This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

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Found problems: 92

2016 CCA Math Bonanza, L3.3
Tags: CCA Math Bonanza
Triangle $ABC$ has side length $AB = 5$, $BC = 12$, and $CA = 13$. Circle $\Gamma$ is centered at point $X$ exterior to triangle $ABC$ and passes through points $A$ and $C$. Let $D$ be the second intersection of $\Gamma$ with segment $\overline{BC}$. If $\angle BDA = \angle CAB$, the radius of $\Gamma$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m+n$. [i]2016 CCA Math Bonanza Lightning #3.3[/i]

2016 CCA Math Bonanza, I7
Tags: CCA Math Bonanza
Simon is playing chess. He wins with probability 1/4, loses with probability 1/4, and draws with probability 1/2. What is the probability that, after Simon has played 5 games, he has won strictly more games than he has lost? [i]2016 CCA Math Bonanza Individual #7[/i]

2023 CCA Math Bonanza, T1
Tags: CCA Math Bonanza
Evan's bike lock has been stolen by Jonathan, and he has changed the passcode. Jonathan is refusing to tell Evan the passcode. All Evan knows is it is a five-digit number with following properties: (a) It can be written as $a\cdot \overline{ab}\cdot\overline{abc}$ where $a, b, c$ are pairwise different digits and $a$, $\overline{ab}$, $\overline{abc}$ are prime. (b) The sum of its digits is 21. (c) The passcode's last digit is $c$. Find the bike passcode. [i]Team #1[/i]

2016 CCA Math Bonanza, I1
Tags: CCA Math Bonanza
Compute the integer $$\frac{2^{\left(2^5-2\right)/5-1}-2}{5}.$$ [i]2016 CCA Math Bonanza Individual Round #1[/i]

2016 CCA Math Bonanza, I6
Tags: geometry , geometric transformation , rotation , CCA Math Bonanza
Let $a,b,c$ be non-zero real numbers. The lines $ax + by = c$ and $bx + cy = a$ are perpendicular and intersect at a point $P$ such that $P$ also lies on the line $y=2x$. Compute the coordinates of point $P$. [i]2016 CCA Math Bonanza Individual #6[/i]

2023 CCA Math Bonanza, I5
Tags: CCA Math Bonanza
Find the sum of all distinct prime factors of $2023^3 - 2000^3 - 23^3$. [i]Individual #5[/i]

2016 CCA Math Bonanza, T3
Tags: CCA Math Bonanza
Find the sum of all integers $n$ not less than $3$ such that the measure, in degrees, of an interior angle of a regular $n$-gon is an integer. [i]2016 CCA Math Bonanza Team #3[/i]

2023 CCA Math Bonanza, T3
Tags: CCA Math Bonanza
There are exactly 3 distinct 4-digit factors of 3212005. Find their sum. [i]Team #3[/i]

2016 CCA Math Bonanza, L5.3
Tags: CCA Math Bonanza , function
Let $A(x)=\lfloor\frac{x^2-20x+16}{4}\rfloor$, $B(x)=\sin\left(e^{\cos\sqrt{x^2+2x+2}}\right)$, $C(x)=x^3-6x^2+5x+15$, $H(x)=x^4+2x^3+3x^2+4x+5$, $M(x)=\frac{x}{2}-2\lfloor\frac{x}{2}\rfloor+\frac{x}{2^2}+\frac{x}{2^3}+\frac{x}{2^4}+\ldots$, $N(x)=\textrm{the number of integers that divide }\left\lfloor x\right\rfloor$, $O(x)=|x|\log |x|\log\log |x|$, $T(x)=\sum_{n=1}^{\infty}\frac{n^x}{\left(n!\right)^3}$, and $Z(x)=\frac{x^{21}}{2016+20x^{16}+16x^{20}}$ for any real number $x$ such that the functions are defined. Determine $$C(C(A(M(A(T(H(B(O(N(A(N(Z(A(2016)))))))))))))).$$ [i]2016 CCA Math Bonanza Lightning #5.3[/i]

2023 CCA Math Bonanza, T6
Tags: CCA Math Bonanza
$ABC$ is an equilateral triangle and $l$ is a line such that the distances from $A, B,$ and $C$ to $l$ are $39, 35,$ and $13$, respectively. Find the largest possible value of $AB$. [i]Team #6[/i]

2016 CCA Math Bonanza, T1
Tags: CCA Math Bonanza
It takes $3$ rabbits $5$ hours to dig $9$ holes. It takes $5$ beavers $36$ minutes to build $2$ dams. At this rate, how many more minutes does it take $1$ rabbit to dig $1$ hole than it takes $1$ beaver to build $1$ dam? [i]2016 CCA Math Bonanza Team #1[/i]

2023 CCA Math Bonanza, I9
Tags: geometry , angle bisector , CCA Math Bonanza
Let $ABC$ be a triangle with $AB=3, BC=4, CA=5$. Let $M$ be the midpoint of $BC$, and $\Gamma$ be a circle through $A$ and $M$ that intersects $AB$ and $AC$ again at $D$ and $E$, respectively. Given that $AD=AE$, find the area of quadrilateral $MEAD$. [i]Individual #9[/i]

2016 CCA Math Bonanza, I11
Tags: CCA Math Bonanza
How many ways are there to place 8 1s and 8 0s in a $4\times 4$ array such that the sum in every row and column is 2? \begin{tabular}{|c|c|c|c|} \hline 1 & 0 & 0 & 1 \\ \hline 0 & 1 & 1 & 0 \\ \hline 0 & 1 & 1 & 0 \\ \hline 1 & 0 & 0 & 1 \\ \hline \end{tabular} [i]2016 CCA Math Bonanza Individual #11[/i]

2016 CCA Math Bonanza, T7
Tags: CCA Math Bonanza , geometry , 3D geometry , octahedron
A [i]cuboctahedron[/i], shown below, is a polyhedron with 8 equilateral triangle faces and 6 square faces. Each edge has the same length and each of the 24 vertices borders 2 squares and 2 triangles. An \textit{octahedron} is a regular polyhedron with 6 vertices and 8 equilateral triangle faces. Compute the sum of the volumes of an octahedron with side length 5 and a cuboctahedron with side length 5. [img]http://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvMi82LzBmNjM1OTM2M2ExYTQzOTFhODEwODkwM2FiYmM1MTljOGQzNmJhLmpwZw==&rn=Q3Vib2N0YWhlZHJvbi5qcGc=[/img] [i]2016 CCA Math Bonanza Team #7[/i]

2016 CCA Math Bonanza, I10
Tags: CCA Math Bonanza
Let $ABC$ be a triangle with $AC = 28$, $BC = 33$, and $\angle ABC = 2\angle ACB$. Compute the length of side $AB$. [i]2016 CCA Math Bonanza #10[/i]

2023 CCA Math Bonanza, L5.4
Tags: CCA Math Bonanza
Submit a positive integer $N$ between 1 and 20, inclusive. If $C$ is the total number of teams that submit $N$ for this question, your score will be $\lfloor\frac{N}{C}\rfloor$ [i]Lightning 5.4[/i]

2023 CCA Math Bonanza, TB1
Tags: CCA Math Bonanza
$\text{Find }\left(\sum_{k=1}^{2023}{(k^{42432})}\right)\text{ mod 2023}$ [i]Tiebreaker #1[/i]