Suppose that a planet contains $\left(CCAMATHBONANZA_{71}\right)^{100}$ people ($100$ in decimal), where in base $71$ the digits $A,B,C,\ldots,Z$ represent the decimal numbers $10,11,12,\ldots,35$, respectively. Suppose that one person on this planet is snapping, and each time they snap, at least half of the current population disappears. Estimate the largest number of times that this person can snap without disappearing. An estimate of $E$ earns $2^{1-\frac{1}{200}\left|A-E\right|}$ points, where $A$ is the actual answer. [i]2019 CCA Math Bonanza Lightning Round #5.2[/i]

Professor Conway collects a total of $58$ midterms from the two sections of his introductory linear algebra course. He notices that the number of midterms from the smaller section is equal to the product of the digits of the number of midterms from his larger section. Assuming that every student handed in a midterm, how many students are there in the smaller section?

If $a,b \ge 0$ are real numbers, prove the inequality $$\left(\frac{\sqrt{a}+\sqrt{b}}{2}\right)^2\leq\frac{a+\sqrt[3] {a^2b}+\sqrt[3] {ab^2}+b}{4}\leq\frac{a+\sqrt{ab}+b}{3} \leq \sqrt{\left(\frac{a^{2/3}+b^{2/3}}{2}\right)^{3}}$$ For each of the inequalities, find the cases of equality.

It is given $5$ numbers $1$, $3$, $5$, $7$, $9$. We get the new $5$ numbers such that we take arbitrary $4$ numbers(out of current $5$ numbers) $a$, $b$, $c$ and $d$ and replace them with $\frac{a+b+c-d}{2}$, $\frac{a+b-c+d}{2}$, $\frac{a-b+c+d}{2}$ and $\frac{-a+b+c+d}{2}$. Can we, with repeated iterations, get numbers: $a)$ $0$, $2$, $4$, $6$ and $8$ $b)$ $3$, $4$, $5$, $6$ and $7$

Prove that there exist infinitely many positive integers $n$ such that $\frac{4^n+2^n+1}{n^2+n+1}$ is a positive integer.

Let $ABCD$ be a isosceles trapezoid with $AB \| CD $ , $AD=BC$, $ AC \cap BD = $ { $O$ }. $ M $ is the midpoint of the side $AD$ . The circumcircle of triangle $ BCM $ intersects again the side $AD$ in $K$. Prove that $OK \| AB $ .

How many positive integers $n$ are there such that $3n^2 + 3n + 7$ is a perfect cube? $ \textbf{a)}\ 0 \qquad\textbf{b)}\ 1 \qquad\textbf{c)}\ 3 \qquad\textbf{d)}\ 7 \qquad\textbf{e)}\ \text{Infinitely many} $

Solve in non-negative integers the equation $125.2^n-3^m=271$

If $a_1,\cdots,a_n$ and $b_1,\cdots,b_n$ are real numbers satisfying $a_1^2+\cdots+a_n^2 \le 1$ and $b_1^2+\cdots+b_n^2 \le 1$ , show that: $$(1-(a_1^2+\cdots+a_n^2))(1-(b_1^2+\cdots+b_n^2)) \le (1-(a_1b_1+\cdots+a_nb_n))^2$$

Let $ABC$ be a triangle such that $AB = AC$. Let $D,E$ be points on $AB,AC$ respectively such that $DE = AC$. Let $DE$ meet the circumcircle of triangle $ABC$ at point $T$. Let $P$ be a point on $AT$. Prove that $PD + PE = AT$ if and only if $P$ lies on the circumcircle of triangle $ADE$.

In the adjoining diagram, $BO$ bisects $\angle CBA$, $CO$ bisects $\angle ACB$, and $MN$ is parallel to $BC$. If $AB=12$, $BC=24$, and $AC=18$, then the perimeter of $\triangle AMN$ is [asy] size(200); defaultpen(linewidth(0.7)+fontsize(10)); pair B=origin, C=(24,0), A=intersectionpoints(Circle(B,12), Circle(C,18))[0], O=incenter(A,B,C), M=intersectionpoint(A--B, O--O+40*dir(180)), N=intersectionpoint(A--C, O--O+40*dir(0)); draw(B--M--O--B--C--O--N--C^^N--A--M); label("$A$", A, dir(90)); label("$B$", B, dir(O--B)); label("$C$", C, dir(O--C)); label("$M$", M, dir(90)*dir(B--A)); label("$N$", N, dir(90)*dir(A--C)); label("$O$", O, dir(90));[/asy] $\textbf {(A) } 30 \qquad \textbf {(B) } 33 \qquad \textbf {(C) } 36 \qquad \textbf {(D) } 39 \qquad \textbf {(E) } 42$

Let $ A\equal{}A_1A_2A_3A_4$ be a tetrahedron, and suppose that for each $ j \not\equal{} k, [A_j,A_{jk}]$ is a segment of length $ \rho$ extending from $ A_j$ in the direction of $ A_k$. Let $ p_j$ be the intersection line of the planes $ [A_{jk}A_{jl}A_{jm}]$ and $ [A_kA_lA_m]$. Show that there are infinitely many straight lines that intersect the straight lines $ p_1,p_2,p_3,p_4$ simultaneously.

For how many ordered pairs of positive integers $(x,y),$ with $y<x\le 100,$ are both $\frac xy$ and $\frac{x+1}{y+1}$ integers?

If on $ \triangle ABC$, trinagles $ AEB$ and $ AFC$ are constructed externally such that $ \angle AEB\equal{}2 \alpha$, $ \angle AFB\equal{} 2 \beta$. $ AE\equal{}EB$, $ AF\equal{}FC$. COnstructed externally on $ BC$ is triangle $ BDC$ with $ \angle DBC\equal{} \beta$ , $ \angle BCD\equal{} \alpha$. Prove that 1. $ DA$ is perpendicular to $ EF$. 2. If $ T$ is the projection of $ D$ on $ BC$, then prove that $ \frac{DA}{EF}\equal{} 2 \frac{DT}{BC}$.

Let a triangle $ABC$ be given. Consider the circle touching its circumcircle at $A$ and touching externally its incircle at some point $A_1$. Points $B_1$ and $C_1$ are defined similarly. a) Prove that lines $AA_1, BB_1$ and $CC1$ concur. b) Let $A_2$ be the touching point of the incircle with $BC$. Prove that lines $AA_1$ and $AA_2$ are symmetric about the bisector of angle $\angle A$.

Square $ ABCD$ has side length $ 2$. A semicircle with diameter $ \overline{AB}$ is constructed inside the square, and the tangent to the semicricle from $ C$ intersects side $ \overline{AD}$ at $ E$. What is the length of $ \overline{CE}$? [asy] defaultpen(linewidth(0.8)); pair A=origin, B=(1,0), C=(1,1), D=(0,1), X=tangent(C, (0.5,0), 0.5, 1), F=C+2*dir(C--X), E=intersectionpoint(C--F, A--D); draw(C--D--A--B--C--E); draw(Arc((0.5,0), 0.5, 0, 180)); pair point=(0.5,0.5); label("$A$", A, dir(point--A)); label("$B$", B, dir(point--B)); label("$C$", C, dir(point--C)); label("$D$", D, dir(point--D)); label("$E$", E, dir(point--E));[/asy] $ \textbf{(A)}\ \frac {2 \plus{} \sqrt5}{2} \qquad \textbf{(B)}\ \sqrt 5 \qquad \textbf{(C)}\ \sqrt 6 \qquad \textbf{(D)}\ \frac52 \qquad \textbf{(E)}\ 5 \minus{} \sqrt5$

Circle $\Omega(O,R)$ and its chord $AB$ is given. Suppose $C$ is midpoint of arc $AB$. $X$ is an arbitrary point on the cirlce. Perpendicular from $B$ to $CX$ intersects circle again in $D$. Perpendicular from $C$ to $DX$ intersects circle again in $E$. We draw three lines $\ell_{1},\ell_{2},\ell_{3}$ from $A,B,E$ parralell to $OX,OD,OC$. Prove that these lines are concurrent and find locus of concurrncy point.

\begin{quote} Ted quite likes haikus, \\ poems with five-seven-five, \\ but Ted knows few words. He knows $2n$ words \\ that contain $n$ syllables \\ for every int $n$. Ted can only write \\ $N$ distinct haikus. Find $N$. \\ Take mod one hundred. \end{quote} Ted loves creating haikus (Japanese three-line poems with $5$, $7$, $5$ syllables each), but his vocabulary is rather limited. In particular, for integers $1 \le n \le 7$, he knows $2n$ words with $n$ syllables. Furthermore, words cannot cross between lines, but may be repeated. If Ted can make $N$ distinct haikus, compute the remainder when $N$ is divided by $100$. [i]Proposed by Lewis Chen[/i]

Suppose that $a$ and $ b$ are distinct positive integers satisfying $20a + 17b = p$ and $17a + 20b = q$ for certain primes $p$ and $ q$. Determine the minimum value of $p + q$.

The CCA Math Banana$^{\text{TM}}$ costs \$$100$. The cost rises $10$% then drops $10$%. Now what is the cost of the CCA Math Banana$^{\text{TM}}$? [i]2018 CCA Math Bonanza Lightning Round #1.2[/i]

The following four statements, and only these are found on a card: [asy] pair A,B,C,D,E,F,G; A=(0,1); B=(0,5); C=(11,5); D=(11,1); E=(0,4); F=(0,3); G=(0,2); draw(A--B--C--D--cycle); label("On this card exactly one statement is false.", B, SE); label("On this card exactly two statements are false.", E, SE); label("On this card exactly three statements are false.", F, SE); label("On this card exactly four statements are false.", G, SE); [/asy] (Assume each statement is either true or false.) Among them the number of false statements is exactly $\textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 3 \qquad \textbf{(E)}\ 4$

Let m be a positive integer, and let $S=\{a_1=1, a_2, ..., a_m\}$ be a set of positive integers. Prove that there exists a positive integer $n$ with $n\le m$ and a set $T={b_1, b_2, ..., b_n}$ of positive integers such that (a) all the subsets of $T$ have distinct sums of elements; (b) each of the numbers $a_1$, $a_2$, ..., $a_m$ is the sum of the elements of a subset of $T$.

Quadrilateral $ABCD$ is such that $AB \perp CD$ and $AD \perp BC$. Prove that there exist a point such that the distances from it to the sidelines are proportional to the lengths of the corresponding sides.

[b]H[/b]orizontal parallel segments $AB=10$ and $CD=15$ are the bases of trapezoid $ABCD$. Circle $\gamma$ of radius $6$ has center within the trapezoid and is tangent to sides $AB$, $BC$, and $DA$. If side $CD$ cuts out an arc of $\gamma$ measuring $120^{\circ}$, find the area of $ABCD$.

A polynomial $ Q (x) = k_n x ^ n + k_ {n-1} x ^ {n-1} + \ldots + k_1 x + k_0 $ with real coefficients is called [i]powerful[/i] if the equality $ | k_0 | = | k_1 | + | k_2 | + \ldots + | k_ {n-1} | + | k_n | $, and [i]non-increasing[/i] , if $ k_0 \geq k_1 \geq \ldots \geq k_ {n-1} \geq k_n $. Let for the polynomial $ P (x) = a_d x ^ d + a_ {d-1} x ^ {d-1} + \ldots + a_1 x + a_0 $ with nonzero real coefficients, where $ a_d> 0 $, the polynomial $ P (x) (x-1) ^ t (x + 1) ^ s $ is [i]powerful[/i] for some non-negative integers $ s $ and $ t $ ($ s + t> 0 $). Prove that at least one of the polynomials $ P (x) $ and $ (- 1) ^ d P (-x) $ is [i]nonincreasing[/i].