Olympiad problems

2009 South africa National Olympiad, 6

Let $A$ denote the set of real numbers $x$ such that $0\le x<1$. A function $f:A\to \mathbb{R}$ has the properties: (i) $f(x)=2f(\frac{x}{2})$ for all $x\in A$; (ii) $f(x)=1-f(x-\frac{1}{2})$ if $\frac{1}{2}\le x<1$. Prove that (a) $f(x)+f(1-x)\ge \frac{2}{3}$ if $x$ is rational and $0<x<1$. (b) There are infinitely many odd positive integers $q$ such that equality holds in (a) when $x=\frac{1}{q}$.

2018 AMC 12/AHSME, 10

Tags: absolute value

How many ordered pairs of real numbers $(x,y)$ satisfy the following system of equations? \begin{align*}x+3y&=3\\ \big||x|-|y|\big|&=1\end{align*} $\textbf{(A) } 1 \qquad \textbf{(B) } 2 \qquad \textbf{(C) } 3 \qquad \textbf{(D) } 4 \qquad \textbf{(E) } 8 $

2003 Iran MO (3rd Round), 26

Tags: trig identities , absolute value , vector , trapezoid , law of cosines , trigonometry , geometry

Circles $ C_1,C_2$ intersect at $ P$. A line $ \Delta$ is drawn arbitrarily from $ P$ and intersects with $ C_1,C_2$ at $ B,C$. What is locus of $ A$ such that the median of $ AM$ of triangle $ ABC$ has fixed length $ k$.

1999 Swedish Mathematical Competition, 1

Tags: algebra , absolute value , equation

Solve $|||||x^2-x-1| - 2| - 3| - 4| - 5| = x^2 + x - 30$.

2012 ELMO Shortlist, 8

Tags: polynomial , binomial coefficient , algebra , absolute value , geometric series , inequalities , ceiling function

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]

2022 AMC 10, 6

Tags: algebra , absolute value

Which expression is equal to $\left | a-2-\sqrt{(a-1)^2} \right|$ for $a<0$? $\textbf{(A) } 3-2a \qquad \textbf{(B) } 1-a \qquad \textbf{(C) } 1 \qquad \textbf{(D) } a+1 \qquad \textbf{(E) } 3$

2021 AMC 10 Fall, 16

Tags: absolute value

The graph of $f(x) = |\lfloor x \rfloor| - |\lfloor 1 - x \rfloor|$ is symmetric about which of the following? (Here $\lfloor x \rfloor$ is the greatest integer not exceeding $x$.) $\textbf{(A) }$ the $y$-axis $\qquad \textbf{(B) }$ the line $x = 1$ $\qquad \textbf{(C) }$ the origin $\qquad \textbf{(D) }$ the point $\left(\dfrac12, 0\right)$ $\qquad \textbf{(E) }$ the point $(1,0)$

2000 Harvard-MIT Mathematics Tournament, 1

Tags: inequalities , absolute value

How many integers $x$ satisfy $|x|+5<7$ and $|x-3|>2$?

2011 Junior Balkan Team Selection Tests - Moldova, 1

Tags: absolute value , min , algebra

The absolute value of the difference of the solutions of the equation $x^2 + px + q = 0$, with $p, q \in R$, is equal to $4$. Find the solutions of the equation if it is known that $(q + 1) p^2 + q^2$ takes the minimum value.

2015 VTRMC, Problem 6

Tags: absolute value , functional equation , fe

Let $(a_1,b_1),\ldots,(a_n,b_n)$ be $n$ points in $\mathbb R^2$ (where $\mathbb R$ denotes the real numbers), and let $\epsilon>0$ be a positive number. Can we find a real-valued function $f(x,y)$ that satisfies the following three conditions? 1. $f(0,0)=1$; 2. $f(x,y)\ne0$ for only finitely many $(x,y)\in\mathbb R^2$; 3. $\sum_{r=1}^n\left|f(x+a_r,y+b_r)-f(x,y)\right|<\epsilon$ for every $(x,y)\in\mathbb R^2$. Justify your answer.

2008 Harvard-MIT Mathematics Tournament, 6

Tags: inequalities , conic , quadratic , parabola , analytic geometry , absolute value , function

Determine all real numbers $ a$ such that the inequality $ |x^2 \plus{} 2ax \plus{} 3a|\le2$ has exactly one solution in $ x$.

2014 Singapore Senior Math Olympiad, 9

Tags: absolute value

Find the number of real numbers which satisfy the equation $x|x-1|-4|x|+3=0$. $ \textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }3\qquad\textbf{(E) }4 $

2007 South africa National Olympiad, 2

Tags: polynomial , system of equations , absolute value , algebra

Consider the equation $ x^4 \equal{} ax^3 \plus{} bx^2 \plus{} cx \plus{} 2007$, where $ a,b,c$ are real numbers. Determine the largest value of $ b$ for which this equation has exactly three distinct solutions, all of which are integers.

2007 Federal Competition For Advanced Students, Part 2, 1

Tags: pigeonhole principle , algebra , absolute value , polynomial

Let $ M$ be the set of all polynomials $ P(x)$ with pairwise distinct integer roots, integer coefficients and all absolut values of the coefficients less than $ 2007$. Which is the highest degree among all the polynomials of the set $ M$?

2015 Tournament of Towns, 5

Tags: absolute value , polynomial , algebra

Do there exist two polynomials with integer coefficients such that each polynomial has a coefficient with an absolute value exceeding $2015$ but all coefficients of their product have absolute values not exceeding $1$? [i]($10$ points)[/i]

2011 Czech and Slovak Olympiad III A, 3

Tags: inequalities , absolute value

Suppose that $x$, $y$, $z$ are real numbers satisfying \[x+y+z=12,\qquad\text{and}\qquad x^2+y^2+z^2=54.\] Prove that:[list](a) Each of the numbers $xy$, $yz$, $zx$ is at least $9$, but at most $25$. (b) One of the numbers $x$, $y$, $z$ is at most $3$, and another one is at least $5$.[/list]

2009 District Olympiad, 2

Tags: algebra , complex number , absolute value

Find the complex numbers $ z_1,z_2,z_3 $ of same absolute value having the property that: $$ 1=z_1z_2z_3=z_1+z_2+z_3. $$

2007 China Girls Math Olympiad, 7

Tags: polynomial , algebra , absolute value

Let $ a$, $ b$, $ c$ be integers each with absolute value less than or equal to $ 10$. The cubic polynomial $ f(x) \equal{} x^3 \plus{} ax^2 \plus{} bx \plus{} c$ satisfies the property \[ \Big|f\left(2 \plus{} \sqrt 3\right)\Big| < 0.0001. \] Determine if $ 2 \plus{} \sqrt 3$ is a root of $ f$.

1963 Miklós Schweitzer, 7

Tags: real analysis , integration , function , absolute value

Prove that for every convex function $ f(x)$ defined on the interval $ \minus{}1\leq x \leq 1$ and having absolute value at most $ 1$, there is a linear function $ h(x)$ such that \[ \int_{\minus{}1}^1|f(x)\minus{}h(x)|dx\leq 4\minus{}\sqrt{8}.\] [L. Fejes-Toth]

2009 Math Prize For Girls Problems, 1

Tags: floor function , absolute value , conic , hyperbola

How many ordered pairs of integers $ (x, y)$ are there such that \[ 0 < \left\vert xy \right\vert < 36?\]

ICMC 6, 1

Tags: algebra , absolute value

The city of Atlantis is built on an island represented by $[ -1, 1]$, with skyline initially given by $f(x) = 1 - |x| $. The sea level is currently $y=0$, but due to global warming, it is rising at a rate of $0.01$ a year. For any position $-1 < x < 1$, while the building at $x$ is not completely submerged, then it is instantaneously being built upward at a rate of $r$ per year, where $r$ is the distance (along the $x$-axis) from this building to the nearest completely submerged building. How long will it be until Atlantis becomes completely submerged? [i]Proposed by Ethan Tan[/i]

2004 AMC 10, 4

Tags: function , absolute value

What is the value of $ x$ if $ |x \minus{} 1| \equal{} |x \minus{} 2|$? $ \textbf{(A)}\ \minus{}\!\frac {1}{2}\qquad \textbf{(B)}\ \frac {1}{2}\qquad \textbf{(C)}\ 1\qquad \textbf{(D)}\ \frac {3}{2}\qquad \textbf{(E)}\ 2$

1951 Miklós Schweitzer, 11

Tags: algebra , absolute value , polynomial

Prove that, for every pair $ n$, $r$ of positive integers, there can be found a polynomial $ f(x)$ of degree $ n$ with integer coefficients, so that every polynomial $ g(x)$ of degree at most $ n$, for which the coefficients of the polynomial $ f(x)\minus{}g(x)$ are integers with absolute value not greater than $ r$, is irreducible over the field of rational numbers.

2013 IPhOO, 6

Tags: trigonometry , absolute value , magnetic field , centripetal force

A particle with charge $8.0 \, \mu\text{C}$ and mass $17 \, \text{g}$ enters a magnetic field of magnitude $\text{7.8 mT}$ perpendicular to its non-zero velocity. After 30 seconds, let the absolute value of the angle between its initial velocity and its current velocity, in radians, be $\theta$. Find $100\theta$. [i](B. Dejean, 5 points)[/i]

2008 Paraguay Mathematical Olympiad, 2

Tags: parabola , absolute value , conic

Find for which values of $n$, an integer larger than $1$ but smaller than $100$, the following expression has its minimum value: $S = |n-1| + |n-2| + \ldots + |n-100|$