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, 5
Tags: algebra test
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 $$\sum_{k=1}^{10}\frac{F_k}{F_{k+1}F_{k+2}}.$$

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.

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

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?

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 $\tfrac{a_9}{a_{20}}$.

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

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 $$\sum_{n=0}^{\infty}\frac{a_n}{2^n}=10$$

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

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 $$16\cos^2\sqrt{x^2+y^2}+24\sin\sqrt{x^2+y^2}?$$

2016 ASDAN Math Tournament, 6
Tags: 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*}

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 $a_1$.

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

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 $(a,b)$.

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.

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.

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

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 $x+y+z$.

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, 5
Tags: 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 $f(a)=f^{-1}(a)$.

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 $a$.

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 $m$?

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.

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

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 $(a,b)$.