Estimate the number of times a one-digit answer (0, 1, 2, 3, 4, 5, 6, 7, 8, or 9) has been submitted as an answer for any question by any team in the first 4 sets of this competition's lightning round. An estimate $E$ earns $\frac{2}{1+|log_2(A)-log_2(E)|}$ points, where $A$ is the actual answer. [i]2022 CCA Math Bonanza Lightning Round 5.3[/i]

Let $ABC$ be a triangle with $\angle{A} = 60^\circ$. The midperpendicular of segment $AB$ meets line $AC$ at point $C_1$. The midperpendicular of segment $AC$ meets line $AB$ at point $B_1$. Prove that line $B_1C_1$ touches the incircle of triangle $ABC$.

Let $P$ be the parabola with equation $y = x^2$ and let $Q = (20, 14)$ There are real numbers $r$ and $s$ such that the line through $Q$ with slope $m$ does not intersect $P$ if and only if $r < m < s$. What is $r + s?$ $ \textbf{(A)} 1 \qquad \textbf{(B)} 26 \qquad \textbf{(C)} 40 \qquad \textbf{(D)} 52 \qquad \textbf{(E)} 80 \qquad $

Suppose a triangle $\vartriangle ABC$ with $\angle C = 90^o$ is given. Let $D$ be the midpoint of $AC$, and let $E$ be the foot of the altitude through $C$ on $BD$. Show that the tangent in $C$ of the circumcircle of $\vartriangle AEC$ is perpendicular to $AB$.

Let $C_1,C_2$ be two circles intersecting at points $A,P$ with centers $O,K$ respectively. Let $B,C$ be the symmetric of $A$ wrt $O,K$ in circles $C_1,C_2 $ respectively. A random line passing through $A$ intersects circles $C_1,C_2$ at $D,E$ respectively. Prove that the center of circumcircle of triangle $DEP$ lies on the circumcircle of triangle $OKP$.

Let $\mathbb R^+$ denote the set of positive real numbers.Find all functions $f:\mathbb R^+\rightarrow \mathbb R^+$ satisfying the following equation for all $x,y\in \mathbb R^+$: $$f(xy+f(y)^2)=f(x)f(y)+yf(y)$$

Determine all real solutions of the system \begin{align*} x+y+z &=3, \\ \frac1x+\frac1y+\frac1z &= \frac{5}{12}, \\ x^3+y^3+z^3 &=45. \end{align*}

What is the smallest amount that may be invested at interest rate $i$, compounded annually, in order that one may withdraw $1$ dollar at the end of the first year, $4$ dollars at the end of the second year, $\ldots$ , $n^2$ dollars at the end of the $n$-th year, in perpetuity?

A complex quartic polynomial $Q$ is [i]quirky [/i] if it has four distinct roots, one of which is the sum of the other three. There are four complex values of $k$ for which the polynomial $Q(x) = x^4-kx^3-x^2-x-45$ is quirky. Compute the product of these four values of $k$.

In concurrent computing, two processes may have their steps interwoven in an unknown order, as long as the steps of each process occur in order. Consider the following two processes: \begin{tabular}{c|cc} Process & $A$ & $B$\\ \hline Step 1 & $x\leftarrow x-4$ & $x\leftarrow x-5$\\ Step 2 & $x\leftarrow x\cdot3$ & $x\leftarrow x\cdot4$\\ Step 3 & $x\leftarrow x-4$ & $x\leftarrow x-5$\\ Step 4 & $x\leftarrow x\cdot3$ & $x\leftarrow x\cdot4$ \end{tabular} One such interweaving is $A1$, $B1$, $A2$, $B2$, $A3$, $B3$, $B4$, $A4$, but $A1$, $A3$, $A2$, $A4$, $B1$, $B2$, $B3$, $B4$ is not since the steps of $A$ do not occur in order. We run $A$ and $B$ concurrently with $x$ initially valued at $6$. Find the minimal possible value of $x$ among all interweavings.

Does there exist non-zero complex numbers $a, b, c$ and natural number $h$ such that if integers $k, l, m$ satisfy $|k| + |l| + |m| \geq 1996$, then $|ka + lb + mc| > \frac {1}{h}$ is true?

Let $ABC$ be a triangle with $\angle A=60^\circ$; $AD$, $BE$, and $CF$ be its bisectors; $P, Q$ be the projections of $A$ to $EF$ and $BC$ respectively; and $R$ be the second common point of the circle $DEF$ with $AD$. Prove that $P, Q, R$ are collinear.

In an acute-angled triangle $ABC$, the points $M$ and $N$ are the midpoints of the sides $AB$ and $AC$, respectively. For an arbitrary point $S$ lying on the side of $BC$ prove that the condition holds $(MB- MS)(NC-NS) \le 0$

Show that given any nine integers, we can find four, $a, b, c, d$ such that $a + b - c - d$is divisible by $20$. Show that this is not always true for eight integers.

A neighborhood consists of $10 \times 10$ squares. On New Year's Eve it snowed for the first time and since then exactly $10$ cm of snow fell on each square every night (and snow fell only at night). Every morning, the janitor selects one row or column and shovels all the snow from there onto one of the adjacent rows or columns (from each cell to the adjacent side). For example, he can select the seventh column and from each of its cells shovel all the snow into the cell of the left of it. You cannot shovel snow outside the neighborhood. On the evening of the 100th day of the year, an inspector will come to the city and find the cell with the snowdrift of maximal height. The goal of the janitor is to ensure that this height is minimal. What height of snowdrift will the inspector find? [i]Proposed by A. Solynin[/i]

Find the set of all quadruplets $(x,y, z,w)$ of non-zero real numbers which satisfy $$1 +\frac{1}{x}+\frac{2(x + 1)}{xy}+\frac{3(x + 1)(y + 2)}{xyz}+\frac{4(x + 1)(y + 2)(z + 3)}{xyzw}= 0$$