Found problems: 15
2017 ASDAN Math Tournament, 3
For some integers $b$ and $c$, neither of the equations below have real solutions:
\begin{align*}
2x^2+bx+c&=0\\
2x^2+cx+b&=0.
\end{align*}
What is the largest possible value of $b+c$?
2015 ASDAN Math Tournament, 3
Let $f(x)$ be a polynomial of finite degree satisfying
$$(x+9)f(x+1)=(x+3)f(x+3)$$
for all real $x$. If $f(0)=1$, find the value of $f(1)$.
2014 ASDAN Math Tournament, 3
Compute
$$\sin\left(\frac{\pi}{9}\right)\sin\left(\frac{2\pi}{9}\right)\sin\left(\frac{4\pi}{9}\right).$$
2017 ASDAN Math Tournament, 2
Let $a$ and $b$ be positive integers that satisfy $ab-7a-11b+13=0$. What is the minimum possible value of $a+b$?
2018 ASDAN Math Tournament, 3
Compute $ax^{2018}+by^{2018}$, given that there exist real $a$, $b$, $x$, and $y$ which satisfy the following four equations:
\begin{align*}
ax^{2014}+by^{2014}&=6\\
ax^{2015}+by^{2015}&=7\\
ax^{2016}+by^{2016}&=3\\
ax^{2017}+by^{2017}&=50.
\end{align*}
2018 ASDAN Math Tournament, 2
Given that $\sec x+\tan x=2018$, compute $\csc x+\cot x$.
2015 ASDAN Math Tournament, 2
Compute
$$\sum_{n=0}^\infty\frac{n+1}{2^n}.$$
2016 ASDAN Math Tournament, 2
Simplify the expression
$$\sqrt[3]{20+14\sqrt{2}}+\sqrt[3]{20-14\sqrt{2}}.$$
2018 ASDAN Math Tournament, 1
Given that $x\ge0$, $y\ge0$, $x+2y\le6$, and $2x+y\le6$, compute the maximum possible value of $x+y$.
2016 ASDAN Math Tournament, 3
Denote the dot product of two sequences $\langle x_1,\dots,x_n\rangle$ and $\langle y_1,\dots,y_n\rangle$ to be
$$x_1y_1+x_2y_2+\dots+x_ny_n.$$
Let $\langle a_1,\dots,a_n\rangle$ and $\langle b_1,\dots,b_n\rangle$ be two sequences of consecutive integers (i.e. for $1\leq i,i+1\leq n$, $a_i+1=a_{i+1}$ and similarly for $b_i$). Minnie permutes the two sequences so that their dot product, $m$, is minimized. Maximilian permutes the two sequences so that their dot product, $M$, is maximized. Given that $m=4410$ and $M=4865$, compute $n$, the number of terms in each sequence.
2014 ASDAN Math Tournament, 2
Let $a$ and $b$ be the roots of the quadratic $x^2-7x+c$. Given that $a^2+b^2=17$, compute $c$.
2017 ASDAN Math Tournament, 1
If $a$, $6$, and $b$, in that order, form an arithmetic sequence, compute $a+b$.
2016 ASDAN Math Tournament, 1
Let $x$ and $y$ be positive real numbers such that $x+y=\tfrac{1}{x}+\tfrac{1}{y}=5$. Compute $x^2+y^2$.
2014 ASDAN Math Tournament, 1
Kevin is running $1000$ meters. He wants to have an average speed of $10$ meters a second. He runs the first $100$ meters at a speed of $4$ meters a second. Compute how quickly, in meters per second, he must run the last $900$ meters to attain his desired average speed of $10$ meters a second.
2015 ASDAN Math Tournament, 1
Let $a_n$ be a sequence defined as $a_1=1$, $a_2=2$, and $a_n=a_{n-1}-a_{n-2}$. Compute $a_{2015}$.