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

2018 ASDAN Math Tournament, 8
Tags: algebra test
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.

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

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?

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.

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

2017 ASDAN Math Tournament, 7
Tags: algebra test
For real numbers $x,y$ satisfying $x^2+y^2-4x-2y+4=0$, what is the greatest value of $$16\cos^2\sqrt{x^2+y^2}+24\sin\sqrt{x^2+y^2}?$$

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

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 $f(8)$.

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, 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 $x_1+\frac{1}{x_1}$.

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 $(a,b)$.

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

2014 ASDAN Math Tournament, 4
Tags: 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.

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 $$\sum_{i=0}^\infty(-1)^i\sum_{j=i}^\infty(-1)^j\frac{2}{j^2+4j+3}.$$

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

2015 ASDAN Math Tournament, 3
Tags: algebra test
Let $a_1,a_2,a_3,\dots,a_6$ be an arithmetic sequence with common difference $3$. Suppose that $a_1$, $a_3$, and $a_6$ also form a geometric sequence. Compute $a_1$.

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, 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?

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

2018 ASDAN Math Tournament, 4
Tags: algebra test
Given that $4^{x_1} = 5, 5^{x_2} = 6, \dots , 2047^{x_{2044}} = 2048$, compute the product $x_1 \dots x_{2044}$.