Found problems: 60
2015 ASDAN Math Tournament, 12
Find the smallest positive integer solution to the equation $2^{2^k}\equiv k\pmod{29}$.
2015 ASDAN Math Tournament, 1
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 $x$?
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2015 ASDAN Math Tournament, 7
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 $r$.
2015 ASDAN Math Tournament, 8
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 $x\neq0$.
2015 ASDAN Math Tournament, 9
A coin is tossed $10$ times. Compute the probability that two heads will turn up in succession somewhere in the sequence of throws.
2015 ASDAN Math Tournament, 6
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, 3
Simplify $\sqrt{7+\sqrt{33}}-\sqrt{7-\sqrt{33}}$.
2020 ASDAN Math Tournament, 8
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, 1
Compute the remainder when $2^{30}$ is divided by $1000$.
2016 ASDAN Math Tournament, 9
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?
2014 ASDAN Math Tournament, 13
Let $\alpha,\beta,\gamma$ be the three real roots of the polynomial $x^3-x^2-2x+1=0$. Find all possible values of $\tfrac{\alpha}{\beta}+\tfrac{\beta}{\gamma}+\tfrac{\gamma}{\alpha}$.
2015 ASDAN Math Tournament, 13
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 $2015$?
2020 ASDAN Math Tournament, 11
$\vartriangle ABC$ is right with $\angle C = 90^o$. The internal angle bisectors of $\angle A$ and $\angle B$ meet at point $D$, while the external angle bisectors of $\angle A$ and $\angle B$ meet at point $E$. Suppose that $AD = 1$ and $BD = 2$. The value of $DE^2$ can be expressed as $x+y \sqrt{z}$ for integers $x$, $y$, and $z$, where $z$ is greater than $1$ and not divisible by the square of any prime. Compute $100x + 10y + z$.
Note: For a generic triangle $\vartriangle PQR$, if we let $Q'$ be the reflection of $Q$ over $P$, then the external angle bisector of $\angle P$ is the line that contains the internal angle bisector of $\angle Q'PR$.
2014 ASDAN Math Tournament, 12
Find the last two digits of $\tbinom{200}{100}$. Express the answer as an integer between $0$ and $99$. (e.g. if the last two digits are $05$, just write $5$.)
2020 ASDAN Math Tournament, 6
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 $\overline{AD}$?
2015 ASDAN Math Tournament, 11
Let $ABCDEF$ be a regular hexagon, and let $G$, $H$, $I$, $J$, $K$, and $L$ be the midpoints of sides $AB$, $BC$, $CD$, $DE$, $EF$, and $FA$, respectively. The intersection of lines $\overline{AH}$, $\overline{BI}$, $\overline{CJ}$, $\overline{DK}$, $\overline{EL}$, and $\overline{FG}$ bound a smaller regular hexagon. Find the ratio of the area of the smaller hexagon to the area of $ABCDEF$.
2020 ASDAN Math Tournament, 15
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 $12$)?
2015 ASDAN Math Tournament, 14
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 $a+b$.
2020 ASDAN Math Tournament, 4
There are $2$ ways to write $2020$ as a sum of $2$ squares: $2020 = a^2 + b^2$ and $2020 = c^2 + d^2$, where $a$, $b$, $c$, and $d$ are distinct positive integers with $a < b$ and $c < d$. Compute $a+b+c+d$.
2016 ASDAN Math Tournament, 3
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?
2014 ASDAN Math Tournament, 7
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?
2014 ASDAN Math Tournament, 11
In the following system of equations
$$|x+y|+|y|=|x-1|+|y-1|=2,$$
find the sum of all possible $x$.
2016 ASDAN Math Tournament, 10
Compute the smallest positive integer $x$ which satisfies $x^2-8x+1\equiv0\pmod{22}$ and $x^2-22x+1\equiv0\pmod{8}$.
2016 ASDAN Math Tournament, 14
Suppose that $x,y,z$ are positive real numbers that satisfy
\begin{align*}
x+y+z&=xyz\\
\frac{x^2}{16(1+x^2)}=\frac{y^2}{25(1+y^2)}&=\frac{z^2}{36(1+z^2)}.
\end{align*}
Compute
$$\frac{x^2(1+x^2)^2}{x^2(1+z^2)^2}.$$
2020 ASDAN Math Tournament, 13
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 $a + b + c + d$.