Found problems: 60
2014 ASDAN Math Tournament, 5
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 $\triangle ABC$.
2016 ASDAN Math Tournament, 8
Let $ABC$ be a triangle with $AB=24$, $BC=30$, and $AC=36$. Point $M$ lies on $BC$ such that $BM=12$, and point $N$ lies on $AC$ such that $CN=20$. Let $X$ be the intersection of $AM$ and $BN$ and let line $CX$ intersect $AB$ at point $L$. Compute
$$\frac{AX}{XM}+\frac{BX}{XN}+\frac{CX}{XL}.$$
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 $\tfrac{\alpha}{\beta}+\tfrac{\beta}{\gamma}+\tfrac{\gamma}{\alpha}$.
2016 ASDAN Math Tournament, 13
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, 5
Given that $x$ and $y$ are real numbers, compute the minimum value of
$$x^4+4x^3+8x^2+4xy+6x+4y^2+10.$$
2020 ASDAN Math Tournament, 1
Consider triangle $\vartriangle ABC$ with $\angle C = 90^o$. Let $P$ be the midpoint of $\overline{AC}$ so that $AP = PC = 1$, and suppose $\angle BAC = \angle CBP$. Compute $AB^2$.
2020 ASDAN Math Tournament, 14
If $f$ is a permutation of $S = \{0, 1,..., 14\}$, then for integers $k \ge 1$, define $$f^k(x) =\underbrace{f(f...(f(x))... ))}_{k\,\,\, applications \,\,\, of \,\,\, f}$$ Compute the number of permutations $f$ of $S$ such that, for some $k \ge 1$, $f^k(x) = (x + 5) \mod \,\,\, 15$ for all $x \in S$.
2020 ASDAN Math Tournament, 10
Let $r = 1-\sqrt[5]{2}+ \sqrt[5]{4}-\sqrt[5]{8}+ \sqrt[5]{16}$. There exists a unique fifth-degree polynomial $P$ such that its leading coefficient is positive, all of its coefficients are integers whose greatest common factor (among all of them) is $1$, and $P(r) = 0$. Evaluate $P(10)$.
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 $5$.)
2015 ASDAN Math Tournament, 5
Laurie loves multiplying numbers in her head. One day she decides to multiply two $2$-digit numbers $x$ and $y$ such that $x\leq y$ and the two numbers collectively have at least three distinct digits. Unfortunately, she accidentally remembers the digits of each number in the opposite order (for example, instead of remembering $51$ she remembers $15$). Surprisingly, the product of the two numbers after flipping the digits is the same as the product of the two original numbers. How many possible pairs of numbers could Laurie have tried to multiply?