Found problems: 190
2016 CMIMC, 3
Sophia writes an algorithm to solve the graph isomorphism problem. Given a graph $G=(V,E)$, her algorithm iterates through all permutations of the set $\{v_1, \dots, v_{|V|}\}$, each time examining all ordered pairs $(v_i,v_j)\in V\times V$ to see if an edge exists. When $|V|=8$, her algorithm makes $N$ such examinations. What is the largest power of two that divides $N$?
2016 ASDAN Math Tournament, 22
An $n\times n$ Latin square is a $n\times n$ grid that is filled with $n$ $1$'s, $n$ $2$'s, $\dots$, and $n$ $n$'s such that each column and row of the grid contains exactly one of each $1$, $2$, $\dots$, $n$. For example, the following is a valid $2\times2$ Latin square: $\textstyle\begin{bmatrix}2&1\\1&2\end{bmatrix}$, but this is not: $\textstyle\begin{bmatrix}2&1\\2&1\end{bmatrix}$. How many $4\times4$ Latin squares are there?
2016 ASDAN Math Tournament, 2
Define a $\textit{subsequence}$ of a string $\mathcal{S}$ of letters to be a positive-lenght string using any number of the letters in $\mathcal{S}$ in order. For example, a subsequence of $HARRISON$ is $ARRON$. Compute the number of subsequences in $HARRISON$.
2016 ASDAN Math Tournament, 10
Compute the smallest positive integer $x$ which satisfies $x^2-8x+1\equiv0\pmod{22}$ and $x^2-22x+1\equiv0\pmod{8}$.
2016 CMIMC, 2
For each integer $n\geq 1$, let $S_n$ be the set of integers $k > n$ such that $k$ divides $30n-1$. How many elements of the set \[\mathcal{S} = \bigcup_{i\geq 1}S_i = S_1\cup S_2\cup S_3\cup\ldots\] are less than $2016$?
2016 ASDAN Math Tournament, 2
Suppose $a$ and $b$ are two variables that satisfy $\textstyle\int_0^2(-ax^2+b)dx=0$. What is $\tfrac{a}{b}$?
2016 ASDAN Math Tournament, 5
$ABCD$ is a four digit number ($A\neq0$) such that both $ABC$ and $BCD$ are divisible by $9$ ($ABCD$ is not necessarily divisible by $9$, and $B,C,D$ may be $0$). Compute the number of four digit numbers satisfying this property.
2016 ASDAN Math Tournament, 1
Compute the area of the trapezoid $ABCD$ with right angles $BAD$ and $ADC$ and side lengths of $AB=3$, $BC=5$, and $CD=7$.
2016 ASDAN Math Tournament, 1
Moor owns $3$ shirts, one each of black, red, and green. Moor also owns $3$ pairs of pants, one each of white, red, and green. Being stylish, he decides to wear an outfit consisting of one shirt and one pair of pants that are different colors. How many combinations of shirts and pants can Moor choose?
2016 ASDAN Math Tournament, 3
Find the $2016$th smallest positive integer that is a solution to $x^x\equiv x\pmod{5}$.
2017 NIMO Problems, 2
Let $\{a_n\}$ be a sequence of integers such that $a_1=2016$ and \[\dfrac{a_{n-1}+a_n}2=n^2-n+1\] for all $n\geq 1$. Compute $a_{100}$.
[i]Proposed by David Altizio[/i]
2016 ASDAN Math Tournament, 5
Let $\Gamma_1$ be a circle of radius $6$, and let $\Gamma_2$ be a circle of radius $1$. Next, let the circles be internally tangent at point $P$, and let $AP$ be a diameter of circle $\Gamma_1$. Finally, let $Y$ be a point on $\Gamma_2$ such that $AY$ is tangent to it. Compute the length of $PY$.
2016 ASDAN Math Tournament, 2
A four-pointed star is formed by placing for equilateral triangles of side length $4$ in a coordinate grid. The triangles are placed such that their bases lie along one of the coordinate axes, with the midpoint of the bases lying at the origin, and such that the vertices opposite the bases lie at four distinct points. Compute the area contained within the star.
2016 ASDAN Math Tournament, 6
Rectangle $ABCD$ has $AB=20$ and $BC=15$. $2$ circles with diameters $AB$ and $AC$ intersect again at point $E$. What is the length of $DE$?
2016 ASDAN Math Tournament, 1
$ABCDE$ is a pentagon where $AB=12$, $BC=20$, $CD=7$, $DE=24$, $EA=9$, and $\angle EAB=\angle CDE=90^\circ$. Compute the area of the pentagon.
2016 ASDAN Math Tournament, 21
Suppose that we have a $2\times5$ grid, and we wish to write $0$'s and $1$'s inside such that for any $2\times2$ sub-block, the $\textit{determinant}$ is $0$. The determinant of a $2\times2$ block $\textstyle\begin{bmatrix}a&b\\c&d\end{bmatrix}$ is $ad-bc$. For example, the following is a valid configuration:
[center]<see attached>[/center]
However, the following is not valid because the last $2\times2$ sub-block has determinant $1$:
[center]<see attached>[/center]
How many such valid $2\times5$ configurations are there?
2016 ASDAN Math Tournament, 3
Let $ABCD$ be a unit square, and let there be two unit circles centered at $C$ and $D$. Let $P$ be the point of intersection of the two circles inside the square. Compute $\angle APB$ in degrees.
2016 CMIMC, 8
Let $r_1$, $r_2$, $\ldots$, $r_{20}$ be the roots of the polynomial $x^{20}-7x^3+1$. If \[\dfrac{1}{r_1^2+1}+\dfrac{1}{r_2^2+1}+\cdots+\dfrac{1}{r_{20}^2+1}\] can be written in the form $\tfrac mn$ where $m$ and $n$ are positive coprime integers, find $m+n$.
2016 NIMO Summer Contest, 13
The area of the region in the $xy$-plane satisfying the inequality \[\min_{1 \le n \le 10} \max\left(\frac{x^2+y^2}{4n^2}, \, 2 - \frac{x^2+y^2}{4n^2-4n+1}\right) \le 1\] is $k\pi$, for some integer $k$. Find $k$.
[i]Proposed by Michael Tang[/i]
2016 ASDAN Math Tournament, 2
The largest factor of $n$ not equal to $n$ is $35$. Compute the largest possible value of $n$.
2016 ASDAN Math Tournament, 2
Let $f(x)=ax^3+bx^2+cx+d$ be some cubic polynomial. Given that $f(1)=20$ and $f(-1)=16$, what is $b+d$?
2016 CMIMC, 4
Given a list $A$, let $f(A) = [A[0] + A[1], A[0] - A[1]]$. Alef makes two programs to compute $f(f(...(f(A))))$, where the function is composed $n$ times:
\begin{tabular}{l|l}
1: \textbf{FUNCTION} $T_1(A, n)$ & 1: \textbf{FUNCTION} $T_2(A, n)$ \\
2: $\quad$ \textbf{IF} $n = 0$ & 2: $\quad$ \textbf{IF} $n = 0$ \\
3: $\quad$ $\quad$ \textbf{RETURN} $A$ & 3: $\quad$ $\quad$ \textbf{RETURN} $A$ \\
4: $\quad$ \textbf{ELSE} & 4: $\quad$ \textbf{ELSE} \\
5: $\quad$ $\quad$ \textbf{RETURN} $[T_1(A, n - 1)[0] + T_1(A, n - 1)[1],$ & 5: $\quad$ $\quad$ $B \leftarrow T_2(A, n - 1)$ \\
$\quad$ $\quad$ $\quad$ $T_1(A, n - 1)[0] - T_1(A, n - 1)[1]]$ & 6: $\quad$ $\quad$ \textbf{RETURN} $[B[0] + B[1], B[0] - B[1]]$ \\
\end{tabular}
Each time $T_1$ or $T_2$ is called, Alef has to pay one dollar. How much money does he save by calling $T_2([13, 37], 4)$ instead of $T_1([13, 37], 4)$?
2016 CMIMC, 5
Let $\mathcal{P}$ be a parallelepiped with side lengths $x$, $y$, and $z$. Suppose that the four space diagonals of $\mathcal{P}$ have lengths $15$, $17$, $21$, and $23$. Compute $x^2+y^2+z^2$.
2016 ASDAN Math Tournament, 9
Define $\phi_n(x)$ to be the number of integers $y$ less than or equal to $n$ such that $\gcd(x,y)=1$. Also, define $m=\text{lcm}(2016,6102)$. Compute
$$\frac{\phi_{m^m}(2016)}{\phi_{m^m}(6102)}.$$
2016 CMIMC, 2
Identical spherical tennis balls of radius 1 are placed inside a cylindrical container of radius 2 and height 19. Compute the maximum number of tennis balls that can fit entirely inside this container.