Let's take a $ABCD$ rectangle of paper; the side $AB$ measures $5$ cm and the side $BC$ measures $9$ cm. We do three folds: 1.We take the $AB$ side on the $BC$ side and call $P$ the point on the $BC$ side that coincides with $A$. A right trapezoid $BCDQ$ is then formed. 2. We fold so that $B$ and $Q$ coincide. A $5$-sided polygon $RPCDQ$ is formed. 3. We fold again by matching $D$ with $C$ and $Q$ with $P$. A new right trapezoid $RPCS$. After making these folds, we make a cut perpendicular to $SC$ by its midpoint $T$, obtaining the right trapezoid $RUTS$. Calculate the area of the figure that appears as we unfold the last trapezoid $RUTS$.

(a) Prove that from $2007$ given positive integers, one of them can be chosen so the product of the remaining numbers is expressible in the form $a^2 - b^2$ for some positive integers $a$ and $b$. (2) (b) One of $2007$ given positive integers is $2006$. Prove that if there is a unique number among them such that the product of the remaining numbers is expressible in the form $a^2 - b^2$ for some positive integers $a$ and $b$, then this unique number is $2006$. (2)

Let $a$ be a given natural number and $x_1, x_2, x_3, ...$ the sequence with $x_n = \frac{n}{n+a}$ ($n \in N^*$ ). Prove that for every $n \in N^*$ , the term $x_n$ can be represented as the product of two terms of this sequence , and determine the number of representations depending on $n$ and $a$.

[b]a)[/b] Do there exist $2$-element subsets $A_1,A_2,A_3,...$ of natural numbers such that each natural number appears in exactly one of these sets and also for each natural number $n$, sum of the elements of $A_n$ equals $1391+n$? [b]b)[/b] Do there exist $2$-element subsets $A_1,A_2,A_3,...$ of natural numbers such that each natural number appears in exactly one of these sets and also for each natural number $n$, sum of the elements of $A_n$ equals $1391+n^2$? [i]Proposed by Morteza Saghafian[/i]

Let $C$ and $D$ be two points on the semicricle with diameter $AB$ such that $B$ and $C$ are on distinct sides of the line $AD$. Denote by $M$, $N$ and $P$ the midpoints of $AC$, $BD$ and $CD$ respectively. Let $O_A$ and $O_B$ the circumcentres of the triangles $ACP$ and $BDP$. Show that the lines $O_AO_B$ and $MN$ are parallel.

Let $ABC$ be a triangle with $AB = 26$, $BC = 51$, and $CA = 73$, and let $O$ be an arbitrary point in the interior of $\vartriangle ABC$. Lines $\ell_1$, $\ell_2$, and $\ell_3$ pass through $O$ and are parallel to $\overline{AB}$, $\overline{BC}$, and $\overline{CA}$, respectively. The intersections of $\ell_1$, $\ell_2$, and $\ell_3$ and the sides of $\vartriangle ABC$ form a hexagon whose area is $A$. Compute the minimum value of $A$.

The figure below contains two squares which share an edge, one with side length $200$ units and the other with side length $289$ units. The figure is divided into a whole number of regions, each with an equal whole number area but not necessarily of the same shape. Given that there is more than one region and each region has an area greater than $1$, find the sum of the number of regions and the area of each region. [asy] size(4cm); draw((0,0)--(200,0)--(200,200)--(0,200)--cycle); label("$200$",(0,0)--(200,0)); label("$289$",(200,0)--(489,0)); draw((200,0)--(489,0)--(489,289)--(200,289)--cycle); [/asy] $\textbf{(A) } 704\qquad\textbf{(B) } 874\qquad\textbf{(C) } 924\qquad\textbf{(D) } 978\qquad\textbf{(E) } 1028$

Find all integer solutions to $m^{m+n} = n^{12}, n^{m+n} = m^3$.

A tennis tournament has at least three participants. Every participant plays exactly one match against every other participant. Moreover, every participant wins at least one of the matches he plays. (Draws do not occur in tennis matches.) Show that there are three participants $A, B $ and $C$ for which the following holds: $A$ wins against $B, B$ wins against $C$, and $C$ wins against $A$.

Island Hopping Holidays offer short holidays to $64$ islands, labeled Island $i, 1 \le i \le 64$. A guest chooses any Island $a$ for the first night of the holiday, moves to Island $b$ for the second night, and finally moves to Island $c$ for the third night. Due to the limited number of boats, we must have $b \in T_a$ and $c \in T_b$, where the sets $T_i$ are chosen so that (a) each $T_i$ is non-empty, and $i \notin T_i$, (b) $\sum^{64}_{i=1} |T_i| = 128$, where $|T_i|$ is the number of elements of $T_i$. Exhibit a choice of sets $T_i$ giving at least $63\cdot 64 + 6 = 4038$ possible holidays. Note that c = a is allowed, and holiday choices $(a, b, c)$ and $(a',b',c')$ are considered distinct if $a \ne a'$ or $b \ne b'$ or $c \ne c'$.

Are there $14$ consecutive positive integers, each of which has a divisor other than $1$ and not exceeding $11$?

For what value of $x$ does the following determinant have the value $2013$? \[\left|\begin{array}{ccc}5+x & 8 & 2+x \\ 1 & 1+x & 3 \\ 2 & 1 & 2\end{array}\right|\]

Let the function $f(x,y,t)=\frac{x^2-y^2}{2}-\frac{(x-yt)^2}{1-t^2}$ for all real values $x,y$ and $t\not=\pm1$ a) Evaluate $f(2,0,3)$ and $f(0,2,3)$. b) Show that $f(x,y,0)=f(y,x,0)$ for any values of $(x,y)$. c) Show that $f(x,y,t)=f(y,x,t)$ for any values of $(x,y)$ and $t\not=\pm1$. d) Given $$g(x,y,s)=\frac{(x^2-y^2)(1+\sin(s))}{2} -\frac{(x-y\sin(s))^2}{1-\sin(s)}$$ for all real values $x,y$ and $s\not=\frac{\pi}{2}+2\pi k$, where $k$ is an integer number, show that $g(x,y,s)=g(y,x,s)$ for any values of $(x,y)$ and $s$ in the domain of $g(x,y,s)$.

For how many integers $N$ between $1$ and $1990$ is the improper fraction $\frac{N^2+7}{N+4}$ not in lowest terms? $\text{(A)} \ 0 \qquad \text{(B)} \ 86 \qquad \text{(C)} \ 90 \qquad \text{(D)} \ 104 \qquad \text{(E)} \ 105$

It is known that diagonals of a square, as well as a regular pentagon, are all equal. Find the bigeest natural $n$ such that a convex $n$-gon has all it's diagonals equal.

Barbara and Jenna play the following game, in which they take turns. A number of coins lie on a table. When it is Barbara's turn, she must remove $2$ or $4$ coins, unless only one coin remains, in which case she loses her turn. When it is Jenna's turn, she must remove $1$ or $3$ coins. A coin flip determines who goes first. Whoever removes the last coin wins the game. Assume both players use their best strategy. Who will win when the game starts with $2013$ coins and when the game starts with $2014$ coins? $\textbf{(A)}$ Barbara will win with $2013$ coins, and Jenna will win with $2014$ coins. $\textbf{(B)}$ Jenna will win with $2013$ coins, and whoever goes first will win with $2014$ coins. $\textbf{(C)}$ Barbara will win with $2013$ coins, and whoever goes second will win with $2014$ coins. $\textbf{(D)}$ Jenna will win with $2013$ coins, and Barbara will win with $2014$ coins. $\textbf{(E)}$ Whoever goes first will win with $2013$ coins, and whoever goes second will win with $2014$ coins.

South Adrican Magical Flights (SAMF) operates flights between South Adrican airports. If there is a flight from airport $A$ to airpost $B$, there will be also a flight from $B$ to $A$. The SAMF headquarters are located in Kimberley. Every airport that is served by Kimberley can be reached from Kimberley in precisely one way. This way of reaching Kimberley may involve stopping at other airports on the way. (For example, it may happen that you can get to Kimberley by flying from Durban to Bloemfontein and then from to Bloemfontein to Kimberley. In that case there is no other way to get from Durban to Kimberley. For example, there would be no direct Hight from Durban to Kimberley.) An airport (other than Kimberley) is called terminal if there are flights to (and from) precisely one other airport. Suppose that there are $t$ terminal airports. Due to budget cuts, SAMF decides to close down $k$ of the airports. It should still be possible to reach each of the remaining airports from Kimberley. Let $C$ be the number of choices for the $k$ destinations that are discontinued. Prove that $$\frac{t!}{k!(t-k)} \le C \le \frac{(t+k-1)!}{k!(t-1)!} .$$

$3$ uncoordinated aliens launch a $3$-day attack on $4$ galaxies. Each day, each of the three aliens chooses a galaxy uniformly at random from the remaining galaxies and destroys it. They make their choice simultaneously and independently, so two aliens could destroy the same galaxy. If the probability that every galaxy is destroyed by the end of the attack can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m,n$, what is $m+n$? [i]2020 CCA Math Bonanza Lightning Round #2.3[/i]

Let $ n \geq 2$ be a natural number and $ p(x)$ a real polynomial of degree at most $ n$ for which \[ \max _{ \minus{}1 \leq x \leq 1} |p(x)| \leq 1, \; p(\minus{}1)\equal{}p(1)\equal{}0 \ .\] Prove that then \[ |p'(x)| \leq \frac{n \cos \frac{\pi}{2n}}{\sqrt{1\minus{}x^2 \cos^2 \frac{\pi}{2n}}} \;\;\;\;\; \left( \minus{}\frac{1}{\cos \frac{\pi}{2n}} < x < \frac{1}{\cos \frac{\pi}{2n}} \\\\\ \right).\] [i]J. Szabados[/i]

Triangle $ABC$ has side length $AB = 5$, $BC = 12$, and $CA = 13$. Circle $\Gamma$ is centered at point $X$ exterior to triangle $ABC$ and passes through points $A$ and $C$. Let $D$ be the second intersection of $\Gamma$ with segment $\overline{BC}$. If $\angle BDA = \angle CAB$, the radius of $\Gamma$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m+n$. [i]2016 CCA Math Bonanza Lightning #3.3[/i]

Compute the real solution for$ x$ to the equation $$(4^x + 8)^4 - (8^x - 4)^4 = (4 + 8^x + 4^x)^4.$$

Consider all finite sequences of positive real numbers each of whose terms is at most $3$ and the sum of whose terms is more than $100$. For each such sequence, let $S$ denote the sum of the subsequence whose sum is the closest to $100$, and define the [i]defect[/i] of this sequence to be the value $|S-100|$. Find the maximum possible value of the defect.

Let $S$ be a nonempty set of positive integers such that, for any (not necessarily distinct) integers $a$ and $b$ in $S$, the number $ab+1$ is also in $S$. Show that the set of primes that do not divide any element of $S$ is finite. [i]Proposed by Carl Schildkraut[/i]

Given an acute triangle $ABC$ with $O$ as its circumcenter. Line $AO$ intersects $BC$ at $D$. Points $E$, $F$ are on $AB$, $AC$ respectively such that $A$, $E$, $D$, $F$ are concyclic. Prove that the length of the projection of line segment $EF$ on side $BC$ does not depend on the positions of $E$ and $F$.

Let $a_{1}, \ldots, a_{n}$ be an infinite sequence of strictly positive integers, so that $a_{k} < a_{k+1}$ for any $k.$ Prove that there exists an infinity of terms $ a_{m},$ which can be written like $a_m = x \cdot a_p + y \cdot a_q$ with $x,y$ strictly positive integers and $p \neq q.$