Let $ a_{0} \equal{} 1994$ and $ a_{n \plus{} 1} \equal{} \frac {a_{n}^{2}}{a_{n} \plus{} 1}$ for each nonnegative integer $ n$. Prove that $ 1994 \minus{} n$ is the greatest integer less than or equal to $ a_{n}$, $ 0 \leq n \leq 998$

As shown below, convex pentagon $ ABCDE$ has sides $ AB \equal{} 3$, $ BC \equal{} 4$, $ CD \equal{} 6$, $ DE \equal{} 3$, and $ EA \equal{} 7$. The pentagon is originally positioned in the plane with vertex $ A$ at the origin and vertex $ B$ on the positive $ x$-axis. The pentagon is then rolled clockwise to the right along the $ x$-axis. Which side will touch the point $ x \equal{} 2009$ on the $ x$-axis? [asy]size(250); defaultpen(linewidth(.8pt)+fontsize(8pt)); dotfactor=4; pair A=(0,0), Ep=7*dir(105), B=3*dir(0); pair D=Ep+B; pair C=intersectionpoints(Circle(D,6),Circle(B,4))[1]; pair[] ds={A,B,C,D,Ep}; dot(ds); draw(B--C--D--Ep--A); draw((6,6)..(8,4)..(8,3),EndArrow(3)); xaxis("$x$",-8,14,EndArrow(3)); label("$E$",Ep,NW); label("$D$",D,NE); label("$C$",C,E); label("$B$",B+(.2,.1),ENE); label("$A$",A+(-.1,.1),WNW); label("$(0,0)$",A,S); label("$3$",midpoint(A--B),N); label("$4$",midpoint(B--C),NW); label("$6$",midpoint(C--D),NE); label("$3$",midpoint(D--Ep),S); label("$7$",midpoint(Ep--A),W);[/asy]$ \textbf{(A)}\ \overline{AB} \qquad \textbf{(B)}\ \overline{BC} \qquad \textbf{(C)}\ \overline{CD} \qquad \textbf{(D)}\ \overline{DE} \qquad \textbf{(E)}\ \overline{EA}$

Positive integers $a$ and $b$ satisfy the equality $a+d(a)=b^2+2$ where $d(n)$ denotes the number of divisors of $n$. Prove that $a+b$ is even.

Find the side sizes of an isosceles trapezoid that has longest side $13$ cm, perimeter $28$ cm and area $27$ cm$^2$. Is there such a trapezoid, if we we ask for area $27.001$ cm$^2$ ?

Prove that the polynomial $P_n(x)=1+x+\frac{x^2}{2!}+\cdots +\frac{x^n}{n!}$ has no real zeros if $n$ is even and has exatly one real zero if $n$ is odd

In a non-isosceles triangle $ABC$ the bisectors of angles $A$ and $B$ are inversely proportional to the respective sidelengths. Find angle $C$.

A Boy Scouts camp holds a campfire. The camp has scarfs of m colors with n scarves of each color, and gives each of its $mn$ scouts a scarf, where $m, n \ge 2$ are integers. The camp then divides its scouts into troops by the color of their scarfs. At the beginning of the campfire, the scouts are seated in a circle so that scouts in the same troop are seated next to each other. The camp organizer then proceeds to select, round by round, representatives to perform a show, with the following conditions: there must be at least two representatives in each round, they must come from the same troop, and any specific set of representatives can only perform once. (For example, if $\{A, B\}$ has performed, then $\{A, B\}$ cannot perform again, but $\{A, B, C\}$ can still perform.) This process is repeated until all valid sets of representatives have performed. At this point, the organizers order each scout to hand their scarfs to the scout to the left, and re-group the scouts into troops, again according to their scarf color, and the process above is resumed, until the set of valid sets of representatives is exhausted again. (The sets of representatives after re-grouping must also be distinct from the sets before re-grouping.) When that happens, the organizers order another re-group, and resumes the process, and this repeats until there can be no further performances. Find, in simple form, the total number of performances that will be performed.

Naomi has three colors of paint which she uses to paint the pattern below. She paints each region a solid color, and each of the three colors is used at least once. If Naomi is willing to paint two adjacent regions with the same color, how many color patterns could Naomi paint? [asy] size(150); defaultpen(linewidth(2)); draw(origin--(37,0)--(37,26)--(0,26)--cycle^^(12,0)--(12,26)^^(0,17)--(37,17)^^(20,0)--(20,17)^^(20,11)--(37,11)); [/asy]

$n$ lines are given in the plane so that no three of them concur and no two are parallel. Show that there is a non-self-intersecting path consisting of $n$ straight segments so that each of the given lines contains exactly one of the segments of the path.

Amy places positive integers in each of these cells so each row and column contains each of $1,2,3,4,5$ exactly once. Find the sum of the numbers in the gray cells. [asy] import graph; size(4cm); fill((4,0)--(10,0)--(10,6)--(4,6)--cycle, gray(.7)); draw((0,0)--(10,0)); draw((0,2)--(10,2)); draw((0,4)--(10,4)); draw((0,6)--(10,6)); draw((0,8)--(10,8)); draw((0,10)--(10,10)); draw((0,0)--(0,10)); draw((2,0)--(2,10)); draw((4,0)--(4,10)); draw((6,0)--(6,10)); draw((8,0)--(8,10)); draw((10,0)--(10,10)); label("1",(1,9)); label("2",(3,9)); label("3",(1,7)); label("4",(3,7)); [/asy]

Define $L(x) = x - \frac{x^2}{2}$ for every real number $x$. If $n$ is a positive integer, define $a_n$ by \[ a_n = L \Bigl( L \Bigl( L \Bigl( \cdots L \Bigl( \frac{17}{n} \Bigr) \cdots \Bigr) \Bigr) \Bigr), \] where there are $n$ iterations of $L$. For example, \[ a_4 = L \Bigl( L \Bigl( L \Bigl( L \Bigl( \frac{17}{4} \Bigr) \Bigr) \Bigr) \Bigr). \] As $n$ approaches infinity, what value does $n a_n$ approach?

Determine the smallest positive integer $A$ with an odd number of digits and this property, that both $A$ and the number $B$ created by removing the middle digit of the number $A$ are divisible by $2018$.

What is the smallest positive integer (in base 10) that has a digit sum of 23 in base 20, and a digit sum of 20 in base 23? (The digit sums are in base 10.) [i]Team #8[/i]

Call a positive integer [i]monotonous[/i] if it is a one-digit number or its digits, when read from left to right, form either a strictly increasing or a strictly decreasing sequence. For example, 3, 23578, and 987620 are monotonous, but 88, 7434, and 23557 are not. How many monotonous positive integers are there? $\textbf{(A)} \text{ 1024} \qquad \textbf{(B)} \text{ 1524} \qquad \textbf{(C)} \text{ 1533} \qquad \textbf{(D)} \text{ 1536} \qquad \textbf{(E)} \text{ 2048}$

Let $a_1=3$, $a_2=7$, and $a_3=1$. Let $b_0=0$ and for all positive integers $n$, let $b_n=10b_{n-1}+a_n$. Compute $b_1\times b_2\times b_3$. [i]2020 CCA Math Bonanza Lightning Round #1.2[/i]

The two circles $\Gamma_1$ and $\Gamma_2$ intersect at $P$ and $Q$. The common tangent that's on the same side as $P$, intersects the circles at $A$ and $B$,respectively. Let $C$ be the second intersection with $\Gamma_2$ of the tangent to $\Gamma_1$ at $P$, and let $D$ be the second intersection with $\Gamma_1$ of the tangent to $\Gamma_2$ at $Q$. Let $E$ be the intersection of $AP$ and $BC$, and let $F$ be the intersection of $BP$ and $AD$. Let $M$ be the image of $P$ under point reflection with respect to the midpoint of $AB$. Prove that $AMBEQF$ is a cyclic hexagon.

Three integers are given. $A$ denotes the sum of the integers, $B$ denotes the sum of the square of the integers and $C$ denotes the sum of cubes of the integers(that is, if the three integers are $x, y, z$, then $A=x+y+z$, $B=x^2+y^2+z^2$, $C=x^3+y^3+z^3$). If $9A \geq B+60$ and $C \geq 360$, find $A, B, C$.

Do there exist 100 points on the plane such that the pairwise distances between them are pairwise distinct consecutive integer numbers larger than 2022?

For $ x,y,z \geq 0,$ establish the inequality \[ x(x\minus{}z)^2 \plus{} y(y\minus{}z)^2 \geq (x\minus{}z)(y\minus{}z)(x\plus{}y\minus{}z)\] and determine when equality holds.

Consider two circles $k_{1},k_{2}$ touching externally at point $T$. a line touches $k_{2}$ at point $X$ and intersects $k_{1}$ at points $A$ and $B$. Let $S$ be the second intersection point of $k_{1}$ with the line $XT$ . On the arc $\widehat{TS}$ not containing $A$ and $B$ is chosen a point $C$ . Let $\ CY$ be the tangent line to $k_{2}$ with $Y\in k_{2}$ , such that the segment $CY$ does not intersect the segment $ST$ . If $I=XY\cap SC$ . Prove that : (a) the points $C,T,Y,I$ are concyclic. (b) $I$ is the excenter of triangle $ABC$ with respect to the side $BC$.

Let $S_n=1+\frac{1}{2}+\frac{1}{3}+\ldots+\frac{1}{n}$, where $n$ is a positive integer. Prove that for any real numbers $a,b,0\le a\le b\le 1$, there exist infinite many $n\in\mathbb{N}$ such that \[a<S_n-[S_n]<b\] where $[x]$ represents the largest integer not exceeding $x$.

How many positive integers $n<2024$ are divisible by $\lfloor \sqrt{n}\rfloor-1$?

Let $A,B,M,C,D$ be distinct points on a line such that $AB=BM=MC=CD=6.$ Circles $\omega_1$ and $\omega_2$ with centers $O_1$ and $O_2$ and radius $4$ and $9$ are tangent to line $AD$ at $A$ and $D$ respectively such that $O_1,O_2$ lie on the same side of line $AD.$ Let $P$ be the point such that $PB\perp O_1M$ and $PC\perp O_2M.$ Determine the value of $PO_2^2-PO_1^2.$ [i]Proposed by Ray Li[/i]

In the plane let a finite family of circles be given satisfying the condition: every two circles, either are outside each other, either touch each other from outside and each circle touch at most 6 other circles. Suppose that every circle which does not touch 6 other circles be assigned a real number. Show that there exist at most one assignment to each remaining circle a real number equal to arithmetic mean of 6 numbers assigned to 6 circles which touch it.

Is it possible to cut a square into nine squares and colour one of them white, three of them grey and ve of them black, such that squares of the same colour have the same size and squares of different colours will have different sizes? [i](3 points)[/i]