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: 60

2014 ASDAN Math Tournament, 10
Tags: team test
Three real numbers $x$, $y$, and $z$ are chosen independently and uniformly at random from the interval $[0,1]$. Compute the probability that $x$, $y$, and $z$ can be the side lengths of a triangle.

2014 ASDAN Math Tournament, 2
Tags: team test
Consider all right triangles with integer side lengths that form an arithmetic sequence. Compute the $2014$th smallest perimeter of all such right triangles.

2015 ASDAN Math Tournament, 13
Tags: team test
The incircle of triangle $\triangle ABC$ is the unique inscribed circle that is internally tangent to the sides $\overline{AB}$, $\overline{BC}$, and $\overline{CA}$. How many non-congruent right triangles with integer side lengths have incircles of radius $2015$?

2014 ASDAN Math Tournament, 11
Tags: team test
In the following system of equations $$|x+y|+|y|=|x-1|+|y-1|=2,$$ find the sum of all possible $x$.

2020 ASDAN Math Tournament, 8
Tags: team test
For nonzero integers $n$, let $f(n)$ be the sum of all positive integers $b$ for which all solutions $x$ to $x^2 +bx+n = 0$ are integers, and let $g(n)$ be the sum of all positive integers $c$ for which all solutions $x$ to $cx + n = 0$ are integers. Compute $\sum^{2020}_{n=1} (f(n) - g(n))$.

2014 ASDAN Math Tournament, 7
Tags: team test
Eddy draws $6$ cards from a standard $52$-card deck. What is the probability that four of the cards that he draws have the same value?

2015 ASDAN Math Tournament, 7
Tags: team test
Nine identical spheres of radius $r$ are packed into a unit cube. One sphere is centered at the center of the cube and is tangent to the other eight spheres, each of which is located in a corner of the cube and is tangent to three faces of the cube. Compute the radius of the spheres $r$.

2014 ASDAN Math Tournament, 4
Tags: team test
A frog is hopping from $(0,0)$ to $(8,8)$. The frog can hop from $(x,y)$ to either $(x+1,y)$ or $(x,y+1)$. The frog is only allowed to hop to point $(x,y)$ if $|y-x|\leq1$. Compute the number of distinct valid paths the frog can take.

2016 ASDAN Math Tournament, 3
Tags: team test
Moor has $2016$ white rabbit candies. He and his $n$ friends split the candies equally amongst themselves, and they find that they each have an integer number of candies. Given that $n$ is a positive integer (Moor has at least $1$ friend), how many possible values of $n$ exist?

2020 ASDAN Math Tournament, 6
Tags: team test , geometry
Triangle $\vartriangle ABC$ has side lengths $AB = 26$, $BC = 34$, and $CA = 24\sqrt2$. A fourth point $D$ makes a right angle $\angle BDC$. What is the smallest possible length of $\overline{AD}$?