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

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*}

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 $$\max\{a+b-c,a-b+c,-a+b+c\}.$$

2014 ASDAN Math Tournament, 2
Tags: algebra test
Let $a$ and $b$ be positive integers such that $a>b$ and the difference between $a^2+b$ and $a+b^2$ is prime. Compute all possible pairs $(a,b)$.

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, 3
Tags: 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 $a + b + c$?

2016 ASDAN Math Tournament, 9
Tags: algebra test
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 $m$?

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

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

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, 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 $$\frac{a_{2018}\cdot a_{2016}}{a_{2017}^2}.$$

2016 ASDAN Math Tournament, 10
Tags: algebra test
Let $a_1,a_2,\dots$ be a sequence of real numbers satisfying $$\frac{a_{n+1}}{a_n}-\frac{a_{n+2}}{a_n}-\frac{a_{n+1}a_{n+2}}{a_n^2}=\frac{na_{n+2}a_{n+1}}{a_n}.$$ Given that $a_1=-1$ and $a_2=-\tfrac{1}{2}$, find the value of $\tfrac{a_9}{a_{20}}$.

2016 ASDAN Math Tournament, 2
Tags: 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, 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, 4
Tags: algebra test
Let $f(x)=(x-a)^3$. If the sum of all $x$ satisfying $f(x)=f(x-a)$ is $42$, find $a$.

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.

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, 5
Tags: 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.

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

2014 ASDAN Math Tournament, 1
Tags: algebra test
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.

2018 ASDAN Math Tournament, 9
Tags: algebra test
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 $x_1+\frac{1}{x_1}$.

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?

2014 ASDAN Math Tournament, 9
Tags: 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, 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.

2016 ASDAN Math Tournament, 8
Tags: algebra test
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 $(a,b)$.

2017 ASDAN Math Tournament, 4
Tags: 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?