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: 84

2014 ASDAN Math Tournament, 5
Tags: 2014 , team test
Given a triangle $ABC$ with integer side lengths, where $BD$ is an angle bisector of $\angle ABC$, $AD=4$, $DC=6$, and $D$ is on $AC$, compute the minimum possible perimeter of $\triangle ABC$.

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, 7
Tags: 2014 , Advanced Topics Test
Two math students play a game with $k$ sticks. Alternating turns, each one chooses a number from the set $\{1,3,4\}$ and removes exactly that number of sticks from the pile (so if the pile only has $2$ sticks remaining the next player must take $1$). The winner is the player who takes the last stick. For $1\leq k\leq100$, determine the number of cases in which the first player can guarantee that he will win.

2014 ASDAN Math Tournament, 1
Tags: 2014 , Algebra Tiebreaker
Kevin is running $1000$ meters. He wants to have an average speed of $10$ meters a second. He runs the first $100$ meters at a speed of $4$ meters a second. Compute how quickly, in meters per second, he must run the last $900$ meters to attain his desired average speed of $10$ meters a second.

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 $2$.

2014 ASDAN Math Tournament, 2
Tags: 2014 , Algebra Test
Let $a$ and $b$ be positive integers such that $a>b$ and the difference between $a^2+b$ and $a+b^2$ is prime. Compute all possible pairs $(a,b)$.

2014 ASDAN Math Tournament, 3
Tags: 2014 , Advanced Topics Test
A mouse is playing a game of mouse hopscotch. In mouse hopscotch there is a straight line of $11$ squares, and starting on the first square the mouse must reach the last square by jumping forward $1$, $2$, or $3$ squares at a time (so in particular the mouse’s first jump can be to the second, third, or fourth square). The mouse cannot jump past the last square. Compute the number of ways there are to complete mouse hopscotch.

2015 Canada National Olympiad, 4
Tags: geometry , circumcircle , Canada , 2014 , National Olympiads
Let $ABC$ be an acute-angled triangle with circumcenter $O$. Let $I$ be a circle with center on the altitude from $A$ in $ABC$, passing through vertex $A$ and points $P$ and $Q$ on sides $AB$ and $AC$. Assume that \[BP\cdot CQ = AP\cdot AQ.\] Prove that $I$ is tangent to the circumcircle of triangle $BOC$.