Found problems: 50
Let $p(x)=c_1+c_2\cdot2^x+c_3\cdot3^x+c_4\cdot5^x+c_5\cdot8^x$. Given that $p(k)=k$ for $k=1,2,3,4,5$, compute $p(6)$.
Consider the sequence of real numbers $a_n$ satisfying the recurrence
$$a_na_{n+2}-a_{n+1}^2-(n+1)a_na_{n+1}=0.$$
Given that $a_1=1$ and $a_2=2018$, compute
$$\frac{a_{2018}\cdot a_{2016}}{a_{2017}^2}.$$
The Fibonacci numbers are a sequence of numbers defined recursively as follows: $F_1=1$, $F_2=1$, and $F_n=F_{n-1}+F_{n-2}$. Using this definition, compute the sum
$$\sum_{k=1}^{10}\frac{F_k}{F_{k+1}F_{k+2}}.$$
$f(x)$ is a quartic polynomial with a leading coefficient $1$ where $f(2)=4$, $f(3)=9$, $f(4)=16$, and $f(5)=25$. Compute $f(8)$.
Consider the recurrence relation
$$a_{n+3}=\frac{a_{n+2}a_{n+1}-2}{a_n}$$
with initial condition $(a_0,a_1,a_2)=(1,2,5)$. Let $b_n=a_{2n}$ for nonnegative integral $n$. It turns out that $b_{n+2}+xb_{n+1}+yb_n=0$ for some pair of real numbers $(x,y)$. Compute $(x,y)$.
Let $f(x)=\sum_{i=1}^{2014}|x-i|$. Compute the length of the longest interval $[a,b]$ such that $f(x)$ is constant on that interval.
Find all triples of integers $(x,y,z)$ which satisfy the equations
\begin{align*}
x^2-y-2z&=4\\
y^2-2z-3x&=-2\\
2z^2-3x-5y&=-22.\\
\end{align*}
For real numbers $x,y$ satisfying $x^2+y^2-4x-2y+4=0$, what is the greatest value of
$$16\cos^2\sqrt{x^2+y^2}+24\sin\sqrt{x^2+y^2}?$$
Given $2017$ positive numbers $x_1,\dots,x_{2017}$ such that
$$\sum_{i=1}^{2017}x_i=\sum_{i=1}^{2017}\frac{1}{x_i}=2018,$$
compute the maximum possible value of $x_1+\frac{1}{x_1}$.
If $x=14$ and $y=6$, then compute $\tfrac{x^2-y^2}{x-y}$.
Suppose that $f(x)=x^2-10x+21$. Find all distinct real roots of $f(f(x)+7)$.
What is the maximum possible value for the sum of the squares of the roots of $x^4+ax^3+bx^2+cx+d$ where $a$, $b$, $c$, and $d$ are $2$, $0$, $1$, and $7$ in some order?
If $x+y^{-99}=3$ and $x+y=-2$, find the sum of all possible values of $x$.
Let $x$, $y$, and $z$ be real numbers satisfying the equations
\begin{align*}
4x+2yz-6z+9xz^2&=4\\
xyz&=1.
\end{align*}
Find all possible values of $x+y+z$.
Let $f(n)$ be the integer closest to $\sqrt{n}$. Compute the largest $N$ less than or equal to $2018$ such that $\sum_{i=1}^N\frac{1}{f(i)}$ is integral.
Given that $x$ is a real number, compute the minimum possible value of $(x-20)^2 + (x-18)^2$.
Compute the unique value of $\theta$, in degrees, where $0^\circ<\theta<90^\circ$, such that
$$\csc\theta=\sum_{i=3}^{11}\csc(2^i)^\circ.$$
It is possible to express the sum
$$\sum_{n=1}^{24}\frac{1}{\sqrt{n+\sqrt{n^2-1}}}$$
as $a\sqrt{2}+b\sqrt{3}$, for some integers $a$ and $b$. Compute the ordered pair $(a,b)$.
Compute $\cos(\tfrac{\pi}{9})-\cos(\tfrac{2\pi}{9})+\cos(\tfrac{3\pi}{9})-\cos(\tfrac{4\pi}{9})$.
Compute all real solutions $(x,y)$ with $x\geq y$ that satisfy the pair of equations
\begin{align*}
xy&=5\\
\frac{x^2+y^2}{x+y}&=3.
\end{align*}
Let $\zeta=e^{2\pi i/36}$. Compute
$$\prod_{\stackrel{a=1}{\gcd(a,36)=1}}^{35}(\zeta^a-2).$$
Find the sum of the squares of the roots of $x^2-5x-7$.
A college math class has $N$ teaching assistants. It takes the teaching assistants $5$ hours to grade homework assignments. One day, another teaching assistant joins them in grading and all homework assignments take only $4$ hours to grade. Assuming everyone did the same amount of work, compute the number of hours it would take for $1$ teaching assistant to grade all the homework assignments.
Given that $x > 1$, compute $x$ such that
$$\log_{16}(x) + \log_x(2)$$
is minimal.
Let $P(x)$ be a monic cubic polynomial. The line $y=0$ and $y=m$ intersect $P(x)$ at points $A,C,E$ and $B,D<F$ from left to right for a positive real number $m$. If $AB=\sqrt{7}$, $CD=\sqrt{15}$, and $EF=\sqrt{10}$, what is the value of $m$?