Olympiad problems

2017 ASDAN Math Tournament, 15

Each face of a regular tetrahedron can be colored one of red, purple, blue, or orange. How many distinct ways can we color the faces of the tetrahedron? Colorings are considered distinct if they cannot reach one another by rotation.

2016 ASDAN Math Tournament, 8

Tags: 2016 , Guts Round

There are $n$ integers $a$ such that $0\leq a<91$ and $a$ is a solution to $x^3+8x^2-x+83\equiv0\pmod{91}$. What is $n$?

2016 ASDAN Math Tournament, 14

Tags: 2016 , Guts Round

In the diagram to the right, squares are drawn on the side of the triangle with side lengths $5$, $6$, and $7$ as shown below. The corners of adjacent squares are then connected. What is the area of the resulting hexagon?

2017 ASDAN Math Tournament, 11

Tags: 2017 , Guts Round

If $a+b+c=12$ and $a^2+b^2+c^2=62$, what is $ab+bc+ac$?

2017 ASDAN Math Tournament, 12

Tags: 2017 , Guts Round

Anna has a magical compass which can point only in four directions: North, East, South, West. Initially, the compass points North. After each minute, the compass can either turn left, turn right, or stay at its current orientation, with each action occurring with equal probability. What is the probability that the compass points South after $6$ minutes?

2017 ASDAN Math Tournament, 16

Tags: 2017 , Guts Round

Let $x$ and $y$ be real numbers satisfying $9x^2+16y^2=144$. What is the maximum possible value of $xy$?

2017 ASDAN Math Tournament, 22

Tags: 2017 , Guts Round

Let $x=2\sin8^\circ+2\sin16^\circ+\dots+2\sin176^\circ$. What is $\arctan(x)$?

2016 ASDAN Math Tournament, 15

Tags: 2016 , Guts Round

Let $a$ be the least positive integer with $20$ positive divisors and $b$ be the least positive integer with $16$ positive divisors. What is $a+b$? (Note that for any integer $n$, both $1$ and $n$ are considered divisors of $n$.)

2017 ASDAN Math Tournament, 4

Tags: 2017 , Guts Round

Alice and Bob are painting a house. Alice can paint a house in $20$ hours by herself. Bob can paint a house in $40$ hours by himself. Both people start at the same time, paint at their own constant rate, and work together to paint one house. When the house is fully painted, what fraction of the house was painted by Alice?

2016 ASDAN Math Tournament, 1

Tags: 2016 , Guts Round

Bill is buying cans of soup. Cans come in $2$ shapes. Can $A$ is a rectangular prism shaped can with dimensions $20\times16\times10$, and can $B$ is a cylinder shaped can with radius $10$ and height $10$. Let $\alpha$ be the volume of the larger can, and $\beta$ be the volume of the smaller can. What is $\alpha-\beta$?

2017 ASDAN Math Tournament, 8

Tags: 2017 , Guts Round

How many integer solutions are there to $y^2=x^2-2017$?

2016 ASDAN Math Tournament, 22

Tags: 2016 , Guts Round

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?

2017 ASDAN Math Tournament, 10

Tags: 2017 , Guts Round

The perimeter of an isosceles trapezoid is $24$. If each of the legs is two times the length of the shorter base and is two-thirds the length of the longer base, what is the area of the trapezoid?

2017 ASDAN Math Tournament, 9

Tags: 2017 , Guts Round

Eddy owns $5$ different cats, and has $9$ fish to distribute among the cats. Each cat gets at least $1$ fish and at most $3$ fish. If the fish are indistinguishable, how many ways can Eddy distribute the $9$ fish among the $5$ cats?

2016 ASDAN Math Tournament, 21

Tags: 2016 , Guts Round

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, 2

Tags: 2016 , Guts Round

Let $f(x)=ax^3+bx^2+cx+d$ be some cubic polynomial. Given that $f(1)=20$ and $f(-1)=16$, what is $b+d$?

2017 ASDAN Math Tournament, 13

Tags: 2017 , Guts Round

Let $S_1$ be a square of side length $3$. For $i=2,3,4,\dots$, inscribe a square $S_i$ inside $S_{i-1}$ such that the sides of the inner square form four $30^\circ-60^\circ-90^\circ$ triangles with the outer square. Compute the total sum $$\sum_{i=1}^\infty\text{area}(S_i).$$

2017 ASDAN Math Tournament, 5

Tags: 2017 , Guts Round

Let $\alpha$ and $\beta$ be the two roots of $x^2+2017x+k$. What is the sum of the possible values of $k$ so that the lines \begin{align*} y&=2\alpha x+2017^2\\ y&=3\alpha x+2017^3 \end{align*} are perpendicular?

2016 ASDAN Math Tournament, 17

Tags: 2016 , Guts Round

Consider triangle $ABC$ with sides $AB=4$, $BC=11$, and $CA=9$. The triangle is spun around a line that passes through $B$ and the interior of the triangle (including the edges $BC$ and $BA$). Of all possible lines with these constraints, what is the largest possible volume of the resulting solid?

2017 ASDAN Math Tournament, 25

Tags: 2017 , Guts Round

Consider the sequence $\{a_n\}$ defined so that $a_n$ is the leftmost digit of $2^n$. The first few terms of this sequence are $1,2,4,8,1,3,6,\dots$. For how many $0\le n\le100000$ is $a_n=1$? If $C$ is the correct answer and $A$ is your answer, then your score will be rounded up from $\max\left(0,25-\tfrac{1}{6}\sqrt{|A-C|}\right)$.

2017 ASDAN Math Tournament, 14

Tags: 2017 , Guts Round

What are the last two digits of $2017^{2017}$?

2017 ASDAN Math Tournament, 19

Tags: 2017 , Guts Round

How many ways can you tile a $2\times5$ rectangle with $2\times1$ dominoes of $4$ different colors if no two dominoes of the same color may be adjacent?

2016 ASDAN Math Tournament, 10

Tags: 2016 , Guts Round

A point $P$ and a segment $AB$ with length $20$ are randomly drawn on a plane. Suppose that the probability that a randomly selected line passing through $P$ intersects segment $AB$ is $\tfrac{1}{2}$. Next, randomly choose point $Q$ on segment $AB$. What is the probability with respect to choosing $Q$ that a circle centered at $Q$ passing through $P$ contains both $A$ and $B$ in its interior?

2017 ASDAN Math Tournament, 18

Tags: 2017 , Guts Round

Find the sum of all integers $0\le a \le124$ so that $a^3-2$ is a multiple of $125$.

2016 ASDAN Math Tournament, 3

Tags: 2016 , Guts Round

A number $n$ is $\textit{almost prime}$ if any of $n-2$, $n-1$, $n$, $n+1$, or $n+2$ is prime. Compute the smallest positive integer that is not $\textit{almost prime}$.