Found problems: 92
2023 CCA Math Bonanza, L4.2
A mouse is on the below grid:
\begin{center}
\begin{asy}
unitsize(2cm);
filldraw(circle((0,0),0.07), black);
filldraw(circle((0,1),0.07), black);
filldraw(circle((1,0),0.07), black);
filldraw(circle((0.5,0.5),0.07), black);
filldraw(circle((1,1),0.07), black);
draw((0,0)--(1,0));
draw((0,0)--(0,1));
draw((1,0)--(1,1));
draw((0,1)--(1,1));
draw((0,1)--(0.5,0.5));
draw((1,0)--(0.5,0.5));
draw((1,1)--(0.5,0.5));
draw((0,0)--(0.5,0.5));
\end{asy}
\end{center}
The paths connecting each node are the possible paths the mouse can take to walk from a node to another node. Call a ``turn" the action of a walk from one node to another. Given the mouse starts off on an arbitrary node, what is the expected number of turns it takes for the mouse to return to its original node?
[i]Lightning 4.2[/i]
2023 CCA Math Bonanza, TB3
Triangle $ABC$ has incircle centered at $I.$ Define $M$ and $N$ the midpoints of $BC$ and $CA,$ respectively. Extend $BI$ and $MN$ to meet at a point $K.$ The circumcircle of $\triangle BKC$ intersects the incircle at two points $D$ and $G,$ where $D$ is closer to $AB$ than $G.$ Line $BK$ intersects the incircle at two points $E$ and $F,$ where $FK<EK.$ Let $H$ be $DC \cap BK$. Given that $BD=3$ and $DF=4,$ compute $\tfrac{BE}{EF} \cdot \tfrac{BH}{HF}.$
[i]Tiebreaker #3[/i]
2016 CCA Math Bonanza, I3
Amanda has the list of even numbers $2, 4, 6, \dots 100$ and Billy has the list of odd numbers $1, 3, 5, \dots 99$. Carlos creates a list by adding the square of each number in Amanda's list to the square of the corresponding number in Billy's list. Daisy creates a list by taking twice the product of corresponding numbers in Amanda's list and Billy's list. What is the positive difference between the sum of the numbers in Carlos's list and the sum of the numbers in Daisy's list?
[i]2016 CCA Math Bonanza Individual #3[/i]
2016 CCA Math Bonanza, T8
As $a$, $b$ and $c$ range over [i]all[/i] real numbers, let $m$ be the smallest possible value of $$2\left(a+b+c\right)^2+\left(ab-4\right)^2+\left(bc-4\right)^2+\left(ca-4\right)^2$$ and $n$ be the number of ordered triplets $\left(a,b,c\right)$ such that the above quantity is minimized. Compute $m+n$.
[i]2016 CCA Math Bonanza Team #8[/i]
2016 CCA Math Bonanza, T5
How many permutations of the word ``ACADEMY'' have that there exist two vowels that are separated by an odd distance? For example, the X and Y in XAY are separated by an even distance, while the X and Y in XABY are separated by an odd distance. Note: the vowels are A, E, I, O, and U. Y is [b]NOT[/b] a vowel.
[i]2016 CCA Math Bonanza Team #5[/i]
2016 CCA Math Bonanza, L5.2
In this problem, the symbol $0$ represents the number zero, the symbol $1$ represents the number seven, the symbol $2$ represents the number five, the symbol $3$ represents the number three, the symbol $4$ represents the number four, the symbol $5$ represents the number two, the symbol $6$ represents the number nine, the symbol $7$ represents the number one, the symbol $8$ represents an arbitrarily large positive integer, the symbol $9$ represents the number six, and the symbol $\infty$ represents the number eight. Compute the value of $\left|0-1+2-3^4-5+6-7^8\times9-\infty\right|$.
[i]2016 CCA Math Bonanza Lightning #5.2[/i]
2023 CCA Math Bonanza, L4.1
A pack of MIT students are holding an escape room, where students may compete in teams of 4, 5, or 6. There is \$60 dollars worth of prize money in Amazon gift cards for the winning team. If each gift card can contain any whole number of dollars, what is the minimum number of gift cards required so that the prize money can be distributed evenly among any team?
[i]Lightning 4.1[/i]
2016 CCA Math Bonanza, L1.3
If the GCD of $a$ and $b$ is $12$ and the LCM of $a$ and $b$ is $168$, what is the value of $a\times b$?
[i]2016 CCA Math Bonanza L1.3[/i]
2023 CCA Math Bonanza, I1
How many positive integers have digits whose product is 20 and sum is 23?
[i]Individual #1[/i]
2023 CCA Math Bonanza, L1.1
If 100 dice are rolled, what is the probability that the sum of the numbers rolled is even?
[i]Lightning 1.1[/i]
2023 CCA Math Bonanza, I4
What is the minimum possible perimeter of a right triangle with integer side lengths whose perimeter is equal to its area?
[i]Individual #4[/i]
2016 CCA Math Bonanza, I2
Rectangle $ABCD$ has perimeter $178$ and area $1848$. What is the length of the diagonal of the rectangle?
[i]2016 CCA Math Bonanza Individual Round #2[/i]
2023 CCA Math Bonanza, L4.4
Let $ABC$ be a triangle with side lengths $AB=6, BC=7, CA=8$ and circumcircle $\omega.$ Denote $M$ to be the midpoint of $BC.$ Let $P$ be the intersection of the tangent to $\omega$ at $A$ and $BC.$ The line parallel to $BC$ passing through $A$ intersects $\omega$ at another point $D.$ The tangent to $\omega$ passing through $P$ that is not $PA$ intersects $DM$ at a point $Q.$ Denote $J$ to be the intersection of $(BMQ)$ and $AQ.$ Extend $BJ$ to intersect $AC$ at $E.$ Compute $\tfrac{BJ}{JE}.$
[i]Lightning 4.4[/i]
2023 CCA Math Bonanza, I2
Derek rolls three $20$-sided dice. Given their sum is $23,$ find the probability one of them shows a 20.
[i]Individual #2[/i]
2023 CCA Math Bonanza, T8
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]
2016 CCA Math Bonanza, L4.3
Let $ABC$ be a non-degenerate triangle with perimeter $4$ such that $a=bc\sin^2A$. If $M$ is the maximum possible area of $ABC$ and $m$ is the minimum possible area of $ABC$, then $M^2+m^2$ can be expressed in the form $\frac{a}{b}$ for relatively prime positive integers $a$ and $b$. Compute $a+b$.
[i]2016 CCA Math Bonanza Lightning #4.3[/i]
2016 CCA Math Bonanza, L5.1
The first question was asked in Set 4. The second question was asked in Set 5.
Question) Eshaan the Elephant has a long memory. He remembers that out of the integers $0, 1, 2, \dots, 15$, one of them is special. Submit to the grader an ordered 4-tuple of subsets of $0, 1, 2, \dots, 15$ and they will tell you whether the special number is in each. You can then submit your guess for the special number on the next round for points. (You might want to write down a copy of your submission somewhere other than your answer sheet. Note that this question itself is not worth any points, though the corresponding problem in Set 5 is.)
Question) Eshaan the Elephant has a long memory. He remembers that out of the integers $0, 1, 2, \dots, 15$, one of them is special. You have submitted an ordered 4-tuple of subsets of $0, 1, 2, \dots, 15$. Here is your reply from the grader.
\begin{tabular}{|c|c|c|c|}
\hline
1 & 2 & 3 & 4 \\ \hline
Y/N & Y/N & Y/N & Y/N \\ \hline
\end{tabular}
What is the special number?
[i]2016 CCA Math Bonanza Lightning #5.1[/i]
2023 CCA Math Bonanza, I11
Let $ABC$ be a triangle such that $AB=\sqrt{10}, BC=4,$ and $CA=3\sqrt{2}.$ Circle $\omega$ has diameter $BC,$ with center at $O.$ Extend the altitude from $A$ to $BC$ to hit $\omega$ at $P$ and $P',$ where $AP < AP'.$ Suppose line $P'O$ intersects $AC$ at $X.$ Given that $PX$ can be expressed as $m\sqrt{n}-\sqrt{p},$ where $n$ and $p$ are squarefree, find $m+n+p.$
[i]Individual #11[/i]
2016 CCA Math Bonanza, L2.2
In triangle $ABC$, $AB=7$, $AC=9$, and $BC=8$. The angle bisector of $\angle{BAC}$ intersects side $BC$ at $D$, and the angle bisector of $\angle{ABC}$ intersects $AD$ at $E$. Compute $AE^2$.
[i]2016 CCA Math Bonanza Lightning #2.2[/i]
2023 CCA Math Bonanza, L2.2
For a positive integer $n$ let $f(n)$ denote the number of ways to put $n$ objects into pairs if the only thing that matters is which object each object gets paired with. Find the sum of all $f(f(2k))$, where $k$ ranges from 1 to 2023.
[i]Lightning 2.2[/i]
2016 CCA Math Bonanza, I12
Let $f$ be a function from the set $X = \{1,2, \dots, 10\}$ to itself. Call a partition ($S$, $T$) of $X$ $f-balanced$ if for all $s \in S$ we have $f(s) \in T$ and for all $t \in T$ we have $f(t) \in S$. (A partition $(S, T)$ is a pair of subsets $S$ and $T$ of $X$ such that $S\cap T = \emptyset$ and $S \cup T = X$. Note that $(S, T)$ and $(T, S)$ are considered the same partition).
Let $g(f)$ be the number of $f-balanced$ partitions, and let $m$ equal the maximum value of $g$ over all functions $f$ from $X$ to itself. If there are $k$ functions satisfying $g(f) = m$, determine $m+k$.
[i]2016 CCA Math Bonanza Individual #12[/i]
2023 CCA Math Bonanza, L1.4
Find the area of the shaded region.
[i]Lightning 1.4[/i]
2023 CCA Math Bonanza, T2
How many ways are there to fill an $8\times8\times8$ cube with $1\times1\times8$ sticks? Rotations and reflections are considered distinct.
[i]Team #2[/i]
2016 CCA Math Bonanza, L1.4
A triangle has a perimeter of $4$ [i]yards[/i] and an area of $6$ square [i]feet[/i]. If one of the angles of the triangle is right, what is the length of the largest side of the triangle, in feet?
[i]2016 CCA Math Bonanza Lightning #1.4[/i]
2016 CCA Math Bonanza, T6
Consider the polynomials $P\left(x\right)=16x^4+40x^3+41x^2+20x+16$ and $Q\left(x\right)=4x^2+5x+2$. If $a$ is a real number, what is the smallest possible value of $\frac{P\left(a\right)}{Q\left(a\right)}$?
[i]2016 CCA Math Bonanza Team #6[/i]