Ali has $6$ types of $2\times2$ squares with cells colored in white or black, and has presented them to Mohammad as forbidden tiles. $a)$ Prove that Mohammad can color the cells of the infinite table (from each $4$ sides.) in black or white such that there's no forbidden tiles in the table. $b)$ Prove that Ali can present $7$ forbidden tiles such that Mohammad cannot achieve his goal.

A positive integer is called [i]beautiful[/i] if it can be represented in the form $\dfrac{x^2+y^2}{x+y}$ for two distinct positive integers $x,y$. A positive integer that is not beautiful is [i]ugly[/i]. a) Prove that $2014$ is a product of a beautiful number and an ugly number. b) Prove that the product of two ugly numbers is also ugly.

Let $\mathbb{R}^+$ be the set of all positive real numbers. Show that there is no function $f:\mathbb{R}^+ \to \mathbb{R}^+$ satisfying \[f(x+y)=f(x)+f(y)+\dfrac{1}{2012}\] for all positive real numbers $x$ and $y$. [i]Proposer: Fajar Yuliawan[/i]

The circumcircle of triangle $ABC$ has centre $O$. $P$ is the midpoint of $\widehat{BAC}$ and $QP$ is the diameter. Let $I$ be the incentre of $\triangle ABC$ and let $D$ be the intersection of $PI$ and $BC$. The circumcircle of $\triangle AID$ and the extension of $PA$ meet at $F$. The point $E$ lies on the line segment $PD$ such that $DE=DQ$. Let $R,r$ be the radius of the inscribed circle and circumcircle of $\triangle ABC$, respectively. Show that if $\angle AEF=\angle APE$, then $\sin^2\angle BAC=\dfrac{2r}R$

A $\vartriangle ABC$ is given with $\angle A = 70^o$. The angle bisectors of $\vartriangle ABC$ intersect at $I$. Suppose that $CA + AI=BC$. Find, with proof, the value of $\angle B$.

Let $\mathbb N = B_1\cup\cdots \cup B_q$ be a partition of the set $\mathbb N$ of all positive integers and let an integer $l \in \mathbb N$ be given. Prove that there exist a set $X \subset \mathbb N$ of cardinality $l$, an infinite set $T \subset \mathbb N$, and an integer $k$ with $1 \leq k \leq q$ such that for any $t \in T$ and any finite set $Y \subset X$, the sum $t+ \sum_{y \in Y} y$ belongs to $B_k.$

Let $n$ be a positive integer. Yang the Saltant Sanguivorous Shearling is on the side of a very steep mountain that is embedded in the coordinate plane. There is a blood river along the line $y=x$, which Yang may reach but is not permitted to go above (i.e. Yang is allowed to be located at $(2016,2015)$ and $(2016,2016)$, but not at $(2016,2017)$). Yang is currently located at $(0,0)$ and wishes to reach $(n,0)$. Yang is permitted only to make the following moves: (a) Yang may [i]spring[/i], which consists of going from a point $(x,y)$ to the point $(x,y+1)$. (b) Yang may [i]stroll[/i], which consists of going from a point $(x,y)$ to the point $(x+1,y)$. (c) Yang may [i]sink[/i], which consists of going from a point $(x,y)$ to the point $(x,y-1)$. In addition, whenever Yang does a [i]sink[/i], he breaks his tiny little legs and may no longer do a [i]spring[/i] at any time afterwards. Yang also expends a lot of energy doing a [i]spring[/i] and gets bloodthirsty, so he must visit the blood river at least once afterwards to quench his bloodthirst. (So Yang may still [i]spring[/i] while bloodthirsty, but he may not finish his journey while bloodthirsty.) Let there be $a_n$ different ways for which Yang can reach $(n,0)$, given that Yang is permitted to pass by $(n,0)$ in the middle of his journey. Find the $2016$th smallest positive integer $n$ for which $a_n\equiv 1\pmod 5$. [i]Proposed by James Lin[/i]

Let $ABCD$ be a rectangle such that $AB = 20$ and $AD = 24.$ Point $P$ lies inside $ABCD$ such that triangles $PAC$ and $PBD$ have areas $20$ and $24,$ respectively. Compute all possible areas of triangle $PAB.$

Find all positive integers $n$ for which there exists a set of exactly $n$ distinct positive integers, none of which exceed $n^2$, whose reciprocals add up to $1$.

A [i]word[/i] is a sequence of 0 and 1 of length 8. Let $ x$ and $ y$ be two words differing in exactly three places. Prove that the number of words differing from each of $ x$ and $ y$ in at least five places is 188.

Given a circle $\omega$ with radius $1$. Let $T$ be a set of triangles good, if the following conditions apply: (a) the circumcircle of each triangle in the set $T$ is $\omega$; (b) The interior of any two triangles in the set $T$ has no common point. Find all positive real numbers $t$, for which for each positive integer $n$ there is a good set of $n$ triangles, where the perimeter of each triangle is greater than $t$.

We are given $k$ colors and we have to assign a single color to every vertex. An edge is [u][b]satisfied[/b][/u] if the vertices on that edge, are of different colors. [list] [*]Prove that you can always find an algorithm which assigns colors to vertices so that at least $\frac{k - 1}{k}|E|$ edges are satisfied where \(|E|\) is the cardinality of the edges in the graph.[/*] [*]Prove that there is a poly time deterministic algorithm for this [/*] [/list]

Show that it is possible to cut any triangle into several pieces, so that a rectangle is formed when they are joined together.

Segment equal to median $AA_0$ of triangle $ABC$ is drawn from point $A_0$ perpendicular to side $BC$ to the outer side of the triangle. Let's denote the second end of the constructed segment as $A_1$. Points $B_1$ and $C_1$ are constructed similarly. Find the angles of triangle $A_1B_1C_1$ if the angles of the triangle $ABC$ are $30^o$, $30^o$ and $120^o$. [hide=original wording]Медиану AA0 треугольника ABC отложили от точки A0 перпендикулярно стороне BC во внешнюю сторону треугольника. Обозначим второй конец построенного отрезка через A1. Аналогично строятся точки B1 и C1. Найдите углы треугольника A1B1C1, если углы треугольника ABC равны 30^o, 30^o и 120^o.[/hide]

Let $a, b, c, d$ be positive integers such that $a+b+c+d = 2011$. Prove that $2011$ is not a divisor of $ab - cd$.

Cyclic quadrilateral $ABCD$ has $AC\perp BD$, $AB+CD=12$, and $BC+AD=13$. FInd the greatest possible area of $ABCD$.

Let $(a_n)$ be sequnce of positive integers such that first $k$ members $a_1,a_2,...,a_k$ are distinct positive integers, and for each $n>k$, number $a_n$ is the smallest positive integer that can't be represented as a sum of several (possibly one) of the numbers $a_1,a_2,...,a_{n-1}$. Prove that $a_n=2a_{n-1}$ for all sufficently large $n$.

Let $P(x)=a_{0}+a_{1}x+...+a_{n}x^{n}\in\mathbb{C}[x]$ , where $a_{n}=1$. The roots of $P(x)$ are $b_{1},b_{2},...,b_{n}$, where $|b_{1}|,|b_{2}|,...,|b_{j}|>1$ and $|b_{j+1}|,...,|b_{n}|\leq 1$. Prove that $\prod_{i=1}^{j}|b_{i}|\leq\sqrt{|a_{0}|^{2}+|a_{1}|^{2}+...+|a_{n}|^{2}}$.

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