Find the locus of the points that are the midpoints of the chords of the secant to the given circle and passing through a given point.

The image below consists of a large triangle divided into $13$ smaller triangles. Let $N$ be the number of ways to color each smaller triangle one of red, green, and blue such that if $T_1$ and $T_2$ are smaller triangles whose perimeters intersect at more than one point, $T_1$ and $T_2$ have two different colors. Compute the number of positive integer divisors of $N$. [asy] size(5 cm); draw((-4,0)--(4,0)--(0,6.928)--cycle); draw((0,0)--(2,3.464)--(-2,3.464)--cycle); draw((-2,0)--(-1,1.732)--(-3,1.732)--cycle); draw((2,0)--(1,1.732)--(3,1.732)--cycle); draw((0,3.464)--(1,5.196)--(-1,5.196)--cycle); [/asy] [i]2021 CCA Math Bonanza Individual Round #7[/i]

Sarah pours four ounces of coffee into an eight-ounce cup and four ounces of cream into a second cup of the same size. She then transfers half the coffee from the first cup to the second and, after stirring thoroughly, transfers half the liquid in the second cup back to the first. What fraction of the liquid in the first cup is now cream? $ \textbf{(A)}\ 1/4 \qquad \textbf{(B)}\ 1/3 \qquad \textbf{(C)}\ 3/8 \qquad \textbf{(D)}\ 2/5 \qquad \textbf{(E)}\ 1/2$

A figure $\Phi$ composed of unit squares has the following property: if the squares of an $m \times n$ rectangle ($m,n$ are fixed) are filled with numbers whose sum is positive, the figure $\Phi$ can be placed within the rectangle (possibly after being rotated) so that the sum of the covered numbers is also positive. Prove that a number of such figures can be put on the $m\times n$ rectangle so that each square is covered by the same number of figures.

Let the number $ p $ be a prime divisor of the number $ 2 ^ {2 ^ k} + 1 $. Prove that $ p-1 $ is divisible by $ 2 ^ {k + 1} $.

Prove that for any given positive integer $m$ and $n$, there is always a positive integer $k$ so that $2^k-m$ has at least $n$ different prime divisors.

Let $A$ and $C$ be two points on a circle $X$ so that $AC$ is not diameter and $P$ a segment point on $AC$ different from its middle. The circles $c_1$ and $c_2$, inner tangents in $A$, respectively $C$, to circle $X$, pass through the point $P$ ¸ and intersect a second time at point $Q$. The line $PQ$ intersects the circle $X$ in points $B$ and $D$. The circle $c_1$ intersects the segments $AB$ and $AD$ in $K$, respectively $N$, and circle $c_2$ intersects segments $CB$ and ¸ $CD $ in $L$, respectively $M$. Show that: a) the $KLMN$ quadrilateral is isosceles trapezoid; b) $Q$ is the middle of the segment $BD$. Proposed by Thanos Kalogerakis

Pairwise distinct points $P_1,\dots,P_{1024}$, which lie on a circle, are marked by distinct reals $a_1,\dots,a_{1024}$. Let $P_i$ be $Q-$good for a $Q$ on the circle different than $P_1,\dots,P_{1024}$, if and only if $a_i$ is the greatest number on at least one of the two arcs $P_iQ$. Let the score of $Q$ be the number of $Q-$good points on the circle. Determine the greatest $k$ such that regardless of the values of $a_1,\dots,a_{1024}$, there exists a point $Q$ with score at least $k$.

In the convex quadrilateral $ABCD$, $AD = BD$ and $\angle ACD = 3 \angle BAC$. Let $M$ be the midpoint of side $AD$. If the lines $CM$ and $AB$ are parallel, prove that the angle $\angle ACB$ is right.

For any natural number $n= \overline{a_k...a_1a_0}$ $(a_k \ne 0)$ in decimal system write $p(n)=a_0 \cdot a_1 \cdot ... \cdot a_k$, $s(n)=a_0+ a_1+ ... + a_k$, $n^*= \overline{a_0a_1...a_k}$. Consider $P=\{n | n=n^*, \frac{1}{3} p(n)= s(n)-1\}$ and let $Q$ be the set of numbers in $P$ with all digits greater than $1$. (a) Show that $P$ is infinite. (b) Show that $Q$ is finite. (c) Write down all the elements of $Q$.

A round robin tournament is held with $2016$ participants. Each player plays each other player once and no games result in ties. We say a pair of players $A$ and $B$ is a [i]dominant pair[/i] if all other players either defeat $A$ and $B$ or are defeated by both $A$ and $B$. Find the maximum number dominant pairs. [i]Proposed by Nathan Ramesh

In an acute-angled triangle $ ABC $ consider $ A_1,B_1,C_1 $ to be the symmetric points of the orthocenter of $ ABC $ to the sides $ BC,AC,AB, $ respectively. Show that if the centroids of the triangles $ ABC,A_1B_1C_1 $ are the same, then $ ABC $ is equilateral. [i]Carmen Botea[/i]

For a point $P$ on the plane, denote by $\lVert P \rVert$ the distance to its nearest lattice point. Prove that there exists a real number $L > 0$ satisfying the following condition: For every $\ell > L$, there exists an equilateral triangle $ABC$ with side-length $\ell$ and $\lVert A \rVert, \lVert B \rVert, \lVert C \rVert < 10^{-2017}$.

Does it exist a permutation of the numbers $1,2,\ldots,1992$ such that the arithmetic mean of arbitrary two of the numbers is not equal to any of the numbers which is placed between these two numbers in the permutation?

Positive integers $x_1, x_2, \dots, x_n$ ($n \ge 4$) are arranged in a circle such that each $x_i$ divides the sum of the neighbors; that is \[ \frac{x_{i-1}+x_{i+1}}{x_i} = k_i \] is an integer for each $i$, where $x_0 = x_n$, $x_{n+1} = x_1$. Prove that \[ 2n \le k_1 + k_2 + \dots + k_n < 3n. \]

Let $ABC$ be a triangle. Prove that there exists a point $D$ on the side $AB$ of the triangle $ABC$, such that $CD$ is the geometric mean of $AD$ and $DB$, iff the triangle $ABC$ satisfies the inequality $\sin A\sin B\le\sin^2\frac{C}{2}$. [hide="Comment"][i]Alternative formulation, from IMO ShortList 1974, Finland 2:[/i] We consider a triangle $ABC$. Prove that: $\sin(A) \sin(B) \leq \sin^2 \left( \frac{C}{2} \right)$ is a necessary and sufficient condition for the existence of a point $D$ on the segment $AB$ so that $CD$ is the geometrical mean of $AD$ and $BD$.[/hide]

Let $ABC$ be an acute triangle with $BC = 48$. Let $M$ be the midpoint of $BC$, and let $D$ and $E$ be the feet of the altitudes drawn from $B$ and $C$ to $AC$ and $AB$ respectively. Let $P$ be the intersection between the line through $A$ parallel to $BC$ and line $DE$. If $AP = 10$, compute the length of $PM$,

Let $p$ be a prime number. For each $k$, $1\le k\le p-1$, there exists a unique integer denoted by $k^{-1}$ such that $1\le k^{-1}\le p-1$ and $k^{-1}\cdot k=1\pmod{p}$. Prove that the sequence \[1^{-1},\quad 1^{-1}+2^{-1},\quad 1^{-1}+2^{-1}+3^{-1},\quad \ldots ,\quad 1^{-1}+2^{-1}+\ldots +(p-1)^{-1} \] (addition modulo $p$) contains at most $\frac{p+1}{2}$ distinct elements.

A game consists in pushing a flat stone along a sequence of squares $S_0, S_1, S_2, . . .$ that are arranged in linear order. The stone is initially placed on square $S_0$. When the stone stops on a square $S_k$ it is pushed again in the same direction and so on until it reaches $S_{1987}$ or goes beyond it; then the game stops. Each time the stone is pushed, the probability that it will advance exactly $n$ squares is $\frac{1}{2^n}$. Determine the probability that the stone will stop exactly on square $S_{1987}.$

One day the Beverage Barn sold $252$ cans of soda to $100$ customers, and every customer bought at least one can of soda. What is the maximum possible median number of cans of soda bought per customer on that day? $\textbf{(A) }2.5\qquad\textbf{(B) }3.0\qquad\textbf{(C) }3.5\qquad\textbf{(D) }4.0\qquad \textbf{(E) }4.5$

In a convex trapezoid $ABCD$, side $AD$ is twice the length of the other sides. Let $E$ and $F$ be points on segments $AC$ and $BD$, respectively, such that $\angle BEC = 70^\circ$ and $\angle BFC = 80^\circ$. Determine the ratio of the areas of quadrilaterals $BEFC$ and $ABCD$. [i]Proposed by Zaza Meliqidze, Georgia [/i]

Solve the following Diophantines. [b]a)[/b] $ x^2+y^2=6z^2 $ [b]b)[/b] $ x^2+y^2-2x+4y-1=0 $ [i]Dan Negulescu[/i]

Parallelograms $ABGF$, $CDGB$ and $EFGD$ are drawn so that $ABCDEF$ is a convex hexagon, as shown. If $\angle ABG = 53^o$ and $\angle CDG = 56^o$, what is the measure of $\angle EFG$, in degrees? [img]https://cdn.artofproblemsolving.com/attachments/9/f/79d163662e02bc40d2636a76b73f632e59d584.png[/img]

Find the least three digit number that is equal to the sum of its digits plus twice the product of its digits.

An integer $x$ has the property that the sums of the digits of $x$ and of $3x$ are the same. Prove that $x$ is divisible by $9$.