This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

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Found problems: 252

2007 India IMO Training Camp, 3
Tags: combinatorics , absolute value
Given a finite string $S$ of symbols $X$ and $O$, we denote $\Delta(s)$ as the number of$X'$s in $S$ minus the number of $O'$s (For example, $\Delta(XOOXOOX)=-1$). We call a string $S$ [b]balanced[/b] if every sub-string $T$ of (consecutive symbols) $S$ has the property $-1\leq \Delta(T)\leq 2.$ (Thus $XOOXOOX$ is not balanced, since it contains the sub-string $OOXOO$ whose $\Delta$ value is $-3.$ Find, with proof, the number of balanced strings of length $n$.

2021 Polish Junior MO First Round, 3
Tags: absolute value , algebra
The numbers $a, b, c$ satisfy the condition $| a - b | = 2 | b - c | = 3 | c - a |$. Prove that $a = b = c$.

2012 ELMO Shortlist, 8
Tags: inequalities , absolute value , binomial coefficient , ceiling function , geometric series , polynomial , algebra
Fix two positive integers $a,k\ge2$, and let $f\in\mathbb{Z}[x]$ be a nonconstant polynomial. Suppose that for all sufficiently large positive integers $n$, there exists a rational number $x$ satisfying $f(x)=f(a^n)^k$. Prove that there exists a polynomial $g\in\mathbb{Q}[x]$ such that $f(g(x))=f(x)^k$ for all real $x$. [i]Victor Wang.[/i]

2024 Ukraine National Mathematical Olympiad, Problem 2
Tags: construction , absolute value , combinatorics
There is a table with $n > 2$ cells in the first row, $n-1$ cells in the second row is a cell, $n-2$ in the third row, $\ldots$, $1$ cell in the $n$-th row. The cells are arranged as shown below. [img]https://i.ibb.co/0Z1CR0c/UMO24-8-2.png[/img] In each cell of the top row Petryk writes a number from $1$ to $n$, so that each number is written exactly once. For each other cell, if the cells directly above it contains numbers $a, b$, it contains number $|a-b|$. What is the largest number that can be written in a single cell of the bottom row? [i]Proposed by Bogdan Rublov[/i]

2015 Romania National Olympiad, 1
Tags: complex number , absolute value , algebra
Find all triplets $ (a,b,c) $ of nonzero complex numbers having the same absolute value and which verify the equality: $$ \frac{a}{b} +\frac{b}{c}+\frac{c}{a} =-1 $$

1958 AMC 12/AHSME, 39
Tags: function , quadratic , absolute value
We may say concerning the solution of \[ |x|^2 \plus{} |x| \minus{} 6 \equal{} 0 \] that: $ \textbf{(A)}\ \text{there is only one root}\qquad \textbf{(B)}\ \text{the sum of the roots is }{\plus{}1}\qquad \textbf{(C)}\ \text{the sum of the roots is }{0}\qquad \\ \textbf{(D)}\ \text{the product of the roots is }{\plus{}4}\qquad \textbf{(E)}\ \text{the product of the roots is }{\minus{}6}$

1955 AMC 12/AHSME, 19
Tags: algebra , absolute value , polynomial
Two numbers whose sum is $ 6$ and the absolute value of whose difference is $ 8$ are roots of the equation: $ \textbf{(A)}\ x^2\minus{}6x\plus{}7\equal{}0 \qquad \textbf{(B)}\ x^2\minus{}6x\minus{}7\equal{}0 \qquad \textbf{(C)}\ x^2\plus{}6x\minus{}8\equal{}0 \\ \textbf{(D)}\ x^2\minus{}6x\plus{}8\equal{}0 \qquad \textbf{(E)}\ x^2\plus{}6x\minus{}7\equal{}0$

2011 Bosnia Herzegovina Team Selection Test, 3
Tags: absolute value , algebra
Numbers $1,2, ..., 2n$ are partitioned into two sequences $a_1<a_2<...<a_n$ and $b_1>b_2>...>b_n$. Prove that number \[W= |a_1-b_1|+|a_2-b_2|+...+|a_n-b_n|\] is a perfect square.

2014 Putnam, 2
Tags: function , absolute value , real analysis , integration
Suppose that $f$ is a function on the interval $[1,3]$ such that $-1\le f(x)\le 1$ for all $x$ and $\displaystyle \int_1^3f(x)\,dx=0.$ How large can $\displaystyle\int_1^3\frac{f(x)}x\,dx$ be?

2011 Hanoi Open Mathematics Competitions, 8
Tags: inequalities , absolute value , minimum , algebra
Find the minimum value of $S = |x + 1| + |x + 5|+ |x + 14| + |x + 97| + |x + 1920|$.

2016 Thailand Mathematical Olympiad, 7
Tags: absolute value , algebra , polynomial
Given $P(x)=a_{2016}x^{2016}+a_{2015}x^{2015}+...+a_1x+a_0$ be a polynomial with real coefficients and $a_{2016} \neq 0$ satisfies $|a_1+a_3+...+a_{2015}| > |a_0+a_2+...+a_{2016}|$ Prove that $P(x)$ has an odd number of complex roots with absolute value less than $1$ (count multiple roots also) edited: complex roots

2011 AMC 10, 19
Tags: absolute value , quadratic
What is the product of all the roots of the equation \[\sqrt{5|x|+8} = \sqrt{x^2-16}. \] $ \textbf{(A)}\ -64 \qquad \textbf{(B)}\ -24 \qquad \textbf{(C)}\ -9 \qquad \textbf{(D)}\ 24 \qquad \textbf{(E)}\ 576 $

2013 Philippine MO, 3
Tags: absolute value
3. Let n be a positive integer. The numbers 1, 2, 3,....., 2n are randomly assigned to 2n distinct points on a circle. To each chord joining two of these points, a value is assigned equal to the absolute value of the difference between the assigned numbers at its endpoints. Show that one can choose n pairwise non-intersecting chords such that the sum of the values assigned to them is $n^2$ .

2008 China Team Selection Test, 2
Tags: modular arithmetic , absolute value , polynomial , algebra
Prove that for all $ n\geq 2,$ there exists $ n$-degree polynomial $ f(x) \equal{} x^n \plus{} a_{1}x^{n \minus{} 1} \plus{} \cdots \plus{} a_{n}$ such that (1) $ a_{1},a_{2},\cdots, a_{n}$ all are unequal to $ 0$; (2) $ f(x)$ can't be factorized into the product of two polynomials having integer coefficients and positive degrees; (3) for any integers $ x, |f(x)|$ isn't prime numbers.

2007 India IMO Training Camp, 3
Tags: absolute value , combinatorics
Given a finite string $S$ of symbols $X$ and $O$, we denote $\Delta(s)$ as the number of$X'$s in $S$ minus the number of $O'$s (For example, $\Delta(XOOXOOX)=-1$). We call a string $S$ [b]balanced[/b] if every sub-string $T$ of (consecutive symbols) $S$ has the property $-1\leq \Delta(T)\leq 2.$ (Thus $XOOXOOX$ is not balanced, since it contains the sub-string $OOXOO$ whose $\Delta$ value is $-3.$ Find, with proof, the number of balanced strings of length $n$.

2002 France Team Selection Test, 3
Tags: combinatorics , absolute value
Let $n$ be a positive integer and let $(a_1,a_2,\ldots ,a_{2n})$ be a permutation of $1,2,\ldots ,2n$ such that the numbers $|a_{i+1}-a_i|$ are pairwise distinct for $i=1,\ldots ,2n-1$. Prove that $\{a_2,a_4,\ldots ,a_{2n}\}=\{1,2,\ldots ,n\}$ if and only if $a_1-a_{2n}=n$.

2024 Indonesia TST, A
Tags: absolute value , inequalities , algebra
Given real numbers $x,y,z$ which satisfies $$|x+y+z|+|xy+yz+zx|+|xyz| \le 1$$ Show that $max\{ |x|,|y|,|z|\} \le 1$.

2005 Iran MO (3rd Round), 2
Tags: geometric transformation , group theory , abstract algebra , gauss , topology , absolute value , geometry
We define a relation between subsets of $\mathbb R ^n$. $A \sim B\Longleftrightarrow$ we can partition $A,B$ in sets $A_1,\dots,A_n$ and $B_1,\dots,B_n$(i.e $\displaystyle A=\bigcup_{i=1} ^n A_i,\ B=\bigcup_{i=1} ^n B_i, A_i\cap A_j=\emptyset,\ B_i\cap B_j=\emptyset$) and $A_i\simeq B_i$. Say the the following sets have the relation $\sim$ or not ? a) Natural numbers and composite numbers. b) Rational numbers and rational numbers with finite digits in base 10. c) $\{x\in\mathbb Q|x<\sqrt 2\}$ and $\{x\in\mathbb Q|x<\sqrt 3\}$ d) $A=\{(x,y)\in\mathbb R^2|x^2+y^2<1\}$ and $A\setminus \{(0,0)\}$

2010 Tournament Of Towns, 3
Tags: ratio , absolute value
From a police station situated on a straight road in nite in both directions, a thief has stolen a police car. Its maximal speed equals $90$% of the maximal speed of a police cruiser. When the theft is discovered some time later, a policeman starts to pursue the thief on a cruiser. However, he does not know in which direction along the road the thief has gone, nor does he know how long ago the car has been stolen. Is it possible for the policeman to catch the thief?

2013 Princeton University Math Competition, 8
Tags: algebra , absolute value , polynomial
If $x,y$ are real, then the $\textit{absolute value}$ of the complex number $z=x+yi$ is \[|z|=\sqrt{x^2+y^2}.\] Find the number of polynomials $f(t)=A_0+A_1t+A_2t^2+A_3t^3+t^4$ such that $A_0,\ldots,A_3$ are integers and all roots of $f$ in the complex plane have absolute value $\leq 1$.

2021 OMpD, 3
Tags: n-variable inequality , absolute value , algebra , inequalities
Let $a$ and $b$ be positive real numbers, with $a < b$ and let $n$ be a positive integer. Prove that for all real numbers $x_1, x_2, \ldots , x_n \in [a, b]$: $$ |x_1 - x_2| + |x_2 - x_3| + \cdots + |x_{n-1} - x_n| + |x_n - x_1| \leq \frac{2(b - a)}{b + a}(x_1 + x_2 + \cdots + x_n)$$ And determine for what values of $n$ and $x_1, x_2, \ldots , x_n$ the equality holds.

2022 Belarusian National Olympiad, 11.7
Tags: absolute value , algebra , sequence
Numbers $-1011, -1010, \ldots, -1, 1, \ldots, 1011$ in some order form the sequence $a_1,a_2,\ldots, a_{2022}$. Find the maximum possible value of the sum $$|a_1|+|a_1+a_2|+\ldots+|a_1+\ldots+a_{2022}|$$

2000 Slovenia National Olympiad, Problem 2
Tags: parameterization , absolute value
Find all real numbers $a$ for which the following equation has a unique real solution: $$|x-1|+|x-2|+\ldots+|x-99|=a.$$

2009 Croatia Team Selection Test, 1
Tags: algebra , absolute value , polynomial
Determine the lowest positive integer n such that following statement is true: If polynomial with integer coefficients gets value 2 for n different integers, then it can't take value 4 for any integer.

2004 Junior Balkan Team Selection Tests - Romania, 3
Tags: absolute value
Let $A$ be a $8\times 8$ array with entries from the set $\{-1,1\}$ such that any $2\times 2$ sub-square of the array has the absolute value of the sum of its element equal with 2. Prove that the array must have at least two identical lines.