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

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, 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, 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 $f(8)$.

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

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

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

2017 ASDAN Math Tournament, 9
Tags: 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 $a+b$.

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, 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, 7
Tags: algebra test
Let $s$ and $t$ be the solutions to $x^2-10x+10=0$. Compute $\tfrac{1}{s^5}+\tfrac{1}{t^5}$.

2017 ASDAN Math Tournament, 10
Tags: algebra test
Let $\zeta=e^{2\pi i/36}$. Compute $$\prod_{\stackrel{a=1}{\gcd(a,36)=1}}^{35}(\zeta^a-2).$$

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.

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 $a_1$.

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

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

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

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 $x_1 \dots x_{2044}$.

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

2017 ASDAN Math Tournament, 2
Tags: 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?

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

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

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 $f(a)=f^{-1}(a)$.

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

2014 ASDAN Math Tournament, 8
Tags: 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 $(x,y)$.

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