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, 5
Tags: 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, 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.

2016 ASDAN Math Tournament, 12
Tags: team test
Let $$f(x)=\frac{2016^x}{2016^x+\sqrt{2016}}.$$ Evaluate $$\sum_{k=0}^{2016}f\left(\frac{k}{2016}\right).$$

2016 ASDAN Math Tournament, 13
Tags: team test
Ash writes the positive integers from $1$ to $2016$ inclusive as a single positive integer $n=1234567891011\dots2016$. What is the result obtained by successively adding and subtracting the digits of $n$? (In other words, compute $1-2+3-4+5-6+7-8+9-1+0-1+\dots$.)

2016 ASDAN Math Tournament, 4
Tags: team test
Three roots of the quartic polynomial $f(x)=x^4+ax^3+bx+c$ are $-1$, $3$, and $5$. What is $a+b-c$?

2015 ASDAN Math Tournament, 15
Tags: team test
In a given acute triangle $\triangle ABC$ with the values of angles given (known as $a$, $b$, and $c$), the inscribed circle has points of tangency $D,E,F$ where $D$ is on $BC$, $E$ is on $AB$, and $F$ is on $AC$. Circle $\gamma$ has diameter $BC$, and intersects $\overline{EF}$ at points $X$ and $Y$. Find $\tfrac{XY}{BC}$ in terms of the angles $a$, $b$, and $c$.

2015 ASDAN Math Tournament, 2
Tags: team test
Jonah recently harvested a large number of lychees and wants to split them into groups. Unfortunately, for all $n$ where $3\leq n\leq8$, when the lychees are distributed evenly into $n$ groups, $n-1$ lychees remain. What is the smallest possible number of lychees that Jonah could have?

2020 ASDAN Math Tournament, 3
Tags: team , team test
A fair coin is flipped $6$ times. The probability that the coin lands on the same side $3$ flips in a row at some point can be expressed as a common fraction $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Compute $100m + n$.

2016 ASDAN Math Tournament, 15
Tags: team test
Circles $\omega_1$ and $\omega_2$ have radii $r_1<r_2$ respectively and intersect at distinct points $X$ and $Y$. The common external tangents intersect at point $Z$. The common tangent closer to $X$ touches $\omega_1$ and $\omega_2$ at $P$ and $Q$ respectively. Line $ZX$ intersects $\omega_1$ and $\omega_2$ again at points $R$ and $S$ and lines $RP$ and $SQ$ intersect again at point $T$. If $XT=8$, $XZ=15$, and $XY=12$, then what is $\tfrac{r_1}{r_2}$?

2020 ASDAN Math Tournament, 7
Tags: team test
Alex scans the list of integers between $1$ and $2020$ inclusive using the following algorithm. First, he reads off perfect squares between $1$ and $2020$ in ascending order and removes these numbers from the list. Next, he reads off numbers now at perfect square indices in ascending order, which are $2$, $6$, $12$, $...$, and removes these numbers from the list. He repeats this algorithm until he reads off $2020$, which is the nth number he has read o so far. Compute $n$.