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: 2017 , 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 $a+b$.

2015 ASDAN Math Tournament, 10
Tags: 2015 , 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 $$\max\{a+b-c,a-b+c,-a+b+c\}.$$

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

2017 ASDAN Math Tournament, 6
Tags: 2016 , Algebra Test
If $x+y^{-99}=3$ and $x+y=-2$, find the sum of all possible values of $x$.

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

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

2016 ASDAN Math Tournament, 2
Tags: 2016 , Algebra Test
A pet shop sells cats and two types of birds: ducks and parrots. In the shop, $\tfrac{1}{12}$ of animals are ducks, and $\tfrac{1}{4}$ of birds are ducks. Given that there are $56$ cats in the pet shop, how many ducks are there in the pet shop?

2017 ASDAN Math Tournament, 4
Tags: 2017 , Algebra Test
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?

2016 ASDAN Math Tournament, 6
Tags: 2016 , Algebra Test
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*}

2014 ASDAN Math Tournament, 4
Tags: 2014 , Algebra Test
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.

2014 ASDAN Math Tournament, 9
Tags: 2014 , Algebra Test
A sequence $\{a_n\}_{n\geq0}$ obeys the recurrence $a_n=1+a_{n-1}+\alpha a_{n-2}$ for all $n\geq2$ and for some $\alpha>0$. Given that $a_0=1$ and $a_1=2$, compute the value of $\alpha$ for which $$\sum_{n=0}^{\infty}\frac{a_n}{2^n}=10$$

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

2018 ASDAN Math Tournament, 3
Tags: 2018 , Algebra Test
The integers $a, b,$ and $c$ form a strictly increasing geometric sequence. Suppose that $abc = 216$. What is the maximum possible value of $a + b + c$?

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

2016 ASDAN Math Tournament, 7
Tags: 2016 , Algebra Test
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 $x+y+z$.

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

2014 ASDAN Math Tournament, 3
Tags: 2014 , Algebra Test
Compute all prime numbers $p$ such that $8p+1$ is a perfect square.

2015 ASDAN Math Tournament, 9
Tags: 2015 , Algebra Test
Compute all pairs of nonzero real numbers $(x,y)$ such that $$\frac{x}{x^2+y}+\frac{y}{x+y^2}=-1\qquad\text{and}\qquad\frac{1}{x}+\frac{1}{y}=1.$$

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

2016 ASDAN Math Tournament, 5
Tags: 2016 , Algebra Test
Let $f(x)$ be a real valued function. Recall that if the inverse function $f^{-1}(x)$ exists, then $f^{-1}(x)$ satisfies $f(f^{-1}(x))=f^{-1}(f(x))=x$. Given that the inverse of the function $f(x)=x^3-12x^2+48x-60$ exists, find all real $a$ that satisfy $f(a)=f^{-1}(a)$.

2014 ASDAN Math Tournament, 5
Tags: 2014 , Algebra Test
A positive integer $k$ is $2014$-ambiguous if the quadratics $x^2+kx+2014$ and $x^2+kx-2014$ both have two integer roots. Compute the number of integers which are $2014$-ambiguous.

2014 ASDAN Math Tournament, 7
Tags: 2014 , 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, 4
Tags: 2016 , Algebra Test
Suppose that $f(x)=x^2-10x+21$. Find all distinct real roots of $f(f(x)+7)$.

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

2018 ASDAN Math Tournament, 2
Tags: 2018 , Algebra Test
Given that $x$ is a real number, compute the minimum possible value of $(x-20)^2 + (x-18)^2$.