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: 50

2017 ASDAN Math Tournament, 9
Tags: algebra test
Let $f(x)=x^3+ax^2+bx$ for some $a,b$. For some $c$, $f(c)$ achieves a local maximum of $539$ (in other words, $f(c)$ is the maximum value of $f$ for some open interval around $c$). In addition, at some $d$, $f(d)$ achieves a local minimum of $-325$. Given that $c$ and $d$ are integers, compute $a+b$.

2018 ASDAN Math Tournament, 6
Tags: algebra test
Given that $x > 1$, compute $x$ such that $$\log_{16}(x) + \log_x(2)$$ is minimal.

2017 ASDAN Math Tournament, 2
Tags: algebra test
Eric has $2$ boxes of apples, with the first box containing red and yellow apples and the second box containing green apples. Eric observes that the red apples make up $\tfrac{1}{2}$ of the apples in the first box. He then moves all of the red apples to the second box, and observes that the red apples now make up $\tfrac{1}{3}$ of the apples in the second box. Suppose that Eric has $28$ apples in total. How many red apples does Eric have?

2017 ASDAN Math Tournament, 3
Tags: algebra test
Let $a$ and $b$ be real numbers such that $a^5b^8=12$ and $a^8b^{13}=18$. Find $ab$.

2015 ASDAN Math Tournament, 1
Tags: algebra test
Given that $xy+x+y=5$ and $x+1=2$, compute $y+1$.

2014 ASDAN Math Tournament, 10
Tags: algebra test
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 $p(6)$.

2017 ASDAN Math Tournament, 10
Tags: algebra test
Let $\zeta=e^{2\pi i/36}$. Compute $$\prod_{\stackrel{a=1}{\gcd(a,36)=1}}^{35}(\zeta^a-2).$$

2017 ASDAN Math Tournament, 5
Tags: algebra test
Compute $$\sum_{i=0}^\infty(-1)^i\sum_{j=i}^\infty(-1)^j\frac{2}{j^2+4j+3}.$$

2014 ASDAN Math Tournament, 7
Tags: algebra test
$f(x)$ is a quartic polynomial with a leading coefficient $1$ where $f(2)=4$, $f(3)=9$, $f(4)=16$, and $f(5)=25$. Compute $f(8)$.

2016 ASDAN Math Tournament, 3
Tags: algebra test
Real numbers $x,y,z$ form an arithmetic sequence satisfying \begin{align*} x+y+z&=6\\ xy+yz+zx&=10. \end{align*} What is the absolute value of their common difference?

2017 ASDAN Math Tournament, 6
Tags: algebra test
If $x+y^{-99}=3$ and $x+y=-2$, find the sum of all possible values of $x$.

2015 ASDAN Math Tournament, 2
Tags: algebra test
Find the sum of the squares of the roots of $x^2-5x-7$.

2017 ASDAN Math Tournament, 1
Tags: algebra test
Suppose $(x+y)^2=25$ and $(x-y)^2=1$. Compute $xy$.

2016 ASDAN Math Tournament, 1
Tags: algebra test
If $x=14$ and $y=6$, then compute $\tfrac{x^2-y^2}{x-y}$.

2017 ASDAN Math Tournament, 8
Tags: algebra test
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 $$\frac{a_{2018}\cdot a_{2016}}{a_{2017}^2}.$$

2018 ASDAN Math Tournament, 7
Tags: algebra test
Let $s$ and $t$ be the solutions to $x^2-10x+10=0$. Compute $\tfrac{1}{s^5}+\tfrac{1}{t^5}$.

2015 ASDAN Math Tournament, 6
Tags: algebra test
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*}

2018 ASDAN Math Tournament, 1
Tags: algebra test
Alice’s age in years is twice Eve’s age in years. In $10$ years, Eve will be as old as Alice is now. Compute Alice’s age in years now.

2018 ASDAN Math Tournament, 10
Tags: algebra test
Compute the unique value of $\theta$, in degrees, where $0^\circ<\theta<90^\circ$, such that $$\csc\theta=\sum_{i=3}^{11}\csc(2^i)^\circ.$$

2015 ASDAN Math Tournament, 10
Tags: algebra test
The polynomial $f(x)=x^3-4\sqrt{3}x^2+13x-2\sqrt{3}$ has three real roots, $a$, $b$, and $c$. Find $$\max\{a+b-c,a-b+c,-a+b+c\}.$$

2015 ASDAN Math Tournament, 7
Tags: algebra test
Compute the minimum value of $$\frac{x^4+2x^3+3x^2+2x+10}{x^2+x+1}$$ where $x$ can be any real number.

2014 ASDAN Math Tournament, 6
Tags: algebra test
Compute $\cos(\tfrac{\pi}{9})-\cos(\tfrac{2\pi}{9})+\cos(\tfrac{3\pi}{9})-\cos(\tfrac{4\pi}{9})$.

2015 ASDAN Math Tournament, 8
Tags: algebra test
Let $\{x\}$ denote the fractional part of $x$, which means the unique real $0\leq\{x\}<1$ such that $x-\{x\}$ is an integer. Let $f_{a,b}(x)=\{x+a\}+2\{x+b\}$ and let its range be $[m_{a,b},M_{a,b})$. Find the minimum value of $M_{a,b}$ as $a$ and $b$ range along all real numbers.

2014 ASDAN Math Tournament, 8
Tags: algebra test
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 $(x,y)$.

2018 ASDAN Math Tournament, 5
Tags: algebra test
In the expansion of $(x + b)^{2018}$, the coefficients of $x^2$ and $x^3$ are equal. Compute $b$.