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, 7

Tags: 2015 , 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, 3

Tags: 2016 , 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?

2018 ASDAN Math Tournament, 4

Tags: 2018 , Algebra Test
Given that $4^{x_1} = 5, 5^{x_2} = 6, \dots , 2047^{x_{2044}} = 2048$, compute the product $x_1 \dots x_{2044}$.

2017 ASDAN Math Tournament, 1

Tags: 2017 , Algebra Test
Suppose $(x+y)^2=25$ and $(x-y)^2=1$. Compute $xy$.

2015 ASDAN Math Tournament, 5

Tags: 2015 , 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 $$\sum_{k=1}^{10}\frac{F_k}{F_{k+1}F_{k+2}}.$$

2017 ASDAN Math Tournament, 7

Tags: 2017 , 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, 1

Tags: 2014 , 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: 2018 , 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 $x_1+\frac{1}{x_1}$.

2015 ASDAN Math Tournament, 6

Tags: 2015 , 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, 7

Tags: 2018 , Algebra Test
Let $s$ and $t$ be the solutions to $x^2-10x+10=0$. Compute $\tfrac{1}{s^5}+\tfrac{1}{t^5}$.

2017 ASDAN Math Tournament, 2

Tags: 2017 , 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?

2014 ASDAN Math Tournament, 8

Tags: 2014 , 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 $(x,y)$.

2014 ASDAN Math Tournament, 6

Tags: 2014 , Algebra Test
Compute $\cos(\tfrac{\pi}{9})-\cos(\tfrac{2\pi}{9})+\cos(\tfrac{3\pi}{9})-\cos(\tfrac{4\pi}{9})$.

2017 ASDAN Math Tournament, 3

Tags: 2017 , Algebra Test
Let $a$ and $b$ be real numbers such that $a^5b^8=12$ and $a^8b^{13}=18$. Find $ab$.

2018 ASDAN Math Tournament, 8

Tags: 2018 , 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, 10

Tags: 2017 , Algebra Test
Let $\zeta=e^{2\pi i/36}$. Compute $$\prod_{\stackrel{a=1}{\gcd(a,36)=1}}^{35}(\zeta^a-2).$$

2015 ASDAN Math Tournament, 8

Tags: 2015 , 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.

2015 ASDAN Math Tournament, 4

Tags: 2015 , 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, 8

Tags: 2016 , 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 $(a,b)$.

2016 ASDAN Math Tournament, 10

Tags: 2016 , 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 $\tfrac{a_9}{a_{20}}$.

2015 ASDAN Math Tournament, 3

Tags: 2015 , 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: 2016 , 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 $m$?

2017 ASDAN Math Tournament, 5

Tags: 2017 , 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, 2

Tags: 2014 , 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: 2018 , 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.