5$ chocolates from her box, where order matters?

Given a circle centered at $ O $ and two points $ A $ and $ B $ lying on it. $ A $ and $ B $ do not form a diameter. The point $ C $ is chosen on the circle so that the line $ AC $ divides the segment $ OB $ in half. Let lines $ AB $ and $ OC $ intersect at $ D $, and let lines $ BC $ and $ AO $ intersect at $ F $. Prove that $ AF = CD $.

Let $G = (S, T; E)$ be a finite bipartite graph with a perfect matching. Prove that there exists an injective edge weighting $w: E \to \mathbb{R}$ satisfying the following: 1. If $e_s$ is the edge with the smallest weight among the edges incident to $s$ for all $s \in S$, then $\{ e_s \mid s \in S \}$ forms a perfect matching in $G$. 2. If $e_t$ is the edge with the largest weight among the edges incident to $t$ for all $t \in T$, then $\{ e_t \mid t \in T \}$ forms a perfect matching in $G$.

In a convex $n$-gon, several diagonals are drawn. Among these diagonals, a diagonal is called [i]good[/i] if it intersects exactly one other diagonal drawn (in the interior of the $n$-gon). Find the maximum number of good diagonals.

Through the midpoint $M$ of the side $BC$ of the triangle $ABC$ passes a line which intersects the rays $AB$ and $AC$ at $D$ and $E$, respectively, such that $AD=AE$. Let $F$ be the foot of the perpendicular from $A$ onto $BC$ and $P{}$ the circumcenter of triangle $ADE$. Prove that $PF=PM$.

$ABCD$ is inscribed in unit circle $\Gamma$. Let $\Omega_1$, $\Omega_2$ be the circumcircles of $\vartriangle ABD$, $\vartriangle CBD$ respectively. Circles $\Omega_1$, $\Omega_2$ are tangent to segment $BD$ at $M$,$N$ respectively. Line A$M$ intersects $\Gamma$, $\Omega_1$ again at points $X_1$,$X_2$ respectively (different from $A$, $M$). Let $\omega_1$ be the circle passing through $X_1$, $X_2$ and tangent to $\Omega_1$. Line $CN$ intersects $\Gamma$, $\Omega_2$ again at points $Y_1$, $Y_2$ respectively (different from $C$, $N$). Let $\omega_2$ be the circle passing through $Y_1$, $Y_2$ and tangent to $\Omega_2$. Circles $\Omega_1$,$\Omega_2$, $\omega_1$, $\omega_2$ have radii $R_1$, $R_2$, $r_1$, $r_2$ respectively. Prove that $$r_1+r_2-R_1-R_2=1.$$ [img]https://cdn.artofproblemsolving.com/attachments/1/5/70471f2419fadc4b2183f5fe74f0c7a2e69ed4.png[/img] [url=https://www.geogebra.org/m/vxx8ghww]geogebra file[/url]

Find all functions $f : R \to R$ with $f (x + yf(x + y))= y^2 + f(x)f(y)$ for all $x, y \in R$.

All cells of the checkered plane are painted in $5$ colors so that in any figure of the species [img]https://cdn.artofproblemsolving.com/attachments/f/f/49b8d6db20a7e9cca7420e4b51112656e37e81.png[/img] all colors are different. Prove that in any figure of the species $ \begin{tabular}{ | l | c| c | c | r| } \hline & & & &\\ \hline \end{tabular}$, all colors are different..

Let $n$ and $k$ be two integers with $n>k\geqslant 1$. There are $2n+1$ students standing in a circle. Each student $S$ has $2k$ [i]neighbors[/i] - namely, the $k$ students closest to $S$ on the left, and the $k$ students closest to $S$ on the right. Suppose that $n+1$ of the students are girls, and the other $n$ are boys. Prove that there is a girl with at least $k$ girls among her neighbors. [i]Proposed by Gurgen Asatryan, Armenia[/i]

A $7\times 7$ square is cut into pieces following types: [img]https://cdn.artofproblemsolving.com/attachments/e/d/458b252c719946062b655340cbe8415d1bdaf9.png[/img] Show that exactly one of the pieces is of type (b). [img]https://cdn.artofproblemsolving.com/attachments/4/9/f3dd0e13fed9838969335c82f5fe866edc83e8.png[/img]

Let $O$ be the center and $[AB]$ be the diameter of a semicircle. $E$ is a point between $O$ and $B$. The perpendicular to $[AB]$ at $E$ meets the semicircle at $D$. A circle which is internally tangent to the arc $\overarc{BD}$ is also tangent to $[DE]$ and $[EB]$ at $K$ and $C$, respectively. Prove that $\widehat{EDC}=\widehat{BDC}$.

Let be a point $ P $ in the interior of a triangle $ ABC $ such that $ BP=AC, M $ be the middlepoint of the segment $ AP, R $ be the middlepoint of $ BC $ and $ E $ be the intersection of $ BP $ with $ AC. $ Prove that the bisector of $ \angle BEA $ is perpendicular on $ MR $

A function $ f: \mathbb{N} \rightarrow \mathbb{N}$ satisfies: $ (a)$ $ f(ab)\equal{}f(a)f(b)$ whenever $ a$ and $ b$ are coprime; $ (b)$ $ f(p\plus{}q)\equal{}f(p)\plus{}f(q)$ for all prime numbers $ p$ and $ q$. Prove that $ f(2)\equal{}2,f(3)\equal{}3$ and $ f(1999)\equal{}1999.$

A square $ABCD$ of side length $2a$ is given on a plane $\Pi$. Let $S$ be a point on the ray $Ax$ perpendicular to $\Pi$ such that $AS = 2a.$ $(a)$ Let $M \in BC$ and $N \in CD$ be two variable points. $i$. Find the positions of $M,N$ such that $BM + DN \ge \frac{3}{2}$, planes $SAM$ and $SMN$ are perpendicular and $BM \cdot DN$ is minimum. $ii$. Find $M$ and $N$ such that $\angle MAN = 45^{\circ}$ and the volume of $SAMN$ attains an extremum value. Find these values. $(b)$ Let $Q$ be a point such that $\angle AQB = \angle AQD = 90^{\circ}$. The line $DQ$ intersects the plane $\pi$ through $AB$ perpendicular to $\Pi$ at $Q'$. $i$. Find the locus of $Q'$. $ii$. Let $K$ be the locus of points $Q$ and let $CQ$ meet $K$ again at $R$. Let $DR$ meets $\Pi$ at $R'$. Prove that $sin^2 \angle Q'DB + sin^2 \angle R'DB$ is independent of $Q$.

Given a convex pentagon $ABCDE$, with $\angle BAC = \angle ABE = \angle DEA - 90^o$, $\angle BCA = \angle ADE$ and also $BC = ED$. Prove that $BCDE$ is parallelogram.

What is the largest integer $n$ such that $2^n + 65$ is equal to square of an integer? $ \textbf{(A)}\ 1024 \qquad\textbf{(B)}\ 268 \qquad\textbf{(C)}\ 10 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ \text{None of the preceding} $

Give the conditions under which the equation $$ \sqrt{x - a} + \sqrt{x - b} = \sqrt{x - c }$$ has roots, assuming that the numbers $ a $, $ b $, $ c $ are pairs of differences

Find all polynomials $f(x)$ with real coefficients for which \[f(x)f(2x^2) = f(2x^3 + x).\]

The base $4$ repeating decimal $0.\overline{12}_4$ can be expressed in the form $\frac{a}{b}$ in base 10, where $a$ and $b$ are relatively prime positive integers. Compute the sum of $a$ and $b$. [i]2020 CCA Math Bonanza Team Round #2[/i]

Does there exist a positive integer $n$, such that the number of divisors of $n!$ is divisible by $2019$?

Consider a rectangle $A_1A_2A_3A_4$ and a circle $\mathcal{C}_i$ centered at $A_i$ with radius $r_i$ for $i=1,2,3,4$. Suppose that $r_1+r_3=r_2+r_4<d$, where $d$ is the diagonal of the rectangle. The two pairs of common outer tangents of $\mathcal{C}_1$ and $\mathcal{C}_3$, and of $\mathcal{C}_2$ and $\mathcal{C}_4$ form a quadrangle. Prove that this quadrangle has an inscribed circle.

With expressions containing the symbol $*$, the following transformations can be performed: 1) rewrite the expression in the form $x * (y * z) as ((1 * x) * y) * z$; 2) rewrite the expression in the form $x * 1$ as $x$. Conversions can only be performed with an integer expression, but not with its parts. For example, $(1 *1) * (1 *1)$ can be rewritten according to the first rule as $((1 * (1 * 1)) * 1) * 1$ (taking $x = 1 * 1$, $y = 1$ and $z = 1$), but not as $1 * (1 * 1)$ or $(1* 1) * 1$ (in the last two cases, the second rule would be applied separately to the left or right side $1 * 1$). Find all positive integers $n$ for which the expression $\underbrace{1 * (1 * (1 * (...* (1 * 1)...))}_{n units}$ it is possible to lead to a form in which there is not a single asterisk. Note. The expressions $(x * y) * $z and $x * (y * z)$ are considered different, also, in the general case, the expressions $x * y$ and $y * x$ are different.

Find all pairs of positive integers $(a,b)$ such that $\sqrt{a-1}+\sqrt{b-1}=\sqrt{ab-1}.$