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

Tags were heavily modified to better represent problems.

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Found problems: 25

2014 ASDAN Math Tournament, 24
Tags: 2014 , General Test
It's pouring down rain, and the amount of rain hitting point $(x,y)$ is given by $$f(x,y)=|x^3+2x^2y-5xy^2-6y^3|.$$ If you start at the origin $(0,0)$, find all the possibilities for $m$ such that $y=mx$ is a straight line along which you could walk without any rain falling on you.

2014 ASDAN Math Tournament, 15
Tags: 2014 , General Test
Antoine, Benoît, Claude, Didier, Étienne, and Françoise go to the cinéma together to see a movie. The six of them want to sit in a single row of six seats. But Antoine, Benoît, and Claude are mortal enemies and refuse to sit next to either of the other two. How many different arrangements are possible?

2014 ASDAN Math Tournament, 20
Tags: 2014 , General Test
$ABCD$ is a parallelogram, and circle $S$ (with radius $2$) is inscribed insider $ABCD$ such that $S$ is tangent to all four line segments $AB$, $BC$, $CD$, and $DA$. One of the internal angles of the parallelogram is $60^\circ$. What is the maximum possible area of $ABCD$?

2014 ASDAN Math Tournament, 10
Tags: 2014 , General Test
Find the area of the smallest possible square that contains the points $(2,-1)$ and $(4,4)$.

2014 ASDAN Math Tournament, 17
Tags: 2014 , General Test
Given that the line $y=mx+k$ intersects the parabola $y=ax^2+bx+c$ at two points, compute the product of the two $x$-coordinates of these points in terms of $a$, $b$, $c$, $k$, and $m$.

2014 ASDAN Math Tournament, 1
Tags: 2014 , General Test
Alex gets $8$ points on an exam, while his friend gets $3$ times as many points as Alex. What is the average of their scores?

2014 ASDAN Math Tournament, 7
Tags: 2014 , General Test
Ben works quickly on his homework, but tires quickly. The first problem takes him $1$ minute to solve, and the second problem takes him $2$ minutes to solve. It takes him $N$ minutes to solve problem $N$ on his homework. If he works for an hour on his homework, compute the maximum number of problems he can solve.

2014 ASDAN Math Tournament, 8
Tags: 2014 , General Test
George and two of his friends go to a famous jiaozi restaurant, which serves only two kinds of jiaozi: pork jiaozi, and vegetable jiaozi. Each person orders exactly $15$ jiaozi. How many different ways could the three of them order? Two ways of ordering are different if one person orders a different number of pork jiaozi in both orders.

2014 ASDAN Math Tournament, 23
Tags: 2014 , General Test
Let triangle $ABC$ have side lengths $AB=11$, $BC=7$, and $AC=12$. Let $D$ be a point on $AC$ and $E$ be a point on $AB$ such that $\angle CDE=90^\circ$ and the area of triangle $CDE$ is maximized. Find the area of triangle $CDE$.

2014 ASDAN Math Tournament, 9
Tags: 2014 , General Test
The operation $\oslash$, called "reciprocal sum," is useful in many areas of physics. If we say that $x=a\oslash b$, this means that $x$ is the solution to $$\frac{1}{x}=\frac{1}{a}+\frac{1}{b}$$ Compute $4\oslash2\oslash4\oslash3\oslash4\oslash4\oslash2\oslash3\oslash2\oslash4\oslash4\oslash3$.

2014 ASDAN Math Tournament, 5
Tags: 2014 , General Test
Screws are sold in packs of $10$ and $12$. Harry and Sam independently go to the hardware store, and by coincidence each of them buys exactly $k$ screws. However, the number of packs of screws Harry buys is different than the number of packs Sam buys. What is the smallest possible value of $k$?

2014 ASDAN Math Tournament, 22
Tags: 2014 , General Test
Apples cost $2$ dollars. Bananas cost $3$ dollars. Oranges cost $5$ dollars. Compute the number of distinct baskets of fruit such that there are $100$ pieces of fruit and the basket costs $300$ dollars. Two baskets are distinct if and only if, for some type of fruit, the two baskets have differing amounts of that fruit.

2014 ASDAN Math Tournament, 21
Tags: 2014 , General Test
A bitstring of length $\ell$ is a sequence of $\ell$ $0$'s or $1$'s in a row. How many bitstrings of length $2014$ have at least $2012$ consecutive $0$'s or $1$'s?

2014 ASDAN Math Tournament, 13
Tags: 2014 , General Test
Square $S_1$ is inscribed inside circle $C_1$, which is inscribed inside square $S_2$, which is inscribed inside circle $C_2$, which is inscribed inside square $S_3$, which is inscribed inside circle $C_3$, which is inscribed inside square $S_4$. [center]<see attached>[/center] Let $a$ be the side length of $S_4$, and let $b$ be the side length of $S_1$. What is $\tfrac{a}{b}$?

2014 ASDAN Math Tournament, 19
Tags: 2014 , General Test
Given that $f(x)+2f(4-x)=x+8$, compute $f(16)$.

2014 ASDAN Math Tournament, 11
Tags: 2014 , General Test
Mr. Ambulando is at the intersection of $5^{\text{th}}$ and $\text{A St}$, and needs to walk to the intersection of $1^{\text{st}}$ and $\text{F St}$. There's an accident at the intersection of $4^{\text{th}}$ and $\text{B St}$, which he'd like to avoid. [center]<see attached>[/center] Given that Mr. Ambulando wants to walk the shortest distance possible, how many different routes through downtown can he take?

2014 ASDAN Math Tournament, 4
Tags: 2014 , General Test
If Bobby’s age is increased by $6$, it’s a number with an integral (positive) square root. If his age is decreased by $6$, it’s that square root. How old is Bobby?

2014 ASDAN Math Tournament, 25
Tags: 2014 , General Test
$300$ couples (one man, one woman) are invited to a party. Everyone at the party either always tells the truth or always lies. Exactly $2/3$ of the men say their partner always tells the truth and the remaining $1/3$ say their partner always lies. Exactly $2/3$ of the women say their partner is the same type as themselves and the remaining $1/3$ say their partner is different. Find $a$, the maximum possible number of people who tell the truth, and $b$, the minimum possible number of people who tell the truth. Express your answer as $(a,b)$.

2014 ASDAN Math Tournament, 18
Tags: 2014 , General Test
A two-digit positive integer is $\textit{primeable}$ if one of its digits can be deleted to produce a prime number. A two-digit positive integer that is prime, yet not primeable, is $\textit{unripe}$. Compute the total number of unripe integers.

2014 ASDAN Math Tournament, 16
Tags: 2014 , General Test
Compute the number of geometric sequences of length $3$ where each number is a positive integer no larger than $10$.

2014 ASDAN Math Tournament, 3
Tags: 2014 , General Test
Boris is driving on a remote highway. His car’s odometer reads $24942\text{ km}$, which Boris notices is a palindromic number, meaning it is not changed when it is reversed. “Hm,” he thinks, “it should be a long time before I see that again.” But it takes only $1$ hour for the odometer to once again show a palindromic number! How fast is Boris driving in $\text{km/h}$?

2014 ASDAN Math Tournament, 6
Tags: 2014 , General Test
In triangle $ABC$, we have that $AB=AC$, $BC=16$, and that the area of $\triangle ABC$ is $120$. Compute the length of $AB$.

2014 ASDAN Math Tournament, 12
Tags: 2014 , General Test
Consider a rectangular tiled room with dimensions $m\times n$, where the tiles are $1\times1$ in size. Compute all ordered pairs $(m,n)$ with $m\leq n$ such that the number of tiles on the perimeter is equal to the number of tiles in the interior (i.e. not on the perimeter).

2014 ASDAN Math Tournament, 14
Tags: 2014 , General Test
Patricia has a rectangular painting that she wishes to frame. The frame must also be rectangular and will extend $3\text{ cm}$ outward from each of the four sides of the painting. When the painting is framed, the area of the frame not covered by the painting is $108\text{ cm}^2$. What is the perimeter of the painting alone (without the frame)?

2014 ASDAN Math Tournament, 2
Tags: 2014 , General Test
Sally rolls an $8$-sided die with faces numbered $1$ through $8$. Compute the probability that she gets a power of $2$.