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

2020 ASDAN Math Tournament, 6
Tags: geometry , team test
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 $\overline{AD}$?

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 $\tfrac{r_1}{r_2}$?

2015 ASDAN Math Tournament, 14
Tags: team test
For a given positive integer $m$, the series $$\sum_{k=1,k\neq m}^{\infty}\frac{1}{(k+m)(k-m)}$$ evaluates to $\frac{a}{bm^2}$, where $a$ and $b$ are positive integers. Compute $a+b$.

2014 ASDAN Math Tournament, 6
Tags: team test
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 $1$.

2020 ASDAN Math Tournament, 13
Tags: team test , geometry
Let $ABCD$ be a convex quadrilateral such that $\vartriangle ABC$ is an equilateral triangle. Let $P$ be a point inside the quadrilateral such that $\vartriangle APD$ is an equilateral triangle and $\angle PCD = 30^o$. Suppose $CP = 6$ and $CD = 8$. The area of the triangle formed by $P$, the midpoint of $\overline{BC}$, and the midpoint of $\overline{AB}$ can be expressed in simplest radical form as $\frac{a+b\sqrt{c}}{d}$ , where $a$, $b$, $c$, and $d$ are positive integers with $gcd(a, b, d) = 1$ and with $c$ not divisible by the square of any prime. Compute $a + b + c + d$.

2015 ASDAN Math Tournament, 6
Tags: team test
Let $f(x)=x^4-4x^3-3x^2-4x+1$. Compute the sum of the real roots of $f(x)$.

2015 ASDAN Math Tournament, 8
Tags: team test
Let $f(x)=\tfrac{x+a}{x+b}$ for real numbers $x$ such that $x\neq -b$. Compute all pairs of real numbers $(a,b)$ such that $f(f(x))=-\tfrac{1}{x}$ for $x\neq0$.

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, 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, 2
Tags: team test
Sam's cup has a $400$ mL mixture of coffee and milk tea. He pours $200$ mL into Ben's empty cup. Ben then adds 100mL of co ee to his cup and stirs well. Finally, Ben pours $200$ mL out of his cup back into Sam's cup. If the mixture in Sam's cup is now $50\%$ milk tea, then how many milliliters of milk tea were in it originally?

2015 ASDAN Math Tournament, 3
Tags: team test
Simplify $\sqrt{7+\sqrt{33}}-\sqrt{7-\sqrt{33}}$.

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 $x$.

2014 ASDAN Math Tournament, 3
Tags: team test
A segment of length $1$ is drawn such that its endpoints lie on a unit circle, dividing the circle into two parts. Compute the area of the larger region.

2015 ASDAN Math Tournament, 12
Tags: team test
Find the smallest positive integer solution to the equation $2^{2^k}\equiv k\pmod{29}$.

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?

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.

2020 ASDAN Math Tournament, 9
Tags: team test
A positive integer $n$ has the property that, for any $2$ integers $a$ and $b$, if $ab + 1$ is divisible by $n$, then $a + b$ is also divisible by $n$. What is the largest possible value of $n$?

2020 ASDAN Math Tournament, 12
Tags: team test
Let $S_n$ be the number of subsets of the first $n$ positive integers that have the same number of even values and odd values; the empty set counts as one of these subsets. Compute the smallest positive integer $n$ such that $S_n$ is a multiple of $2020$.

2015 ASDAN Math Tournament, 1
Tags: team test
In his spare time, Thomas likes making rectangular windows. He builds windows by taking four $30\text{ cm}\times20\text{ cm}$ rectangles of glass and arranging them in a larger rectangle in wood. The window has an $x\text{ cm}$ wide strip of wood between adjacent glass pieces and an $x\text{ cm}$ wide strip of wood between each glass piece and the adjacent edge of the window. Given that the total area of the glass is equivalent to the total area of the wood, what is $x$? [center]<see attached>[/center]

2014 ASDAN Math Tournament, 1
Tags: team test
Compute the remainder when $2^{30}$ is divided by $1000$.

2016 ASDAN Math Tournament, 1
Tags: team test
Pooh has an unlimited supply of $1\times1$, $2\times2$, $3\times3$, and $4\times4$ squares. What is the minimum number of squares he needs to use in order to fully cover a $5\times5$ with no $2$ squares overlapping?

2014 ASDAN Math Tournament, 14
Tags: team test
Consider a round table on which $2014$ people are seated. Suppose that the person at the head of the table receives a giant plate containing all the food for supper. He then serves himself and passes the plate either right or left with equal probability. Each person, upon receiving the plate, will serve himself if necessary and similarly pass the plate either left or right with equal probability. Compute the probability that you are served last if you are seated $2$ seats away from the person at the head of the table.

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, 9
Tags: team test
A cake in the shape of a rectangular prism has dimensions $6\text{ cm}\times14\text{ cm}\times21\text{ cm}$. It is cut into $1764$ equally sized cubes such that each cube is $1\text{ cm}^3$. Andy the ant starts at one corner of the cake and eats through the cake in a straight line to the opposite corner of the cake. How many of the $1\text{ cm}^3$ cubes does Andy bite through?

2020 ASDAN Math Tournament, 15
Tags: team test
For integers $z$, let $\#(z)$ denote the number of integer ordered pairs $(x, y)$ that satisfy $x^2 - xy + y^2 = z$. How many integers $z$ between $0$ and $150$ inclusive satisfy $\#(z) \equiv 6$ (mod $12$)?