Found problems: 84
2014 ASDAN Math Tournament, 11
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.
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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
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, 8
Nick has a $3\times3$ grid and wants to color each square in the grid one of three colors such that no two squares that are adjacent horizontally or vertically are the same color. Compute the number of distinct grids that Nick can create.
2014 ASDAN Math Tournament, 2
Let $a$ and $b$ be the roots of the quadratic $x^2-7x+c$. Given that $a^2+b^2=17$, compute $c$.
2014 ASDAN Math Tournament, 4
Cynthia and Lynnelle are collaborating on a problem set. Over a $24$-hour period, Cynthia and Lynnelle each independently pick a random, contiguous $6$-hour interval to work on the problem set. Compute the probability that Cynthia and Lynnelle work on the problem set during completely disjoint intervals of time.
2014 ASDAN Math Tournament, 1
Consider a square of side length $1$ and erect equilateral triangles of side length $1$ on all four sides of the square such that one triangle lies inside the square and the remaining three lie outside. Going clockwise around the square, let $A$, $B$, $C$, $D$ be the circumcenters of the four equilateral triangles. Compute the area of $ABCD$.
2014 Taiwan TST Round 3, 1
Positive integers $x_1, x_2, \dots, x_n$ ($n \ge 4$) are arranged in a circle such that each $x_i$ divides the sum of the neighbors; that is \[ \frac{x_{i-1}+x_{i+1}}{x_i} = k_i \] is an integer for each $i$, where $x_0 = x_n$, $x_{n+1} = x_1$. Prove that \[ 2n \le k_1 + k_2 + \dots + k_n < 3n. \]
2014 ASDAN Math Tournament, 7
Let $ABCD$ be a square piece of paper with side length $4$. Let $E$ be a point on $AB$ such that $AE=3$ and let $F$ be a point on $CD$ such that $DF=1$. Now, fold $AEFD$ over the line $EF$. Compute the area of the resulting shape.
2014 ASDAN Math Tournament, 25
$300$ 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, 2
Let $RICE$ be a quadrilateral with an inscribed circle $O$ such that every side of $RICE$ is tangent to $O$. Given taht $RI=3$, $CE=8$, and $ER=7$, compute $IC$.
2014 ASDAN Math Tournament, 18
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, 1
Compute the smallest positive integer that is $3$ more than a multiple of $5$, and twice a multiple of $6$.
2014 ASDAN Math Tournament, 3
A robot is standing on the bottom left vertex $(0,0)$ of a $5\times5$ grid, and wants to go to $(5,5)$, only moving to the right $(a,b)\mapsto(a+1,b)$ or upward $(a,b)\mapsto(a,b+1)$. However this robot is not programmed perfectly, and sometimes takes the upper-left diagonal path $(a,b)\mapsto(a-1,b+1)$. As the grid is surrounded by walls, the robot cannot go outside the region $0\leq a,b\leq5$. Supposing that the robot takes the diagonal path exactly once, compute the number of different routes the robot can take.
2014 ASDAN Math Tournament, 1
A college math class has $N$ teaching assistants. It takes the teaching assistants $5$ hours to grade homework assignments. One day, another teaching assistant joins them in grading and all homework assignments take only $4$ hours to grade. Assuming everyone did the same amount of work, compute the number of hours it would take for $1$ teaching assistant to grade all the homework assignments.
2014 ASDAN Math Tournament, 16
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
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, 8
Consider the recurrence relation
$$a_{n+3}=\frac{a_{n+2}a_{n+1}-2}{a_n}$$
with initial condition $(a_0,a_1,a_2)=(1,2,5)$. Let $b_n=a_{2n}$ for nonnegative integral $n$. It turns out that $b_{n+2}+xb_{n+1}+yb_n=0$ for some pair of real numbers $(x,y)$. Compute $(x,y)$.
2014 ASDAN Math Tournament, 6
Compute $\cos(\tfrac{\pi}{9})-\cos(\tfrac{2\pi}{9})+\cos(\tfrac{3\pi}{9})-\cos(\tfrac{4\pi}{9})$.
2014 ASDAN Math Tournament, 6
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 Vietnam National Olympiad, 1
Let $({{x}_{n}}),({{y}_{n}})$ be two positive sequences defined by ${{x}_{1}}=1,{{y}_{1}}=\sqrt{3}$ and
\[ \begin{cases} {{x}_{n+1}}{{y}_{n+1}}-{{x}_{n}}=0 \\ x_{n+1}^{2}+{{y}_{n}}=2 \end{cases} \] for all $n=1,2,3,\ldots$.
Prove that they are converges and find their limits.
2014 ASDAN Math Tournament, 2
Compute the number of positive integers less than or equal to $10000$ which are relatively prime to $2014$.
2014 ASDAN Math Tournament, 2
Consider all right triangles with integer side lengths that form an arithmetic sequence. Compute the $2014$th smallest perimeter of all such right triangles.
2014 ASDAN Math Tournament, 6
Consider $7$ points on a circle. Compute the number of ways there are to draw chords between pairs of points such that two chords never intersect and one point can only belong to one chord. It is acceptable to draw no chords.
2014 ASDAN Math Tournament, 6
Compute the largest integer $N$ such that one can select $N$ different positive integers, none of which is larger than $17$, and no two of which share a common divisor greater than $1$.
2014 ASDAN Math Tournament, 10
Find the remainder when $(1^2+1)(2^2+1)(3^2+1)\dots(42^2+1)$ is divided by $43$. Your answer should be an integer between $0$ and $42$.