In the triangle $ ABC $ on the sides $ BC $, $ CA $, $ AB $, the points $ D $, $ E $, $ F $ are chosen respectively in such a way that $$ BD \colon DC = CE \colon EA = AF \colon FB = k,$$ where $k$ is a given positive number. Given the area $ S $ of the triangle $ ABC $, calculate the area of the triangle $ DEF $

How many sequences of $5$ positive integers $(a,b,c,d,e)$ satisfy $abcde\leq a+b+c+d+e\leq10$?

Given a positive integer $n$, let $d(n)$ be the largest positive divisor of $n$ less than $n$. For example, $d(8) = 4$ and $d(13) = 1$. A sequence of positive integers $a_1, a_2,\dots$ satisfies \[a_{i+1} = a_i +d(a_i),\] for all positive integers $i$. Prove that regardless of the choice of $a_1$, there are infinitely many terms in the sequence divisible by $3^{2011}$.

Ximena lists the whole numbers $1$ through $30$ once. Emilio copies Ximena's numbers, replacing each occurrence of the digit $2$ by the digit $1$. Ximena adds her numbers and Emilio adds his numbers. How much larger is Ximena's sum than Emilio's? $\textbf{(A)}\ 13\qquad\textbf{(B)}\ 26\qquad\textbf{(C)}\ 102\qquad\textbf{(D)}\ 103\qquad\textbf{(E)}\ 110$

A quadrilateral that has consecutive sides of lengths $70, 90, 130$ and $110$ is inscribed in a circle and also has a circle inscribed in it. The point of tangency of the inscribed circle to the side of length $130$ divides that side into segments of lengths $x$ and $y$. Find $|x-y|$. $ \textbf{(A)}\ 12 \qquad\textbf{(B)}\ 13 \qquad\textbf{(C)}\ 14 \qquad\textbf{(D)}\ 15 \qquad\textbf{(E)}\ 16 $

On a plane are given finitely many red and blue lines, no two parallel, such that any intersection point of two lines of the same color also lies on another line of the other color. Prove that all the lines pass through a single point.

Prove that in any polygon, there exist two sides whose radio is less than $2$.(Essentialy if $a_1\geq a_2\geq...\geq a_n$ are the sides of a polygon prove that there exist $i,j\in\{1,2,..,n\}$ so that $i<j$ and $\frac {a_i}{a_j}<2$).

Given that $5^{2018}$ has $1411$ digits and starts with $3$ (the leftmost non-zero digit is $3$), for how many integers $1\leq n\leq2017$ does $5^n$ start with $1$? [i]2018 CCA Math Bonanza Tiebreaker Round #3[/i]

Let $ABCD$ be a cyclic quadrilateral with $AB = 7$ and $CD = 8$. Point $P$ and $Q$ are selected on segment $AB$ such that $AP = BQ = 3$. Points $R$ and $S$ are selected on segment $CD$ such that $CR = DS = 2$. Prove that $PQRS$ is a cyclic quadrilateral. [i]Proposed by Evan O'Dorney[/i]

Prove that for any positive real numbers $x,y$ and $z,$ $xyz(x+2)(y+2)(z+2)\le(1+\frac{2(xy+yz+zx)}{3})^3$

Let $f(x)=1-|x|$. Let \begin{align*}f_n(x)&=(\overbrace{f\circ \cdots\circ f}^{n\text{ copies}})(x)\\g_n(x)&=|n-|x| |\end{align*} Determine the area of the region bounded by the $x$-axis and the graph of the function $\textstyle\sum_{n=1}^{10}f(x)+\textstyle\sum_{n=1}^{10}g(x).$

In the figure below, the centers of the six congruent circles form a regular hexagon with side length 2. [asy] import graph; pair A = 2dir(0); pair B = 2dir(60); pair C = 2dir(120); pair D = 2dir(180); pair E = 2dir(240); pair F = 2dir(300); path hexagon = A -- B -- C -- D -- E -- F -- cycle; fill(hexagon, gray); filldraw(Circle(A, 1), white); filldraw(Circle(B, 1), white); filldraw(Circle(C, 1), white); filldraw(Circle(D, 1), white); filldraw(Circle(E, 1), white); filldraw(Circle(F, 1), white); draw(hexagon); [/asy] Adjacent circles are tangent to each other. What is the area of the shaded region?

Consider two points $Y$ and $X$ in a plane and a variable point $P$ which is not on $XY$. Let the parallel line to $YP$ through $X$ intersect the internal angle bisector of $\angle XYP$ in $A$, and let the parallel line to $XP$ through $Y$ intersect the internal angle bisector of $\angle YXP$ in $B$. Let $AB$ intersect $XP$ and $YP$ in $S$ and $T$ respectively. Show that the product $|XS|*|YT|$ does not depend on the position of $P$.

In the cells of the $2024\times 2024$ board, integers are arranged so that in any $2 \times 2023$ rectangle (vertical or horizontal) with one cut corner cell that does not go beyond the board, the sum of the numbers is divided by $13$. Prove that the sum of all the numbers on the board is divisible by $13$.

Circle $C$ center $O$ touches externally circle $C'$ center $O'$. A line touches $C$ at $A$ and $C'$ at $B$. $P$ is the midpoint of $AB$. Show that $\angle OPO' = 90^o$.

An acute-angled triangle $ABC$ is fixed on a plane with largest side $BC$. Let $PQ$ be an arbitrary diameter of its circumscribed circle, and the point $P$ lies on the smaller arc $AB$, and the point $Q$ is on the smaller arc $AC$. Points $X, Y, Z$ are feet of perpendiculars dropped from point $P$ to the line $AB$, from point $Q$ to the line $AC$ and from point $A$ to line $PQ$. Prove that the center of the circumscribed circle of triangle $XYZ$ lies on a fixed circle.

How many coins can be placed on a $10 \times 10$ board (each at the center of its square, at most one per square) so that no four coins form a rectangle with sides parallel to the sides of the board?

In a triangle $ABC$, $P$ and $Q$ are the feet of the altitudes from $B$ and $A$ respectively. Find the locus of the circumcentre of triangle $PQC$, when point $C$ varies (with $A$ and $B$ fixed) in such a way that $\angle ACB$ is equal to $60^{\circ}$.

Andy and Bethany have a rectangular array of numbers with $ 40$ rows and $ 75$ columns. Andy adds the numbers in each row. The average of his $ 40$ sums is $ A$. Bethany adds the numbers in each column. The average of her $ 75$ sums is $ B$. What is the value of $ \frac{A}{B}$? $ \textbf{(A)}\ \frac{64}{225} \qquad \textbf{(B)}\ \frac{8}{15} \qquad \textbf{(C)}\ 1 \qquad \textbf{(D)}\ \frac{15}{8} \qquad \textbf{(E)}\ \frac{225}{64}$

Suppose that $a_0, a_1, \cdots $ and $b_0, b_1, \cdots$ are two sequences of positive integers such that $a_0, b_0 \ge 2$ and \[ a_{n+1} = \gcd{(a_n, b_n)} + 1, \qquad b_{n+1} = \operatorname{lcm}{(a_n, b_n)} - 1. \] Show that the sequence $a_n$ is eventually periodic; in other words, there exist integers $N \ge 0$ and $t > 0$ such that $a_{n+t} = a_n$ for all $n \ge N$.

$ABCDE$ is a convex pentagon with $BD=CD=AC$, and $B$, $C$, $D$, $E$ are concyclic. If $\angle BAC+\angle AED=180^{\circ}$ and $\angle DCA=\angle BDE$, prove that $AB=DE$ or $AB=2AE$.

Two players, $B$ and $R$, play the following game on an infinite grid of unit squares, all initially colored white. The players take turns starting with $B$. On $B$'s turn, $B$ selects one white unit square and colors it blue. On $R$'s turn, $R$ selects two white unit squares and colors them red. The players alternate until $B$ decides to end the game. At this point, $B$ gets a score, given by the number of unit squares in the largest (in terms of area) simple polygon containing only blue unit squares. What is the largest score $B$ can guarantee? (A [i]simple polygon[/i] is a polygon (not necessarily convex) that does not intersect itself and has no holes.) [i]Proposed by David Torres[/i]

In triangle $ABC$, the altitude $AH$ passes through midpoint of the median $BM$. Prove that in the triangle $BMC$ also one of the altitudes passes through the midpoint of one of the medians.

The number $2.5081081081081 \ldots$ can be written as $m/n$ where $m$ and $n$ are natural numbers with no common factors. Find $m + n$.

Determine all non-constant monic polynomials $P(x)$ with integer coefficients such that no prime $p>10^{100}$ divides any number of the form $P(2^n)$