Found problems: 92
2023 CCA Math Bonanza, L5.1
Estimate the number of ordered pairs $(a,b)$ of relatively prime positive integers such that $a+b<1412.$ Your score is determined by the function $max\{0, 20 - \lfloor \frac{|A - E|}{500}\rfloor\}$where $A$ is the actual answer, and $E$ is your estimate.
[i]Lightning 5.1[/i]
2016 CCA Math Bonanza, L4.4
Real numbers $X_1, X_2, \dots, X_{10}$ are chosen uniformly at random from the interval $[0,1]$. If the expected value of $\min(X_1,X_2,\dots, X_{10})^4$ can be expressed as a rational number $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$, what is $m+n$?
[i]2016 CCA Math Bonanza Lightning #4.4[/i]
2023 CCA Math Bonanza, T7
The positive integer equal to the expression
\[ \sum_{i=0}^{9} \left(i+(-9)^i\right)8^{9-i} \binom{9}{i}\]
is divisible by exactly six distinct primes. Find the sum of these six distinct prime factors.
[i]Team #7[/i]
2023 CCA Math Bonanza, L2.3
A frog starts at origin (0,0). At each minute it picks a random integer $x$, turns $x$ degrees counterclockwise, and jumps exactly 1 unit. After 2 minutes what is the probability that the frog is exactly one unit from the origin?
[i]Lightning 2.3[/i]
2023 CCA Math Bonanza, T5
Find the sum of all positive integers $k$ such that the sum of $k$ consecutive integers, starting from $20$, is a triangular number. \\(A triangular number is of the form $1+2+\dots+j$ for some integer $j$.)
[i]Team #5[/i]
2016 CCA Math Bonanza, L2.1
Bhairav runs a 15-mile race at 28 miles per hour, while Daniel runs at 15 miles per hour and Tristan runs at 10 miles per hour. What is the greatest length of time, in [i]minutes[/i], between consecutive runners' finishes?
[i]2016 CCA Math Bonanza Lightning #2.1[/i]
2016 CCA Math Bonanza, I14
Compute \[\sum_{k=1}^{420} \gcd(k,420).\]
[i]2016 CCA Math Bonanza Individual #14[/i]
2016 CCA Math Bonanza, L3.4
Let $S$ be the set of the reciprocals of the first $2016$ positive integers and $T$ the set of all subsets of $S$ that form arithmetic progressions. What is the largest possible number of terms in a member of $T$?
[i]2016 CCA Math Bonanza Lightning #3.4[/i]
2016 CCA Math Bonanza, L5.4
In the game of Colonel Blotto, you have 100 troops to distribute among 10 castles. Submit a 10-tuple $(x_1, x_2, \dots x_{10})$ of nonnegative integers such that $x_1 + x_2 + \dots + x_{10} = 100$, where each $x_i$ represent the number of troops you want to send to castle $i$. Your troop distribution will be matched up against each opponent's and you will win 10 points for each castle that you send more troops to (if you send the same number, you get 5 points, and if you send fewer, you get none). Your aim is to score the most points possible averaged over all opponents.
For example, if team $A$ submits $(90,10,0,\dots,0)$, team B submits $(11,11,11,11,11,11,11,11,11,1)$, and team C submits $(10,10,10,\dots 10)$, then team A will win 10 points against team B and 15 points against team C, while team B wins 90 points against team C. Team A averages 12.5 points, team B averages 90 points, and team C averages 47.5 points.
[i]2016 CCA Math Bonanza Lightning #5.4[/i]
2023 CCA Math Bonanza, L1.2
Find all positive integer solutions $a, b, c$ such that $(a - 1)\cdot(b - 2)\cdot(c - 3) = abc$
[i]Lightning 1.2[/i]
2016 CCA Math Bonanza, L1.2
What is the largest prime factor of $729-64$?
[i]2016 CCA Math Bonanza Lightning #1.2[/i]
2016 CCA Math Bonanza, I5
Let $ABC$ be a triangle with $AB = 3$, $BC = 4$, and $AC =5$. If $D$ is the projection from $B$ onto $AC$, $E$ is the projection from $D$ onto $BC$, and $F$ is the projection from $E$ onto $AC$, compute the length of the segment $DF$.
[i]2016 CCA Math Bonanza Individual #5[/i]
2023 CCA Math Bonanza, I3
A particle is moving randomly around a plane. It starts at $(0,0)$. Every second, it moves one unit randomly in a direction parallel to the $x$ or $y$ axis. At some time in the first hour, the particle was at the point $(2023,23)$. After $4092$ seconds, the particle is at $(x,y)$. Find the expected value of $x+y$.
[i]Individual #3[/i]
2023 CCA Math Bonanza, L5.3
Estimate the number of characters, excluding spaces, in the \LaTeX~source file for this Lightning Round, which includes the answer sheets and exactly one Asymptote diagram. Your score is determined by the function $max\{0, 20 - \lfloor \frac{|A - E|}{20}\rfloor\}$where $A$ is the actual answer, and $E$ is your estimate?
[i]Lightning 5.3[/i]
2016 CCA Math Bonanza, T4
In the [i]minesweeper[/i] game below, each unopened square (for example, the one in the top left corner) is either empty or contains a mine. The other squares are empty and display the number of mines in the neighboring 8 squares (if this is 0, the square is unmarked). What is the minimum possible number of mines present on the field?
[img]http://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvZS9hLzNlYTNhMjI2YWYyNmFkZGFiNWFmODBhNzA3YjA3OWM5MTZlNDlkLnBuZw==&rn=bWluZXN3ZWVwZXIucG5n[/img]
[i]2016 CCA Math Bonanza Team #4[/i]
2023 CCA Math Bonanza, T4
Triangle $ABC$ has side lengths $AB=7, BC=8, CA=9.$ Let $E$ be the foot from $B$ to $AC$ and $F$ be the foot from $C$ to $AB.$ Denote $M$ the midpoint of $BC.$ The circumcircles of $\triangle BMF$ and $\triangle CME$ meet at another point $G.$ Compute the length of $GC.$
[i]Team #4[/i]
2023 CCA Math Bonanza, I6
What’s the smallest integer $n>1$ such that $p \mid \left(n^{p-1}-1\right)$ for all integers $2 \leq p \leq 10?$
[i]Individual #6[/i]
2023 CCA Math Bonanza, T9
How many permutations $p$ of $\{1, 2, ..., 8\}$ satisfy $|p(p(a)) - a| \leq 1$ for all $a$?
[i]Team #9[/i]
2023 CCA Math Bonanza, L4.3
Define a rod to be a 1 by $n$ rectangle for any integer $n$. An $8 \times 8$ board is tiled with 13 rods so that all of it is covered without overlap. Find the maximum possible value of the product of the lengths of the 13 rods.
[i]Lightning 4.3[/i]
2016 CCA Math Bonanza, I9
Let $P\left(X\right)=X^5+3X^4-4X^3-X^2-3X+4$. Determine the number of monic polynomials $Q\left(x\right)$ with integer coefficients such that $\frac{P\left(X\right)}{Q\left(X\right)}$ is a polynomial with integer coefficients. Note: a monic polynomial is one with leading coefficient $1$ (so $x^3-4x+5$ is one but not $5x^3-4x^2+1$ or $x^2+3x^3$).
[i]2016 CCA Math Bonanza Individual #9[/i]
2016 CCA Math Bonanza, L2.4
What is the largest integer that must divide $n^5-5n^3+4n$ for all integers $n$?
[i]2016 CCA Math Bonanza Lightning #2.4[/i]
2023 CCA Math Bonanza, I14
The decimal expansion of $37^9$ is $129A617B979C077$ for digits $A, B,$ and $C$. Find the three digit number $ABC$.
[i]Individual #14[/i]
2016 CCA Math Bonanza, L3.2
Let $a_0 = 1$ and define the sequence $\{a_n\}$ by \[a_{n+1} = \frac{\sqrt{3}a_n - 1}{a_n + \sqrt{3}}.\] If $a_{2017}$ can be expressed in the form $a+b\sqrt{c}$ in simplest radical form, compute $a+b+c$.
[i]2016 CCA Math Bonanza Lightning #3.2[/i]
2023 CCA Math Bonanza, L3.2
Evan is a BONANZARANT addict. He wants to buy skins in BONANZARANT; however, he is broke (\$0 in his bank account). For the next ten days, Evan can either make \$25, or spend \$25 on a skin. He cannot buy a skin if he has no money. Calculate the amount of different ways Evan can buy 5 skins at the end of the 10 days assuming the skins each day are unique.
[i]Lightning 3.2[/i]
2023 CCA Math Bonanza, L1.3
Let $P$ and $Q$ be two concentric circles, and let $p_1 \dots p_{20}$ be equally spaced points around $P$ and $q_1 \dots q_{23}$ be equally spaced points around $Q$. How many ways are there to connect each $p_i$ to a distinct $q_j$ with some curve (not necessarily a straight line) so that no two curves cross and no curve crosses either circle?
[i]Lightning 1.3[/i]