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

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 $b$.

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, 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.

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

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 $a$.

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 $ab$.

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 $a+b$.

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

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

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 $p(6)$.

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

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

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, 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*}

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?

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 $x$.

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 $a + b + c$?

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

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 $(x,y)$.

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

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?

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

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

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 $(a,b)$.