Olympiad problems

2018 AMC 10, 12

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 $

2013-2014 SDML (High School), 4

Tags: absolute value

If $\left|x\right|-x+y=42$ and $x+\left|y\right|+y=24$, then what is the value of $x+y$? Express your answer in simplest terms. $\text{(A) }-4\qquad\text{(B) }\frac{26}{5}\qquad\text{(C) }6\qquad\text{(D) }10\qquad\text{(E) }18$

KoMaL A Problems 2020/2021, A. 789

Tags: algebra , polynomial , absolute value

Let $p(x) = a_{21} x^{21} + a_{20} x^{20} + \dots + a_1 x + 1$ be a polynomial with integer coefficients and real roots such that the absolute value of all of its roots are less than $1/3$, and all the coefficients of $p(x)$ are lying in the interval $[-2019a,2019a]$ for some positive integer $a$. Prove that if this polynomial is reducible in $\mathbb{Z}[x]$, then the coefficients of one of its factors are less than $a$. [i]Submitted by Navid Safaei, Tehran, Iran[/i]

2015 AMC 10, 12

Tags: quadratic , algebra , polynomial , vieta , quadratic formula , absolute value

Points $(\sqrt{\pi}, a)$ and $(\sqrt{\pi}, b)$ are distinct points on the graph of $y^2+x^4=2x^2y+1$. What is $|a-b|$? $ \textbf{(A) }1\qquad\textbf{(B) }\dfrac{\pi}{2}\qquad\textbf{(C) }2\qquad\textbf{(D) }\sqrt{1+\pi}\qquad\textbf{(E) }1+\sqrt{\pi} $

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]

2006 AIME Problems, 15

Tags: function , inequalities , absolute value

Given that a sequence satisfies $x_0=0$ and $|x_k|=|x_{k-1}+3|$ for all integers $k\ge 1,$ find the minimum possible value of $|x_1+x_2+\cdots+x_{2006}|$.

IV Soros Olympiad 1997 - 98 (Russia), 9.4

Tags: absolute value , inequalities , algebra

Find the smallest and largest values of the expression $$\frac{ \left| ...\left| |x-1|-1\right| ... -1\right| +1}{\left| |x-2|-1 \right|+1}$$ (The number of units in the numerator of a fraction, including the last one, is eleven, of which ten are under the absolute value sign.)

2006 German National Olympiad, 5

Tags: inequalities , absolute value , polynomial , root

Let $x \neq 0$ be a real number satisfying $ax^2+bx+c=0$ with $a,b,c \in \mathbb{Z}$ obeying $|a|+|b|+|c| > 1$. Then prove \[ |x| \geq \frac{1}{|a|+|b|+|c|-1}. \]

2023 Ecuador NMO (OMEC), 1

Tags: absolute value , algebra

Find all reals $(a, b, c)$ such that $$\begin{cases}a^2+b^2+c^2=1\\ |a+b|=\sqrt{2}\end{cases}$$

2009 District Olympiad, 2

Tags: complex number , absolute value , algebra

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

2001 Estonia National Olympiad, 4

Tags: algebra , equation , absolute value

It is known that the equation$ |x - 1| + |x - 2| +... + |x - 2001| = a$ has exactly one solution. Find $a$.

2007 National Olympiad First Round, 18

Tags: absolute value

How many integers $n$ are there such that $n^3+8$ has at most $3$ positive divisors? $ \textbf{(A)}\ 4 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 1 \qquad\textbf{(E)}\ \text{None of the above} $

2012 EGMO, 4

Tags: algorithm , induction , absolute value , combinatorics

A set $A$ of integers is called [i]sum-full[/i] if $A \subseteq A + A$, i.e. each element $a \in A$ is the sum of some pair of (not necessarily different) elements $b,c \in A$. A set $A$ of integers is said to be [i]zero-sum-free[/i] if $0$ is the only integer that cannot be expressed as the sum of the elements of a finite nonempty subset of $A$. Does there exist a sum-full zero-sum-free set of integers? [i]Romania (Dan Schwarz)[/i]

2005 Today's Calculation Of Integral, 9

Tags: calculus , integration , trigonometry , absolute value , calculus computations , logarithm

Calculate the following indefinite integrals. [1] $\int (x^2+4x-3)^2(x+2)dx$ [2] $\int \frac{\ln x}{x(\ln x+1)}dx$ [3] $\int \frac{\sin \ (\pi \log _2 x)}{x}dx$ [4] $\int \frac{dx}{\sin x\cos ^ 2 x}$ [5] $\int \sqrt{1-3x}\ dx$

2012 Today's Calculation Of Integral, 818

Tags: calculus , integration , function , limit , absolute value , logarithm

For a function $f(x)=x^3-x^2+x$, find the limit $\lim_{n\to\infty} \int_{n}^{2n}\frac{1}{f^{-1}(x)^3+|f^{-1}(x)|}\ dx.$

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 $\textbf{(A) }48\qquad\textbf{(B) }52\qquad\textbf{(C) }56\qquad\textbf{(D) }59\qquad \textbf{(E) }64$

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.

1952 Moscow Mathematical Olympiad, 214

Tags: inequalities , algebra , absolute value , moscow

Prove that if $|x| < 1$ and $|y| < 1$, then $\left|\frac{x - y}{1 -xy}\right|< 1$.

2009 China Western Mathematical Olympiad, 1

Tags: polynomial , geometry , geometric transformation , absolute value , 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 $M$.

1990 AMC 8, 7

Tags: absolute value

When three different numbers from the set $ \{-3,-2,-1, 4, 5\} $ are multiplied, the largest possible product is $ \text{(A)}\ 10\qquad\text{(B)}\ 20\qquad\text{(C)}\ 30\qquad\text{(D)}\ 40\qquad\text{(E)}\ 60 $

1985 All Soviet Union Mathematical Olympiad, 410

Tags: algebra , set , absolute value

Numbers $1,2,3,...,2n$ are divided onto two equal groups. Let $a_1,a_2,...,a_n$ be the first group numbers in the increasing order, and $b_1,b_2,...,b_n$ -- the second group numbers in the decreasing order. Prove that $$|a_1 - b_1| + |a_2 - b_2| + ... + |a_n - b_n| = n^2$$

2011 Dutch BxMO TST, 3

Tags: absolute value , equation , algebra

Find all triples $(x, y, z)$ of real numbers that satisfy $x^2 + y^2 + z^2 + 1 = xy + yz + zx +|x - 2y + z|$.

2005 Taiwan TST Round 1, 2

Tags: absolute value , algebra

The absolute value of every number in the sequence $\{a_n\}$ is smaller than 2005, and \[a_{n+6}=a_{n+4}+a_{n+2}-a_n.\] holds for all positive integers n. Prove that $\{a_n\}$ is periodic. Incredibly, this was probably the most difficult problem of our independent study problems in the 1st TST (excluding the final exam).

2005 Putnam, A4

Tags: linear algebra , matrix , inequalities , vector , ratio , absolute value

Let $H$ be an $n\times n$ matrix all of whose entries are $\pm1$ and whose rows are mutually orthogonal. Suppose $H$ has an $a\times b$ submatrix whose entries are all $1.$ Show that $ab\le n.$

2004 National Olympiad First Round, 35

Tags: absolute value

We are placing $n$ integers whose sum is $94$ over a circle such that each number is equal to the absolute value of the difference of (clockwise) next two numbers. What is the largest $n$ that makes such placing possible? $ \textbf{(A)}\ 188 \qquad\textbf{(B)}\ 186 \qquad\textbf{(C)}\ 141 \qquad\textbf{(D)}\ 100 \qquad\textbf{(E)}\ 47 $