Olympiad problems

2014 Vietnam National Olympiad, 3

Find all sets of not necessary distinct 2014 rationals such that:if we remove an arbitrary number in the set, we can divide remaining 2013 numbers into three sets such that each set has exactly 671 elements and the product of all elements in each set are the same.

2010 Romanian Master of Mathematics, 2

Tags: function , inequalities , cauchy inequality , 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]

2018 Macedonia National Olympiad, Problem 2

Tags: absolute value , combinatorics

Let $n$ be a natural number and $C$ a non-negative real number. Determine the number of sequences of real numbers $1, x_{2}, ..., x_{n}, 1$ such that the absolute value of the difference between any two adjacent terms is equal to $C$.

2013 ITAMO, 3

Tags: absolute value , combinatorics

Each integer is colored with one of two colors, red or blue. It is known that, for every finite set $A$ of consecutive integers, the absolute value of the difference between the number of red and blue integers in the set $A$ is at most $1000$. Prove that there exists a set of $2000$ consecutive integers in which there are exactly $1000$ red numbers and $1000$ numbers blue.

2024 CCA Math Bonanza, I9

Tags: absolute value

Find the median value of $m$ over all integers $m$ where $|m^2 + 8m - 65|$ is a perfect power. A perfect power is any integer at least $2$ which can be written as $a^b$, where $a$, $b$ are integers and $b \ge 2$. [i]Individual #9[/i]

2012 Serbia Team Selection Test, 1

Tags: polynomial , function , inequalities , absolute value , algebra

Let $P(x)$ be a polynomial of degree $2012$ with real coefficients satisfying the condition \[P(a)^3 + P(b)^3 + P(c)^3 \geq 3P(a)P(b)P(c),\] for all real numbers $a,b,c$ such that $a+b+c=0$. Is it possible for $P(x)$ to have exactly $2012$ distinct real roots?

2024 Indonesia TST, A

Tags: algebra , absolute value , inequalities

Given real numbers $x,y,z$ which satisfies $$|x+y+z|+|xy+yz+zx|+|xyz| \le 1$$ Show that $max\{ |x|,|y|,|z|\} \le 1$.

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: $S = |n-1| + |n-2| + \ldots + |n-100|$

2010 Tournament Of Towns, 3

Tags: ratio , absolute value

From a police station situated on a straight road innite 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?

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 China National Olympiad, 3

Tags: absolute value , combinatorics

Let $n \geq 5$ be a positive integer and let $A$ and $B$ be sets of integers satisfying the following conditions: i) $|A| = n$, $|B| = m$ and $A$ is a subset of $B$ ii) For any distinct $x,y \in B$, $x+y \in B$ iff $x,y \in A$ Determine the minimum value of $m$.

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. $\text{(A) }-4\qquad\text{(B) }\frac{26}{5}\qquad\text{(C) }6\qquad\text{(D) }10\qquad\text{(E) }18$

2018 Bosnia and Herzegovina Team Selection Test, 4

Tags: combinatorics , board , maximum , absolute value

Every square of $1000 \times 1000$ board is colored black or white. It is known that exists one square $10 \times 10$ such that all squares inside it are black and one square $10 \times 10$ such that all squares inside are white. For every square $K$ $10 \times 10$ we define its power $m(K)$ as an absolute value of difference between number of white and black squares $1 \times 1$ in square $K$. Let $T$ be a square $10 \times 10$ which has minimum power among all squares $10 \times 10$ in this board. Determine maximal possible value of $m(T)$

2011 Hanoi Open Mathematics Competitions, 8

Tags: minimum , algebra , inequalities , absolute value

Find the minimum value of $S = |x + 1| + |x + 5|+ |x + 14| + |x + 97| + |x + 1920|$.

2024 Macedonian Balkan MO TST, Problem 4

Tags: inequalities , absolute value

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.

2000 All-Russian Olympiad, 5

Tags: absolute value , combinatorics

Let $M$ be a finite sum of numbers, such that among any three of its elements there are two whose sum belongs to $M$. Find the greatest possible number of elements of $M$.

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 $$|a_1 - b_1| + |a_2 - b_2| + ... + |a_n - b_n| = n^2$$

2009 Croatia Team Selection Test, 1

Tags: polynomial , absolute value , algebra

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.

2015 AMC 12/AHSME, 25

Tags: algebra , polynomial , geometric series , absolute value

A bee starts flying from point $P_0$. She flies 1 inch due east to point $P_1$. For $j \ge 1$, once the bee reaches point $P_j$, she turns $30^{\circ}$ counterclockwise and then flies $j+1$ inches straight to point $P_{j+1}$. When the bee reaches $P_{2015}$ she is exactly $a\sqrt{b} + c\sqrt{d}$ inches away from $P_0$, where $a$, $b$, $c$ and $d$ are positive integers and $b$ and $d$ are not divisible by the square of any prime. What is $a+b+c+d$? $ \textbf{(A)}\ 2016 \qquad\textbf{(B)}\ 2024 \qquad\textbf{(C)}\ 2032 \qquad\textbf{(D)}\ 2040 \qquad\textbf{(E)}\ 2048$

2005 Polish MO Finals, 3

Tags: linear algebra , matrix , geometry , rectangle , rotation , absolute value , combinatorics

In a matrix $2n \times 2n$, $n \in N$, are $4n^2$ real numbers with a sum equal zero. The absolute value of each of these numbers is not greater than $1$. Prove that the absolute value of a sum of all the numbers from one column or a row doesn't exceed $n$.

2007 Germany Team Selection Test, 2

Tags: absolute value , algebra

Determine the sum of absolute values for the complex roots of $ 20 x^8 \plus{} 7i x^7 \minus{}7ix \plus{} 20.$

2013 Bogdan Stan, 2

Tags: function , calculus , integration , absolute value

Let be a sequence of continuous functions $ \left( f_n \right)_{n\ge 1} :[0,1]\longrightarrow\mathbb{R} $ satisfying the following properties: $ \text{a) } $ for any natural $ n $ and $ x\in [1/n,1] ,$ it follows $ \left| f_n(x) \right|\leqslant 1/n. $ $ \text{b) } $ for any natural $ n, $ it follows $ \int_0^1 f_n^2(t)dt\leqslant 1. $ Then, $\lim_{n\to 0} \int_0^1\left| f_n(t) \right| dt=0 $ [i]Cristinel Mortici[/i]

2005 All-Russian Olympiad, 1

Tags: function , abstract algebra , modular arithmetic , algebra , absolute value

Find the maximal possible finite number of roots of the equation $|x-a_1|+\dots+|x-a_{50}|=|x-b_1|+\dots+|x-b_{50}|$, where $a_1,\,a_2,\,\dots,a_{50},\,b_1,\dots,\,b_{50}$ are distinct reals.

2002 France Team Selection Test, 3

Tags: absolute value , combinatorics

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 $a_1-a_{2n}=n$.

PEN H Problems, 9

Tags: diophantine equation , algebra , polynomial , vieta , absolute value

Determine all integers $a$ for which the equation \[x^{2}+axy+y^{2}=1\] has infinitely many distinct integer solutions $x, \;y$.