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: 252

1975 Putnam, A2
Tags: analytic geometry , absolute value , algebra
Describe the region $R$ consisting of the points $(a,b)$ of the cartesian plane for which both (possibly complex) roots of the polynomial $z^2+az+b$ have absolute value smaller than $1$.

2016 AMC 12/AHSME, 3
Tags: absolute value
Let $x=-2016$. What is the value of $\left| \ \bigl \lvert { \ \lvert x\rvert -x }\bigr\rvert -|x|{\frac{}{}}^{}_{}\right|-x$? $\textbf{(A)}\ -2016\qquad\textbf{(B)}\ 0\qquad\textbf{(C)}\ 2016\qquad\textbf{(D)}\ 4032\qquad\textbf{(E)}\ 6048$

2004 AIME Problems, 2
Tags: absolute value , p2 catalyst , wilkinson catalyst , birch reduction
Set $A$ consists of $m$ consecutive integers whose sum is $2m$, and set $B$ consists of $2m$ consecutive integers whose sum is $m$. The absolute value of the difference between the greatest element of $A$ and the greatest element of $B$ is $99$. Find $m$.

2009 China Western Mathematical Olympiad, 1
Tags: polynomial , absolute value , geometric transformation , geometry , algebra
Let $M$ be the set of the real numbers except for finitely many elements. Prove that for every positive integer $n$ there exists a polynomial $f(x)$ with $\deg f = n$, such that all the coefficients and the $n$ real roots of $f$ are all in $M$.

2019 Azerbaijan Junior NMO, 2
Tags: ratio , area , absolute value , geometry , algebra
Alice creates the graphs $y=|x-a|$ and $y=c-|x-b|$ , where $a,b,c\in\mathbb{R^+}$. She observes that these two graphs and $x$ axis divides the positive side of the plane ($x,y>0$) into two triangles and a quadrilateral. Find the ratio of sums of two triangles' areas to the area of quadrilateral. [hide=There might be a translation error] In the original statement,it says $XOY$ plane,instead of positive side of the plane. I think these 2 are the same,but I might be wrong [/hide]

2012 Today's Calculation Of Integral, 818
Tags: logarithm , absolute value , function , calculus , integration , limit
For a function $f(x)=x^3-x^2+x$, find the limit $\lim_{n\to\infty} \int_{n}^{2n}\frac{1}{f^{-1}(x)^3+|f^{-1}(x)|}\ dx.$

2012 Today's Calculation Of Integral, 801
Tags: calculus computations , absolute value , function , integration , calculus
Answer the following questions: (1) Let $f(x)$ be a function such that $f''(x)$ is continuous and $f'(a)=f'(b)=0$ for some $a<b$. Prove that $f(b)-f(a)=\int_a^b \left(\frac{a+b}{2}-x\right)f''(x)dx$. (2) Consider the running a car on straight road. After a car which is at standstill at a traffic light started at time 0, it stopped again at the next traffic light apart a distance $L$ at time $T$. During the period, prove that there is an instant　for which the absolute value of the acceleration of the car is more than or equal to $\frac{4L}{T^2}.$

2013 All-Russian Olympiad, 1
Tags: pigeonhole principle , absolute value , combinatorics
$101$ distinct numbers are chosen among the integers between $0$ and $1000$. Prove that, among the absolute values ​​of their pairwise differences, there are ten different numbers not exceeding $100$.

2013 Romania Team Selection Test, 1
Tags: algebra , inequalities , triangle inequality , absolute value , polynomial , number theory
Let $a$ and $b$ be two square-free, distinct natural numbers. Show that there exist $c>0$ such that \[ \left | \{n\sqrt{a}\}-\{n\sqrt{b}\} \right |>\frac{c}{n^3}\] for every positive integer $n$.

2010 Belarus Team Selection Test, 7.1
Tags: number theory , minimum , absolute value , minimum value
Find the smallest value of the expression $|3 \cdot 5^m - 11 \cdot 13^n|$ for all $m,n \in N$. (Folklore)

2010 Tuymaada Olympiad, 3
Tags: quadratic , vieta , absolute value , polynomial , algebra
Let $f(x) = ax^2+bx+c$ be a quadratic trinomial with $a$,$b$,$c$ reals such that any quadratic trinomial obtained by a permutation of $f$'s coefficients has an integer root (including $f$ itself). Show that $f(1)=0$.

2014 Turkey MO (2nd round), 1
Tags: absolute value , combinatorics
In a bag there are $1007$ black and $1007$ white balls, which are randomly numbered $1$ to $2014$. In every step we draw one ball and put it on the table; also if we want to, we may choose two different colored balls from the table and put them in a different bag. If we do that we earn points equal to the absolute value of their differences. How many points can we guarantee to earn after $2014$ steps?

2009 China National Olympiad, 1
Tags: algebra , symmetry , polynomial , absolute value , inequalities , function
Given an integer $ n > 3.$ Let $ a_{1},a_{2},\cdots,a_{n}$ be real numbers satisfying $ min |a_{i} \minus{} a_{j}| \equal{} 1, 1\le i\le j\le n.$ Find the minimum value of $ \sum_{k \equal{} 1}^n|a_{k}|^3.$

2007 Turkey Team Selection Test, 3
Tags: absolute value , induction , matrix , linear algebra , combinatorics
We write $1$ or $-1$ on each unit square of a $2007 \times 2007$ board. Find the number of writings such that for every square on the board the absolute value of the sum of numbers on the square is less then or equal to $1$.

2005 ISI B.Stat Entrance Exam, 8
Tags: algebra , function , absolute value
A function $f(n)$ is defined on the set of positive integers is said to be multiplicative if $f(mn)=f(m)f(n)$ whenever $m$ and $n$ have no common factors greater than $1$. Are the following functions multiplicative? Justify your answer. (a) $g(n)=5^k$ where $k$ is the number of distinct primes which divide $n$. (b) $h(n)=\begin{cases} 0 & \text{if} \ n \ \text{is divisible by} \ k^2 \ \text{for some integer} \ k>1 \\ 1 & \text{otherwise} \end{cases}$

2007 District Olympiad, 4
Tags: inequalities , absolute value , vector
[b]a)[/b] Let $ \bold {u,v,w,} $ be three coplanar vectors of absolute value $ 1. $ Show that there exist $ \varepsilon_1 ,\varepsilon_2, \varepsilon_3\in \{ \pm 1\} $ such that $$ \big| \varepsilon_1\bold u +\varepsilon_2\bold v +\varepsilon_3\bold w \big|\le 1. $$ [b]b)[/b] Give an example of three vectors such that the inequality above does not work for any sclaras from $ \{ \pm 1\} . $

2001 AIME Problems, 14
Tags: complex number , trigonometry , absolute value
There are $2n$ complex numbers that satisfy both $z^{28}-z^{8}-1=0$ and $|z|=1$. These numbers have the form $z_{m}=\cos\theta_{m}+i\sin\theta_{m}$, where $0\leq\theta_{1}<\theta_{2}< \dots <\theta_{2n}<360$ and angles are measured in degrees. Find the value of $\theta_{2}+\theta_{4}+\dots+\theta_{2n}$.

2022 China Team Selection Test, 4
Tags: inequalities , function , absolute value , algebra
Given a positive integer $n$, find all $n$-tuples of real number $(x_1,x_2,\ldots,x_n)$ such that \[ f(x_1,x_2,\cdots,x_n)=\sum_{k_1=0}^{2} \sum_{k_2=0}^{2} \cdots \sum_{k_n=0}^{2} \big| k_1x_1+k_2x_2+\cdots+k_nx_n-1 \big| \] attains its minimum.

2024 Macedonian Balkan MO TST, Problem 4
Tags: absolute value , inequalities
Let $x_1, ..., x_n$ $(n \geq 2)$ be real numbers from the interval $[1,2]$. Prove that $$|x_1-x_2|+...+|x_n-x_1| + \frac{1}{3} (|x_1-x_3|+...+|x_n-x_2|) \leq \frac{2}{3} (x_1+...+x_n)$$ and determine all cases of equality.

2016 AMC 10, 3
Tags: absolute value
Let $x=-2016$. What is the value of $\left| \ \bigl \lvert { \ \lvert x\rvert -x }\bigr\rvert -|x|{\frac{}{}}^{}_{}\right|-x$? $\textbf{(A)}\ -2016\qquad\textbf{(B)}\ 0\qquad\textbf{(C)}\ 2016\qquad\textbf{(D)}\ 4032\qquad\textbf{(E)}\ 6048$

1969 AMC 12/AHSME, 25
Tags: logarithm , absolute value , inequalities
If it is known that $\log_2a+\log_2b\geq 6$, then the least value that can be taken on by $a+b$ is: $\textbf{(A) }2\sqrt6\qquad \textbf{(B) }6\qquad \textbf{(C) }8\sqrt2\qquad \textbf{(D) }16\qquad \textbf{(E) }\text{none of these.}$

2012 Grigore Moisil Intercounty, 1
Tags: matrix , absolute value , linear algebra , inequalities
The absolute value of the sum of the elements of a real orthogonal matrix is at most the order of the matrix.

MathLinks Contest 7th, 5.1
Tags: product of roots , inequalities , vieta , algebra , absolute value , polynomial
Find all real polynomials $ g(x)$ of degree at most $ n \minus{} 3$, $ n\geq 3$, knowing that all the roots of the polynomial $ f(x) \equal{} x^n \plus{} nx^{n \minus{} 1} \plus{} \frac {n(n \minus{} 1)}2 x^{n \minus{} 2} \plus{} g(x)$ are real.

1971 AMC 12/AHSME, 25
Tags: absolute value
A teen age boy wrote his own age after his father's. From this new four place number, he subtracted the absolute value of the difference of their ages to get $4,289$. The sum of their ages was $\textbf{(A) }48\qquad\textbf{(B) }52\qquad\textbf{(C) }56\qquad\textbf{(D) }59\qquad \textbf{(E) }64$

2010 Contests, 2
Tags: function , cauchy inequality , inequalities , absolute value , algebra
For each positive integer $n$, find the largest real number $C_n$ with the following property. Given any $n$ real-valued functions $f_1(x), f_2(x), \cdots, f_n(x)$ defined on the closed interval $0 \le x \le 1$, one can find numbers $x_1, x_2, \cdots x_n$, such that $0 \le x_i \le 1$ satisfying \[|f_1(x_1)+f_2(x_2)+\cdots f_n(x_n)-x_1x_2\cdots x_n| \ge C_n\] [i]Marko Radovanović, Serbia[/i]